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IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Every normal logic programhas a 2-valued Minimal Hypotheses semantics

Alexandre Miguel Pinto and Luís Moniz Pereira

Faculdade de Ciências e Tecnologia

Universidade Nova de Lisboa

October 11th, 2011

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Outline

1 Introduction

2 Building a semantics

3 Minimal Hypotheses semantics

4 Conclusions and Future Work

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Outline

1 Introduction

2 Building a semantics

3 Minimal Hypotheses semantics

4 Conclusions and Future Work

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (1/2)

Knowledge can be written in sentencesSentences can be translated into logic formalismsLogic Programs are one such formalism for KRLP = Normal Logic Rules + Integrity Constraints

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (1/2)

Knowledge can be written in sentencesSentences can be translated into logic formalismsLogic Programs are one such formalism for KRLP = Normal Logic Rules + Integrity Constraints

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (1/2)

Knowledge can be written in sentencesSentences can be translated into logic formalismsLogic Programs are one such formalism for KRLP = Normal Logic Rules + Integrity Constraints

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (1/2)

Knowledge can be written in sentencesSentences can be translated into logic formalismsLogic Programs are one such formalism for KRLP = Normal Logic Rules + Integrity Constraints

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (2/2)

Logic Rule (LR)

A LR is of the form h← b1, . . . ,bn,not c1, . . . ,not cm withm,n ≥ 0 and finite, and where bi , cj are ground atoms.

Notation: head(r) = h and body(r) = {b1, . . . , bn, not c1, . . . , not cm}.

Normal Logic Program (NLP)

A NLP is a (countable) set of Normal Logic Rules (NLRs),where a NLR r is s.t. head(r) is a ground atom.

Integrity Constraint (IC)

A IC is a LR r such that head(r) = ⊥.

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (2/2)

Logic Rule (LR)

A LR is of the form h← b1, . . . ,bn,not c1, . . . ,not cm withm,n ≥ 0 and finite, and where bi , cj are ground atoms.

Notation: head(r) = h and body(r) = {b1, . . . , bn, not c1, . . . , not cm}.

Normal Logic Program (NLP)

A NLP is a (countable) set of Normal Logic Rules (NLRs),where a NLR r is s.t. head(r) is a ground atom.

Integrity Constraint (IC)

A IC is a LR r such that head(r) = ⊥.

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Background (2/2)

Logic Rule (LR)

A LR is of the form h← b1, . . . ,bn,not c1, . . . ,not cm withm,n ≥ 0 and finite, and where bi , cj are ground atoms.

Notation: head(r) = h and body(r) = {b1, . . . , bn, not c1, . . . , not cm}.

Normal Logic Program (NLP)

A NLP is a (countable) set of Normal Logic Rules (NLRs),where a NLR r is s.t. head(r) is a ground atom.

Integrity Constraint (IC)

A IC is a LR r such that head(r) = ⊥.

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (1/2)

Ideally, KR Declarativity with LPs would allowFull separation of concerns of rules, i.e.,KR role of NLRs 6= KR role of ICs, i.e.,

Normal Logic Rules⇔ Define space of candidate solutionsIntegrity Constraints⇔ Prune out undesired candidates

Semantics is what determines which sets of beliefs(models) are candidates

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (1/2)

Ideally, KR Declarativity with LPs would allowFull separation of concerns of rules, i.e.,KR role of NLRs 6= KR role of ICs, i.e.,

Normal Logic Rules⇔ Define space of candidate solutionsIntegrity Constraints⇔ Prune out undesired candidates

Semantics is what determines which sets of beliefs(models) are candidates

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (1/2)

Ideally, KR Declarativity with LPs would allowFull separation of concerns of rules, i.e.,KR role of NLRs 6= KR role of ICs, i.e.,

Normal Logic Rules⇔ Define space of candidate solutionsIntegrity Constraints⇔ Prune out undesired candidates

Semantics is what determines which sets of beliefs(models) are candidates

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (1/2)

