Theorem Provers for Every Normal Modal Logic 1 T obias Gleißner Alexander Steen Christoph Benzmüller Freie Universität Berlin 21st Int. Conf. on Logic for Programming, Artificial Intelligence and Reasoning 1This work has been supported by the DFG under grant BE 2501/11-1 (Leo-III).
T alk outline 1. Higher-Order Modal Logic (HOML) 2. Automating HOML 3. Evaluation 4. Example / Demo , Theorem Provers for Every Normal Modal Logic, LPAR-21 2
Introduction Reasoning in Non-Classical Logics ◮ Increasing interest various fields ◮ Artificial Intelligence (e.g. Agents, Knowledge) ◮ Computer Linguistics (e.g. Semantics) ◮ Mathematics (e.g. Geometry, Category theory) ◮ Theoretical Philsophy (e.g. Metaphysics) ◮ Most powerful ATP/ITP: Classical logic only Our focus: Modal logics ◮ Prover for (propositional) modal logics exist ◮ ModLeanTAP, Molle, Bliksem, FaCT++, ◮ MOLTAP, KtSeqC, ST eP, TRP ◮ ... ◮ Only few for quantified variants ◮ MleanTAP, MleanCoP, MleanSeP (J. Otten) ◮ f2p+MSPASS , Theorem Provers for Every Normal Modal Logic, LPAR-21 3
Introduction Reasoning in Non-Classical Logics ◮ Increasing interest various fields ◮ Artificial Intelligence (e.g. Agents, Knowledge) ◮ Computer Linguistics (e.g. Semantics) ◮ Mathematics (e.g. Geometry, Category theory) ◮ Theoretical Philsophy (e.g. Metaphysics) ◮ Most powerful ATP/ITP: Classical logic only Our focus: Modal logics ◮ Prover for (propositional) modal logics exist ◮ ModLeanTAP, Molle, Bliksem, FaCT++, ◮ MOLTAP, KtSeqC, ST eP, TRP ◮ ... ◮ Only few for quantified variants ◮ MleanTAP, MleanCoP, MleanSeP (J. Otten) ◮ f2p+MSPASS , Theorem Provers for Every Normal Modal Logic, LPAR-21 3
Introduction Reasoning in Non-Classical Logics ◮ Increasing interest various fields ◮ Artificial Intelligence (e.g. Agents, Knowledge) ◮ Computer Linguistics (e.g. Semantics) ◮ Mathematics (e.g. Geometry, Category theory) ◮ Theoretical Philsophy (e.g. Metaphysics) ◮ Most powerful ATP/ITP: Classical logic only Our focus: Modal logics ◮ Prover for (propositional) modal logics exist ◮ ModLeanTAP, Molle, Bliksem, FaCT++, ◮ MOLTAP, KtSeqC, ST eP, TRP ◮ ... ◮ Only few for quantified variants ◮ MleanTAP, MleanCoP, MleanSeP (J. Otten) ◮ f2p+MSPASS , Theorem Provers for Every Normal Modal Logic, LPAR-21 3
Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed , Theorem Provers for Every Normal Modal Logic, LPAR-21 4
Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed Studies in Metaphysics (e.g. Ontological Argument), Studies in Computer Ethics , Theorem Provers for Every Normal Modal Logic, LPAR-21 4
Automation of Quantified Modal Logic Motivation 1. First-order quantification is (sometimes) not enough 2. Semantic diversity/flexibility needed , Theorem Provers for Every Normal Modal Logic, LPAR-21 4
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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 4
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 a proving system ◮ Semantic variations with minor adjustments ◮ Axiomatization ◮ Quantification semantics ◮ ... , Theorem Provers for Every Normal Modal Logic, LPAR-21 4
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 a proving system ◮ Semantic variations with minor adjustments ◮ Axiomatization ◮ Quantification semantics ◮ ... , Theorem Provers for Every Normal Modal Logic, LPAR-21 4
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 → ◮ T ypically, base types are o and ι , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι T ype of truth-values , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι T ype of individuals , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι ◮ T erms 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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι ◮ T erms defined by ( α , β ∈ T , c α ∈ Σ , X α ∈ V , i ∈ I ) s , t ::= c α | X α | ( λ X 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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι ◮ T erms defined by ( α , β ∈ T , c α ∈ Σ , X α ∈ V , i ∈ I ) s , t ::= c α | X α | ( λ X 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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι ◮ T erms defined by ( α , β ∈ T , c α ∈ Σ , X α ∈ V , i ∈ I ) s , t ::= c α | X α | ( λ X 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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
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 → ◮ T ypically, base types are o and ι ◮ T erms defined by ( α , β ∈ T , c α ∈ Σ , X α ∈ V , i ∈ I ) s , t ::= c α | X α | ( λ X 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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 5
Higher Order Modal Logic – Semantics Extend HOL models with Kripke structures W , { R i } i ∈ I , { D w } w ∈ W , { I w } w ∈ W � � M = , Theorem Provers for Every Normal Modal Logic, LPAR-21 6
Higher Order Modal Logic – Semantics Extend HOL models with Kripke structures W , { R i } i ∈ I , { D w } w ∈ W , { I w } w ∈ W � � M = Set of possible worlds , Theorem Provers for Every Normal Modal Logic, LPAR-21 6
Higher Order Modal Logic – Semantics Extend HOL models with Kripke structures W , { R i } i ∈ I , { D w } w ∈ W , { I w } w ∈ W � � M = Family of accessibility relations R i ⊆ W × W , Theorem Provers for Every Normal Modal Logic, LPAR-21 6
Higher Order Modal Logic – Semantics Extend HOL models with Kripke structures W , { R i } i ∈ I , { D w } w ∈ W , { I w } w ∈ W � � M = 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 , Theorem Provers for Every Normal Modal Logic, LPAR-21 6
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