Formal foundations for situation awareness based on dependent type theory

Information Fusion 14 (2013) 87–107

Contents lists available at SciVerse ScienceDirect

Information Fusion

journal homepage: www.elsevier .com/locate / inf fus

Formal foundations for situation awareness based on dependent type theory

Richard Dapoigny ⇑, Patrick BarlatierUniversity of Savoie, Laboratory of Computer Science, Systems, Information and Knowledge Processing, PO Box 80439, 74944 Annecy-le-vieux cedex, France

a r t i c l e i n f o

Article history:Received 1 July 2010Received in revised form 28 February 2012Accepted 29 February 2012Available online 9 March 2012

Keywords:Situation awarenessOntologiesDependent typesAggregationSubsumptionType inhabitation

1566-2535/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.inffus.2012.02.006

⇑ Corresponding author.E-mail address: [email protected] (R

a b s t r a c t

Cognitive situation awareness has recently caught the attention of the information fusion community.Some approaches have developed formalizations that are both ontology-based and underpinned withSituation Theory. While the semantics of Situation Theory is very attractive from the cognitive point ofview, the languages that are used to express knowledge and to reason with suffer from a number oflimitations concerning both expressiveness and reasoning capabilities. In this paper we propose a moregeneral formal foundation denoted S-DTT (Situation-based Dependent Type Theory) that is expressedwith the language of the Extended Calculus of Constructions (ECC), a widely used theory in mathematicalformalization and in software validation. Situation awareness relies on small blocks of knowledge calledsituation fragment types whose composition leads to a very expressive and unifying theory. The semanticpart is provided by an ontology that is rooted in the S-DTT theory and, on which higher-order reasoningcan be performed. The basis of the theory is summarized and its expressing power is illustrated withnumerous examples. A scenario in the healthcare context for patient safety issues is detailed and acomparison with well-known approaches is discussed.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In the data fusion process, what is termed situation awarenessincreasingly appears as a central component. Situation awarenessoriginates from the JDL Data Fusion Model [1]. As advocated in[2], it is ‘‘concerned with the perception of elements, of theirmeaning and the projection of their status in the near future’’.The present work is grounded on this model since it has beenwidely used in various approaches to serve as a basis for structur-ing situation awareness. More precisely, the meaning of the ele-ments of the situation refers to the situation assessment. Theinformation fusion community has emphasized the growingimportance of high-level information fusion, where the situationassessment appears to be a challenging topic. For example, deriv-ing higher order relations and identifying meaningful events andactivities are not trivial tasks. The critical role of backgroundknowledge as a mechanism for improving both current and futureapproaches to information fusion has been highlighted and it hasbeen argued that knowledge-based approaches have general appli-cability in solving a number of problems.

On the one hand, a number of issues that are related to theknowledge-based approach include knowledge-based awarenesssupport for contextual relevance reasoning, information elicitation,representation of situation state, etc. Other issues arise in semanticinteroperability where the ambiguities are inherent in the lan-

ll rights reserved.

. Dapoigny).

guage terms used by diverse user-communities. For example, inthe perspective of knowledge acquisition, semantic web contenthas to be taken into account. To address this problem, recent pa-pers promote ontologies as an essential component of informationfusion [3,4]. However, they have pointed out some difficulties, suchas the (structured) textual content of documents on the web that isnot sufficient to solve ambiguities and their semantic integration.Meta-data annotations are also needed to specify the semanticsof these information sources based on an ontologically wellfounded approach. Therefore a problem which arises is how to ex-press meta-data within ontologies since they have a distinctabstraction level?

On the other hand, the awareness concept means that the sys-tem is able to draw inferences from observations. As a conse-quence, reasoning appears to be a crucial component ofinformation fusion at any level of the fusion process. A formal the-ory is required in which the meaning of each symbol is identified interms of some other primitive symbols or in terms of axioms. Sym-bols represent the internal state of the system (i.e., the system’sview about the situation) which interacts with the outside world.This internal state itself describes the system’s beliefs. Beliefs en-code the explanations of the world and can be expressed with Hornclauses [5]. The inference engine then computes the logical conse-quences of arbitrarily many beliefs. Inferences are the basis forhigh-level information fusion and decision support. They resulteither from the formal semantics of the knowledge representationlanguage, or from implication rules expressing more complex typesof inference. Whereas traditional approaches for low-level

http://dx.doi.org/10.1016/j.inffus.2012.02.006

mailto:[email protected]

http://dx.doi.org/10.1016/j.inffus.2012.02.006

http://www.sciencedirect.com/science/journal/15662535

http://www.elsevier.com/locate/inffus

88 R. Dapoigny, P. Barlatier / Information Fusion 14 (2013) 87–107

information fusion are probabilistic, the high-level approach israther symbolic and focuses on relations. In that perspective, a stillopen research challenge is to integrate domain specific knowledgeand to automatically draw conclusions that have otherwise to bedrawn by a user.

From these aspects, it appears that the knowledge-based ap-proach for situation awareness should both require the semanticsprovided by a domain ontology and a powerful reasoning system(e.g., SWRL, RuleML, etc.). A consensus has emerged from manyworks that have investigated the role of ontologies in informationfusion [3,6–8]: ontologies provide an effective means to addressmany of the issues highlighted above. Given this assumption, theontology language OWL could be used since it allows a form ofautomated reasoning as determined by the formal semantics ofthe language. It is a language built on Description Logics (DLs) thatis appropriate for knowledge representation and reasoning. How-ever, since DLs are conceptually oriented, they lack a rule-basedreasoning mechanism such as the one found in logic programming(e.g., Horn clauses). Therefore, various approaches have been pro-posed for integrating logic programming and description logicssuch as [9–11]. Nevertheless, these reasoning techniques sufferfrom some limitations such as the restriction of DLs to ‘‘safe rules’’[12] and complicate the reasoning process by adding a translationmechanism between DLs and logic programming. More precisely,using these languages for practical applications raises several chal-lenges [3,13]. The restriction to binary predicates in both SWRL andOWL is a first difficulty leading to violate safe rules for expressingthe higher-order nature of the rules that have to be constructed.The safety condition in SWRL makes it difficult to write rules thatresult in the assertion of complex relations. The SWRL built-ins aredefined as relations (instead of functions) with no explicit input/output designations assigned to their arguments which complicatethe codification. Dealing with time under the RDFs monotonicityassumption does not receive a clear solution. Furthermore, despitesignificant progress in research on information fusion, it lacks aunified theoretical framework taking in account both the ontolog-ical and the logical aspects.

This paper provides a language for expressing and reasoningabout knowledge with the perspective of (i) apply it to high-levelsituation awareness and (ii) to go beyond the state-of-the-art.The approach is rooted both in the domains of artificial intelligenceand in theoretical computer science, and makes use of dependenttype theory. The fundamental problem of situation awareness,i.e., whether a relation holds or not, is addressed from a logical per-spective that is equivalent (through the Curry-Howard isomor-phism) to a type inhabitation problem.1 The resulting decrease inthe search space of relations relies on the typing of domain ontolo-gies and on the use of dependent types replacing Cartesian products,i.e., a (set � set) space search with a (set � subset) space search.Along these lines, the contribution of this proposal is a first concretestep towards a unified, rigorous and expressive theory centered onthe notion of dependent type in constructive logic.

Dependent type theory has been described in [14] as a founda-tion for constructive mathematics. The underlying logic is con-structive, and epistemologically unclear steps in proofs areforbidden. For languages supporting the construction and verifica-tion of large modular systems, it is well known that the concept ofdependent type is inevitable. Then, multiple dependently-typedtheories were developed such as the Logical Framework or the Cal-culus of Constructions [15] and several of them were implementedgiving rise to theorem provers [16] such as Coq and LEGO. In com-puter science, dependent types have been applied in building proofassistants and automated theorem provers. Dependent type sys-

1 The problem of finding a term having a particular type.

tems mix types and expressions to produce code that is provento be correct with respect to its expected behavior. Dependenttypes have also been applied to write elegant and precise formal-izations within a strongly typed specification language for securityprotocols [17,18]. Surprisingly, there is a lack of applications inArtificial Intelligence (AI) using type theory except in Natural Lan-guage Processing (NLP) [19–21]. Since a basic task in NLP is tomodel situations, it seems coherent to use type theory with regardto situation modeling. It is worth to mention that the syntax oftype theory is more uniform than the syntax of first-order logicsince expressions serve as both terms and formulas. In addition,type theory can be used to reason about an infinite hierarchy ofhigher-order functions built over a domain of individuals and a do-main of truth values. The proposed type theory precisely includes anumber of properties such as a simple set of rules, a single lan-guage both allowing a knowledge representation with types andincluding a higher-order logic to reason about this knowledge. Itprovides major improvements in situation awareness and fusionprocesses due to the following points: (i) its high-level expressive-ness, (ii) its precision by allowing types to depend on values, (iii)its ability to cope with partial knowledge, (iv) the computationalproperty resulting from the Curry-Howard isomorphism and (v)the logical power of the internal higher-order (constructive) logic.We propose here the first (up to the author’s knowledge) situationawareness theory based on an intensional type theory. By doingthis, we have the double motivation to help fill the gap betweenontologies and logical theories and to use type semantics to getnew insight on situation assessment. We believe that the proposi-tion we present here is a first step in these directions.

2. General assumptions for situation modeling

The object of the present section is not to define a new philo-sophical perspective on situation modeling, but rather to explainthe coherence of the ontological and logical choices w.r.t. the philo-sophical assumptions (i.e., intuitionism). In order to use situationawareness, situations must be recognized and characterized andmore precisely, situations must be first-order citizens of a theoryas it is defined in the calculus of situations. Furthermore, as a partof a knowledge-based system, they must be related to a number ofontological notions.

2.1. Ontological assumptions

In the following, the term ‘‘type’’ will only refer to mathematicalcategories rather than to ontological categories. The notion ofontology offers two perspectives, i.e., from philosophy and fromcomputer science. The objectives of the former is to identify theentities of the world and to classify them into categories. The latterrather focuses on artifacts or engineering models of reality that areinterpreted and reasoned over by software (inference engines).These software are endowed with human level semantics. Mostontologies in computer science which describe applications areknown as domain ontologies. The problem is that these ontologiesare often developed independently for different purposes and willoften differ greatly from each other. A common way to overcomethis difficulty is to relate a particular domain description to a for-mal (or upper) ontology which is not tied to a particular problemdomain but describes general entities.

Formal ontologies provide a rigorous account of notions that arefundamental in any domain such as the categories of universalsand particulars, categories like endurants and perdurants2 and

2 They reflect different modes of existence in time.

R. Dapoigny, P. Barlatier / Information Fusion 14 (2013) 87–107 89

partonomic relations. However, the ways to formally characterizethese fundamental ontological divides are much more controversial.It is needed therefore, to motivate the choices that have beendecided for S-DTT. We follow here the definitions which are preva-lent in Artificial Intelligence [22].

Definition 1 (Formal Ontology). A formal ontology refers to anengineering artifact constituted (i) by a specific vocabulary used todescribe a certain reality, and (ii) a set of explicit assumptions(written e.g., in a first-order logical theory) regarding the intendedmeaning of the vocabulary terms. In the case of first-order logicaltheory, vocabulary terms appear as predicate names giving rise toconcepts and relations.

For example, a formal ontology of situations does not tell whichsituations there are, but it might tell under what operations situa-tions are closed and what structure all situations exhibit. A majorbenefit is that information represented with a formal ontologycan be more easily accessible to automated informationprocessing.

Definition 2 (Domain Ontology). A domain ontology describes thevocabulary related to a generic domain (like medicine or law) byspecializing the terms introduced in the formal ontology.

Each scientific field will for example, have its own domainontology, defined by the vocabulary inherent to that field and bythe canonical formulations of its theories. Gradually, however, itis realized that the provision of a common backbone taxonomyof relevant entities whose purpose is to be included in an applica-tion domain would provide significant advantages over the case-by-case resolution of incompatibilities between domainontologies.

The distinction between universals and particulars is a philo-sophical debate that we do not address here. We assume the fol-lowing definitions commonly in use in computer science.

Definition 3 (Instance). Anything of which a category T can bepredicated is called an instance for T.

Definition 4 (Particular). A particular is anything that instantiatessome universal and which cannot be predicated of anything, i.e., ithas no instance.

Examples of particulars are: JohnDoe (which represents the per-son John Doe), the planet Earth, this piece of cheese, your eating ofthis piece of cheese, etc.

Definition 5 (Universal). Universals are expressed by categorieswhich have the following two characteristics:

– First, a category is distinguished from a particular by thefact that it can have instances and that it can be predi-cated of things. A category may also be an instance ofanother category.

– Second, a category can be predicated of many things andshould represent what is common to many cases.

Universals are required to explain relations of resemblance be-tween particulars. They come in different categories and are con-ceived as that in reality to which the general terms used inmaking scientific assertions correspond. Examples of universalsare: human being, enzyme, butterfly, flight connection. From aphilosophical point of view, we assume a realist view, i.e., that uni-versals exist in a domain of quantification distinct from that of par-ticulars. A major consequence is that universals are represented inthe ontological domain of discourse and that the knowledge lan-guage should support higher-order quantification, i.e., quantifica-

tion over universals since universals can, e.g., stand for relationswhich themselves quantify over universals. In computational sci-ence, universals are represented by classes (or types), propertiesand relations. The instantiation of properties corresponds to aquality or a characteristic of the particular(s) while properties rep-resent relative qualities (i.e., they express a universal from anotheruniversal). Relations can be hierarchical or non-hierarchical. Themajor hierarchical relations that we address here are the subsump-tion and the partonomic relations. From the ontological viewpoint,subsumption is a relation between universals.

Definition 6 (Subsumption [23]). A universal A subsumes auniversal A0 if A0 is more specific than A, then A0 inherits theproperties of A.

In other words, a category A subsumes a category A0 if the exten-sion of the properties characterizing A is included in the extensionof the properties which hold for A0. Subsumption relations organizeuniversals into hierarchies of subcategories (e.g., ‘‘all dogs are ani-mals’’). Therefore, we assume that subsumption organizes univer-sals according to their intension (their properties).

The well-known part-of relation has been the subject of manydiscussions. Whereas the individual part-of relation refers to par-ticulars, the partonomic relation rather relates universals. Giventhat the extension of a universal is the collection of individualswhich instantiates the universal, we can define partonomic rela-tions as follows:

Definition 7 (Partonomic Relations [24]). A universal A0 is parto-nomically included in the universal A if and only if A and A0 haveextensions at the same times and, at those times, the extension ofA0 is partonomically included in the extension of A.

Most works in knowledge-based systems (e.g., in the biologicaldomain) has focused almost exclusively on classes which are nottemporally conceived and considering instances which exist intime and space ensures that the data used as input to these sys-tems is organized formally [25]. Despite their practical importance(e.g., in biomedical ontologies), partonomic relations are rarelyconsidered as primitives in the representation languages. Further-more, there is a need to clearly distinguish the partonomic inclu-sion between universals with the subsumption (i.e., taxonomicinclusion). As underlined in [25] a coherent treatment of theserelations must rely on ‘‘explicit formal definitions which take intoaccount not only the classes involved as terms of these relationsbut also the instances of these classes’’. We will show in Section4.4 how the present formalism will fulfill these requirements.

The distinction between endurants and perdurants is alsoan open issue in philosophy. Usually, an endurant is an entity whichexists wholly in every instant at which it exists at all, while a perdu-rant unfolds itself over time into consecutive temporal parts(phases).

2.2. Logical assumptions

As underlined above, the logical foundations of the theory mustbe coherent w.r.t. the above ontological choices and we have to de-cide what logic the present theory is committed to. It has been ex-plained in [26] that the philosophical viewpoints about universals,i.e., realism, conceptualism and nominalism correspond respec-tively to logicism, intuitionism (or constructivism) and formalismin the philosophy of mathematics. Therefore, the acceptance of aconceptualist ontology would appear as a coherent choice in linewith the intuitionistic type theory that the theory is made of. How-ever, another choice for the ontology is possible, but requires somejustifications.