Ideally, KR Declarativity with LPs would allowFull separation of concerns of rules, i.e.,KR role of NLRs 6= KR role of ICs, i.e.,

Normal Logic Rules⇔ Define space of candidate solutionsIntegrity Constraints⇔ Prune out undesired candidates

Semantics is what determines which sets of beliefs(models) are candidates

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (1/2)

Ideally, KR Declarativity with LPs would allowFull separation of concerns of rules, i.e.,KR role of NLRs 6= KR role of ICs, i.e.,

Normal Logic Rules⇔ Define space of candidate solutionsIntegrity Constraints⇔ Prune out undesired candidates

Semantics is what determines which sets of beliefs(models) are candidates

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (1/2)

Ideally, KR Declarativity with LPs would allowFull separation of concerns of rules, i.e.,KR role of NLRs 6= KR role of ICs, i.e.,

Normal Logic Rules⇔ Define space of candidate solutionsIntegrity Constraints⇔ Prune out undesired candidates

Semantics is what determines which sets of beliefs(models) are candidates

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (2/2)

According to this Ideal KR Declarativity view. . .. . . since only ICs are allowed to prune out candidatemodels. . .. . . LPs with no ICs (i.e., NLPs) should always allow, atleast, one model. . .. . . otherwise NLRs would be allowed to play the role of ICstooThis means: a semantics for NLPs should guaranteemodel existence

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (2/2)

According to this Ideal KR Declarativity view. . .. . . since only ICs are allowed to prune out candidatemodels. . .. . . LPs with no ICs (i.e., NLPs) should always allow, atleast, one model. . .. . . otherwise NLRs would be allowed to play the role of ICstooThis means: a semantics for NLPs should guaranteemodel existence

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (2/2)

According to this Ideal KR Declarativity view. . .. . . since only ICs are allowed to prune out candidatemodels. . .. . . LPs with no ICs (i.e., NLPs) should always allow, atleast, one model. . .. . . otherwise NLRs would be allowed to play the role of ICstooThis means: a semantics for NLPs should guaranteemodel existence

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (2/2)

According to this Ideal KR Declarativity view. . .. . . since only ICs are allowed to prune out candidatemodels. . .. . . LPs with no ICs (i.e., NLPs) should always allow, atleast, one model. . .. . . otherwise NLRs would be allowed to play the role of ICstooThis means: a semantics for NLPs should guaranteemodel existence

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

The Problem (2/2)

According to this Ideal KR Declarativity view. . .. . . since only ICs are allowed to prune out candidatemodels. . .. . . LPs with no ICs (i.e., NLPs) should always allow, atleast, one model. . .. . . otherwise NLRs would be allowed to play the role of ICstooThis means: a semantics for NLPs should guaranteemodel existence

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Common semantics for NLPs

2-valued: Stable ModelsClassically Supported, but. . .Lacks guarantee of model existence — undermining livenesscannot provide semantics to arbitrarily updated/merged programsLacks relevancedoes not allow for top-down query-answering proof-proceduresLacks cumulativitydoes not allow use of tabling methods to speed up computations

3-valued: Well-Founded SemanticsClassically SupportedExistence of ModelRelevanceCumulativity, but. . .it is 3-valued

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Goal: Desired Semantics (1/2)

2-valued semanticsGuarantee of model existence(e.g., for liveness in the face of updates)

RelevanceTruth of a literal is determined solely by the literals in its call-graph

CumulativityAtoms true in all models can be safely added to program as facts

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Goal: Desired Semantics (1/2)

2-valued semanticsGuarantee of model existence(e.g., for liveness in the face of updates)

RelevanceTruth of a literal is determined solely by the literals in its call-graph

CumulativityAtoms true in all models can be safely added to program as facts

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Goal: Desired Semantics (1/2)

2-valued semanticsGuarantee of model existence(e.g., for liveness in the face of updates)

RelevanceTruth of a literal is determined solely by the literals in its call-graph

CumulativityAtoms true in all models can be safely added to program as facts

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

BackgroundThe ProblemCommon semantics for NLPsGoal: Desired Semantics

Goal: Desired Semantics (1/2)

