Monodic epistemic predicate logicHolger Sturm, Frank Wolter, and Michael ZakharyaschevInstitut f�ur Informatik, Universit�at LeipzigAugustus-Platz 10-11, 04109 Leipzig, GermanyDivision of Arti�cial Intelligence, School of Computer StudiesUniversity of Leeds, Leeds LS2 9JT, UK.(e-mails: fhsturm,wolterg@informatik.uni-leipzig.de, mz@scs.leeds.ac.uk)AbstractWe consider the monodic formulas of common knowledge predicate logic, which allowapplications of epistemic operators to formulas with at most one free variable. We provide�nite axiomatizations of the monodic fragment of the most important common knowledgepredicate logics (the full logics are known to be not recursively enumerable) and single outa number of their decidable fragments. On the other hand, it is proved that the additionof the equality symbol to the monodic fragment makes it not recursively enumerable.1 IntroductionEver since it became common knowledge that intelligent behaviour of an agent is based notonly on her knowledge about the world but also on knowledge about both her own and otheragents' knowledge, logical formalisms designed for reasoning about knowledge have attractedattention in arti�cial intelligence, computer science, economic theory, and philosophy (cf. e.g.the books [5, 16, 13] and the seminal works [8, 1]). In all these areas, one of the most successfulapproaches is to supply classical|propositional or �rst-order|logic with an explicit epistemicoperator Ki for each agent i under consideration. Ki' means that agent i knows (or believes)', K1K2' says then that agent 1 knows that agent 2 knows ', and the schema of positiveintrospection Ki ! KiKi states that agent i knows what she knows. In the �rst-order casethis language is capable of formalizing the distinction between `knowing that' and `knowingwhat' (i.e., modalities de dicto and de re): the formula Ki9x name(x; y) stands for `i knowsthat y has a name,' while 9xKiname(x; y) means `i knows a name of y.'There can be di�erent interpretations of the knowledge operators (e.g. with or withoutpositive or negative introspection), and for many of them transparent axiomatic representa-tions have been found (cf. e.g. [7, 5]). On the other hand, the possible worlds semantics [8]provided a framework to interpret this language: in a world w agent i knows ' if and only if 'holds in all worlds that i regards possible in w (the di�erence between various understandingsof Ki is re ected by di�erent accessibility relations among the worlds).The situation becomes much more complicated when|in order to describe the behaviorof multi-agent systems|we extend the language with one more modal operator, C, to capturethe common knowledge of a group of agents. Such an operator was required for analyzingconventions [14], coordinizations in multi-agent systems [5], common sense reasoning [15],1

agreement [1, 2], etc.1 Although the intended meaning of the common knowledge operatorinvolves in�nity: C' stands for the in�nite conjunction of the formK1' ^K1K2' ^K2K1K2' ^ : : : ;both natural possible worlds semantics and clear inductive axiomatizations have been foundfor propositional common knowledge logics [7]. (The new operator, however, considerablyincreases the computational complexity of these logics|from PSPACE to EXPTIME; consult[5].)But real problems arise when we try to combine the common knowledge operator with the�rst-order quanti�ers. First, no common knowledge predicate logic with both a �nitary (orat least recursive) axiomatization and a reasonable semantics has ever been constructed! Andsecond, the common knowledge predicate logics determined by the standard possible worldssemantics are known to be not recursively axiomatizable (and so not recursively enumerable)[17]. Thus, similar to second-order logic or �rst-order temporal logic, it is impossible tocharacterize common knowledge predicate logics syntactically. In some sense this means thatneither we nor the Turing machine have the capacity of understanding the interaction betweencommon knowledge and quanti�ers. Moreover, this is true of even very small fragments ofthe logics, say, the monadic or two-variable fragments (see [17]).Does it mean that we should completely abandon the idea of using common knowledgepredicate logic? Still there exist manageable fragments with non-trivial interaction betweenthe common knowledge operator and quanti�ers.A promising approach to singling out non-trivial decidable fragments of �rst-order modaland temporal logics has been proposed in [9, 20]. The idea is to restrict attention to theclass of monodic2 formulas which allow applications of modal or temporal operators only toformulas with at most one free variable. In the epistemic context, monodicity means, inparticular, that� we have the full expressive power of �rst-order logic as far as we do not apply epistemicoperators to open formulas;� we can reason about agents' knowledge of properties, for instance,8x (C loves(John; x) _ C :loves(John; x))(`for every object x, it is a common knowledge whether John loves x'); however, we arenot permitted to reason about agents' knowledge of relations, say8x; y (C loves(x; y) _ C :loves(x; y))(`for all pairs x, y, it is a common knowledge whether x loves y').The main aim of this paper is to show that the monodic fragment of common knowledgepredicate logic turns out to be quite manageable. First, we show that for almost all interestinginterpretations of the operators Ki the monodic fragment of the valid formulas (without equal-ity) can be �nitely axiomatized. Moreover, we observe that a number of natural subclasses ofthe monodic fragment, say, with only monadic predicates or two variables, are decidable. Onthe other hand, it is proved that the addition of the equality symbol to the monodic fragmentmakes it not recursively enumerable.1An alternative approach adds in�nitary operators to the language, see [10, 11].2Monody is a composition with only one melodic line.2