On the one hand, from Gödel’s second incompleteness theoremand from Tarski’s theorem on undefinability of truth, it is proved


that no sufficiently rich interpreted language can represent its ownsemantics. As a corollary, any metalanguage expressing the seman-tics of some target language must have an expressive power whichexceeds that of the target language. It follows that having selecteda language for logic, an upper level (metalanguage) is required toexpress its semantics. The metalanguage includes primitive no-tions, axioms, and rules absent from the target language, so thatthere are theorems provable in the metalanguage that are notprovable in the object language.

On the other hand, in intuitionistic logic a statement is only true(resp. false) if one can construct a proof showing that it is true(resp. false) and as such, it preserves justification rather than truth(anti-realism in truth value). It follows that (i) the law of excludemiddle, i.e., for any first order proposition P, the axiom P _ :P isno longer valid and (ii) neither is the double negation elimination,i.e., ::P ! P. Some authors [27,28] have advocated for consider-ing different levels (in line with the need of a metalanguage) cor-responding to possibly different philosophical positions, e.g.,realism in ontology vs intuitionism in truth values. If such a posi-tion is tenable, then there need not be a domain of objectively trueor false statements prior to our discovery or assertion of them. Inaddition, it has been shown [29] that semantical realism andanti-realism (e.g., intuitionism) are not contradictory from anintuitionistic point of view.

Therefore, assuming ontological realism with intuitionistic logicyields that universals are seen as pure artifacts which exist in peo-ple’s minds and as such they may result from a realizable construc-tion process at the proof level. Then, the metalanguage can use arealist version of the propositions-as-types principle and theBHK3 interpretation to define the provability relation inductively.As a consequence, if we assume a lower level for the logic adoptingintuitionism and a higher level for the language adopting realismshould result in an expressive while coherent theory. In the lower le-vel language, basic notions are predicates and facts.

Definition 8 (Predicates). A predicate is a function that yields atruth value. Predicates are true of objects and refer to properties aswell as relations.

Definition 9 (Facts). A fact can denote the existence of someobject(s), and/or the existence of a relation between several objectsand/or the holding of some property for an object.

Typical facts include specific instantiated relations or proper-ties. Facts can be nothing other than cognitive constructs, depend-ing strongly on a symbol system.

2.3. Towards situation semantics

Having explained the ontological choices, the next issue to solveis, what do we mean by ‘‘situation’’. From a philosophical perspec-tive, a real situation is the complete state of the world at a given in-stant of time. In fact, what is called a situation in the scope of thispaper refers to a part of the real situation, that is, all the relevantinformation to solve the problem at hand. More precisely, it hasbeen noticed that it is the intention that we have in mind [30](e.g., a goal) which delimits the relevant components of the situa-tion. Most approaches in knowledge representation and reasoningconsider a situation as a set of explicitly known facts, i.e., relationsinvolving objects of the situation. Then, the question arises of whata situation is made of. Some relevant advice can be found in the sit-uation theoretic model [31], such as the treatment of partiality andthe ability to cope with dynamic information. The first formal no-tion of situation has been introduced in what is termed Situation

3 Brouwer, Heyting and Kolmogorov.

Theory (ST) [32,33] further extended in [34,35]. ST is a mathemat-ical theory of meaning in which the basic components are individ-uals, properties, spatio-temporal locations, relations and situations.Individuals are considered as invariants having properties andstanding in relations. The striking feature of ST is that situationsare first class objects on which one can reason about. Invariantsand uniformities express particular views of the reality across situ-ations. Situation Theory has served as a foundation for many worksin situation assessment [30,36–38]. The significant results of situa-tion theory suggest that a situation could be both composed withfacts involving objects of the current situation and constraintsinvolving what is called ‘‘regularities’’ (i.e., types).

In natural language processing, a number of approachesdescribe discourse situations with formal structures known asDiscourse Representation Structures (DRSs) [39]. Discourse repre-sentation theory is based on model-theoretical approaches to lan-guage and formal semantics. In the box notation, DRS consists oftwo parts, an upper part describing the universe of discourse(i.e., a set of variables), and a lower part dedicated to a set of con-ditions. In addition to simple DRS, the theory allows for DRS as partof conditions, i.e., inside the body of a DRS. This possibility of usingnested discourse representation structures is a central property ofDRS since it expresses extensibility without enumerating all theatomic elements that compose the condition. This kind of (implic-itly) nested knowledge structure also appears in the dependentrecord types [40] whose purposes are the same as for DRS. Then,given some knowledge structure abstracting situations, the avail-ability of nested structures appears a significant part of the knowl-edge representation mechanism.

According to this analysis, a small number of properties that asituation theory should comply with can be summarized asfollows:

� From Situation Theory, it follows that the specification ofa situation structure should support a typed frameworkinvolving both types and objects.

� Nested structures also appear as a powerful ingredient tocope with the problem of dependence between knowl-edge structures.

� An ontological framework should be required for a share-able and reusable situation description.

From these guidelines, it seems reasonable to think of the con-cept of situation as a knowledge structure having a grain of appro-priate size to be machine readable in a given application. For thatpurpose, a situation can be divided into parts denoted SituationFragments (SFs) where each SF must be considered as a universaland contains a number of facts, each of them involving objects ofthe domain that are relevant at hand. An intensional theory is wellfounded to fulfill these properties. After this informal presentationof what a situation should express in a modeling perspective, weanalyze in the next section how a formalism based on type theorycan be an appropriate solution to the problem of situation model-ing in intelligent systems.

3. Requirements for a type theoretical approach

3.1. A need for a computable theory

In the Fregean perspective, a concept is a function whose inputvalue is an object and whose output value is always a truth value.Concepts are proved if and only if the object falls under the con-cept. For example the meaning of ‘‘John drinks’’ can be representedby a function argument expression A(x) where A denotes a function(‘‘drinks’’) and x an argument to that function (e.g., ‘‘john’’). It is

6 Impredicativity occurs whenever type definitions are circular, i.e., they areinvolved themselves in the definition.


well known that one way to make this syntax computational is bymeans of lambda notation. The function expression of this exampleis then written as kx � A[x], in which [x] denotes the usual b-conver-sion.4 Therefore, it seems promising to consider concepts, and a for-tiori knowledge, as the result of a lambda calculus. We follow thatline and consider concepts, and relations, as the result of a lambdacalculus. In a more concrete way, it means that the theoreticalframework should be supported by a functional language (i.e., animplementation of the lambda-calculus). Whereas the idea of typingfirst appears in the Principia Mathematica of Russell, a typed versionof the lambda calculus has been proposed by Church in 1940.Church’s type theory is a formal logical language which includesfirst-order logic, but is more expressive in a practical sense. Alterna-tively, a typed version of the lambda calculus with two basic types,i.e., e for individuals and t for truth values developed in [41] has beenwidely used in natural language processing [42–45]. However, it hasbeen noticed that the expressiveness of such a theory could be sig-nificantly enhanced with the help of an extended collection of types[46], and more attractively, with dependent types [47,40,19]. Assum-ing a type-theoretic framework for concepts and relations, type the-ories seem an adequate tool to express situation components sincethey precisely include a computational basis (lambda calculus) to-gether with a sophisticated typing mechanism with dependent typesuseful for expressiveness.

3.2. A need for strong logical foundations

The most influential and foundational idea is known as theCurry-Howard isomorphism. It states an analogy between two for-malisms, i.e., (i) a formalism for expressing effective functions (thek-calculus) and (ii) a formalism for expressing proofs (naturaldeduction for intuitionistic logic). Type theories make use of thisisomorphism stating that propositions are seen as types whileproofs are seen as objects. In other words they establish an ‘‘equiv-alence’’ between a computation system (i.e., a lambda-calculus)and a (intuitionistic) logical theory. Type theories have explicitproof objects which are terms in an extension of the typed lamb-da-calculus while at the same time, provability corresponds to typeinhabitation.5 Proofs in a computational setting can be the result of adatabase lookup, the existence of a function performing a given ac-tion or the output of a theorem prover, given assumptions aboutentities, properties or constraints. Having a proof for a propositioncomes very close to say that the proposition is true. However, a proofis built up of premises, and using different contexts, the same asser-tion can have different meanings. A proposition is identified with theset of its proofs rather than verifying worlds as in FOL. Since there isan equivalence between a logic and a typing mechanism, the use oftype theory replaces logical derivations with (computational) typingreductions.

Furthermore, as explained for example in [48], type theoriesmust enjoy two fundamental properties (i) the decidability of typechecking and (ii) the existence of canonical forms. Establishingthese meta-theoretic results are usually not easy, however in S-DTT the property of type checking has been proved for the ECC part(see [49] for more details) while the property (ii) requires no proofsince we have only canonical forms in our system.

3.3. A need for stratified abstraction layers

There are different versions of type theories. The main differ-ences are reflected in the different structures of their conceptualuniverse of types. A type theory can be understood in a hierarchical

4 The process of substituting a bound variable in the body of a lambda-abstractionby the argument passed to the function whenever it is applied.

5 A proposition is true iff its set of proofs is inhabited.

way: one starts by introducing various basic types and using typeconstructors, builds up more complex types. Every object in S-DTThas a type and every type has itself a type but to avoid inconsis-tency, a hierarchical structure is needed. There is not a type of alltypes, instead there is an infinite sequence of universes Type0:-Typel:Type2: . . . , which approximate the type of all types. Noticethat Russell in the Principia Mathematica thought it necessary tointroduce a kind of hierarchy, the so-called ‘‘ramified hierarchy’’.One should make a distinction between the first-order properties,that do not refer to the totality of properties, and consider thatthe second-order properties refer only to the totality of first-orderproperties. This clearly eliminates all circularities connected to im-predicative definitions. The major impredicative6 types theories arethe system F [50] and the Calculus of Constructions (CC) [15].Quantification is allowed over all propositions or types to form anew proposition or a new type. However, the calculus of construc-tions that offers two universes, one for data types (Set) and one forlogic (Prop), has some difficulties to represent data types with prop-ositions and does not provide a subtyping mechanism. Alternatively,in predicative type theories such as the well-known Martin-Löf’stype theory [14], one loses the possibility to incorporate strongerlogical mechanisms such as higher-order logic while in impredica-tive type theories, the impredicative definitions of data types aretoo weak. With respect to these difficulties Z. Luo has developedthe Extended Calculus of Constructions (ECC) [49] which representsa synthesis between CC and Martin-Löf’s type theory. ECC is an im-predicative type theory having an impredicative universe Prop forpropositions and a predicative hierarchy of data types in the spiritof Russell. It provides a higher-order logic with polymorphism (forProp with quantification over all propositions), expressive data typeswith strong sums and a concise system of rules. According to theseproperties, ECC will form the core of our approach. A crucial propertyof the ECC language is that it allows to represent and reason7 aboutknowledge structures in a formal way called universe hierarchy. Thetheory comprises an infinite hierarchy of predicative type universesdenoted Type0, Type1, . . . and an impredicative universe noted Prop.8

The universes Typei are the universes for data types while Prop is theuniverse for logic. The hierarchy is cumulative, that is, Typei is con-tained in Typei+1 for every i. A universe is seen as a type that is closedunder the type-forming operations of the calculus. Since impredic-ativity exists only for Prop,9 Typei may contain only types from uni-verses Prop, Type0, . . . , Typei�1 while Prop may be constructed withtypes from Prop, Type0, . . . , Typei, . . . . In a type system with uni-verses, it is possible that something could be both a type and an in-stance of a type. For instance, a person is mapped to a type Personsince one can define objects of that type, while being at the sametime of the type expressing the category Agentive Physical Objectw.r.t. the terminology of the DOLCE ontology (see Section 3.4).

3.4. A need for an ontological support

Modeling languages should be founded on upper-level ontolo-gies and these ontologies must be themselves logically founded[51]. For this purpose, assuming that there exists a mapping be-tween an ontology and a type theory, a hierarchy of categories cor-responds to a hierarchy of types that assigns each category to atype (more precisely to a universe). Since types correspond tothe result of a categorization procedure, they have a natural ade-quacy with ontologies. According to the Aristotelian conception

7 With a logic that is internal to the language.8 The fact that Prop is impredicative, means that one can quantify over propositions

to form a new proposition.9 Prop is seen as an object of Type0.

Fig. 1. The structure of the S-DTT Theory.


of universals, types are justified in the sense that there are no unin-stantiated universals just as there are no non-inhabited types. Itfollows that universals are represented with types while general-ization is interpreted as formal subsumption. Then, the questionarises what formal ontology will best support our ontologicalrequirements. We have considered the DOLCE ontology of particu-lars [52,53] because (i) it adopts both a descriptive and multiplica-tive approach10 which does not classify universals leaving room forconceptual choices about universal structures, (ii) it is designed to beminimal, in that it includes only the most reusable and widely appli-cable upper-level categories and (iii) it remains neutral about thespatio-temporal properties (see [52] for more details) and providesa significant level of interoperability. However, as stated in Section2, the logical level we are using is constructive which is not the casein DOLCE. Moreover, we have also advocated in the same section, fora common part of the domain ontology that would be provided by aformal ontology. The formal characterization in DOLCE relies on FOLand should require quantifications over properties to enhance itsexpressive power. For instance, we can reason about meta-proper-ties using specifications (see [67] for more details). Due to the lackof space, this aspect is not explored here. The logical part of DOLCEis not considered here (only its taxonomy is used). The obvious con-sequence is that relations are expressed in S-DTT using the core ECC(for example, the hierarchical relation is_a is derived from the sub-typing rule). The hierarchical taxonomy of DOLCE will serve as abackbone, referred to as DOLCE backbone, for defining the stratifica-tion which allow more fine grain categories in the final domainontology. In addition, DOLCE is an ontology of particulars in thesense that, in the domain of quantification there are only particulars.Alternatively, in S-DTT the domain of quantification both considersparticulars and categories of particulars. Excepted in the particularextension DOLCE-CORE, DOLCE does not have an instantiation prim-itive, whereas it is already part of the core theory ECC in S-DTT. Prop-erties and relations in S-DTT are considered as universals like inDOLCE. In S-DTT, universals are organized and characterized bymeans of meta-properties without leaving the theory. They can rep-resent either ontological commitments over the categories whichappear in the DOLCE backbone or user-defined constraints in the do-main ontology. Finally, the DOLCE backbone plus ontological com-mitments on appropriate structures expressed with ECC will formthe S-DTT theory (see Fig. 1). The S-DTT ontological part can be usedto express knowledge as long as the added features respect the corestructures together with their logical constraints.

Once data structures are defined, there is a need to discuss howthe hierarchies are constructed. We have two different perspec-tives to consider, a classification of universals according to theirset of properties (subsumption) and a classification of relationsw.r.t. their structure. Following Definition 6, the mapping fromuniversals to types together with the intensional nature of typesargues for describing the subsumption between universals with

10 Different entities can be co-located in the same spatio-temporal location.

the ordering relation between their corresponding universes, i.e.,the cumulativity relation (see Section 4.2.2).

Due to their extensional nature, partonomic relations addressthe latter using a classification of the dependent types which rep-resent relations with regard to the extent of their structure (i.e., atthe meta-level).

3.5. Towards a situation profile

In summary, the type theory offers several characteristics thatshould be relevant with regard to properties of situation awareness:

� it is computable (e.g., using functional programming) to guaran-tee a result for situation assessment,� it has the power of higher-order logic to provide high-level

reasoning,� it provides a high level of expressiveness with the help of

dependent types,� multiple abstraction layers are defined with the perspective of

knowledge subsumption (and reasoning) at different abstrac-tion levels.