2-valued semanticsGuarantee of model existence(e.g., for liveness in the face of updates)

RelevanceTruth of a literal is determined solely by the literals in its call-graph

CumulativityAtoms true in all models can be safely added to program as facts

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Outline

1 Introduction

2 Building a semantics

3 Minimal Hypotheses semantics

4 Conclusions and Future Work

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics intuitions

Closed World Assumption Programs with stratified negation have only one model. . .. . . which hints that. . .. . . non-determinism (more than one model) comes from“non-stratification” of negationI.e., negated literals in loop play important semantic roleNeed for a different kind of Assumption for non-stratifiednegationA model can be seen as set of assumed hypotheses + theirconclusions

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics principles

Maximal skepticism→ atomic beliefs must have supportCWA is particular case

SupportClassicalbelief in atom requires a rule for it (head) with all body literals true

OR(new) Layered (Stratified)belief in atom requires a rule for it (head) with non-loop body literals trueClassical Support⇒ Layered Support

Maximal skepticism→ minimal atomic beliefsMinimality

Of all atoms true in the modelOR

Of hypotheses assumed true in the model

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics principles

Maximal skepticism→ atomic beliefs must have supportCWA is particular case

SupportClassicalbelief in atom requires a rule for it (head) with all body literals true

OR(new) Layered (Stratified)belief in atom requires a rule for it (head) with non-loop body literals trueClassical Support⇒ Layered Support

Maximal skepticism→ minimal atomic beliefsMinimality

Of all atoms true in the modelOR

Of hypotheses assumed true in the model

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics principles

Maximal skepticism→ atomic beliefs must have supportCWA is particular case

SupportClassicalbelief in atom requires a rule for it (head) with all body literals true

OR(new) Layered (Stratified)belief in atom requires a rule for it (head) with non-loop body literals trueClassical Support⇒ Layered Support

Maximal skepticism→ minimal atomic beliefsMinimality

Of all atoms true in the modelOR

Of hypotheses assumed true in the model

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics principles

Maximal skepticism→ atomic beliefs must have supportCWA is particular case

SupportClassicalbelief in atom requires a rule for it (head) with all body literals true

OR(new) Layered (Stratified)belief in atom requires a rule for it (head) with non-loop body literals trueClassical Support⇒ Layered Support

Maximal skepticism→ minimal atomic beliefsMinimality

Of all atoms true in the modelOR

Of hypotheses assumed true in the model

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics principles

Maximal skepticism→ atomic beliefs must have supportCWA is particular case

SupportClassicalbelief in atom requires a rule for it (head) with all body literals true

OR(new) Layered (Stratified)belief in atom requires a rule for it (head) with non-loop body literals trueClassical Support⇒ Layered Support

Maximal skepticism→ minimal atomic beliefsMinimality

Of all atoms true in the modelOR

Of hypotheses assumed true in the model

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantics principles

Maximal skepticism→ atomic beliefs must have supportCWA is particular case

SupportClassicalbelief in atom requires a rule for it (head) with all body literals true

OR(new) Layered (Stratified)belief in atom requires a rule for it (head) with non-loop body literals trueClassical Support⇒ Layered Support

Maximal skepticism→ minimal atomic beliefsMinimality

Of all atoms true in the modelOR

Of hypotheses assumed true in the model

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantic choices

Positive hypotheses or negative hypotheses?Traditionally, maximum negative hypotheses, but. . .

Problem with negative hypotheses

a ← not a

Assuming neg. hyp. not a leads to contradiction: aAssuming pos. hyp. a does not lead to contradiction!No explicit ¬a can be derived since we are using NLPs

Minimum positive hypotheses it is, then!In the example above a has no Classical Support. . . but it hasLayered Support!

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantic choices

Positive hypotheses or negative hypotheses?Traditionally, maximum negative hypotheses, but. . .