2 First-order logics of common knowledgeThe logics we deal with in this paper are all based on the language we call CL, which extendsthe standard �rst-order language (without equality) with a number of epistemic operators,including the operator expressing common knowledge. The alphabet of CL consists of:� predicate symbols P0; P1; : : : ,� individual variables x0; x1; : : : ,� individual constants c0; c1; : : : ,� the booleans ^, :,� the universal quanti�er 8x for each individual variable x,� a �nite number of knowledge operators K1; : : : ;Kn, n � 1, and� the common knowledge operator C.We assume that the set of predicate symbols is non-empty and that each of them is equippedwith some �xed arity; 0-ary predicates are called propositional variables and denoted byp0; p1; : : : . The individual variables together with the individual constants form the set ofCL-terms. The set of CL-formulas is de�ned as follows:� if P is an n-ary predicate symbol and �1; : : : ; �n are terms, then P (�1; : : : ; �n) is aformula;� if ' and are formulas, then so are ' ^ and :';� if ' is a formula and x a variable, then 8x' is a formula;� if ' is a formula and i � n, then Ki' and C' are formulas.Throughout the paper we make use of the following abbreviations: >, ?, ' _ , ' ! ,' $ , and 9x', which are de�ned as usual, as well as E' (`everyone knows '') whichstands for K1' ^ � � � ^Kn'.The language CL is interpreted in �rst-order Kripke models which are structures of theform M = hF;D; Ii, where F = hW;R1; : : : ; Rni is the underlying Kripke frame (W 6= ; is aset of worlds and the Ri are binary relations on W ), D is a nonempty set, the domain of M,and I a function associating with every world w 2W a �rst-order structureI(w) = DD;P I(w)0 ; : : : ; cI(w)0 ; : : :E ;in which P I(w)i is a predicate on D of the same arity as Pi (for a propositional variable pi,the predicate pI(w)i is either > or ?), and cI(w)i is an element in D such that cI(u)i = cI(v)i forany u; v 2 W . The latter means that constants are treated as rigid designators in the sensethat they designate the same object in every world. To simplify notation we will omit thesuperscript I and write Pwi , pwi , cwi , etc., if I is clear from the context.Remark 1. Note that we assume domains to be constant. Axiomatizations for the case ofexpanding or varying domains can easily be obtained from our results.3

An assignment in D is a function a from the set var of variables to D. The value �M;a (orsimply �a if understood) of a term � under a inM is a(�), if � is a variable, and �w otherwise,where w is some (any) world in W . The truth-relation (M; w) j=a ' (or simply w j=a ') inthe model M in the world w under the assignment a is de�ned inductively as follows:� w j=a Pi(�1; : : : ; �n) i� (�a1 ; : : : ; �an) 2 Pwi ; this fact will also be written as Pwi (�a1 ; : : : ; �an),I(w) j=a Pi(�1; : : : ; �n), or I(w) j= Pi[�a1 ; : : : ; �an];� w j=a ^ � i� w j=a and w j=a �;� w j=a : i� w 6j=a ;� w j=a 8x (x; y1; : : : ; yn) i� w j=b (x; y1; : : : ; yn) for every assignment b in D that maydi�er from a only on x;� w j=a Ki i� v j=a for all v 2W such that wRiv;� w j=a C i� v j=a for all v such that w(Si�nRi)+v, where the superscript + meanstaking the transitive closure of Si�nRi.For a set of formulas �, a model M, a world w and an assignment a, we write w j=a � to saythat w j=a ' for every ' 2 �. In this case � is said to be satis�ed in M. By F j= � we meanthat � is valid in F, i.e., (M; w) j=a � holds for every modelM based on F, every assignmenta in it, and every world w in F.Di�erent epistemic logics correspond to di�erent classes of frames. Usually these classesare determined by combinations of the following properties: re exivity (denoted by r), transi-tivity (t), seriality (s), and euclideanness (e). We denote by Fr the class of all re exive frames,by Fre the class of all re exive and euclidean frames (i.e., the class of frames with equivalencerelations), etc. Fa is the class of all frames.For a class F of frames, we de�ne L(F), the logic of F, to be the set of all CL-formulas thatare valid in all F 2 F. Here is a list of standard logics of common knowledge: KCn = L(Fa),TCn = L(Fr), KDCn = L(Fs), K4Cn = L(Ft), S4Cn = L(Frt), KD45Cn = L(Fste), S5Cn = L(Fre).3 Axiomatizing the monodic fragmentAs was shown in [17], none of the logics listed above is recursively axiomatizable. Moreover,the restriction of these logics to such `orthodox' fragments as the monadic or two-variableformulas does not bring a relief: they are still not recursively enumerable. By analyzing proofsof these `negative' results, one can observe that all of them make use of formulas assertingthat some agents know relations between two objects. On the other hand, the results of[18] establishing decidability of epistemic description logics (in which epistemic operators areapplicable only to unary predicates) give some hope that the fragment without such formulascan be more manageable.De�nition 2 (monodic formulas). Denote by CL1 the set of all CL-formulas ' such thatany subformula of ' of the form Ki or C has at most one free variable. Such formulas willbe called monodic. For a class F of frames, let L1(F) = L(F) \ CL1. In other words, L1(F) isthe monodic fragment of the logic L(F).From now on all formulas are assumed to be monodic.4