We argue that a situation as a whole is too wide for a precisedescription as well as too difficult to delimit and we rather suggestto consider Situation Fragments (SFs) as the bricks for a modular,i.e., a partial description system. One main benefit is that SFs canbe composed to form more complex situations. Furthermore, SFscan be related to an intentional meaning such as a goal, leadingto a more comprehensive model according to the higher layer ofEndsley’s model.

Clear relations must exist between Endsley’s model of situationawareness and the ontological concepts. As underlined in a recentontology-based approach [4], the comprehension layer proposedby Endsley should recognize the role of relations as the basic ingre-dient of situation assessment. In type theory, relations reflectdependencies that are expressed with dependent types. In otherwords, proof objects from the perception layer yield more complexproofs for dependent types reflecting concrete relations. Further-more, new (dependent) types (i.e., new relations) may be provedby derivation from the previous proofs which, thanks to the Cur-ry-Howard isomorphism, corresponds to what is known as infer-ences. This aspect will be detailed in Section 6.

4. Expressing knowledge structures with S-DTT

In the logical part of the S-DTT theory, predicates and functionsare defined for establishing relations between objects of the do-main without any assumption on the nature of these relations.The ontological part characterizes the meaning of basic ontologicalcategories in the domain by restricting the number of possibleinterpretations. The present work includes appropriate structuresexpressed with subtyping definitions giving rise to what is knownas a computational theory [49]. For that purpose, S-DTT uses agrammar in which properties and relations are assigned types ofincreasing levels, assuming that these types are arranged in ahierarchy.

4.1. Preliminaries

Each typing assertion is made with first checking its context.The context is a finite sequence11 of expressions of the form xi:Ti

where xi is a variable and Ti a type. That is, given a context, a type

11 The sequence is ordered because any type in the sequence may depend on theprevious variables.


theory provides rules called judgments for determining whether aparticular variable, say x, belongs to a given type, say T. The contextC , x1:T1, . . . , xn:Tn in a judgment:

C ‘ x : T

contains the prerequisites necessary for establishing the statementx:T. We call x an inhabitant of T, and T, the type of x. A term can beeither a variable, a type or even a universe Typei.

Definition 10. A term M is well-typed if for some context C wehave C ‘M:T for some T.

Definition 11. A term T is inhabited in the context C if C ‘M:T forsome M.

To show that M:T holds in a given context C, one has to showthat either C contains that expression or that it can be obtainedfrom the expressions in C with the help of type deduction rules.

Definition 12. A type T is well-formed if for some context C wehave C ‘T:Typei for some universe Typei,i P 0.

For example, the judgment Phone:Type0‘x:Phone asserts thatthe variable x is of type Phone provided that Phone is a well-formedtype. In the following, we often forget the context for the sake ofclarity, however it must be kept in mind that any assertion is rel-ative to an explicit context involving all the arguments of thenew defined type.

4.2. Representing ontological components

The characteristic feature of dependent type theories is thatthey allow a type to be predicated on a value. This property makesthem much more flexible and expressive than conventional typesystems [54]. The key notion is that a family of types is indexedby a term (having itself a type). Wherever we speak about ‘‘a familyof sets’’ indexed by some other set, we refer to a dependent type.The core ECC of the S-DTT theory admits two dependent types,the dependent product or P-type and the dependent sum or R-type. The non-dependent versions of these types are respectivelynoted ? (functions) and � (non-dependent pair). Types are closedunder dependent products and (strong) sums. Elements are depen-dent functions and dependent pairs, respectively. Dependent sumsmodel data structures while dependent functions model a kind ofinferences. The logical consistency12 of the core theory has beendemonstrated and the decidability property has been deduced as acorollary [49]. We will now see how these dependent types are usedto represent most of the ontological components.

4.2.1. Representing particulars and universalsThe two basic entities of ontologies, i.e., particulars and univer-

sals are respectively understood in terms of proof objects and typesat the logical level. Universals are described (i) with non-depen-dent types for categories, (ii) with product types for propertiesand (iii) with sum-types for relations. For instance, the category‘‘person’’ is described by the type Person in Person:APO‘x:Personwhere APO stands for Agentive Physical Object and is assigned auniverse. The type Person may correspond to the proof (instance):x = JohnDoe while Person:APO asserts that Person is a well-formedtype.13

In FOL, the domain of interest consists of objects and unarypredicates correspond to categories, e.g., Person(x) where x standsfor a given value. If the predicate is true for this value, then it is

12 For a type theory, the logical consistency is identified with termination.13 Notice that the consistency of the universe levels is usually under the respon-

sibility of a proof assistant e.g., Coq.

proved that x represents a person. Here is a significant differencebetween FOL and type theory. In FOL, values are input and thepredicates are checked returning appropriate truth values. In typetheory (and in S-DTT in particular), types come with their own con-structors and checking a type boils down to construct an object ofthat type. In a S-DTT implementation, type constructors wouldread values in a database and output the corresponding values (ifthere are some).

Notice that the relation of instantiation between a universal andits instance corresponds here to the process of type inhabitationinvolving respectively a type and a proof object. Unlike FOL-basedontologies, it describes in a natural and simple way the relation be-tween universals and particulars without the need to introducespecialized ad’hoc formulations such as those of [25]. The S-DTTontology constructed with the DOLCE backbone is also used topopulate the context and to give it a semantics in order to con-strain the semantics of universals or more precisely, of situationtypes, and then to trigger specific assessments when appropriatesituation proofs exist.

4.2.2. Representing subsumptionIn S-DTT, subsumption-based hierarchies take advantage of the

simple mechanism of the cumulative hierarchy of type universes.Each time we are interested in a certain collection of types thatshare some common properties, then universes can be introduced.Every type that belongs to a universe is considered as an object ofupper universes. For example, in the Coq proof assistant [55], thesystem itself generates for each new category which is an instanceof ‘‘Type’’, a new index for the universe and checks that the con-straints between these indexes can be solved. The hierarchy of uni-verses is ordered by the cumulativity relation ^also calledsubtyping in the following part of the paper. The cumulativity rela-tion is a direct consequence of the existence of stratified universes.

Lemma 1 [49]. The cumulativity relation ^is the smallest partialorder over terms such that Prop^Type0^Type1 ^ � � �^Typei.

Where Typei are type universes. The main rule defines theresulting universe of a dependent type as the maximum universebetween the universes of its components. The is_a hierarchy ofDOLCE can be represented by a cumulative hierarchy of categories,where each node in the tree is assigned a universe, the highest uni-verse being PT the category of particulars. It follows that the DOLCEbackbone generates a core hierarchy of universes whose upper uni-verse is always PT while the lowest universe depends on the user’schoices. For instance, let us consider some basic categories such asEating, Reading, and ChemicalReaction that we expect to relate tothe DOLCE backbone of type universes. The categories Eating, Read-ing, ChemicalReaction are specialized types of perdurants. Then, thesmallest level is assumed to be a universe including (at least) Eat-ing, Reading and ChemicalReaction whereas this universe is referredto as PRO (processes). At the same time PRO is a subtype of stativeperdurants (STV) themselves subtypes of perdurant types PD. It fol-lows that:

Eating ^ PRO ^ STV ^ PD

The same relations hold for Reading and ChemicalReaction. Inother words, Eating represents all processes consisting in eatingsomething, PRO denotes all processes, STV focuses on all stativeperdurants while PD addresses all categories of perdurants. How-ever, each new type insertion within this classification preservesthe original order but will automatically increment each universeindex for the upper levels. For example, if one suppose that Eatingbelongs to the universe Typei and that there exist different kinds ofEating, like StuffOneself, Devour, NibbleAt, etc. then the insertion of

14 Unlike Martin Löf type theory, sum types are not logical propositions and cannotrepresent the logical existential quantifier.

15 Usual parenthesis have been removed for improving the readability of complexexpressions.


these categories (which are subsumed by Eating) will assign themin the universe Typei, Eating to Typei+1, etc. One will write:

StuffOneself ^ Eating

Devour ^ Eating

NibbleAt ^ Eating

It results that the previous hierarchy of universes is shifted onelevel high with the introduction of this new level and preserves theprevious classification of DOLCE categories. Subsumption in FOLcan be directly expressed by a logical implication, which is illus-trated in the translation of the following fragment. The logical for-mula "x: (Student(x) ? Person(x)) where the variable x ranges overall domain objects will express that every student is also a person.In S-DTT, subsumption operates at the level of types, that is typesare considered as objects which can be more easily manipulatedand reasoned about. For example if a type represents a complexrelation instead of a simple category like Person, then generic rulescan be applied and enhance the accuracy of the reasoner accord-ingly (see Section 6.3.2).

4.2.3. Representing propertiesOntological properties concern specific attributes of particulars

and we view properties as universals which characterize particu-lars. Therefore, a natural way to express them is to introduce prod-uct types taking these particulars as arguments and yielding eithera type Prop or a type which belongs to a universe Typei within thehierarchy. Since type theory is used to classify terms of the k-calcu-lus, it can also deal with functions. Dependent function types are ageneralization of function types. A dependent function type is afunction type where the range of the function changes accordingto the object to which the function is applied. The notation fordependent function types is P x:A � B. If we apply an inhabitant fof this function type, to an object of type A, then the resulting ob-ject f a is of type B. In B, all free occurrences of x are substitutedwith a. Then, from a logical perspective, we are interested in prop-ositions as the constituents of deductive arguments. In type theory,we reason from the proofs that we (assume to) have for the pre-mises to a proof for the conclusion. As a consequence, the counter-part for universal quantification "x 2 A:P(x) is the dependentfunction type Px:A � P. A proof for this type is a function f which,given any object a:A, returns the proof f a for P a. In other words,given a proof object a of type A, the product type asserts that thereis a corresponding type B[a] whose proofs are the result of its con-structor. For instance, to represent the fact that the property ‘‘asensor provides a physical magnitude’’ holds, one introduces:

Px : InfraRedSensor �Py : Motion � Detectðx; yÞ

in which Detect is a predicate (i.e., a function InfraRedSen-sor ? Motion ? Prop) which for each instance x (e.g., s1276) of thetype InfraRedSensor and each instance of the type Motion (e.g.,Mov2) yields a proof object for the type Detect(s1276, Mov2). Here,it means that all infrared sensors are able to detect movements.

It is commonly accepted in ontology modeling that a property isa universal that is represented as a unary predicate in FOL, e.g.,Round(x) which describes something that has a circular shape. Infact, the variable x in this predicate stands for anything andensures a generic view. In type theory, we are ascribed to providethe type of the object having this property, e.g., Px:Wheel �Round(x). As a consequence, the price to pay for the gain of accu-racy is the loss of generality.

Subtyping must not be confused with product types, even ifthere are some similarities in the objective. For example, the state-ment Eating ^ PRO which asserts that Eating is a subtype of PRO,yields that for each variable x:Eating the type PRO can be used aswell. Now, if one write the product type Px:Eating � PRO, a com-

pletely different result is obtained. It generates a family of types in-dexed on a proof object of type Eating. If Devour is such a proofobject, then a proof object for the type PRO(Devour) describesany process running, provided that there is something which is de-voured at the same time.

4.2.4. Representing relationsAt the ontological level, relations can be hierarchical (e.g., sub-

sumption or partonomic relations) or non-hierarchical (e.g., do-main relations). They denote tuples involving particulars inwhich the last term is generally the proof for a proposition. Sincetuples correspond to nested sum types, they require nested sumtypes structures. Most reasoning with ontological relations isbased on subsumption and partonomic relations. From the logicalperspective, subsumption will use subtyping whereas partonomicrelations rather involve subset relations between proof objects(i.e., particulars).

Binary relations between universals are described with (strong)sum types, i.e., types of pairs of objects. For any type A and anyfamily of types B[x] indexed by an arbitrary object x of type A,

Rx : A � BðxÞ

is the type of pairs ha, bi where a is an object of type A and b is oftype B[a]. Intuitively it represents the set of (dependent) pairs ofelements of A and B[x]:

ha; bija : A; b : B½a�

When B is a predicate over A, it expresses the subset of all objects oftype A satisfying the predicate B.14 Dependent sums model pairs inwhich the second component depends on the first. Let us considerthe sum type Rx:Person � Apartment(x) with Apartment:Person ?NAPO and NAPO, a non agentive physical object. A proof for this typeis given for example by the instance hJohnDoe, q1i indicating that forthe individual JohnDoe, the second type is proved, i.e., q1 is a proof ofApartment(JohnDoe).

hJohnDoe; q1i : ðRx : Person � ApartmentðxÞÞ ð1Þ

The proved pairs express a subset of the set of all persons andcharacterizes the persons having an apartment. In each pair s,appropriate rules [49] extract the first and the second componentwith the respective functions p1s and p2s.15 In the above example,with s , hJohnDoe, q1i, we get p1s = JohnDoe and p2s = Apartment(JohnDoe) where Apartment(JohnDoe) is the type of apartments thatdepend on the particular JohnDoe. It is worth noticing that we candiscriminate between proof objects for the type Apartment(JohnDoe)and that they are countable.

In S-DTT, sum-types are used to construct relation types accord-ing to the following definition.

Definition 13 (Binary relation). A binary generic relation type Relbetween two universes is expressed with a sum type having thesetypes as arguments and whose extension consists of all the proofobjects for that relation [56].

Rel,RT : Typei � RT 0 : Typei � RðT; T0Þ ð2Þ

This definition is general and applies at different levels ofabstraction w.r.t. subtyping. For example to express that someagentive physical object (APO) can be involved in a process (PRO),one introduce a generic relation type:

INVOLV,Rx : APO � Ry : PRO � InvolvedInðx; yÞ ð3Þ

16 They are dependent types whose arguments are given from the outside of thedefinition.

17 Notice that it is the relative durations that are compared: they are specifiedwithin the programming part of the constructors of appropriate types.


while at a lower abstraction level we may have:

S1,Ru : Patient � Rv : WatchTV � InvolvedInðu;vÞ ð4Þ

with Patient^APO and WatchTV^PRO. The types Patient and Watch-TV are at the Typei level while APO and PRO are in Typei+1. Let usmention the relation ParticipateIn described in [57]. Individual qual-ities are elements of the DOLCE backbone so that one can refer tothem directly in formal expressions. In the relation type, the prop-osition ParticipateIn relates a first argument that must be an endu-rant (ED in DOLCE) together with a perdurant (PD). As explained inSection 4.2.2, these arguments belong to Typen�1 with respect to thehighest level for PT (i.e., Typen), then according to the ECC rules, thetype of the whole relation type must be at least in Typen�1.

ðRx : ED � Ry : PD � ParticipatesInðx; yÞÞ : Typen�1

The FOL translation of a n-ary relation R is usually seen as any sub-set of a product of n sets S1, S2, . . . , Sn such that: R � S1 � S2 � � � � �Sn. It follows that binary relations are mapped to binary predicates.For example, the relation (4) would be written Patient(x) ^Watch-TV(y) ^ InvolvedIn(x, y) in FOL. Relations can only be exemplifiedby particulars while in S-DTT relations are types. The first benefitof S-DTT consists in a more powerful reasoning about relationswhich relies on the mechanism of subtyping with relations. A sec-ond advantage of S-DTT over FOL is that the second argument of abinary relation type may depend on the first (see, e.g., relation(1)). If it is the case, it means that instead of looking for all the val-ues corresponding to y, the search is restricted to the values of y re-lated to x, i.e., a subset of the set of values for y. In such a way, thesearch space is decreased as well as the algorithmic complexity.