Problem with negative hypotheses

a ← not a

Assuming neg. hyp. not a leads to contradiction: aAssuming pos. hyp. a does not lead to contradiction!No explicit ¬a can be derived since we are using NLPs

Minimum positive hypotheses it is, then!In the example above a has no Classical Support. . . but it hasLayered Support!

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantic choices

Positive hypotheses or negative hypotheses?Traditionally, maximum negative hypotheses, but. . .

Problem with negative hypotheses

a ← not a

Assuming neg. hyp. not a leads to contradiction: aAssuming pos. hyp. a does not lead to contradiction!No explicit ¬a can be derived since we are using NLPs

Minimum positive hypotheses it is, then!In the example above a has no Classical Support. . . but it hasLayered Support!

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantic choices

Positive hypotheses or negative hypotheses?Traditionally, maximum negative hypotheses, but. . .

Problem with negative hypotheses

a ← not a

Assuming neg. hyp. not a leads to contradiction: aAssuming pos. hyp. a does not lead to contradiction!No explicit ¬a can be derived since we are using NLPs

Minimum positive hypotheses it is, then!In the example above a has no Classical Support. . . but it hasLayered Support!

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantic choices

Positive hypotheses or negative hypotheses?Traditionally, maximum negative hypotheses, but. . .

Problem with negative hypotheses

a ← not a

Assuming neg. hyp. not a leads to contradiction: aAssuming pos. hyp. a does not lead to contradiction!No explicit ¬a can be derived since we are using NLPs

Minimum positive hypotheses it is, then!In the example above a has no Classical Support. . . but it hasLayered Support!

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Semantics intuitionsSemantics principlesSemantic choices

Semantic choices

Positive hypotheses or negative hypotheses?Traditionally, maximum negative hypotheses, but. . .

Problem with negative hypotheses

a ← not a

Assuming neg. hyp. not a leads to contradiction: aAssuming pos. hyp. a does not lead to contradiction!No explicit ¬a can be derived since we are using NLPs

Minimum positive hypotheses it is, then!In the example above a has no Classical Support. . . but it hasLayered Support!

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Outline

1 Introduction

2 Building a semantics

3 Minimal Hypotheses semantics

4 Conclusions and Future Work

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Minimal Hypotheses semantics — Intuitive definition

Assumable hypotheses of program:Hyps = (positive) atoms of negative literals in loopSelect a minimal subset H of Hyps assumed true such thatH is enough to propagate 2-valuedness to all literalsIf so, an MH-model is the M = consequences of HPropagation of truth-values via deterministicpolynomial-time Remainder operator (generalization of TPoperator)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Minimal Hypotheses semantics — Intuitive definition

Assumable hypotheses of program:Hyps = (positive) atoms of negative literals in loopSelect a minimal subset H of Hyps assumed true such thatH is enough to propagate 2-valuedness to all literalsIf so, an MH-model is the M = consequences of HPropagation of truth-values via deterministicpolynomial-time Remainder operator (generalization of TPoperator)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Minimal Hypotheses semantics — Intuitive definition

Assumable hypotheses of program:Hyps = (positive) atoms of negative literals in loopSelect a minimal subset H of Hyps assumed true such thatH is enough to propagate 2-valuedness to all literalsIf so, an MH-model is the M = consequences of HPropagation of truth-values via deterministicpolynomial-time Remainder operator (generalization of TPoperator)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Minimal Hypotheses semantics — Intuitive definition

Assumable hypotheses of program:Hyps = (positive) atoms of negative literals in loopSelect a minimal subset H of Hyps assumed true such thatH is enough to propagate 2-valuedness to all literalsIf so, an MH-model is the M = consequences of HPropagation of truth-values via deterministicpolynomial-time Remainder operator (generalization of TPoperator)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

travel ← not beach,passport_okpassport_ok ← not expired_passport

expired_passport ← not passport_ok

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

travel ← not beach,passport_okpassport_ok ← not expired_passport

expired_passport ← not passport_okexpired_passport

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

travel ← not beach,passport_okexpired_passport

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

expired_passport

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain

expired_passport

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

mountainexpired_passport

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

travel ← not beach,passport_okpassport_ok ← not expired_passport

expired_passport ← not passport_okpassport_ok

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

travel ← not beachpassport_ok

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainmountain ← not travel

travel ← not beachpassport_ok

travel

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beach ← not mountainpassport_ok

travel

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

MH semantics — Example

Vacation example

beachpassport_ok

travel

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Reasoning as Query Answering

In the MH 2-v semantics, models are the deductiveconsequences of a specialized abduction for NLPsExistential query answering (a.k.a. brave reasoning)“is there a model (min. hyps+conseqs.) where query is true?”

