The World s Most Widely Applicable Modal Logic Theorem Prover and its Associated Infrastructure

Transcription

1 The World s Most Widely Applicable Modal Logic Theorem Prover and its Associated Infrastructure Alexander Steen Freie Universität Berlin RuleML Webinar September 29th 0 jww C. Benzmüller, T. Gleißner

2 Talk outline 1. Motivation 2. Flavours of modal logics 3. How it works (roughly) 4. Evaluation The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 2

3 Introduction Reasoning in Non-Classical Logics Increasing interest in various fields Artificial Intelligence (e.g. Agents, Knowledge) Computer Linguistics (e.g. Semantics) Mathematics (e.g. Geometry, Category theory) Theoretical Philosophy (e.g. Metaphysics) Legal Informatics (e.g. Computable/Smart contracts) Most powerful ATP/ITP: Classical logic only Focus here: Modal logics Prover for (propositional) modal logics exist ModLeanTAP, Molle, Bliksem, FaCT++, MOLTAP, KtSeqC, STeP, TRP... Only few for quantified variants MleanTAP, MleanCoP, MleanSeP (J. Otten) f2p+mspass The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 3

4 Introduction Reasoning in Non-Classical Logics Increasing interest in various fields Artificial Intelligence (e.g. Agents, Knowledge) Computer Linguistics (e.g. Semantics) Mathematics (e.g. Geometry, Category theory) Theoretical Philosophy (e.g. Metaphysics) Legal Informatics (e.g. Computable/Smart contracts) Most powerful ATP/ITP: Classical logic only Focus here: Modal logics Prover for (propositional) modal logics exist ModLeanTAP, Molle, Bliksem, FaCT++, MOLTAP, KtSeqC, STeP, TRP... Only few for quantified variants MleanTAP, MleanCoP, MleanSeP (J. Otten) f2p+mspass The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 3

5 Introduction Reasoning in Non-Classical Logics Increasing interest in various fields Artificial Intelligence (e.g. Agents, Knowledge) Computer Linguistics (e.g. Semantics) Mathematics (e.g. Geometry, Category theory) Theoretical Philosophy (e.g. Metaphysics) Legal Informatics (e.g. Computable/Smart contracts) Most powerful ATP/ITP: Classical logic only Focus here: Modal logics Prover for (propositional) modal logics exist ModLeanTAP, Molle, Bliksem, FaCT++, MOLTAP, KtSeqC, STeP, TRP... Only few for quantified variants MleanTAP, MleanCoP, MleanSeP (J. Otten) f2p+mspass The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 3

6 Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed See studies in Metaphysics, e.g. Gödel s Ontological Argument [BenzmüllerW.-Paleo,2017] and several variants of it Anderson-Hájek Controversy [Benzm.WeberW.-Paleo,2017] The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4

7 Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed: Properties of modal operators necessary ( ) and possibly ( )... but that s not all of it! The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4

8 Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed Automation approach Indirect: Via encoding into (classical) HOL Use existing general purpose HOL reasoners The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4

9 Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed Automation approach Indirect: Via encoding into (classical) HOL Use existing general purpose HOL reasoners Advantages Sophisticated existing systems ATPs: TPS, agsyhol, Satallax, LEO-II, Leo-III Further: Isabelle, Nitpick Not fixed to any one proving system Semantic variations with minor adjustments Axiomatization Quantification semantics... The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4

10 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

11 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Type of truth-values The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

12 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Type of individuals The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

13 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Terms defined by (α,β T, c α Σ, X α V, i I) s, t ::= c α X α Allow infix notation for binary logical connectives Remaining logical connectives can be defined as usual Formulae of HOML are those terms with type o The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

14 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Terms defined by (α,β T, c α Σ, X α V, i I) s, t ::= c α X α (λx α.s β ) α β (s α β t α ) β Allow infix notation for binary logical connectives Remaining logical connectives can be defined as usual Formulae of HOML are those terms with type o The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

15 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Terms defined by (α,β T, c α Σ, X α V, i I) s, t ::= c α X α (λx α.s β ) α β (s α β t α ) β ( i o o s o) o Allow infix notation for binary logical connectives Remaining logical connectives can be defined as usual Formulae of HOML are those terms with type o The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

16 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Terms defined by (α,β T, c α Σ, X α V, i I) s, t ::= c α X α (λx α.s β ) α β (s α β t α ) β ( i o o s o) o Allow infix notation for binary logical connectives Remaining logical connectives can be defined as usual Formulae of HOML are those terms with type o The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