Let us illustrate the use of dependent types on a challengingproblem in OWL-DL, i.e., the so-called uncle problem. This problemcan be stated as follows: if the properties hasParent(Bob, Mary) andhasBrother(Mary, Bill) hold, then one should conclude hasUncle(Bob,Bill). The difficulty here is to refer to the right object in the rule, andfor that purpose, let us introduce a first relation:

R1,Rx : Person � Ry : Person � hasParentðx; yÞ

Now, we introduce a second relation R2 but the particularity isthat this relation depends on the first. This aspect is important hereto guarantee that we can refer to the right object.

R2,Rx : R1 � Ry : Male � hasBrotherðp1p2x; yÞ

Finally, using a product type acting on the previous relation, wecan express the quantification over the right values. Given a pairr2:R2, in the first argument of the predicate hasUncle, the termp1r2 refers to the left component of r2, i.e., an object of type R1

and p1p1r2 refers to the left component of p1r2, that is the left com-ponent of the pair having the type R1 which is an object of type Per-son. A similar explanation holds for the second argument ofhasUncle.

Px : R2 � hasUncleðp1p1x;p1p2xÞ

Suppose that a proof (i.e., a triple) for the relation type R1 is hBob,hMary, q1ii with q1 a proof for hasParent(Bob, Mary). Then a prooffor the relation type R2 should be hMary, hBill, q2ii with q2 a prooffor hasBrother(Mary, Bill). Notice that according to the first proof,the first argument value of the predicate hasBrother is necessarilythe same. With respect to these proof objects, one can conclude thatthe predicate hasUncle(Bob, Bill) holds.

4.3. Representing logical assumptions

4.3.1. Representing propositions, facts and predicatesWe follow here the Russelian view in which the logical primi-

tives of our thought about the world correspond to the most basic

ontological categories. At the ontological level, a proposition is asentence expressing that something has a truth value. The termsentence is used instead of proposition to refer to just those stringsof symbols that are truth-bearers. For instance, we can get a proofp0 (in a database) that a computer is connected with Con-nected:Prop‘p0:Connected. At the logical level, they represent typeswhich are of the type universe Prop.

Facts are akin to states of affair in the world. Therefore, in an onto-logical view they are the concrete objects of the world which we canspeak about and they refer to an instantiation of universals. At thelogical level they are proof objects for types (dependent or not).

The other kind of primitive in language and thought are n-placepredicates (with n = 1, 2, . . .), which correspond to properties andrelations. A predicate is the result of combining k names in a k-place relation. They can represent roles at the ontological levelsuch as in Person:APO‘Student:Person ? Prop where Student is atype denoting a role. At the logical level, predicates are dependenttypes whose output is of type Prop.

4.3.2. Expressing constant physical valuesFor physical measures, two types must be introduced, i.e., a type

for the measured quantity (e.g., temperature) and a type for its unit(e.g., ’’Kelvin’’). Furthermore, the measured quantity depends onthe unit and this aspect can easily be captured with dependentsum types as follows. Given the respective types T:Typei, U:Typei

for the measured quantity and its unit, the constant measuredquantity k:T and the unit value u0:U are formalized with parame-terized16 sum types:

Uðu0Þ,Rx : U � Eqðx;u0Þ ð5ÞTðk;u0Þ,Rx : Uðu0Þ � Ry : TðxÞ � Eqðy; kÞ ð6Þ

where Eq(x, u0) and Eq(y, k) refer to the computational equality. Letus prove a situation in which a person is on his bed while the timeinterval since the beginning of this situation is greater than a prede-termined duration (e.g., 8 h). For that purpose, we introduce a con-stant type expressing the time unit (UTC) that will be used both inthe constant type (Time(8, UTC)) and in the type OnBed. The bed sen-sor provides a proof object for the predicate IsPressed witnessingthat there is somebody on the bed.

UðUTCÞ, Ru : U � Eqðu;UTCÞTimeð8;UTCÞ, Rx : UðUTCÞ:

Ry : Timeðp1xÞ � Eqðy;8ÞOnBed, Rx : Apartment � Rb : BedðxÞ � Ru : UðUTCÞ�

Rt : Timeðp1uÞ � IsPressedðb; tÞ

Then a type expressing that the patient remains in its bed morethan a limit duration (8 h) can be written17:

LongLyingDown,Rx : OnBed � Rt : Timeð8;UTCÞ:GreaterThanðp1p2p2p2x;p1p2tÞ

4.4. Expressing situation structures in S-DTT

In the perspective of situation awareness, one can define typicalsituations, i.e., situation types, and check whether the actual situ-ation complies with a given definition. However, it is not possibleand even not desirable, to assess the whole situation in the worldat a given time, but only a part of it, a part that is relevant for ourpurpose. In Situation Theory (ST) [32], the authors take particulars


and universals (i.e., properties, relations and locations) as buildingblocks of the theory. In S-DTT, these blocks are respectivelymapped on proof objects, product types and sum types for the lastpair. In addition and in the spirit of ST, we view a situation type asan abstract state of affairs that may be instantiated during situa-tion assessment.

Definition 14 (Situation Fragment). A situation fragment type,denoted SF type is described by a sum-type having as left term(s)some typed component(s) of the physical environment and rela-tions that can be nested according to the grammar:

SF ::¼ RV1 : T1 � . . . � RVn : Tn � Pð~c1V1; . . . ; ~cnVnÞjSF � SF

where n P 1, Ti are types and P, a proposition. If Ti is a sum-type,then it can be nested up to a finite number of levels and each ~ci

stands for a sequence of p1 or p2 relative to variable V i for nestedsum types or an empty sequence for non-dependent types. In therest of the paper, we assume that expressions such as ~ci will alwaysrefer to finite sequences of p1 or p2. Situation fragments are nothingelse that partial situations that can be easily extended as we will seein the next subsections. Since SFs are formal objects in the seman-tics which denote states, i.e., ST in the DOLCE backbone, they canbe referred to with constants in the logical language and variablescan range over them, e.g., Si:ST for any situation fragment Si. Thereare many ways to supply symbolic information relative to asituation such as natural language (e.g., documents), sensors, orweb-based applications. According to their purpose, SFs are closeto what is termed as ‘‘parametric infons’’ in ST whereas proofobjects for SFs are close to parameter-free infons.

The ontology allows to formally define the SF types of interestand check whether the design is consistent with the agreed con-ceptual model the ontology provides. Furthermore, any situationwhich corresponds to proved SF types must assume that they arenon contradictory.

Proposition 2 (Consistent SF type). Let S a given SF type, then S isconsistent if for any predicate Pi occurring in S we cannot havesimultaneously Pi ? \.18

Proof. Straightforward by induction on the structure of SF types.h

Then, the question arises of what makes a SF relevant for a sit-uation? For example, considering a battlefield environment, thecurrent exchange rate of the yen has nothing to do with the situa-tion that we are interested in. As underlined in [30], a recurrentproblem in situation awareness is determining all the relevantrelations in a situation, that is all the proved relations in that situ-ation. This aspect will be explored in the following subsection. Fur-thermore, the author explains that the goal is a crucial piece ofknowledge since ‘‘it provides us with a handle on what is relevant’’.We fully agree with that idea and claim that SFs could be filteredon the basis of their implication to sustain a global goal. This filter-ing requires that objects present in the goal will be inherited in theSFs of the application. For that purpose, there is a need to introducethe concept of well-formed SF.

Definition 15 (Well-formed Situation Fragment). Given a goal typeexpressed by a proposition depending on typed arguments, a datastructure is a well-formed SF type if these arguments are part ofthe (nested) arguments of the data structure.

Since the structure of a SF is generally nested, the goal actingupon it can refer to any variable occurring in the nested types ofthis SF.

18 i.e., the negation of Pi.

Let s be a given (real) situation in the scope of ST. An abstractsituation is described on set-theoretical basis by {ijs�i} where i de-notes an infon that is true of situation s. In some words, an infon isthe representation of a state of affairs with a n-ary relation. Argu-ments of the relation are filled with relevant objects together witha polarity saying whether the relation holds or not. For example, let_x and _y the respective parameters which hold for a person and ahealth center. Then the parametric infon � Patient; _x; _y;1 de-scribes the facts in which a person is a patient of the health center.Parameters correspond to what we have called types. Moreover,partial functions from the domain of parameters to objects are ap-plied to parametric infons and replace the parameters with the ob-jects of the domain. This mechanism is more naturally taken intoaccount with type inhabitation which relate proof objects to types.To some extent, the semantics of SFs is similar to those of infons. InS-DTT, this infon should be merely described with the proof objectsof the type:

Rx : Person � Ry : HealthCenter � PatientOf ðx; yÞ

Once partial situation structures have been introduced, there is aneed to explain how we can play with them. For that purpose, weintroduce the subsumption and the partonomic relations. With re-spect to these relations, we distinguish two major operations, i.e.,(i) how situation types can be extended and (ii) how reasoningcan be achieved with situation types. We will show in the followingsections that situations can be extended on the basis of partonomicrules and that reasoning about situations requires inferences rules,some of them being based on subsumption (see Section 6.3).

5. Situation assessment with partonomic hierarchies

5.1. Extending relations

One major interest of SFs relies on the possibility to extendthem in order to obtain more general views of a situation and leav-ing room for a compositional model. Since SFs are built on rela-tions, the first step is to introduce the nested sum types whichare required to extend relation types.

Definition 16 (Nested sum type). A sum type A is a nested sumtype if one of the following definitions holds:

(i) A ,Rz:(Rx:A0 � B0) � B or Rz:C0 � B with C0 ,Rx:A0 � B0(ii) A ,Rz:A0 � B0 with B0, a nested sum type

Notice that in the case (i) the left definition involves a local def-inition of a sum type (i.e., the type Rx:A0 � B0 has only a local exis-tence) while the right definition requires an external sum type(i.e., C0 resides in the context). In most cases, a nested sum typeis a sum type having another sum type as the type of its argu-ment(s). For example, the following types express a situation inwhich a fracture is located in the shaft of femur while at the sametime, the shaft of femur is a part of the femur.

S1,Rx : Fracture � Ry : ShaftOfFemur � hasLocationðx; yÞS2,Rx : S1 � Ry : Femur � PartOf ðp1p2x; yÞ

It follows that S2 is a nested sum type depending on the sumtype S1. Notice that the variable y in S2 is independent from thevariable y in S1 due to the variable closure inherent to each sumtype. Nested sum types are a way of propagating the closure,through sequences of pi with i = 1, 2, a mechanism that is usefulfor reasoning. It is also worth noticing that the quantification inS2 is restricted to the shafts of femur in which a fracture holds, withthe following consequences: (i) it reduces significantly the searchspace as advocated before and (ii) we can see that the individuals


addressed by two situations (e.g., S1 and S2) share the same iden-tity. In such a way, relation types can be nested on multiple levelsexpressing ramified dependencies.

5.2. Extending SFs with partonomic relations

SFs can also be classified according to their knowledge contentwith the partonomic relation. The partonomic relation betweenSFs S and S0 says that information that holds in S (i.e., a part), alsoholds in S0 (i.e., its whole). We assume that for the SF partonomy,the partonomic relation is transitive. In other words, in partonomicrelations, situations are constructed incrementally from an existingone as in the nested sum types (see Definition 16). Unlike the indi-vidual part-of relation, the partonomic relation is atemporal. A uni-versal is partonomically included in another universal if and only ifthey have common extensions at the same times and, at thosetimes, the extension of the first is partonomically included in theextension of the second. It is pointed out in [24] that not everytooth is part of a dental arcade and not every cell has a nucleusas one of its parts. It is precisely the property of sum-types to de-fine subsets (the relation is only proved for some values) and moregenerally the property of dependent types to index the second typeon a value of the first.

Definition 17 (SF partonomy). A SF type S is part of another SF typeS0 with the notation: S v S0 if one of the following definitions holds:

– S , S0

– S0,Rx1 : T1 � . . . Rxn : Tn � Pð~c1x1; . . . ; ~cnxnÞ and Ti = S for somei 2 {1, . . . , n}.

As a consequence, the partonomic relation can be seen as anextensional relation in which the set of proof objects of a typeare included within the set of proof objects of its subsumed type.Let us consider first a sum type S1 expressing that some personhas some medical condition (MC) and then another sum type S2

asserting that some of the previous persons take drugs:

S1,Rx : Person � Ry : MC � HasConditionðx; yÞS2,Rx : S1 � Ry : Drug � UsesDrugðp1x; yÞ

Then, since S2 includes a variable x of type S1, we can concludethat S1 v S2. In S-DTT, the partonomic relation may hold betweenquantified terms such as S1 and S2 while at the same time beingintegrated in a full higher-order predicate logic.

Proposition 3. The partonomic relation between SF types is a partialorder.

Proof. A relation is a partial order if it is (i) reflexive, (ii) antisym-metric, and (iii) transitive. From Definition 17, it appears that the‘‘equal’’ part of the relation argues for a reflexive relation. Obvi-ously, if S0 v S and S v S0, it follows that S and S0 describe the sameSF, and then, the partonomic relation is antisymmetric. Finally, itresults from Definition 17 that the transitivity holds since eachSF is constructed incrementally from its parent SF. As a conse-quence, the v relation is a partial order on SF types. h

5.3. Partonomic hierarchies

According to Definition 17, the SF type S0 extends S with newinformation which does not correspond itself to any SF. Thisassumption excludes the aggregation of several SFs, where the no-tion of aggregation refers here to a partonomic relation in which

parts are well-formed SFs. The partial order on SFs generates an or-dered labeled oriented graph whose nodes are SFs and links denotethe partonomic relation. In the following, we assume that eachchild node is deduced from its parent node by adding new informa-tion which excludes another SF.

Proposition 4 (SF tree). Given an ordered labeled oriented graphwhose nodes are SFs and links denote the partonomic relation, if anychild node extends its parent node with new information excludinganother SF, then the graph is a tree.

Proof. In order to prove that it is a tree, we have to first show thatit contains no cycle and then, that it has a single connected compo-nent. The first statement is proved by contradiction. Let us considera node Si describing the situation fragment SFi. By adding someinformation pi and qi to SFi, we get two child nodes, i.e., Sp and Sq

such that their relative informational contents are respectivelyRx:SFi � Ry:pi � /p(x, y) and Rx:SFi � Ry:qi � /q(x, y). Since the twonodes describe knowledge with SFs, any cycle (e.g., aggregatingSp and Sq) would result in a contradiction with the premises.

Since SF structures are nested, there exists an initial node withan empty information called the root node. All nodes, i.e., all SFsderive from this node, and the forest is reduced to a single tree. h

From a practical point of view, the observed knowledge pro-vides proof objects for types, propositions and SFs whereas the as-serted knowledge relies on types. For example, let us consider thetypes S1 and S2 as defined in Section 5.2 with S1 v S2. Assumingthat mc1 denotes John’s cold and d1 the particular drug John takes,a proof hhJohn, hmc1, p1ii, hd1, p2ii of S2 with p1 and p2 the respectiveproofs for HasCondition(John, mc1) and UsesDrug(John, d1) will alsoyield the proof hJohn, hmc1, p1ii for S1.

These aspects are similar to the approach of [58] where theauthor proposes two separate sub-ontologies of body-substanceuniversals which are distinguished according either to structureor to function. The consequence was that these sub-ontologieswere trees with no multiple-inheritance.

5.4. Type inhabitation

Given a context C and a type T such that C ‘T:Typei, type inhab-itation is the problem of either determining a term M such that C‘M:T or showing that no such term exists. The type inhabitation isa kind of inference classifying an object to a type. The classificationtype/object is close in its spirit to the so-called classifications be-tween types and tokens of ST while in DLs, it corresponds to theprocess of satisfiability.

Proposition 5 (Type inhabitation). Let C be a valid context and T atype, then in S-DTT it is effectively decidable whether there exists aterm M such that C‘M:T is derivable.