In this section we give axiomatizations of the monodic fragments of the epistemic logics de-�ned above. (These axiomatizations are �rst-order extensions of those in [7].) To begin with,we axiomatize the monodic fragment of KCn , i.e., L1(Fa). This axiomatic system, denoted byKCn , has the following axiom schemata and inference rules:Axiom schemata (over formulas in CL1):� the set of axiom schemata from some axiomatization of classical �rst-order logic,� Ki('! )! (Ki'! Ki ), for i � n,� C'! E(' ^ C'),� Ki8x $ 8xKi .Inference rules (over formulas in CL1):� the rules of classical �rst-order logic,� 'Ki' , for i � n,� '! E( ^ ')'! C .The monodic fragments of the remaining logics are axiomatized by adding to KCn the corre-sponding standard axiom schemata:AD: Ki'! :Ki:', i � n,AT : Ki'! ', i � n,A4: Ki'! KiKi', i � n,A5: :Ki'! Ki:Ki', i � n.Namely, TCn ,KDCn andK4Cn as the axiomatic systems obtained by adding toKCn the schemataAT , AD, and A4, respectively. S4Cn is K4Cn plus AT . KD45Cn is K4Cn extended by AD andA5, and S5Cn is KD45Cn plus AT .Given an axiomatic system S, we denote by `S its consequence relation. Our aim now isto prove that the de�ned systems indeed axiomatize the monodic fragments of our commonknowledge logics. That is, we are going to show that for every monodic formula ', we have`KCn ' i� ' 2 KCn i� ' is valid in all frames from Fa, and similar claims for the other logics.The easy `only if' part of these claims, i.e., correctness, follows from well-known results(consult e.g. [5, 3]) and the almost obvious fact that the rule '! E( ^')=' ! C preservesvalidity. The `if' part, i.e., completeness, is much more complicated. It will be proved in thenext section.5

4 CompletenessGiven a set � of CL1-formulas, we denote by con(�) and sub(�) the sets of all constants andsubformulas of formulas in �, respectively; subC(�) is de�ned as:subC(�) = sub(�) [ fE( ^ C ); ^ C ;Ki( ^ C ) : C 2 sub(�); i � ng:Let sub:C(�) = f: : 2 subC(�)g and let subn(�) be the subset of sub:C(�) containingonly formulas with � n free variables. For instance, sub0(�) denotes the set of sentences insub:C(�). (Note that subn(�) is not necessarily closed under subformulas and that moduloequivalence we may assume that subn(�) is closed under :.) In what follows we will not bedistinguishing between a �nite set � of formulas and the conjunction V� of formulas in it.Let x be a variable not occurring in �. Putsubx(�) = f fx=yg : (y) 2 sub1(�)g [ f:Ki:?;Ki:?;:?;? : i � ngFor the rest of this section we �x an arbitrary CL1-sentence '.De�nition 3 (type). By a type for ' we mean a boolean-saturated subset t of subx('), i.e.,� ^ � 2 t i� 2 t and � 2 t, for every ^ � 2 subx';� : 2 t i� =2 t, for every 2 subx'.We say that two types t and t0 agree on sub0(') if t \ sub0(') = t0 \ sub0('). Given a type tfor ' and a constant c 2 con('), the pair ht; ci will be called an indexed type for ' (indexedby c) and denoted by tc(x) or simply tc.De�nition 4 (state candidate). Suppose T is a set of types for ' that agree on sub0('),and T con = fht; ci : c 2 con(')g a set of indexed types such that ft : ht; ci 2 T cong � T . Thepair C = hT; T coni is called then a state candidate for '. A pointed state candidate for ' isthe pair P = hC; ti, where t is a type in T , called the point of P. With C and P we associatethe formulas �C = t̂2T 9x t(x) ^ 8x_t2T t(x) ^ ^ht;ci2T con t(c); �P = �C ^ t:In what follows S ranges over the axiomatic systems introduced in Section 3. We remindthe reader that a formula � is said to be S-consistent if 6`S :�.De�nition 5 (suitable pairs). (1) A pair (t1; t2) of types for ' is called i-suitable for S,i � n, if the formula t1 ^ :Ki:t2 is S-consistent.(2) A pair (C1;C2) of state candidates is i-suitable for S, i � n, if �C1 ^ :Ki:�C2 isS-consistent.(3) A pair (P1;P2) of pointed state candidates is i-suitable for S, i � n, if �P1 ^:Ki:�P2is S-consistent. In this case we write P1 �i P2.Lemma 6. (i) For every �nite S-consistent set of CL1-formulas, there is a pointed statecandidate P = (C; t) for ' such that V^ �P is S-consistent. Moreover, if 2 subx(') and 2 , then 2 t.(ii) Suppose is a �nite set of CL1-formulas and :Ki� is a formula in subx' such thatV^:Ki� is S-consistent. Then there exists a pointed state candidate P = (C; t) for ' suchthat :� 2 t and V ^ :Ki:�P is S-consistent.6