More efficient with Relevant semantics, e.g. MH (not SM)1 Get relevant part of P for query; get Hyps for relevant part2 Get MH sub-model satisfying query

Relevance guarantees sub-model is extendible to a full one; no need to compute

full models

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Reasoning as Query Answering

In the MH 2-v semantics, models are the deductiveconsequences of a specialized abduction for NLPsExistential query answering (a.k.a. brave reasoning)“is there a model (min. hyps+conseqs.) where query is true?”

More efficient with Relevant semantics, e.g. MH (not SM)1 Get relevant part of P for query; get Hyps for relevant part2 Get MH sub-model satisfying query

Relevance guarantees sub-model is extendible to a full one; no need to compute

full models

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Reasoning as Query Answering

In the MH 2-v semantics, models are the deductiveconsequences of a specialized abduction for NLPsExistential query answering (a.k.a. brave reasoning)“is there a model (min. hyps+conseqs.) where query is true?”

More efficient with Relevant semantics, e.g. MH (not SM)1 Get relevant part of P for query; get Hyps for relevant part2 Get MH sub-model satisfying query

Relevance guarantees sub-model is extendible to a full one; no need to compute

full models

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

Minimal Hypotheses semanticsReasoning with Minimal Hypotheses semantics

Reasoning as Query Answering

In the MH 2-v semantics, models are the deductiveconsequences of a specialized abduction for NLPsExistential query answering (a.k.a. brave reasoning)“is there a model (min. hyps+conseqs.) where query is true?”

More efficient with Relevant semantics, e.g. MH (not SM)1 Get relevant part of P for query; get Hyps for relevant part2 Get MH sub-model satisfying query

Relevance guarantees sub-model is extendible to a full one; no need to compute

full models

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Outline

1 Introduction

2 Building a semantics

3 Minimal Hypotheses semantics

4 Conclusions and Future Work

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Conclusions

NLR 6= ICsNLPs must have modelMinimum positive hypotheses: generalization ofmaximum negative hypothesesMinimum positive hypotheses→ Layered Support(generalization of Classical Support)Minimal Hypotheses semantics:

2-valued semanticsrelevance, cumulativity, existence (liveness in face ofupdates)generalizes SMs (all SMs are MH models)

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Future Work

Implementation of MH semantics-based system (ongoing)Further comparisons: MH vs RSMs, MH vs PStable, othersGeneralize the MH semantics to Extended Logic ProgramsApplications: Semantic Web, KR (adding regularabduction), combination of 2-v with 3-v

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Future Work

Implementation of MH semantics-based system (ongoing)Further comparisons: MH vs RSMs, MH vs PStable, othersGeneralize the MH semantics to Extended Logic ProgramsApplications: Semantic Web, KR (adding regularabduction), combination of 2-v with 3-v

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Future Work

Implementation of MH semantics-based system (ongoing)Further comparisons: MH vs RSMs, MH vs PStable, othersGeneralize the MH semantics to Extended Logic ProgramsApplications: Semantic Web, KR (adding regularabduction), combination of 2-v with 3-v

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Future Work

Implementation of MH semantics-based system (ongoing)Further comparisons: MH vs RSMs, MH vs PStable, othersGeneralize the MH semantics to Extended Logic ProgramsApplications: Semantic Web, KR (adding regularabduction), combination of 2-v with 3-v

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Future Work

Thank you

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (1/2)