17 Higher Order Modal Logic Syntax Based on Simple type theory [Church, J.Symb.L., 1940] augmented with modalities Simple types T generated by base types and mappings ( ) Usually, base types are o and ι Terms defined by (α,β T, c α Σ, X α V, i I) s, t ::= c α X α (λx α.s β ) α β (s α β t α ) β ( i o o s o) o Allow infix notation for binary logical connectives Remaining logical connectives can be defined as usual Formulae of HOML are those terms with type o The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5

18 Higher Order Modal Logic Semantics Extend HOL models with Kripke structures M = W, {R i } i I, {D w } w W, {I w } w W The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6

19 Higher Order Modal Logic Semantics Extend HOL models with Kripke structures M = W, {R i } i I, {D w } w W, {I w } w W Set of possible worlds The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6

20 Higher Order Modal Logic Semantics Extend HOL models with Kripke structures M = W, {R i } i I, {D w } w W, {I w } w W Family of accessibility relations R i W W The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6

21 Higher Order Modal Logic Semantics Extend HOL models with Kripke structures M = W, {R i } i I, {D w } w W, {I w } w W Family of frames, one for every world Notion of frames D = (D τ ) τ T as in HOL: D ι = D o = {T, F} D τ ν = D D τ ν The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6

22 Higher Order Modal Logic Semantics Extend HOL models with Kripke structures M = W, {R i } i I, {D w } w W, {I w } w W Family of interpretation functions I w I w c τ d D τ D w Assume I w ( ), I w ( )... is standard. The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6

23 Higher Order Modal Logic Semantics Extend HOL models with Kripke structures M = W, {R i } i I, {D w } w W, {I w } w W Value of a term (wrt. var. assignment g): X τ M,g,w = g w (X). (not all shown here...) T if i o o s o M,g,w so M,g,v = T for all v W s.t. (w, v) R i = F otherwise Assume Henkin semantics The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6

24 Semantic variants of HOML 1. Axiomatization of i 2. Quantification 3. Rigidity 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

25 Semantic variants of HOML 1. Axiomatization of i What properties does the box operators have? Depending on the application domain Some popular axiom schemes: Name Axiom scheme Condition on r i Corr. formula K i (s t) ( i s i t) B s i i s symmetric wr i v vr i w D i s i s serial v.wr i v T/M i s s reflexive wr i w 4 i s i i s transitive wr i v vr i u wr i u 5 i s i i s euclidean wr i v wr i u vr i u Quantification 3. Rigidity 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

26 Semantic variants of HOML 1. Axiomatization of i What properties does the box operators have? 2. Quantification What is the meaning of? Several popular choices exist (1) Varying domains: As introduced (unrestricted frames) (2) Constant domains: D w = D v for all worlds w, v W (3) Cumulative domains: D w D v whenever (w, v) R i 3. Rigidity (4) Decreasing domains: D w D v whenever (w, v) R i 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

27 Semantic variants of HOML 1. Axiomatization of i What properties does the box operators have? 2. Quantification What is the meaning of? 3. Rigidity Do all constants c Σ denote the same object at every world? Several popular choices exist (1) Flexible constants: As introduced (unrestricted I w ) (2) Rigid constants: I w (c) = I v (c) for all worlds w, v W and all c Σ 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

28 Semantic variants of HOML 1. Axiomatization of i What properties does the box operators have? 2. Quantification What is the meaning of? 3. Rigidity Do all constants c Σ denote the same object at every world? 4. Consequence What is an appropriate notion of logical consequence = HOML? Many choices exist, two of them are (1) Local consequence:... not displayed here... (2) Global consequence:... not displayed here... The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

29 Semantic variants of HOML 1. Axiomatization of i What properties does the box operators have? 2. Quantification What is the meaning of? 3. Rigidity Do all constants c Σ denote the same object at every world? 4. Consequence What is an appropriate notion of logical consequence = HOML? at least = 160 distinct logics The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

30 Semantic variants of HOML 1. Axiomatization of i What properties does the box operators have? 2. Quantification What is the meaning of? 3. Rigidity Do all constants c Σ denote the same object at every world? 4. Consequence What is an appropriate notion of logical consequence = HOML? Example: Modal logic S5, constant domains, rigid symbols Generated by 5: i s i i s Theorems: all classical tautologies, s i i s, (symmetric r i ) i s s, (reflexive r i ) X. i f X i X.f X (Barcan formula)... The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7