Proof (i) Denote by �' the disjunction of all formulas �P, P a pointed state candidate for'. As �' is classically valid, it is provable in S, whence V ^ �' is S-consistent. It followsthat there is a disjunct �P of �' such that V^�P is S-consistent. Now, if 2 \ subx(')and t is the point of P, then 2 t, for otherwise : 2 t, which is a contradiction.(ii) If V ^ :Ki� is S-consistent, then so is V ^ :Ki:(:� ^ �'). It follows that thereis a pointed state candidate P with point t such that V ^ :Ki:(:� ^ �P) is S-consistent.Clearly, :� 2 t, and we are done. 2Note that Lemma 6 will hold true if we replace x by some constant c.Lemma 7. (i) If a pair (C1;C2) of state candidates for ' is i-suitable for S, i � n, then:1. for every t 2 T1 there exists a t0 2 T2 such that (t; t0) is i-suitable for S;2. for every t0 2 T2 there exists a t 2 T1 such that (t; t0) is i-suitable for S.(ii) Suppose that a pair of types (t; t0) is i-suitable for S. Then:1. 2 t0 whenever Ki 2 t;2. if A4 2 S, then Ki 2 t0 whenever Ki 2 t;3. if fD;A5g � S, then Ki 2 t whenever Ki 2 t0;4. if fD;A4; A5g � S or fT;A5g � S, then Ki 2 t i� Ki 2 t0.(iii) Suppose (t; t0) is i-suitable for S. Then C 2 t implies C 2 t0. If fA4; A5g � S, thenC 2 t i� C 2 t0.Proof (i) Suppose that t 2 T1 but there is no t0 2 T2 for which (t; t0) is i-suitable for S.This means that `S t! Ki:t0, for each t0 2 T2, and so `S t! Ki:Wt02T2 t0. Then we have`S 9xt ! Ki9x:Wt02T2 t0. Since `S 9x:Wt02T2 t0 ! :�C2 and `S �C1 ! 9xt, we �nallyobtain `S �C1 ! Ki:�C2 , contrary to S-consistency of �C1 ^:Ki:�C2 . Claim (i.2) is provedin a similar way.(ii) Suppose that Ki 2 t but =2 t0. Then : 2 t0, `S Ki ! Ki:t0, and so t ^ :Ki:t0is S-inconsistent, which is a contradiction.Now suppose that S contains A4, Ki 2 t, but Ki =2 t0. Then :Ki 2 t0. Hence`S KiKi ! Ki:t0. It follows from A4 that `S Ki ! Ki:t0, and so Ki ^ :Ki:t0 is S-inconsistent, contrary to S-consistency of (t; t0). Claims (i.3) and (i.4) are proved analogously.(iii) Suppose C 2 t. Then E( ^ C ) 2 t and so Ki( ^ C ) 2 t, for i � n. By (ii.1), ^C 2 t0, from which C 2 t0.If C 2 t0 then, as we know, Ki( ^C ) 2 t0, for i � n. So if fA4; A5g � S, then we haveby (ii.4), ^ C 2 t, and so C 2 t. 2De�nition 8 (basic tree). Let T = hW;<1; : : : ; <ni be a structure with pairwise disjointbinary relations <i on W such that DW;Si�n <iE is an intransitive tree.3 By a basic tree for' we mean the pair hT; �i, where � is a map associating with every w 2W a state candidate�(w) = hTw; Twc i for '. A basic tree is called a basic S-tree if ��(w) is S-consistent, for everyw 2W , and the pair (�(w1); �(w2)) is i-suitable for S whenever w1 <i w2.3We remind the reader that G = hW;�i is an intransitive tree if (i) G is rooted, i.e., there is w0 2 W (a rootof G) such that w0 �� w for every w 2 W , where �� is the transitive and re exive closure of �, (ii) for everyw 2 W , the set fv 2 W : v �� wg is �nite and linearly ordered by ��, (iii) every world v in G, save its root,has precisely one predecessor, i.e., jfu 2 W : u� vgj = 1, and (iv) the root w0 is irre exive, i.e., :w0 � w0.7