Proof. The process of type inhabitation boils down to a problem ofdatabase queries since all type constructors generate queries to a(finite) database. Assuming a mapping between types and tablesor table columns in the database, it follows that computed termsare in canonical form up to reduction rules. In relational algebra,all queries can be evaluated in time that is polynomially in the sizeof the database state and query evaluation always terminates.Therefore, if we restrict the queries to a query language equivalentto the relational algebra (e.g., Datalog), each query terminates.Then, the decidability of type inhabitation follows by inductionover the structure of C. h

The proof objects corresponding to a sum type result from a ta-ble reading within a data base. While the usual case requires to ex-plore a number of values that reflect the Cartesian product of the


arguments involved in the table, as underlined above, a significantfeature of dependent types stands in the restriction of this searchspace. The gain originates in the quantification that occurs on sub-sets of proof objects (instead of sets) resulting from the dependen-cies. Let us recall the relation that binds a user to its apartmentwith an argument in NAPO. First, if we add to the context C theassertion Person:APO and if the context C supplemented with avariable x having the type Person yields a type Apartment : Per-son‘NAPO, then a formation rule (see Appendix A) asserts thatwe are able to form the type Rx : Person � Apartment(x). More for-mally, we have:

C ‘ Person : APOC; x : Person ‘ Apartment : Person! NAPO

C ‘ ðRx : Person � ApartmentðxÞÞ : Type0ðR� form1Þ

where Apartment(x) denotes an indexed family of types, i.e., theapartments that are hosted by a given person. Then, an introductionrule explains how to construct pairs having this type:

C ‘ M : PersonC; x : Person ‘ N : Apartment½M=x�

C ‘ hM;Ni : ðRx : Person � ApartmentðxÞÞ ðR� introÞ

If we get proof objects for Person and Apartment then the term Mis inhabited and the search space is limited thanks to the indexedfamilies of types. Finally, the type inhabitation is directly per-formed with appropriate queries to the database (e.g., a table relat-ing persons and the apartment they are living in).

5.5. Situation assessment

The partonomic hierarchy can be exploited to prove situationsin the same way that we have proved context types in [56]. ProvingSFs corresponds to the well-known problem of type inhabitationand proving that SFs are inhabited reduces to find (at least one)proof object(s) for each SF. Since SFs are ordered by the partonomicrelation, we use a situation prover that relies on the following algo-rithm. Its automated reasoning technique exploits well-known andefficient tree-based search algorithms where situation assessmentrelies on finding a search path in the tree of SF types. This algo-rithm constantly runs a loop including five steps and makes surethat the inconsistencies will not occur between SFs.

(i) perception of the current situation (e.g., database updatewith new values incoming from a network and/or fromsensors),

(ii) type inhabitation with exploring the SF tree. The deepest SFin each branch is selected,

(iii) determination of valid type inferences,(iv) derivation of possible compositions from deepest SFs(v) for proved SFs of steps (ii) and (iv), apply their related

domain rules to assess situation.

To prove SFs, the algorithm first explores the SF tree. In eachbranch of the tree, the deepest proved SF is selected since it de-scribes the most precise information. Notice that this does not ruleout the case where several partial situations (SFs) are proved, eachof them being the deepest in its own branch. In a second step, allinferences with product types are performed which can furthervalidate additional SFs (see next section). Then in the third step,the algorithm searches for the deepest SFs that can be composedfrom the already proved SFs. Finally, the situation assessment ap-plies domain rules to proved SFs in order to have the most globalview about the situation. Locally, the situation is unique but itleaves the user the possibility to compose different situations witha more general context. A similar mechanism which focuses on(physical) context types instead of situation types has been imple-

mented [59]. It has shown a polynomial (quadratic) complexity forthe type inhabitation algorithm.

A typical example of a partonomic hierarchy of SF type is givenby a geriatric residence having an administrator whose needs areto speak about the situations of patients. There are several doctorsand each doctor has a situation that includes responsibility for acollection of patient situations. The administrator in turn, isresponsible for a collection of doctor situations. To express this sce-nario, we start with a local SF describing a resident of a health cen-ter having some disease:

PatientSit,Rx : HealthCenter � Ry : ResidentðxÞ:Rz : Disease � SufferFromðy; zÞ

Then one can assert that doctors in such an environment havesome of these situations as part of their own situation:

DoctorSit,Rs : PatientSit � Rd : Doctor: InChargeOf ðd;p1p2sÞ

Finally, the local situations of doctors can be collected by theadministrator of the health center with:

AdminSit,Rs : DoctorSit � Ra : Administrator:

ResponsibleOf ða;p1p2sÞ

In that scenario, the simple partonomic hierarchy PatientSit v Doc-torSit v AdminSit explains how partial situations (i.e., SFs) can benested to form hierarchies. Here we have a single branch, but wecan think of another branch focusing on disease types and lookingfor contagious diseases for which some proofs can be available. Inthat case, the situation is made of the aggregation of the deepestproved SFs in each branch as described in the algorithm. Situationscan be extended as needed depending on the part of the world weare interested in. Now, let us describe two additional exampleswhich apply the SF types for expressing situations hardly accountedfor with ST or OWL-DL.

In ST, infons are considered as truth values on a FOL basis andwhile the semantics adopted in S-DTT is similar on the objectives,we can do more since truth values are replaced with proof objectsproviding more information about situations. Let us consider theexample given in [60] which focuses on the Enemy Course Of Ac-tions (ECOAs) represented on a situation development system.The authors highlight the problem of distinguishing joint objectssuch as boats that are blocking a given harbor. The infon represent-ing a joint of 10 boats requires an operator and is expressed ass�� blockade, {boat1, . . . , boat10}, H1, t, 1 where H1 denotesthe relevant harbor. However, this infon does not entail anyinfon having a proper subset of the set of 10 boats at the sameplace. In S-DTT, we first introduce a type describing the collectionof boats. The type Fleet10 represents the disjoint union of tenboats as:

Fleet10,Rx1 : Boat � . . .x9 : Boat �x10 : Boat:DisjointFromðx1;x2; . . . ;x10Þ

A proof for the type Fleet10 is a tuple which aggregates all the boatsthat are blocking the harbor and its proof, say hboat1, hboat2,h . . . , p1iii with p1 a proof for DisjointFrom(boat1, boat2, . . . , boat10)ensuring that all proofs are distinct. As a result the type Fleet10 be-haves just like the joint entity above, but here we can reason aboutit. Now we can express the partial situation S1 involving Fleet10:

S1,Rx : Fleet10 � Ry : Harbor � Rt : UTC:

BlockadeWithðy; x; tÞ

A proof for S1 could be hhboat1, h . . . , p1ii, hH1, h03:12PM, q1iii,where q1 is a proof of BlockadeWith(H1, F1, 03:12PM). In otherwords, there is always a single proof (i.e., BlockadeWith(H1, hboat1,h . . . , p1ii, 03:12PM)) witnessing for the blockade. Furthermore, ifsome boats do not participate in the blockade, then we get the


new corresponding number of proofs. This aspect proves that S-DTTis more precise in the sense where not only it provides a truthvalue, but it details what are the proofs for a given relation to hold.

Let us now consider the example given in [61] describing a sit-uation where three distinct populations prey on each other: a pop-ulation Px consumes a population Py, which in turn consumes Pz.The authors underline the limitation of OWL-DL to unambiguouslycapture general situations which contain multiple instances of thesame type. This difficulty vanishes in S-DTT with the introductionof the types:

ConsPop,Rx : Population � ConsumedðxÞS3,Rx : Population � Ry : ConsPop � Consumesðx;p1yÞ

The first sum type will restrict the proof objects for the popula-tions that are consumed, i.e., Py and Pz. The situation is expressedby the sum type S3 whose proofs are hPx, hPy, q1ii and hPy, hPz, q2iiwithq1 and q2, the respective proofs for Consumes(Px, Py) and Consumes(Py,Pz). Since SF types are formal objects in the semantics which denotestates of affair, i.e., ST in the DOLCE backbone, they can be referred towith constants in the logical language and variables can range overthem. It follows that the world is partially and constructively de-scribed with situations which are proof objects for SF types.

6. Reasoning about situations: the contribution of S-DTT

Reasoning is an essential ingredient of information fusion at alllevels of the fusion hierarchy. Inferential processes may be used toprovide high-level conceptual abstractions that are implicit in pat-terns of sensor data. Inferences either derive from the formalsemantics of the knowledge representation language, or are basedon more complicated types of inference in the form of implicationrules. Deductive or inductive rules can be applied in S-DTT, how-ever classical deduction rules (as they are used in FOL) are replacedwith sequences of conversion rules transforming types until onegets canonical types, i.e., types structures that cannot be reducedanymore. Alternatively, tactics19 can encode reasoning methods insubtle procedures e.g., in Coq, and are much more complex thatthe single possible tactic used by Prolog. An example will be detailedin Section 7 in order to transform a relation in a SF and insert it auto-matically in the partonomic tree. The inferences in S-DTT are ratherdifferent than inferences in FOL since they may present several as-pects due to the typing mechanism. We will present now these as-pects in detail. Possible inferences in S-DTT are (i) inference withtype inhabitation (see Section 5.4), (ii) inference with partial quanti-fication, (iii) non-monotonic inference and (iv) inference withsubsumption.

6.1. Exploiting dependent quantification

Implication rules are expressed with product types, however,these inferences are more expressive since they allow parameter-ized quantification over types. Let us consider a possible transla-tion in first order logic of a famous sentence due to Geach [62],Every man who owns a donkey beats it:

8 xðDonkeyðxÞ ^ 8 yðManðyÞ ^ ðOwnðy; xÞ � Beatðy; xÞÞÞÞ

However, such an expression is not correct since quantificationis over all donkeys instead of being over men that own a donkey. Amore sophisticated FOL-based solution suggested in [19]:

8 xðManðxÞ ^ 9zðDonkeyðzÞ ^ Ownðx; zÞÞÞ �9yðDonkeyðyÞ ^ Ownðx; yÞ ^ Beatðx; yÞÞ

19 A tactic is a proof procedure that can automatically prove a given theorem.

does not solve the problem since nothing proves that the same don-key is both owned and beaten. In dependent type theory, the mech-anism of dependent quantification solves the problem withdependent types:

Pd : ðRx : Man � Ry : Donkey � Ownðx; yÞÞ � Beatðp1d;p1p2dÞ

in which the domain of quantification of the product (P) is over thepredicate Rx:Man � Ry:Donkey � Own(x, y). This kind of inferencecould be applied to address some issues hardly resolved with ST.Let us consider for example, the situation where a given election oc-curs in a location at a given time while at the same time (and loca-tion) a bombing co-occurs and, as a result will disrupt the election[60]. The authors explain that while ST is able to represent the twoSFs (i.e., infons) it is however unable to express that one SF can im-pact the other. To solve that problem, let us first introduce a SF forexpressing that an election will occur in a given place at a given time:

ElecEvent,Rx : Election � Rl : location � Rt : Time:

OccursAtðx; l; tÞ

Then to relate this event with the bombing event, a partonomicinclusion will provide a simple solution:

Rx : ElecEvent � Re : Bombing � CoOccursAtðe;p1p2x;

p1p2p2xÞ

In such a way, the respective arguments for location and time willrefer to the same values as in the election. Finally, a dependent quan-tification over bombing events which co-occur with election eventswill provide the valid diagnostic, i.e., the election is disrupted:

Pz : ðRx : ElecEvent � Re : Bombing � CoOccursAtðe;p1p2x;p1p2p2xÞÞ � Disruptðp1p2z;p1p1zÞ

6.2. Exploiting contexts for expressing non-monotonic rules

Situation Theory [63] allows a form of non-monotonic reasoningwith the assertion S) S0 jB meaning that, given a situation S, thenwe can assert S0 if the condition B is true. This approach has been ex-tended with the introduction of contexts [31] to give it a more pre-cise formulation. According to that view, it follows that non-monotonic reasoning can be achieved with situation types in S-DTT by applying the algorithm of Section 5.5. In any branch of thepartonomic hierarchy, the most precise information is selected. Anagent knows that we are in a situation S1 because it has some proofobjects for it and then, it can infer e.g., a rule R1. If we add informa-tion (i.e., in the context), we can get some proof object(s) for a moreprecise situation S01 in which e.g., the rule R2 can be derived withS1 v S01, then S01 is selected.20 The consequence is that R2 is preferredto R1. Notice that in S-DTT rules are types and are treated as such.

R1,Px : S1 � P1 and R2,Px : S01 � P2

where P1 and P2 are propositions and S01 extends S1 with new infor-mation. For example, let us consider the fact that a given drug whenused against a medical condition can represent a risk in the pres-ence of some other condition (e.g., a liver problem). The situationtype S1 could be:

S1,Rx : Patient � Ry : DrugðxÞ � Rm : MCðxÞ: ForConditionðy;mÞR1,Px : S1 � TreatableWithðp1x;p1p2xÞ

but if we add information that some person having the medical con-dition (MC) m may have health problems after taking the drug d, weextend the situation such that:

20 It corresponds to the notion of principal type in Type Theory.


S01,Rs : S1 � IncompatibleWithðp1p2x;p1p2p2sÞR2,Px : S01 � AvoidedForðp1p2p2p1x;p1p1xÞ

Above the respective proof objects John, D1 and Cold for thetypes Patient, Drug(John) and MC(John), a proof for S1 is hJohn, hD1,hCold, q1iii where q1 denotes a proof for ForCondition(D1, Cold).Adding the proof object LiverProblem for MC in the database, S01 be-comes proved with hhJohn, hD1, hCold, q1iii,q2i where q2 a proof forIncompatibleWith(D1, LiverProblem).

6.3. Using universe hierarchies for reasoning with subsumption

6.3.1. Subsumption between SFsWhile subsumption appears as a prominent feature for reason-

ing in ontology-related languages such as DLs, the question ariseswhether it is possible to express subsumption in S-DTT. This ques-tion is addressed by first considering the concept of subtyping thathas been introduced in ECC. Subtyping formalizes a subset relationbetween universes, function spaces and Cartesian products built ontop of universes. The rule (Sub) is the basic subtyping relation overterms whereas the rule (Conv) is the conversion rule seen as a spe-cial case of the subtyping rule. Notice that according to the (Sub)rule, a given term may have several types. However, it can beshown that whenever a term is typeable, it has a uniquely deter-mined principal type (see, e.g., [49] for more details). This principaltype is the minimum type of the term with respect to the (Sub) ruleand relatively to a given context.

C ‘ M : AC ‘ A0 : TypeAÂ0

C ‘ M : A0ðSubÞ

C ‘ M : AC ‘ A0 : Type A ’ A0

C ‘ M : A0ðConvÞ

The following meta-theoretic property from ECC, introduced asa corollary of Lemma 1, will be useful for our purpose:

Corollary 6 [49]. The relation îs the smallest partial order overterms with respect to conversion such that:

if A ^ A0 and B ^ B0; then Rx : A � B ^ Rx : A0 � B0:

The notion of subsumption has several readings of which themore important ones are extensional and intensional [64,65].
There are some drawbacks to the extensional interpretation of sub-sumption because (i) determining whether the extension of oneuniversal is included in the extension of another one is often unde-cidable and (ii) observing that two universals have the same exten-sion does not mean that they are identical. Alternatively, in theintensional reading, a universal subsumes another universal onlyif this result can be inferred from the examination of the internalstructure of this universal involving more domain dependent infer-ences. In S-DTT, we only consider the intensional subtyping forexpressing intensional subsumption.
Definition 18 (Subsumption of Universals). A universal A subsumesa universal A0 with A0Â if all the properties of A also hold in A0.

For example, with the non-dependent universals21 Sensor andDevice, we can assert that Sensor^Device and that Device subsumesSensor since a sensor has, at least, all the properties of a deviceand then, contains more precise information than a device. In thatprecise case, properties are implicit, but they can be explicit havingthe form of product types. To investigate this situation, let us firstconsider the subsumption for SF types easily derived from the sub-typing on sum types.