31 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: o o = o o o = Π τ (τ o) o = (4) Meta-logical notions: o o = valid = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 8

32 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: o o = o o o = Π τ (τ o) o = (4) Meta-logical notions: o o = valid = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 8

33 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: o o = o o o = Π τ (τ o) o = (4) Meta-logical notions: o o = valid = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 8

34 Stand-alone tool Embedding procedure implemented as stand-alone tool Semantic specification is analyzed first (Meta-)logical notions are included as axioms/definitions Output format: Plain THF (TH0) Integrated as pre-processor into Leo-III The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 9

35 Evaluation Evaluation setting: Translated all 580 mono-modal QMLTP problems to modal THF Semantic setting: 1. Modal operator axiom system {K, D, T, S4, S5} 2. Quantification semantics {constant, varying, cumul., decreasing} 3. Rigid constants 4. Consequence {local, global} Native modal logic prover: MleanCoP (J. Otten) HOL reasoners: Satallax, LEO-II, Nitpick Timeout 60s (2x AMD Opteron 2376 Quad Core/16 GB RAM) Comments on evaluation result: MleanCoP not applicable to modal logic K MleanCoP not applicable to decreasing domains semantics MleanCoP not applicable to problems with equality symbol MleanCoP not applicable for global consequence Only first-order modal logic problems Embedding approach not restricted to benchmark settings The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 10

36 Evaluation Evaluation setting: Translated all 580 mono-modal QMLTP problems to modal THF Semantic setting: 1. Modal operator axiom system {K, D, T, S4, S5} 2. Quantification semantics {constant, varying, cumul., decreasing} 3. Rigid constants 4. Consequence {local, global} Native modal logic prover: MleanCoP (J. Otten) HOL reasoners: Satallax, LEO-II, Nitpick Timeout 60s (2x AMD Opteron 2376 Quad Core/16 GB RAM) Comments on evaluation result: MleanCoP not applicable to modal logic K MleanCoP not applicable to decreasing domains semantics MleanCoP not applicable to problems with equality symbol MleanCoP not applicable for global consequence Only first-order modal logic problems Embedding approach not restricted to benchmark settings The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 10

37 Evaluation #2 Result excerpt: Theorems 400 LEO-II Satallax MleanCoP Theorems found D vary D const T vary T const S4 vary S4 const S5 vary S5 const The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 11

38 Evaluation #3 Result excerpt: Counter satisfiable (CSA) 250 Nitpick MleanCoP 200 CSA found D vary D const T vary T const S4 vary S4 const S5 vary S5 const The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 12

39 The penultimate slide Related work Generic theorem proving systems: The Logics Workbench,MetTeL2, LoTREC Embedding of further logics: Conditional logics, hybrid logics, many-valued logics, free logic,... Conclusion Provided a quite general semantics for HOML Presented a procedure that automatically converts HOML into HOL Implemented a stand-alone tool (e.g. as preprocessor) standard HOL provers can be used to reason about problems encoded in the modal THF syntax Approach feasible (no evaluation for higher-order problems yet) Many new problems contributed in the modal THF format The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 13

40 The penultimate slide Related work Generic theorem proving systems: The Logics Workbench,MetTeL2, LoTREC Embedding of further logics: Conditional logics, hybrid logics, many-valued logics, free logic,... Conclusion Provided a quite general semantics for HOML Presented a procedure that automatically converts HOML into HOL Implemented a stand-alone tool (e.g. as preprocessor) standard HOL provers can be used to reason about problems encoded in the modal THF syntax Approach feasible (no evaluation for higher-order problems yet) Many new problems contributed in the modal THF format The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 13

41 The ultimate slide Thank you for your attention! The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 14

42 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: (4) Meta-logical notions: = = = = = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15

43 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: (4) Meta-logical notions: = = = = = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15

44 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: (4) Meta-logical notions: = = = = = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15

45 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: (4) Meta-logical notions: o o = λs μ o.λw μ. (S W) o o o = λs μ o.λt μ o.λw μ. (S W) (T W) Π τ (τ o) o = λp τ μ o.λw μ. X τ. P X W o o = λs μ o.λw μ. V μ. (r i W V) S V = The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15