De�nition 9 (run). A run r in a basic S-tree hT; �i is a map associating with every w 2Wa point r(w) 2 Tw such that� the pair (r(w1); r(w2)) is i-suitable for S whenever w1 <i w2;� if :Ki 2 r(w) then 62 r(w0) for some w0 >i w;� if :C 2 r(w) then 62 r(w0) for some w0 such that w(Si�n <i)+w0.De�nition 10 (quasimodel). A basic S-tree hT; �i is called an S-quasimodel for ' if� for all w 2 W and t 2 Tw (�(w) = hTw; Twc i), there exists a run r in T such thatr(w) = t;� for every constant c, the function rc de�ned by rc(w) = t, for ht; ci 2 T conw , w 2W , is arun in hT; �i.We say ' is satis�ed in hT; �i if there exists w 2W such that ��(w) ^ ' is S-consistent.Theorem 11. If ' is satis�able in an S-quasimodel for ', then ' is satis�able in a modelbased on a frame for S.Proof For every monodic formula (x) of the form Ki�(x) or C�(x) with one free variablex, we reserve a unary predicate P (x). Likewise, for every sentence = Ki� or = C� we�x a propositional variable p . P (x) and p will be called the surrogates for (x) and .Given a monodic formula , we denote by the formula that results from by replacingall subformulas of the form Ki�(x), Ki�, C�(x), and C�, which are not within the scopeof another epistemic operator, with their surrogates. Thus, contains no occurrences ofepistemic operators, i.e., it is a purely �rst-order formula; we will call the L-reduct of .For a set of CL1-formulas �, let � = f : 2 �g.Now suppose ' is satis�ed in an S-quasimodel hT; �i, T = hW;<1; : : : ; <ni. So thereis w� 2 W such that ' ^ ��(w�) is S-consistent. It follows that the L-reduct ' ^ ��(w�)is consistent with respect to classical �rst-order logic. Moreover, by De�nition 8, ��(w) isS-consistent and ��(w) is �rst-order consistent, for every w 2W . So, for each w 2W , we can�nd a structure I(w) j= ��(w). We may also assume that I(w�) j=a� ', for some assignmenta�. Take a cardinal � � @0 exceeding the cardinality of the set of all runs in hT; �i and putD = f(r; �) : r 2 ; � < �g:Without loss of generality we can assume that D is the domain of the �rst-order structuresI(w) satisfying the ��(w), that cw = (rc; 0), and thatr(w) = f 2 subx' : I(w) j= [(r; �)]g; (1)for all runs r and � < �. (Note that the underlying �rst-order language does not containequality; for details see [9], Lemma 9.)Let us now de�ne the underlying frame F of the model we are constructing. Its set ofworlds is W . The accessibility relations Ri depend on S. Namely, we de�ne Ri to be� <i if S = KCn or S = KDCn ; 8

� <i [ fhw;wi : w 2Wg if S = TCn ;� the transitive closure of <i if S = K4Cn ;� the re exive and transitive closure of <i if S = S4Cn ;� <+i [ fhw;w0i : 9v 2W (v <+i w& v <+i w0&:9u u <i v)g if S = KD45Cn ;� the re exive, symmetric and transitive closure of <i if S = S5.Note that for S = KDCn the Ri are serial because in this case every S-consistent type for 'contains at least one formula of the form :Ki . For S = KD45Cn the Ri are clearly serial.Suppose w1Riw and wRiw2. If w1 has no <i-predecessor, then w1 <+i w2 and so w1Riw2.Otherwise, there are vj, for j = 1; 2, such that vj <+i wj , vj <+i w and the vj have no <i-predecessors. Since T is an irre exive tree, we get v1 = v2. Thus v1 <+i w1 and v1 <+i w2. Bythe de�nition of Ri, it follows that w1Riw2. Hence Ri is transitive. Similarly one can showthat Ri is euclidean.Thus we have the model M = hF; R1; : : : ; Rn; Ii. By induction on the construction of 2 sub(') we will show now that for every assignment aI(w) j=a i� (M; w) j=a :The basis of induction, i.e., the case where = Pi(�1; : : : ; �m) is clear; for then = . Theinduction step for = 1 ^ 2, = : 1, and = 8x 1 follows by the induction hypothesisfrom the equations 1 ^ 2 = 1 ^ 2, : 1 = : 1, 8x 1 = 8x 1. Let = Ki�(y) andassume that a(y) = hr; �i. (If is a sentence, y is any variable.) We then have:I(w) j=a Ki�(y) ,1 I(w) j=a PKi�(y),2 Ki�(x) 2 r(w),3 8v (wRiv ! �(x) 2 r(v)),4 8v (wRiv ! I(v) j=a �(y)),5 8v (wRiv ! (M; v) j=a �(y)),6 (M; w) j=a Ki�(y):Equivalence ,1 holds by the de�nition of ; ,2 and ,4 are consequences of (1). Theinduction hypothesis yields,5, and,6 holds by de�nition. The only non-trivial case is,3.()3) Suppose Ki� 2 r(w) and wRiw0. So if w <i w0 the claim follows by De�nition 9. IfS = K or S = D then we are done, because Ri =<i.Let S = KD45. By the de�nition of Ri, we have either w <+i w0 or there exists v suchthat v <+i w, v <+i w0 and :9u u <i v. The former case is easy; we leave it to the reader andconsider the latter one here. We have some m 2 ! and worlds v0; : : : ; vm+1 such that v0 = v,vm+1 = w and vj <i vj+1 for every j � m. By De�nition 9, (r(vj); r(vj+1)) is i-suitablefor S whenever j � m. By Lemma 7, Ki�(x) 2 r(v0) = r(v). Similarly, using v <+i w0we obtain worlds u0; : : : ; ul+1 such that u0 = v, ul+1 = w0 and uj <i uj+1 for every j � l.Again, (r(uj); r(uj+1)) is i-suitable for S whenever j � l, and by Lemma 7, Ki�(x) 2 r(ul).Using the same lemma once again, we obtain �(x) 2 r(ul+1) = r(w0). ((3) is an immediateconsequence of De�nition 9. Other cases for S are treated analogously.9