21 They correspond to atomic concepts in DL.

Lemma 7 (Subsumption of SF types). Given a subsumption betweenSF types S,RV1 : T1 � . . . � RVn : Tn � Pð~c1V1; . . . ; ~cnVnÞ and S0,RV1 : T 01 � . . . � RVn : T 0n � P

0ð~c1V1; . . . ; ~cnVnÞ as the smallest partialorder over terms w.r.t. conversion such that T1^T 01; . . . ; Tn^T 0n, andP ^ P0, then S ^ S0.

Proof. Corollary 6 can be easily extended with multiple argu-ments, then the property holds for sum types with multiple argu-ments. Since SFs are composed of sum types, then the result alsoholds for SFs. h

Corollary 8 (Proof inference). Given two SF types S and S0 such thatS ^ S0, then any proof that holds for S also holds for S0.

Proof. Since S^S0, the set of proof objects are in the subset rela-tionship (extensionality of the set of proof objects) and then, anyproof of S is also of type S0 up to the subtyping rule. h

Let us consider for instance a person involved in an activity. Forthat purpose, a relation type Involved can be specified as follows:

Involved,Rx : Person � Ry : Activ ity � InvolvedInðx; yÞ

Since Person ^ APO, Activity ^ PRO, then using the relation definedin (3) and applying Corollary 6, it follows that InvolvÎNVOLV. If weget a proof object JohnDoe for Person together with a proof Take-Shower for Activity, the proof for Involved could be hJohnDoe, hTake-Shower, p1ii with p1 a proof for InvolvedIn(JohnDoe, TakeShower).From Corollary 8, it results that hJohnDoe, hTakeShower, p1ii is alsoa proof for INVOLV.

6.3.2. Reasoning with subsumptionThe subsumption is also a kind of inference between the sub-

sumed type and its super-type. If we are given some general rulesabout SFs, these rules are generally described at the highest levelsof the DOLCE formal ontology. For instance, if we get some prooffor a SF (e.g., a simple relation) at the Typei level and if it is sub-sumed by a Typei+1 SF then, Corollary 8 applies and the SF at theTypei+1 level is proved as well. Now, if the Typei+1 SF is part of a rule(e.g., inference with a product type), a new Typei+1SF can be vali-dated according to Proposition 9 (see Fig. 2). An attractive featureof subsumption is that it can be coupled with rules to provide morepowerful reasoning with SFs.

Proposition 9 (Intensional subsumption with rules). Given:

(i) two SF types S and S0 such that S ^ S0,(ii) a rule defined by a type /0 such that /0,PV : S0 � Pð~c1V; . . . ;

~cnVÞ with P : T 01 ! . . .! T 0n ! Prop, a predicate and T 01; . . . ;

T 0n are types occurring in S0

then, any proof that holds for S also holds for Pð~c1V; . . . ; ~cnVÞ withP:T1 ? � � �? Tn ? Prop and T1^T 01; . . . ; Tn^T 0n.

Proof. From Corollary 8, any proof for S also holds for S0. Since theproduct type /0 describes a property, it follows that any proof for S0

yields a proof for Pð~c1V; . . . ; ~cnVÞ. Then, from Definition 18, itresults that any proof for the universal Pð~c1V; . . . ; ~cnVÞ also holdsfor its subsumed types. h

Rules that are valid at a Typei level also hold for their subsumedrelations at the Typei�1 level. As a result, Typei rules can be stored inan ontology and shared across distributed systems. Let us considerthe following example. It is well-known that some relations areleft- (or right-) distributive over partOf-like relations [66,57,67].The fact that an agentive physical object (APO) is involved within

oc

Fig. 2. Inference mechanism with subsumption and product types.

Fig. 3. The smart apartment.


a process (PRO), with C � APO:Typei, PRO:Typei has been describedby the relation type INVOLV in (3). Then, a generic relation Loc canexpress the fact that the process occurs in a spatio-temporallocation:

Loc,Rx : INVOLV � Ry : Sðp1p2xÞ � Rz : T � LocatedAtðp1p2x; y; zÞð7Þ

Assuming that S:Typei and T:Typei, stand respectively for a spa-tial location type and a temporal type and provided that the rule ofleft distributivity applies to the predicate LocatedAt, then the infer-ence is written:

Pd : Loc � LocatedAtðp1p1d;p1p2d;p1p2p2dÞ ð8Þ

which means that if an agentive physical object is involved in a pro-cess and assuming that this process occurs in a spatio-temporallocation, then the agentive physical object is also located at thislocation. Let us consider now the types Person ^ APO and Take-Shower ^ PRO, then it follows from Corollary 6 that Rx:Per-son � Ry:TakeShower � InvolvedIn(x, y) ^ INVOLV. Now, if we addthe information about the spatio-temporal location such as Bath-Room ^ S and CurrentTime ^ T, the situation fragment is subsumedby the generic relation Loc as follows:

Rx : ðRu : Person � Rv : TakeShower � InvolvedInðu;vÞÞ:Ry : BathRoomðp1p2xÞ � Rz : CurrentTime � LocatedAtðp1p2x; y; zÞ^ L

Given a proof of the left type, e.g.:

hhJohnDoe; hproc1;p1ii; hBR1; h9 : 00PM;p2iii

with JohnDoe a proof for Person, proc1 a proof for the process Take-Shower, p1 a proof for InvolvedIn(JohnDoe, proc1), BR1 a proof forBathRoom(JohnDoe), 9:00 PM a proof for CurrentTime and p2 a prooffor LocatedAt(proc1, BR1, 9:00 PM), then by Corollary 8, it is also aproof for its super-type (i.e., Loc). Then, from the inference rule(8), it follows that we can deduce:

LocatedAtðJohnDoe;BR1;9 : 00PMÞ

This aspect is particularly appealing for an activity recognitionprocess.

7. A case study

Let us consider a scenario extracted from [68] involving a healthcenter with smart apartments for supporting elders aging in place.Within this framework, modeling activities for people sufferingfrom mental disorders such as Alzheimer’s disease is challenging.The starting place is to know which activities the elder is doingin his apartment. Its environment includes a bedroom with a singlebed, a bathroom with a shower and a living room with a TV set (see

Fig. 3). The global goal could use the proposition TakeCare(x, y)with x: HealthCenter and y:Patient. A (partial) tree describing par-ticular situation fragments would require the basic SF type:

ElderInCenter , Rx : HealthCenterRy : ApartmentðxÞ � Rz

: PatientðxÞ � LocatedInðz; yÞ

As mentioned in Section 4.4, this structure is a well-formed SFsince the types of the goal variables are included into its argu-ments. Then, all the nodes from the partonomic tree may derivefrom this SF to compose a well-formed tree. Let us suppose thatthe database contains the proof objects x = AdamsCenter, y = H13,H22, H17 and the respective patients z = JohnDoe, MikeThumb, Ike-Finney, then examples of proofs for the ElderInCenter SF could be:

hAdamsCenter; hH13; hJohnDoe; p01iiihAdamsCenter; hH22; hMikeThumb; p02iiihAdamsCenter; hH17; hIkeFinney;p03iii. . . ;

with p01 a proof for LocatedIn(JohnDoe, H13), p02 a proof for Locat-edIn(MikeThumb, H22) and p03 a proof for LocatedIn(IkeFinney,H17). The suggested scenario will automatically detect the activityfrom information given by sensors or electronic devices and should


automatically assess the resulting situation. This information popu-lates the database and provides updated proof values for appropri-ate objects. Sensors are distributed throughout the environment.For instance, a bed sensor located within the mattress can detectwhether the user is on the bed or not. Presence detectors are alsodistributed within each part of the apartment (e.g., bathroom, toi-lets) and can further be used to monitor where the resident is aspart of a safety plan. Some intelligent appliances such as a TV setprovide information about the TV such as its state (on–off), andthe channel in use.

In the following, any successive sequence of n identical p-elim-inators will be denoted pn

i with i = 1, 2. The type LongLyingDownhas been defined in Section 4.3.2 for expressing that the patientremains in its bed more than a limit duration (8 h). If the databasecontains, in addition with the proofs for ElderInCenter given above,the initial time t0 = 10:00PM used in the constructor, the proofsb = BH13 (in OnBed), u = hUTC,>i and t = 9:10AM, then a proofobject for the LongLyingDown SF could be:

hhhAdamsCenter; hH13; hJohnDoe;p01iii; hBH13; hhUTC;>i;h9 : 10AM;p1iiii; hhhUTC;>i; h8;>ii; p2ii

with p1 and p2 the respective proofs for IsPressed(BH13, 9:10AM)and GreaterThan(11:10, 8). In health care environments for patientsliving in geriatric residences, the information system generally in-cludes a central computer related to distributed sensor systems.This kind of framework does not rule out the possibility of havingseveral proof objects corresponding to the type LongLyingDown. Ifit happens that another patient has the same problem at the sametime, the health care system can collect another proof such as:

hhhAdamsCenter; hH22; hMikeThumb;p02iii; hBH22; hhUTC;>i;h9 : 10AM; q1iiii; hhhUTC;>i; h8;>ii; q2ii

with t0 = 10:25PM, q1 and q2 the respective proofs for IsPres-sed(BH22, 9:10AM) and GreaterThan(10:45, 8). The system is there-fore able to distinguish between them and highlights the accuracyat the proof level that is an important aspect of S-DTT, propertyinherent of (constructive) type theories. It is also worth noticingthat each time the SF are extended with new information, the aver-age number of proofs decreases as well. In other words, for largenested pairs, few proofs (if any), are available.

All variable references will refer to definitions of Section 4.3.2.Now, suppose that we are interested in assessing anomalous activ-ities during the night for patients having a mental disease (e.g., Alz-heimer). For example, if the patient gets up during the night toswitch on the TV set instead of sleeping, this should be interpretedas an unusual situation. A knowledge structure, that is not a SF,

Table 1The derivation from GettingUp to S1 in the tactic SFinsert.

CC ‘ p1S

Rb : Bed

RC ‘ p2

2p

IC

C‘p2p1p22

Rx:Tim

C‘M:(Rx:ElderInCenter � Ry:GoToToilets � InvolvedIn p1p22x; y

� �Þ

C ‘ p1p

C‘N:Toilets(p1p2p1M)C, u:(Rx:ElderInCenter � Ry:GoToToilets � InvolvedIn(x, y)), t:Toilev : UðUTCÞ;w : Timeðp1vÞ ‘ P : LocatedAt p1p2

2p1ðM=uÞ;N=t;p1p2p1p22p1S2=w

� �

C ‘ hM; hN; hp1p2p1p22p1S2; Piii : S1

ðR-

referred to as AtNight, checks the current time and compares it tothe night time (after 10:00PM) with UTC as previously defined.

Timeð10 : 00PM;UTCÞ,Rx : UðUTCÞ�Ry : Timeðp1xÞ � Eqðy;10 : 00PMÞ

AtNight , Ru : UðUTCÞ � Rt : Timeðp1uÞ:Rx : Timeð10 : 00PM;UTCÞ:GreaterThanðt;p1p2xÞ

Then, a SF denoted GettingUp could be defined from the value ofIsPressed at the previous loop (timestamp tb) and add the knowl-edge structure AtNight:

GettingUp,Rg : ðRx : ElderInCenter:Rb : Bedðp1p2xÞ �Ry : AtNight:

Rtb : Timeðp1p1yÞ � IsPressedðb; tbÞÞ: ðIsPressed p1p2g;p1p2p1p22g

� �!?Þ

Now, the previous SF can be extended again with informationinvolving a TV set. The first argument of the proposition On refersto a TV type while the second is a timestamps.

WatchTVAtNight,Rn : ðRx : GettingUp:Ry : TV p1p2p21x

� �:

On y;p1p32p1x

� �!?

� �:On p1p2n;p1p2p1p2

2p1n� ��

The compound predicate means that if the TV was off at the pre-vious instant On y;p1p3

2p1x� �

!?� �

and if we have a proof that it ison at the current time On p1p2n;p1p2p1p2

2p1n� ��

, then the typeWatchTVAtNight is proved as well. Provided that the bed sensorhas a scanning period of 50 and given the following proofs ex-tracted from the database, u = hUTC, >i, t = 1:25AM, z = JohnDoe inElderInCenter, b = BH13 and tb = 1:20AM then a proof object forthe GettingUp SF could be:

hhhAdamsCenter; hH13; hJohnDoe; p01iii; hBH13;hhhUTC;! pi; h1 : 25AM; hhhUTC;! pi;h10 : 00PM;! pii;p1iii; h1 : 20AM;p2iiii;p3i

with p1, p2 and p3 the respective proofs for GreaterThan(1:25AM,10:00PM), IsPressed(BH13, 1:20AM) and IsPressed(BH13, 1:25AM) ? \(there is a proof that the bed sensor is not pressed at1:25AM knowing that it is pressed at 1:20AM). But if it happens thatthe patient wakes up because he is dying for a pee, then we have todistinguish that situation from the previous one. For that purposewe can assume that the knowledge base has the type Loc describedin Section 6. In this situation, the SF WatchTVAtNight is not proved,but the SF GettingUp is. Furthermore, if a presence detector in thetoilets provides a proof saying that there is a process GoToToiletsin progress in the toilets, the following type is proved:

‘ S2 : GettingUp

2 : ðRx : ElderInCenter:ðp1-elimÞ

p1p22x

� �� Ry : AtNight � Rtb : Timeðp1p1yÞ � IsPressedðb; tbÞÞ

C ‘ p2p1S2 : ðRb : Bed p1p22p1S2

� �� Ry : AtNight:

ðp2-elimÞ

tb : Time p21y

� �� IsPressedðb; tbÞÞ

1S2 : ðRy : AtNight � Rtb : Time p21y

� �:ðp2-elimÞ

sPressedðp1p2p1S2; tbÞÞ‘p1p2

2p1S2 :AtNight

p1 S2 :ðRt:Time p31p

22p1 S2 :ð Þ ðp2-elimÞ

ðp1-elimÞ

e(10:00PM, UTC)

GreaterThanðt; xÞÞ2p1p2

2p1S2 : Time p31p

22p1S2

� � ðp1-elimÞ

ts(p1p2p1u),

introÞ

Fig. 4. Situation assessment in the health-care scenario.

22 Notice that AD is not of the type Prop.


S1,Rx : ðRu : ElderInCenter � Rv : GoToToilets:

InvolvedIn p1p22u;v

� �Þ � Ry : Toiletsðp1p2p1xÞ:

Ru : UðUTCÞ � Rt : Timeðp1uÞ:LocatedAt p1p2

2p1x; y; t� �

This type is a restricted version of the relation type (7) sincePatient v ElderInCenter and Patient ^ Person. Therefore the set ofproofs for S1 is a subset of the set of proofs of Eq. (7). Using rule(8), we can deduce that the person is located in the toilets. In addi-tion, we now define a tactic for which it is possible to automaticallyderive a type from the type S1.

Algorithm 1. Algorithm of the SFinsert tactic.

for each branch of the tree dofor each SF node Ni do

if Ni is proved thenfor each argument of the relation R do

try to derive the argument from Ni;if success then

the argument is substituted with Ni andthe resulting relation is stored as SFi

together with the current Ni.;else

next argumentend

endendnext SF node;

endselect the deepest proved node among SFi and insert itabove its related Ni;

end

This tactic is proposed to automatically insert nodes in the par-tonomic tree. Let us consider a relation (e.g., S1) which becomesproved after a sensing or a network operation. We have to decidein what place this relation can be inserted within the partonomictree. The algorithm 1 implements the SFinsert tactic for automati-cally inserting a new SF in the partonomic tree from a relation R.The starting place is the root node in the tree.