46 Embedding of HOML within HOL Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas s o are mapped to HOL predicates s μ o (2) Introduce new constants r i for each i I μ μ o (3) Connectives: (4) Meta-logical notions: o o = λs μ o.λw μ. (S W) o o o = λs μ o.λt μ o.λw μ. (S W) (T W) Π τ (τ o) o = λp τ μ o.λw μ. X τ. P X W o o = λs μ o.λw μ. V μ. (r i W V) S V valid = λs μ o. W μ. s W The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15

47 Embedding of HOML within HOL #2 Embedding semantic variants 1. Axiomatization of i 2. Quantification 3. Rigidity 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16

48 Embedding of HOML within HOL #2 Embedding semantic variants 1. Axiomatization of i Recall correspondences: Name Axiom scheme Condition on r i Corr. formula B s i i s symmetric wr i v vr i w For each desired axiom scheme for i : Postulate frame condition on r i as HOL axiom 2. Quantification 3. Rigidity 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16

49 Embedding of HOML within HOL #2 Embedding semantic variants 1. Axiomatization of i Postulate frame condition on r i as HOL axiom 2. Quantification Choose appropriate definition/axiomatization of quantifier: Constant domains quantifier: Π τ (τ o) o = λp τ μ o.λw μ. X τ. P X W Varying domains quantifier: Π τ (τ o) o,va = λp τ μ o.λw μ. X τ. (eiw X W) (P X W) Cumulative/decreasing domains quantifier: Add axioms on eiw 3. Rigidity 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16

50 Embedding of HOML within HOL #2 Embedding semantic variants 1. Axiomatization of i Postulate frame condition on r i as HOL axiom 2. Quantification Choose appropriate definition/axiomatization of quantifier 3. Rigidity Rigid constants: Only translate Boolean types to predicates: o = μ o Rigid constants: Also translate individuals types to predicates: ι = μ ι 4. Consequence The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16

51 Embedding of HOML within HOL #2 Embedding semantic variants 1. Axiomatization of i Postulate frame condition on r i as HOL axiom 2. Quantification Choose appropriate definition/axiomatization of quantifier 3. Rigidity Appropriate type lifting 4. Consequence Global consequence: Apply valid (μ o) o to all translated s μ o s o = valid (μ o) o s μ o Local consequence: Apply actuality operator A to all translated s μ o s o = A (μ o) o s μ o where A = λs μ o. s w actual and w actual is an uninterpreted symbol The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16

52 Problem representation Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax thf( classical, axiom,! [X:$i]: X)). Extend syntax with modalities thf( modal, axiom,! [X:$i]: X))). thf( multi_modal, axiom,! [X:$i]: X))). (2) Semantics configuration Add logic -annotated statements to the problem: thf(simple_s5, logic, ($modal := [ $constants := $rigid, $quantification := $constant, $consequence := $global, $modalities := $modal_system_s5 ])). Intended semantics is attached to the problem The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 17

53 Problem representation Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax thf( classical, axiom,! [X:$i]: X)). Extend syntax with modalities thf( modal, axiom,! [X:$i]: X))). thf( multi_modal, axiom,! [X:$i]: X))). (2) Semantics configuration Add logic -annotated statements to the problem: thf(simple_s5, logic, ($modal := [ $constants := $rigid, $quantification := $constant, $consequence := $global, $modalities := $modal_system_s5 ])). Intended semantics is attached to the problem The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 17

54 Problem representation Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax thf( classical, axiom,! [X:$i]: X)). Extend syntax with modalities thf( modal, axiom,! [X:$i]: X))). thf( multi_modal, axiom,! [X:$i]: X))). (2) Semantics configuration Add logic -annotated statements to the problem: thf(simple_s5, logic, ($modal := [ $constants := $rigid, $quantification := $constant, $consequence := $global, $modalities := $modal_system_s5 ])). Intended semantics is attached to the problem The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 17

55 Problem representation Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax thf( classical, axiom,! [X:$i]: X)). Extend syntax with modalities thf( modal, axiom,! [X:$i]: X))). thf( multi_modal, axiom,! [X:$i]: X))). (2) Semantics configuration Add logic -annotated statements to the problem: thf( mydomain_type, type, ( human : $ttype ) ). thf( myconstant_declaration, type, ( myconstant : $i ) ). thf( complicated_s5, logic, ( $modal := [ $constants := [ $rigid, myconstant := $flexible ], $quantification := [ $constant, human := $varying ], $consequence := [ $global, myaxiom := $local ], $modalities := [ $modal_system_s5, 1 := $modal_system_t ] ] ) ). Intended semantics is attached to the problem The World s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 18