Finally, let = C�(y) and a(y) = hr; �i. Then we have the following chain of equivalences:I(w) j=a C�(y) ,1 I(w) j=a PC�(y),2 C�(x) 2 r(w),3 8v (w([i�nRi)+v ! �(x) 2 r(v)),4 8v (w([i�nRi)+v ! I(v) j=a �(y)),5 8v (w([i�nRi)+v ! (M; v) j=a �(y)),6 (M; w) j=a C�(y):As before, the only non-trivial one is ,3. ((3) follows from De�nition 9. For the otherdirection, suppose C�(x) 2 r(w) and w(Si�nRi)+v. Then there are v0; : : : ; vm+1, for somem 2 !, such that v0 = w, vm+1 = v and for all j � m there is some ij � n with vjRijvj+1.By induction on j one can show that C� 2 r(vj) for all j � m. (The basis of induction isclear, and the induction step is similar to the proof ()3) above.) Thus we have C� 2 r(vm).By the axiom C�! E(�^C�), we obtain E(�^C�) 2 r(vm). Hence, Kim(�^C�) 2 r(vm),and so, in view of Lemma 7, � 2 r(vm+1) = r(v). 2Thus, to prove completeness of our axiom system S, it su�ces to construct an S-quasimodelsatisfying ' whenever ' is S-consistent.Lemma 12. Let P = hC; ti be a pointed state candidate for ' such that �P is S-consistent.(i) If :Ki 2 t, then there exists P0 = hC0; t0i such that P �i P0 and 62 t0. Moreover, ifht; ci 2 C for some constant c, then we can choose P0 = hC0; t0i with P �i P0 and 62 t0 sothat ht0; ci 2 C0.(ii) If :C 2 t, then there are pointed state candidates Pj = hCj ; tji, j � k, with P0 = Pand P0 �i1 P1 �i2 � � � �ik Pk;for some i1; : : : ; ik � n, such that : 2 tk. Moreover, if ht; ci 2 C, then we can choose sucha sequence with htj ; ci 2 Cj for all j � k.Proof (i) follows from Lemma 6. So let us prove (ii). Suppose that such a sequence doesnot exist. Let T be the minimal set of pointed state candidates such that� P 2 T ,� if D1 2 T and D1 �i D2 for some i, then D2 2 T .Let # = WD2T �D. Then `S # ! Ki#, for all i � n. Indeed, suppose otherwise. Then#^:Ki# is S-consistent for some i � n. But then, by Lemma 6, #^:Ki:�P0 is S-consistentfor some pointed state candidate P0 =2 T . This, however, contradicts the de�nition of T , sincewe would have D �i P0 for some disjunct �D of #. Hence `S # ! E#: Clearly, 2 s forevery (D; s) 2 T , for otherwise we could construct a sequence satisfying condition (ii). Thus,`S #! and so `S #! E( ^ #): By the inference rule for C we obtain `S # ! C , andso `S �P(x) ! C , since P 2 T . But then �P is S-inconsistent, which is a contradiction.2 10

We are in a position now to prove the main result of this section.Theorem 13. If S is one of the axiomatic systems de�ned above and ' an S-consistentmonodic formula, then ' is satis�able in a model based on a frame for S.Proof In view of Theorem 11, it su�ces to construct an S-quasimodel satisfying '. ByLemma 6, we can �nd a state candidate C such that ' ^ �C is S-consistent. We are going toconstruct the required quasimodel as the limit of a sequence(Tm; �m) = (hWm; <m1 ; : : : ; <mn i ; �m);of basic S-trees, m 2 !.Let W0 = fw�g for some point w� and let �0(w�) = C. Suppose now that Tm has beenalready de�ned. For every w 2 Wm � Wm�1 we shall construct a number of new points`saturating' �m(w) (W�1 = ;). Let D = �m(w), D = hT; T coni. Pick some t 2 T and do thefollowing:(a) For every � = :Ki 2 t we take two points a� and b�, add them to Wm, putw <m+1i a�, w <m+1i b�, and �m+1(a�) = �m+1(b�) = D0, for some D0 underlying a pointedstate candidate P0 = (D0; t0) with (D; t) �i P0 and 62 t0(x). That such a P0 exists isguaranteed by Lemma 12. If ht; ci 2 �(w) for some constant c, then we take for t0 the type swith hs; ci 2 D0.(b) For every � = :C 2 t we take two sequences a1�; : : : ; ak� and b1�; : : : ; bk� and putw <m+1i1 a1� <m+1i2 � � � <m+1ik ak�; w <m+1i1 b1� <m+1i2 � � � <m+1ik bk�;and �m+1(aj�) = �m+1(bj�) = Dj; for all j � k;where (Dj; tj) is a sequence of pointed state candidates with(D; t) �i1 (D1; t1) �i2 � � � �ik (Dk; tk)and : 2 tk. Again Lemma 12 ensures the existence of such a sequence. If ht; ci 2 �(w) forsome constant c, then we take a sequence with tj; c� 2 Dj for all j � k.In the same manner we consider the remaining types in T and then the remaining worldsv 2 Wm �Wm�1. Wm+1 is de�ned as the (disjoint) union of Wm and the constructed newpoints. The relations <m+1i coincide with <mi on Wm. For the new points their extension isde�ned above. The function �m+1 coincides with �m on Wm and is de�ned above for the newworlds. Thus we have constructed Tm+1.Finally, put (T; �) = (hW;<m1 ; : : : ; <mn i ; �), whereW = [m<!Wm; <i= [m<! <mi ; � = [m<! �m:It remains to show that (T; �) is an S-quasimodel. It should be clear that the functions rcare runs. So it su�ces to show that, for all w 2 W and t 2 �(w), there exists a run r withr(w) = t.First, using Lemma 7 we �nd a sequencew� = w0 <i1 w1 <i2 � � � <ik wk = w11