Notice that the unit available in the evaluation of the depth isthe number of SF nodes. Within a branch, the deepest node is un-ique. The tactic looks for a sequence of P-eliminations applied toeach proved node until we obtain an argument type of this rela-tion. Applying this algorithm to the relation S1 leads to the follow-ing results. The partonomic tree having, from top to bottom, thesequence ElderInCenter, GettingUp and WatchTVAtNight. First theSF node ElderInCenter is explored but no derivation from this SFto any arguments of S1 can be provided. Then, the following provednode (GettingUp) shows that the relation can be derived from it(see Table 1). The algorithm stops here since the last SF nodeWatchTVAtNight is not proved as indicated above. Then, we can re-place the argument Time in S1 with its antecedent in the derivation,i.e., GettingUp. We obtain a new SF type that will be called UseToilet.

UseToilet,Rx : ðRu : ElderInCenter � Rv : GoToToilets:

InvolvedIn p1p21u;v

� �Þ � Ry : Toiletsðp1p2p1xÞ:

Rz : GettingUp � LocatedAt p1p22p1x; y;

�

p1p2p1p22p1z

�

From Definition 17, it follows that UseToilet v GettingUp and theSF UseToilet is inserted within the hierarchy just below GettingUp.

An efficient situation-awareness system should support themanagement of various information sources such as sensor data,textual information or databases. In addition with the sensor-based information above, we can now consider textual informationto complete the SF tree. Since several authors have used dependenttypes for solving various difficulties of natural language semanticssuch as homonymy, anaphora, ellipsis and co-predication[69,19,70,40,21], it brings out the capabilities of dependent typetheory with regard to NLP. If the patient suffers from Alzheimerdisease, then a sentence extracted from a natural language reportcould state:

Patient identification: John Doe – age: 73 years old – Apartment:H13 – MMSE score: 22 – ADAS-Cog: 19 – treatment: Aricept –10 mg pills – once a day.

From this information, we can extract the patient’s disease to-gether with its level. Patients having an ADAS-Cog score less than35 are said to suffer from Alzheimer disease and if the mental statescore (MMSE) is less than 25, it characterize a mild Alzheimer dis-ease in which the patient is unable to assess situations [71]. Thefollowing rule is the constructor for the type AD (the score is anumber without unit):

AdasCogð35Þ,Rs : AdasCog � Eqðs;35Þ

MMSEð25Þ,Rm : MMSE � Eqðm;25ÞPx : ðRp : ElderInCenter:

Ru : Rx : AdasCog p1p22p

� ��:

Ry : AdasCogð35Þ � LessThanðx; yÞÞ:

Rv : Rx : MMSE p1p22p

� ��:

Ry : MMSEð25Þ � LessThanðx; yÞÞÞ:

AD p1p22p1x

� �

where AD stands for the Alzheimer Disease.22 Then a type of situa-tion fragment saying that the patient having this disease needssupervision can be composed with the previous product type:

Monitoring,Rx : ElderInCenter � Ry : ADðxÞ:SupervisedFor p1p2

2x; y� �

Summarizing all these SFs, one get the SF tree for the domainunder study (see Fig. 4). Then to assess a composition of SFs, a rule(i.e., a product type) must reflect this composition in the domainontology. In the present example, we can compose the SFtypes WatchTVAtNight and Monitoring that should result in the


assessment stating that if the patient suffers from a mental diseaseand he is watching TV late in the night, then he must be the actor ofan abnormal situation:

R1,Px : ðu : WatchTVAtNight � v : MonitoringÞ:AbnormalForðp1x;p1p2xÞ

Another rule can state that the combination of the SF types Use-Toilet and Monitoring leads to assess that the situation is harmless:

R2,Px : ðu : UseToilet � v : MonitoringÞ:HarmlessForðp1x;p1p2xÞ

8. Discussion

According to [6] there is still a deficiency to provide a unifiedtheoretical framework for information fusion: the S-DTT theory isa first step in this direction. Now, we will develop some argumentsexplaining what can be captured in S-DTT.

8.1. Benefits of the type system

Whereas the combination of a (ontologically neutral) logic withan ontology provides a language (e.g., OWL-DL) that can expressrelationships about the entities in the domain of interest, S-DTTrather uses in a same framework a logic, a type system and a back-bone ontology resulting in a richer language. The type systemmakes the difference especially at the level of safe modeling. Forexample, an adequate use of types in a specification (e.g., modelingtime and spatial coordinates as values of abstract sets rather thannatural values) ensures that some forms of error will be automat-ically detected.

The reduction of computational complexity with inferencesillustrates another benefit in using a type theory with subtyping.In [72], the author argues that for many applications, type hierar-chies with subtyping can significantly reduce the complexity class.For example, subsequent reasoning with a typed logic can replace achain of inferences with a type checking judgment.

An information fusion engine should carefully combine the rel-evant information sources (e.g., text and audio, text and image,etc.) in order to construct a comprehensible situation for the user.This aspect is exploited in S-DTT since the logical part, i.e., reason-ing is clearly distinct from the constructive part, i.e., the acquisition(construction) of proofs. For example the type of an image file canbe assessed by a constructor in which the type of the output formatdepends on the result of reading the header of the file.

Current fusion methods are built around the hypothesis of aclosed world which imposes the need to create or constantly mod-ify the models used for situation awareness. Type theory is open-ended, i.e., we can always add new propositions and new typesand even new judgment forms whereas FOL is essentially closed(i.e., when we add new constructs, we are working in other theo-ries or logics that include FOL).

In the building of ontological models in support of informationfusion [73], the author has underlined ‘‘that axioms should be lim-ited to express generic constraints or properties’’. It is an importantaspect with respect to the development of a practical tool, and un-like FOL, the stratified hierarchies of types together with the sub-typing mechanism is a possible answer to this requirement (wecan quantify over properties, properties of properties, etc.).

To the best of our knowledge, no extant logical calculus com-bines all these properties in a single system with well-understoodmeta-mathematical properties.

8.2. Comparison with ST and OWL-DL

We will try here to collect most of the relevant features that asituation aware system should have and compare the ability ofwell-known theories according to these features. What we havein common with ST are: (i) the rejection of the so-called ‘‘possibleworld’’ semantics, (ii) the representation of small blocks ofknowledge about the real world that will be inserted (or not) ina given situation, (iii) non-monotonic capabilities, (iv) the intro-duction of types and (v) a sound logic to reason with. Alternatively,some operations of S-DTT have a direct correspondence withOWL-DL. For instance, satisfiability in OWL-DL is similar to typeinhabitation while subsumption is processed in the same waywith substituting OWL-DL roles with relation types or properties(product types).

In ST, with a real situation s, the set {rjs� r} denotes the corre-sponding abstract situation. In S-DTT, it corresponds to the collec-tion of SFs that are proved in a given situation. While in ST somebasic types are predefined, they are not general enough to beshared among multiple domains unlike ontology-based typeswhich rely on a formal ontology. The part-of relation is availablein ST to provide a partial ordering of the situations. In S-DTT, firstthis relation is introduced as well but at the type level which takesadvantage of more general partonomic relations and second, amechanism of subsumption close to that of OWL-DL allows to en-hance the reasoning power.

Situations in ST are sets of parametric infons, but they may benon-well-founded, i.e., circular as shown in [74]. This problem ofcircularity is logically excluded from the theory S-DTT because of(i) the predicativity of the universe hierarchy in the core ECC(therefore the subsumptive hierarchy remains non-circular) and(ii) the partonomic hierarchy which exhibits a tree structure avoidssuch circularities between SFs. Furthermore, ST does not provide acomplete and single universally accepted version with the conse-quence that any account of ST cannot be both comprehensiveand coherent [75]. In fine, not only S-DTT accommodates the basicfeatures of situation theory but it enhances these features in amore expressive and a more formal way.

In [76] the authors have highlighted the lack of very expressiveconstructors in OWL-DL for modeling complex domains. While inOWL-DL, complex properties can be expressed by composingatomic properties, it is hardly possible to refer to the same partic-ular since there are no variables. For example the properties isE-mployedBy and isEmployerOf can be composed with thedefinition: isColleagueOf � isEmployedBy isEmployerOf. However,the sentence ‘‘if a person a is employed by a person b which isthe employer of c, then a is colleague of c’’ cannot be expressedin OWL-DL. In S-DTT, a first situation S1 could represent the subsetof employees managed by a given manager while a second situa-tion S2 could depend on the first in order to extract employees.

S1,Rx : Employee � Ry : Manager � isEmployedByðx; yÞS2,Rx : S1 � Ry : EmployeeðxÞ � isEmployerOf ðp1p2x; yÞ

Then, an inference acting on the situation type S2 yields the de-sired result. The two arguments of the predicate refer respectivelyto any employee managed by the manager in situation S1 and an-other employee depending on situation S1 (i.e., depending on thesame manager):

Ps : S2 � ColleagueOf ðp1p1s;p1p2sÞ

Another difficulty of OWL-DL is the lack of expressive class con-structors for restricting the membership to a class only to thoseindividual objects that are fillers of two or more properties, i.e.,the role-value-maps. In other words, given two properties P andQ such that P # Q, the class of individuals which are related by

Table 2A comparison between S-DTT, ST and OWL-DL.

Availablilityofn-arypredicates

Subsumption

Partonomic relations

Partialknowledge

Dynamic knowledge

Multiple abstractionlevels

Reasoningincontext

Reasoningabout relations

Non monotonicreasoning

Integrationofmultipledatatypes

Unifyingtheory

Toolsavailability

Userfriendliness

STOWL-DLS-DTT


property P are also related by property Q. For example, let us thinkof a city in a country where there is a civil war. People that have theproperty of being injured in the city have also the property of beingevacuated. This could be captured in S-DTT with:

S1,Rx : Person � Ry : city � isInjuredAtðx; yÞPx : S1 � isEvacuatedFromðp1p1x;p1p2xÞ

It means that all people that are injured are also evacuated fromthe city, but it does not mean that all people that are evacuated areinjured. One can suppose for example that there is also a plague(e.g., cholera), then people having such a disease are also evacu-ated. Moreover, it worth noticing that this problem cannot besolved with subsumption since the property isInjuredAt is not asub-property of isEvacuatedFrom. It follows that the set of proof ob-jects for isInjuredAt is a subset ( # ) of the set of proof objects forisEvacuatedFrom.

In S-DTT, the integration of multiple data types is possible, suchas data from text files and data from numerical sensors. This prop-erty inheres in the functional nature of constructors which can pro-cess words from texts as well as numerical data in any kind ofcomputational task. This is a significant advantage over OWL-DLand ST, especially for concrete applications.

A more formal comparison between the expressive power of aclass of DLs and S-DTT is reported in Appendix B. The consideredDL class offers at least the concept constructors {>, \, u,:; 9; fills; one� ofg and role constructors {role-and, role-not, prod-uct, inver}.

From these results, a small set of relevant features that a situa-tion aware theory should offer can be extracted. These features arereported in Table 2. Subsumption, a very useful relation for reason-ing is not available in ST. Partonomic relation is another kind ofuseful relation that is expressible in OWL-DL while the relation isnot part of the basic predefined set of relations. Whereas partialknowledge is a common property that all theories comply with,dynamic knowledge reveals its epistemological nature. Dynamicknowledge generally refers to what is known as instance check-ing/updates and as such requires some information tagged withtime stamp. Reasoning capabilities are obviously more importantin S-DTT due to its higher-order logic with polymorphism. Noticehowever, that closed world assumption in FOL allows to infer falseproperties from the absence of true statement, while it is not thecase in S-DTT since if a property is not proved, we cannot assertthat it must be false. S-DTT is a unifying theory in the sense whereall features are already included within the theory, while it is notthe case for other that are often acquiring additional functionalitiesthrough add-ons. Finally S-DTT is more difficult to assess than thetwo other theories because it lacks of a user interface. Addressingthis issue is a required step to abstract the user from the burdenof type theory learning.

9. Conclusion

Situation awareness emphasizes symbolic reasoning, involvesmultiple types of domain knowledge, and requires hierarchical rea-soning since multiple levels of abstraction coexist in the reasoningprocess. With respect to these constraints, we have described S-DTT, a novel foundation for situation awareness. It differs fromother approaches by emphasizing the role of higher-order con-structive logic with automated reasoning based on a hierarchy ofuniverses. One central objective of this work was to use an inte-grated semantic approach based on (i) logical inference and rea-soning in which rules are a part of the S-DTT language and (ii) anontological framework for the semantics. A crucial issue men-tioned in [30] is to decide which of the proved relations might beof interest to the user. We argue that a possible answer to this issuerequires a hierarchy of situation types allowing the composition ofthese types, each situation type being related to a predicate repre-senting a goal to achieve (as detailed in the last paragraph of Sec-tion 7). Each proved situation type validates its related goal anddecision criteria could be added to each goal.

Logical representations allow us to make very compact abstrac-tions about the world. Dependencies have two impacts, (i) theyexplicitly state all dependencies between entities of the world (ii)they restrict the search space for proof objects during the type inhab-itation step. A similar work on context modeling using the sametype-theoretical kernel has shown an interesting complexity class(i.e., P) originating mainly through the dependence of data types.

As a matter of fact, the approach advocated here is exactly alongthe lines indicated by Kokar et al. [4]. However, Kokar’s conclusionsmention the difficulty to write procedures for all potential rela-tions since their number can be high. It may be possible to decreasethis number by abstracting on multiple levels, a high level SF cor-responding to several low level SFs.

The architecture is open in the sense that new knowledge canbe added. The ability to express knowledge within an ontologicalcontext is useful. Knowledge-based reasoning often requires factswhich are true in some ontology allowing for some partial evalua-tion. The partiality that is supported by S-DTT is an important fea-ture with the purpose of situation awareness.

Dependent types are also used to facilitate modular reasoning.We can state a property about an arbitrary relation, and theninstantiate it to the property of a particular relation. All the reason-ing mechanisms described here are automated except for the tac-tic-based inferences. However, (i) an increasing number ofautomated tactics becomes available (e.g., for the Coq theoremprover) and (ii) some generic tactics could be easily pre-encodedfor standard situations reducing significantly the task of (domain)ontology designers.

In this paper, it has been shown that derivation of relations insituation awareness can be done within a formal and expressive

Table A.3The ECC rules for sum types.

C ‘ A:TypeiC;x:A ‘ B:TypeiC ‘ Rx:A�B:Typei

ðR-form1Þ

C ‘ A:TypeiC;x:A ‘ B:PropC ‘ Rx:A�B:Typei

ðR-form2Þ

C ‘ M:AC;x:A ‘ N:B½M=x�C ‘ hM;Ni:Rx:A�B ðR-introÞ

C ‘ M:Rx:A�BC ‘ p1 M:A ðp1-elimÞC ‘ M:Rx:A�B

C ‘ p2 M:B½p1M=x� ðp2-elimÞ


framework. Coherence is maintained with dependencies betweensituation fragment types through the nesting process (e.g., proofsalways refer to the right object). In our current experiments, werely on a theorem prover (i.e., Coq), but in general there is no rea-son (except the effort required) that the proofs could not be gener-ated by hand.

However, a major limitation especially concerns the ability ofresearch novices to type theory to grasp the subtleties of the ap-proach. But to quote Pat Hayes, ‘‘A lesson I have drawn from theSemantic web experience is that ugly surface forms sometimeshave to be accepted as useful to machines rather than people’’.Another drawback we saw at the time is the lack of tools, in partic-ular editors and consistency checkers. Ongoing work will soonsolve this problem.

As detailed by the end of the case study, the theory is open,leaving room for adding semantic rules at the Typei level with re-gard to behavior and/or goal structures. This aspect will be ex-plored in future works. When there is a need to assess or‘‘understand’’ a situation we think the goal is not ‘‘perfect under-standing’’ (in the logical sense) but rather ‘‘more precise’’ and weargue that a theory based on type theory such S-DTT is a potentialcandidate in that direction.