Disjunctive LPs (DisjLPs): rules are of the form

h1 ∨ . . . ∨ hq ← b1, . . . ,bn,not c1, . . . ,not cm

with q ≥ 1, m,n ≥ 0 and finite, and where hk ,bi , cj areatomsCan be transformed into NLPs via Shifting Rule

a ∨ b ← Body a← not b,Body

b ← not a,Body

Shifting Rule produces loops over default negationCan be problematic to SMsMH solves any loops by assuming minimal positivehypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (1/2)

Disjunctive LPs (DisjLPs): rules are of the form

h1 ∨ . . . ∨ hq ← b1, . . . ,bn,not c1, . . . ,not cm

with q ≥ 1, m,n ≥ 0 and finite, and where hk ,bi , cj areatomsCan be transformed into NLPs via Shifting Rule

a ∨ b ← Body a← not b,Body

b ← not a,Body

Shifting Rule produces loops over default negationCan be problematic to SMsMH solves any loops by assuming minimal positivehypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (1/2)

Disjunctive LPs (DisjLPs): rules are of the form

h1 ∨ . . . ∨ hq ← b1, . . . ,bn,not c1, . . . ,not cm

with q ≥ 1, m,n ≥ 0 and finite, and where hk ,bi , cj areatomsCan be transformed into NLPs via Shifting Rule

a ∨ b ← Body a← not b,Body

b ← not a,Body

Shifting Rule produces loops over default negationCan be problematic to SMsMH solves any loops by assuming minimal positivehypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (1/2)

Disjunctive LPs (DisjLPs): rules are of the form

h1 ∨ . . . ∨ hq ← b1, . . . ,bn,not c1, . . . ,not cm

with q ≥ 1, m,n ≥ 0 and finite, and where hk ,bi , cj areatomsCan be transformed into NLPs via Shifting Rule

a ∨ b ← Body a← not b,Body

b ← not a,Body

Shifting Rule produces loops over default negationCan be problematic to SMsMH solves any loops by assuming minimal positivehypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (2/2)

Disjunctions / Shifting Rule / Loops

a ∨ b a← not b b ← not ab ∨ c b ← not c c ← not bc ∨ a c ← not a a← not c

Basis of MH intuitition = conceptually reverse the Shifting Rule:Loops (any kind of loops) encode disjunction of hypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (2/2)

Disjunctions / Shifting Rule / Loops

a ∨ b a← not b b ← not ab ∨ c b ← not c c ← not bc ∨ a c ← not a a← not c

Basis of MH intuitition = conceptually reverse the Shifting Rule:Loops (any kind of loops) encode disjunction of hypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Disjunctive Logic Programs (2/2)

Disjunctions / Shifting Rule / Loops

a ∨ b a← not b b ← not ab ∨ c b ← not c c ← not bc ∨ a c ← not a a← not c

Basis of MH intuitition = conceptually reverse the Shifting Rule:Loops (any kind of loops) encode disjunction of hypotheses

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Motivation for the Layered Negative Reduction

Variation of the vacation examplebeach ← not mountain

mountain ← not traveltravel ← not beach

Three MHs: {beach, travel, not mountain}, {beach, not travel,mountain},{not beach, travel,mountain}.

Variation of the vacation example with fourth friendbeach ← not mountain

mountain ← not traveltravel ← not beachbeach

Two MHs: {beach, travel, not mountain}, {beach, not travel,mountain}.

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics

university-logo

IntroductionBuilding a semantics

Minimal Hypotheses semanticsConclusions and Future Work

ConclusionsFuture Work

Motivation for the Layered Negative Reduction

Variation of the vacation examplebeach ← not mountain

mountain ← not traveltravel ← not beach

Three MHs: {beach, travel, not mountain}, {beach, not travel,mountain},{not beach, travel,mountain}.

Variation of the vacation example with fourth friendbeach ← not mountain

mountain ← not traveltravel ← not beachbeach

Two MHs: {beach, travel, not mountain}, {beach, not travel,mountain}.

October 11th, EPIA 2011, Lisbon Every NLP has a 2-valued MH semantics