and types tj 2 �(wj), 0 � j � k, such that tk = t and tj(x) ^ :Kij+1:tj+1(x) is S-consistentfor all j < k.Let r(wj) = tj and V0 = fw0; : : : ; wk+1g. De�ne by induction an increasing chain ofsets Vi � Wi with Vi �Wi � V0, on which we de�ne r. Suppose Vn is de�ned. For everyw 2Wn �Wn�1 with r(w) = t we do the following:� If :Ki 2 t, then take v 2 Wn+1 � Vn with w <i v and t0 2 �(v) such that t ^ :Ki:t0is S-consistent and 62 t0. This can be done because we always took two saturatingworlds in the construction above. Put r(v) = t0.� If :C 2 t, then take a sequence v1; : : : ; vk from Wn+1 � Vn such that w <i0 v1,v1 <i1 � � � <ik�1 vik , and types tj 2 �(vj), 1 � j � k, such that{ (t; t1) is i0-suitable for S,{ (tj ; tj+1) is ij-suitable for S, 1 � j < k,{ 62 tk.Again, this can be done since we always took two saturating sequences. Put r(vj) = tjfor all 1 � j � k.Finally, we have to de�ne r for all v 2 Wn+1 where r was not de�ned above. This can bedone recursively as follows. Suppose r(v) is not de�ned yet for some v 2Wn+1. If r is de�nedalready for the (unique) v0 such that v0 <i v, then take a t 2 �(v) such that (r(v0); t) isi-suitable for S and put r(v) = t (t exists by Lemma 7). Otherwise consider �rst v0 itself.It is now straightforward to see that r is a run. 2As a consequence we obtain:Theorem 14. Let S 2 fKCn ; TCn ;KDCn ;K4Cn ; S4Cn ;KD45Cn ; S5Cn g and let F be the class offrames for S. Then for every monodic formula ', we have `S ' i� ' 2 L1(F).5 DecidabilityAnother important algorithmic feature of the monodic formulas is that if, roughly speaking,we restrict the underlying purely �rst-order formulas to a decidable class, then the resultingmonodic fragments of the epistemic logics under consideration will also be decidable. Inparticular, we have the following:Theorem 15. Let F be any of the frame classes mentioned at the end of Section 2. Then thefollowing fragments are decidable:� the monadic fragment of L1(F),� the two-variable fragment of L1(F),� the guarded fragment of L1(F).12

(Note, however, that the guarded fragment of L(Fa) is undecidable.) For more details and anidea of the proof the reader is referred to [9, 20]. Actually, no non-trivial decidable fragmentsof epistemic predicate logics have been constructed before.It maybe also of interest to note that these decidability results make it possible to constructvarious decidable description logics with common knowledge and other epistemic operatorsapplicable to concepts and formulas (but not to roles; see [19]). Weaker epistemic descriptionlogics were proposed in [6, 12].6 Adding equalityIn this section we show that the addition of equality to the language of monodic formu-las restores the `status quo,' namely, that all the fragments considered above become non-enumerable. Let CL=1 be the language CL1 extended with the equality symbol interpreted in�rst-order structures as identity.Theorem 16. Let F be any of the frame classes de�ned at the end of Section 2. Then thelogic L1(F) in the language CL=1 is not recursively enumerable.Proof De�ne to be the conjunction of the following CL=1 -formulas: 1 = 9xP (x) ^ 8x8y (P (x) ^ P (y)! x = y); 2 = C8x8y (:Ki:P (x) ^ :Ki:P (y) ^ :P (x) ^ :P (y)! x = y); for i � 2, 3 = :C:8x (P (x)$ :C:P (x)); 4 = 8x (:C:P (x)! C:C:P (x)); 5 = 8x (Q(x)$ :C:P (x)):First we notice that for all models M = hF;D; Ii with F = hW;R1; : : : ; Rni and all w 2 W ,if (M; w) j= then QI(w) is �nite. Indeed, by 1, the set P I(w) is a singleton. From 3 weget some w0 for which w(Si�1Ri)+w0 and w0 j= 8x (P (x) $ :C:P (x)). Hence there arew0; : : : ; wm+1 2 W such that w0 = w, wm+1 = w0 and for every j � m there is some ij � 2for which wjRijwj+1 holds. In view of 2, jP I(wj+1) � P I(wj)j � 1 for every j � m, whichyields jP I(w0)j � m + 1. Thus it remains to show that QI(w) � P I(w0). Suppose a 2 QI(w).Then, by 5, w j= :C:P [a] and by 4 we obtain w j= C:C:P [a], from which w0 j= P [a].Second, we show that for every �rst-order sentence � containing neither P nor Q thefollowing are equivalent:(a) � is true in all �nite �rst-order structures;(b) ! �Q is valid in all frames in F.(Here �Q is the relativisation of � to Q, i.e., �Q = � if � is atomic, Q commutes with thebooleans, and (8x�1)Q = 8x (Q(x)! �Q1 ).)(a)) (b). Suppose there is a modelM and a world w in it such that w j= but w 6j= �Q.De�ne a �nite �rst-order structure J with domain E = QI(w) and predicates P Jk = P I(w)k \E.It can be easily shown by induction that for every formula � and every assignment a in E,we have J j=a � i� w j=a �Q. In particular, J 6j= �.13