Appendix A. Inductive data types

An inductive data type is a type whose objects are supplied byterm constructors. For instance, Px:A � B has k for constructor whileRx:A � B has the term pairing h, i as constructor. For an inductivedata type, the set of objects having this type can be seen as theset defined inductively by the constructors. The rules for an induc-tive data type consist of (i) formation rules explaining how the typeis formed, (ii) introduction rules that show how to get an object ofthat type and (iii) elimination rules detailing what can we do withan object of that type. For sum types, the rules are (extracted fromECC) the following:(see Table A.3).

Appendix B. Expressiveness

A measure of the expressive power of a language is fundamentalfor comparing different proposals. The qualitative measure sug-gested here is relative to the well-known Description Logics (DLs).

In [77], Geuvers has proved that one can prove logical formulaein CC that cannot be proved in Higher Order Logic (HOL).

Theorem 10 77. The formulae-as-types embedding of higher orderpredicate logic into CC is not complete.

In other words, derivability in CC implies derivability in higherorder predicate logic.

Corollary 11 (ECC derivability). Derivability in S-DTT implies deriv-ability in higher order predicate logic.

Proof. According to [49], ECC is an extension of CC, therefore The-orem 10 also applies to ECC and derivability in HOL follows from

derivability in ECC. Since S-DTT is a conservative extension ofECC derivability in S-DTT also implies derivability in higher orderpredicate logic. h

It results that the expressiveness of S-DTT is higher than theexpressiveness of HOL. Let us now introduce the language L3, thesubset of first-order predicate calculus with one-place and two-place predicates restricted to three variables symbols and L2 thelanguage limited to two variables. A translated DL is defined witha translation function that maps concepts to formulas with freevariable x and roles to formulas with free variables x and y.

Theorem 12 [78]. The description language with concept construc-tors f>;?;u;:; 9; fills; one-ofg and role constructors {role-and, role-not, product, inver} is as expressive as L2.

Theorem 13. S-DTT is more expressive than DLs having at least theconcept constructors f>;?;u;:; 9; fills; one-ofg and role constructors{role-and, role-not, product, inver}.

Proof. From Theorem 12, it follows that any description logic hav-ing at least the operators enumerated in this theorem are at mostas expressive as L2, that is a subset of first order predicate logic.Since HOL is strictly more expressive than first order logic, itresults than HOL is more expressive than DLs having at least theoperators of Theorem 12. Then, from Theorem 10 it is straightfor-ward to derive that S-DTT is more expressive than DLs having atleast the above operators. h

References

[1] A.N. Steinberg, C.L. Bowman, F.E. White, Revisions to the JDL data fusion model.in: Proceedings of the SPIE Sensor Fusion: Architectures, Algorithms andApplications, 1999, pp. 430–441.

[2] M.R. Endsley, Measurement of situation awareness in dynamic systems,Human Factors 37 (1) (1995) 65–84.

[3] C. Matheus, Using ontology-based rules for situation awareness andinformation fusion, in: Proceedings of the W3C Workshop on Rule Languagesfor Interoperability, Washington DC, 2005.

[4] M.M. Kokar, C.J. Matheus, K. Baclawski, Ontology-based situation awareness,Information Fusion 10 (2009) 83–98.

[5] R.J. Brachman, H.J. Levesque, Knowledge representation and reasoning, MorganKaufman, 2004.

[6] A. Boury-Brisset, Ontology-based approach for information fusion, in:Proceedings of the 6th International Conference on Information Fusion, 2003.

[7] W. Johnson, I.D. Hall, From kinematics to symbolics for situation and threatassessment, in: Proceedings of the Information, Decision and ControlConference, 1999.

[8] K. Sycara, M. Paulucci, M. Lewis, Information discovery and fusion: semanticson the battlefield, in: Proceedings of the 6th International Conference onInformation Fusion, 2003.

[9] T. Eiter, T. Lukasiewicz, R. Schindlauer, H. Tompits, Combining answer setprogramming with description logics for the semantic web, in: Proceedings ofNinth International Conference on the Principles of Knowledge Representationand Reasoning (KR2004), AAAI Press, 2004, pp. 141–151.

[10] R. Rosati, DL+log: tight integration of description logics and disjunctivedatalog, in: Proceedings of Tenth International Conference on Principles ofKnowledge Representation and Reasoning (KR2006), AAAI Press, 2006, pp. 68–78.

[11] M. Krötzsch, P. Hitzler, D. Vrandecic, M. Sintek, How to reason with OWL in alogic programming system, in: Proceedings of RuleML’06, 2006, pp. 17–28.

[12] B. Motik, U. Sattler, R. Studer, Query answering for OWL-DL with rules, in:Proceedings of the International Semantic Web Conference, LNCS 3298,Springer, 2004.

[13] L.F. Pires, M. van Sinderen, E. Munthe-Kaas, S.M.H. Prokaev, D.-J. Plas,Techniques for describing and manipulating context information, Freeband/AMUSE D3.5v2.0, Lucent Technologies, 2005.

[14] P. Martin-Löf, Intuitionistic type theory, Bibliopolis, Napoli, 1984.[15] T. Coquand, G. Huet, The calculus of constructions, Information and

Computation 76 (2–3) (1988) 95–120.[16] H. Barendregt, H. Geuvers, Proof-Assistants Using Dependent Type Systems,

Handbook of Automated Reasoning, Elsevier and MIT Press, 2001.[17] I. Cervesato, M. Stehr, Representing the MSR cryptoprotocol specification

language in an extension of rewriting logic with dependent types, ElectronicNotes in Theoretical Computer Science 117 (2005) 183–207.

[18] N. Oury, W. Swierstra, The Power of Pi, in: Proceedings of the 13th ACMConference on Functional Programming (ICFP’08), 2008, pp. 39–50.


[19] P. Boldini, Formalizing context in intuitionistic type theory, FundamentaInformaticae 42 (2) (2000) 1–23.

[20] R. Cooper, J. Ginzburg, Clarification ellipsis in dependent type theory, in:Proceedings of the 6th Workshop on the Semantics and Pragmatics ofDialogue, 2002.

[21] A. Ranta, Grammatical framework: a type-theoretical grammar formalism,Journal of Functional Programming 14 (2) (2004) 145–189.

[22] N. Guarino, Formal ontology and information systems, Proceedings of FOIS’98,IOS Press, 1998.

[23] D. Nardi, R.J. Brachman, An Introduction to Description Logics, The DescriptionLogic Handbook, Cambridge University Press, 2003 (Chapter 1).

[24] T. Bittner, M. Donnelly, B. Smith, Individuals, universals, collections: on thefoundational relations of ontology, in: Proceedings of the InternationalConference on Formal Ontology in Information Systems, (FOIS04), IOS Press,2004, pp. 37–48.

[25] B. Smith, C. Rosse, The role of foundational relations in the alignment ofbiomedical ontologies, in: Proceedings of MEDINFO 2004, 2004, pp. 444–449.

[26] W.V. Quine, On what there is, Review of Metaphysics 2 (1948).[27] S. Shapiro, Mathematics and philosophy of mathematics, Philosophia

Mathematica 2 (3) (1994) 148–160.[28] M. Sato, Platonism with a flavor of constructivism, in: Workshop on

Constructivism: Mathematics and Logic, 2008.[29] S.A. Rasmussen, J. Ravnkilde, Realism and logic, Synthèse 52 (1982) 379–

437.[30] C. Matheus, K. Baclawski, M. Kokar, Derivation of ontological relations using

formal methods in a situation awareness scenario, in: Proceedings of SPIEConference on Multisensor, Multisource Information Fusion, 2003, pp. 298–309.

[31] V. Akman, M. Surav, The use of situation theory in context modeling,Computational Intelligence 12 (4) (1996) 1–13.

[32] J. Barwise, J. Perry, Situations and Attitudes, MIT Press, Cambridge, MA, 1983.[33] J. Barwise, The Situation in Logic, CSLI Lecture Notes 17, Stanford University,

1989.[34] K. Devlin, Logic and Information, Cambridge University Press, 1991.[35] M. Surav, V. Akman, Modelling context with situation, Technical Report BU-

CEIS-95-07, Department of Computer Engineering and Information Science,Bilkent University, 1995.

[36] N. Baumgartner, W. Retschitzegger, W. Schwinger, Application scenarios ofontology-driven situation awareness systems, in: Proceedings of theInternational Conference on Formal Ontologies Meet Industry, IOS Press,2008, pp. 77–87.

[37] G. Jakobson, J. Buford, L. Lewis, Situation management: basic concepts andapproaches, in: Information Fusion and Geographic Information Systems,Lecture Notes in Geoinformation and Cartography, Springer, 2007, pp. 18–33.

[38] P. O’Brien, An ontology for mobile situation aware systems, AustralasianJournal of Information Systems 15 (2) (2008) 5–33.

[39] H. Kamp, U. Reyle, A calculus for first order discourse representationstructures, Logic, Language and Information 5 (1996) 297–348.

[40] R. Cooper, Records and record types in semantic theory, Journal of Logic andComputation 15 (2) (2005) 99–112.

[41] R. Montague, Pragmatics and intensional logic, Synthèse 22 (1970) 68–94.[42] R.A. Frost, W.S. Saba, A database interface based on Montague’s approach to

the interpretation of natural language, International Journal of Man-MachineStudies 33 (2) (1990) 149–176.

[43] R.A. Muskens, Combining montague semantics and discourse representation,Linguistic and Philosophy 19 (1996) 143–186.

[44] J. Van Eijck, The proper treatment of context in NL, in: ComputationalLinguistics in the Netherlands, UILOTS Utrecht, 2000, pp. 41–51.

[45] P. Cimiano, Translating Wh-questions into F-logic queries, in: Proceedings ofthe 2nd CoLogNET-ElsNET Symposium, 2003, pp. 130–137.

[46] N. Asher, A type driven theory of predication with complex types, FundamentaInformaticae 84 (2) (2008) 151–183.

[47] A. Ranta, Type-Theoretical Grammar, Oxford University Press, 1995.[48] R. Harper, F. Pfenning, On equivalence and canonical forms in the LF type

theory, ACM Transactions on Computational Logic 6 (1) (2005) 61–101.[49] Z. Luo, Computation and Reasoning, vol. 11, Oxford Science Publications, 1994.[50] J.Y. Girard, Y. Lafont, P. Taylor, Proofs and Types, Cambridge University Press,

1989.[51] N. Guarino, Formal ontology, conceptual analysis and knowledge

representation, International Journal of Human–Computer Studies 43 (1995)625–640.

[52] C. Masolo, S. Borgo, A. Gangemi, N. Guarino, A. Oltramari, Ontology library,WonderWeb Deliverable D18 (ver.1.0, 31-12-2003), 2003.

[53] A. Gangemi, N. Guarino, C. Masolo, A. Oltramari, L. Schneider, Sweeteningontologies with DOLCE, in: Proceedings of EKAW 2002, LNAI 2473, 2002, pp.166–181.

[54] J. Mac Kinna, Why dependent types matter, in: 33rd ACM SIGPLAN Notices41(1), 2006.

[55] The Coq Development Team, The Coq Proof Assistant Reference Manual,Version 8.3pl1, 2010.

[56] R. Dapoigny, P. Barlatier, Modeling contexts with dependent types,Fundamenta Informaticae 104 (4) (2010) 293–327.

[57] C.M. Keet, A. Artale, Representing and reasoning over a taxonomy of part-whole relations, Applied Ontology 3 (2008) 91–110.

[58] R. Rector, Modularization of domain ontologies implemented in descriptionlogics and related formalisms including OWL, in: Proceedings of theInternational Conference on Knowledge Capture, 2003, pp. 121–128.

[59] P. Barlatier, Conception et implantation d’un modèle de raisonnement sur lescontextes basé sur une théorie des types et utilisant une ontologie de domaine,Dissertation (in french), University of Savoie, 2009.

[60] B. Ulicny, M. Kokar, C. Matheus, G. Powell, Problems and prospects for formallyrepresenting and reasoning about enemy courses of action, in: Proceedings ofthe 11th International Conference of Information Fusion, (FUSION 2008),Cologne, Germany, 2008, pp. 204–211.

[61] R. Hoekstra, J. Liem, B. Bredeweg, J. Breuker, Requirements for representingsituations, in: CEUR Workshop Proceedings (OWLED’06), 216, 2006.

[62] P.T. Geach, Reference and Generality, An Examination of Medieval and ModernTheories, Cornell University Press, 1962.

[63] E. Tin, V. Akman, Situated non-monotonic temporal reasoning with BABY-SIT,AI Communications 10 (1997) 93–109.

[64] A. Napoli, Subsumption and Classification-Based Reasoning in Object-BasedRepresentations, European Conference on Artificial Intelligence (ECAI’92), JohnWiley & Sons Ltd., 1992.

[65] W.A. Woods, Understanding subsumption and taxonomy: a framework forprogress, in: J. Sowa (Ed.), Principles of Semantic Networks, Morgan Kaufman,San Mateo, CA, 1991, pp. 45–94.

[66] A. Artale, E. Franconi, N. Guarino, L. Pazzi, Part-whole relations in object-centered systems: an overview, Data & Knowledge Engineering 20 (1996)347–383.

[67] R. Dapoigny, P. Barlatier, Towards ontological correctness of part-wholerelations with dependent types, in: A. Galton, R. Mizoguchi (Eds.), Proceedingsof the 6th International Conference on Formal Ontology in InformationSystems, IOS Press, 2010, pp. 45–58.

[68] F. Mastrogiovanni, A. Sgorbissa, R. Zaccaria, On the problem of describingactivities in context-aware environments, International Journal of AssistiveRobotics and Mechatronics 9 (4) (2009) 4–19.

[69] Z. Luo, P. Callaghan, Coercive subtyping and lexical semantics, in: Proceedingsof Logical Aspects of Computational Linguistics (LACL’98), 1998.

[70] J. Ginzburg, R. Cooper, Resolving ellipsis in clarification, in: Proceedings of the39th Annual Meeting on Association for Computational Linguistics, Toulouse,France, 2001, pp. 236–243.

[71] P.M. Doraiswamy, F. Bieber, L. Kaiser, K.R. Krishnan, J. Reuning-Scherer, B.Gulanski, The Alzheimer’s disease assessment scale: patterns and predictors ofbaseline cognitive performance in multicenter Alzheimer’s disease trials,Neurology 48 (6) (1997) 1511–1517.

[72] J. Sowa, Fads and fallacies about logic, IEEE Intelligent Systems 22 (2) (2007)84–87.

[73] A. Boury-Brisset, Ontological engineering for threat evaluation and weaponassignment: a goal-driven approach, in: Proceedings of the 10th InternationalConference on Information Fusion, Quebec, Canada, 2007.

[74] J. Barwise, J. Etchemendy, The Liar: An Essay on Truth and Circularity, OxfordUniversity Press, New York, NY, 1987.

[75] N. Braisby, R. Cooper, Naturalising constraints, in: J. Seligman, D. Westerstahl(Eds.), Language, Logic and Computation, vol. 1, CSLI Lecture Notes Series, vol.58, Stanford, CA, 1996, 93–108.

[76] C. Bettini, O. Brdiczka, K. Henricksen, J. Indulska, D. Nicklas, A. Ranganathanf,D. Riboni, A survey of context modelling and reasoning techniques, Pervasiveand Mobile Computing 6 (2) (2010) 161–180.

[77] J.H. Geuvers, The calculus of constructions and higher order logic, in: The Curry-Howard isomorphism 8, Cahiers du Centre de logique, 1995, pp. 139–191.

[78] A. Borgida, On the relative expressiveness of description logics and predicatelogics, Artificial Intelligence 82 (1–2) (1996) 353–367.

Formal foundations for situation awareness based on dependent type theory

Documents

Transcript of Formal foundations for situation awareness based on dependent type theory