(a) ( (b). Let us show �rst that for every natural number m > 0, there are Mm =hFm;Dm; Imi based on a frame Fm 2 Fre and w in Fm such that jW j =m and w j= . Indeed,put Wm = fw1; : : : ; wmg, wi 6= wj whenever i 6= j, Dm = N, andR1 = fhwk; wk+1i ; hwk+1; wki : k < m & 9l k = 2l + 1g [ fhwk; wki : k � mg;R2 = fhwk; wk+1i ; hwk+1; wki : k < m & 9l k = 2lg [ fhwk; wki : k � mg:Finally, for each k � m, put P I(wk) = f0; : : : ; k� 1g and QI(wk) = f0; : : : ;m� 1g (see Fig. 1).It is easy to see that the model is re exive and euclidean, and w1 j= .e e e e e: : :R1 R2 R1-� -� -� -� -�w1 w2 w3 wm�1 wmPw1 = f0g Pw2 = f0; 1g Pw3 = f0; 1; 2g f0; : : : ;m� 2g f0; : : : ;m� 1gFigure 1:Now, to complete the proof, suppose that there is a �nite �rst-order structure J withdomain D such that jDj = m and J 6j= �. Take the model Mm and the world w 6j= constructed above. Without loss of generality we can assume that QIm(w) = D. Now expandMm to a model M0m by interpreting each predicate symbol Pi as follows: P I0m(w0)i = P Ji , foreach w0 2 Wm. Now, for every �rst-order formula � (without P;Q) and every assignment ain D, we have J j=a � i� (M0m; w) j=a �Q. Therefore, (M0m; w) 6j= �Q, and so ! �Q is notvalid in Fre (which is contained in all our frame classes).It remains to recall that, by Trakhtenbrot's theorem (see [4]), the set of �rst-order sen-tences that are valid in �nite structures is not recursively enumerable. 2References[1] R.J. Aumann. Agreeing to disagree. The Annals of Statistics, 4:1236{1239, 1976.[2] M. Bacharach. The epistemic structure of a theory of game. Theory and Decision,37:7{48, 1994.[3] A.V. Chagrov and M.V. Zakharyaschev. Modal Logic. Clarendon Press, Oxford, 1997.[4] H. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 1995.[5] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Reasoning about Knowledge. MIT Press,1995.[6] A. Gr�aber, H. B�urckert, and A. Laux. Terminological reasoning with knowledge an belief.In A. Laux and H. Wansing, editors, Knowledge and Belief in Philosophy and Arti�cialIntelligence, pages 29{61. Akademie Verlag, 1995.[7] J. Halpern and Yo. Moses. A guide to completeness and complexity for modal logics ofknowledge and belief. Arti�cial Intelligence, 54:319{379, 1992.14

[8] J. Hintikka. Knowledge and Belief: An Introduction to the Logic of Two Notions. CornellUniversity Press, 1962.[9] I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of �rst-ordertemporal logics. Annals of Pure and Applied Logic, 2000.[10] M. Kaneko and T. Nagashima. Game logic and its applications 1. Studia Logica, 57:325{354, 1996.[11] M. Kaneko and T. Nagashima. Game logic and its applications 2. Studia Logica, 58:273{303, 1997.[12] A. Laux. Beliefs in multi-agent worlds: a terminological approach. In Proceedings of the11th European Conference on Arti�cial Intelligence, pages 299{303, Amsterdam, 1994.[13] W. Lenzen. Recent work in epistemic logic. Acta Philosophica Fennica, 30:1{219, 1978.[14] D. Lewis. Convention. A Philosophical Study. Harvard University Press, Cambridge,Massachusets, 1969.[15] J. McCarthy, M. Sato, T. Hayashi, and S. Igarishi. On the model theory of knowledge.Technical Report STAN-CS-78-657, Stanford University, 1979.[16] J.J. Meyer and W. van der Hoek. Epistemic Logic for AI and Computer Science. Cam-brigde University Press, 1995.[17] F. Wolter. Fragments of �rst-order common knowledge logics. Studia Logica, 2000.[18] F. Wolter and M. Zakharyaschev. Satis�ability problem in description logics with modaloperators. In Proceedings of the sixth Conference on Principles of Knowledge Represen-tation and Reasoning, Montreal, Canada, 1998. Morgan Kaufman.[19] F. Wolter and M. Zakharyaschev. Modal description logics: modalizing roles. Funda-menta Informaticae, 39:411{438, 1999.[20] F. Wolter and M. Zakharyaschev. Decidable fragments of �rst-order modal logics. 2000.

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