1. Motivation 2. Uniform matrix characterizations 3. Conversion - PowerPoint PPT Presentation

Converting Non-Classical Matrix Proofs into Sequent-Style Systems Stephan Schmitt Christoph Kreitz � 1. Motivation 2. Uniform matrix characterizations 3. Conversion into prefixed sequent systems 4. Extension to conventional sequent calculi 5. Conclusion & further work

Motivation • Efficient normal form theorem provers for classical logic – Otter (Resolution) / Setheo, KoMeT (Connection method) • Efficient proof procedures for non-classical logics – Extended matrix characterizations of logical validity (Wallen 1990) • Efficiency depends on compact proof representation ❀ Convert “machine proofs” into comprehensible form ⇓ Uniform conversion procedure for non-classical logics – Non-normal form matrix proofs �− → sequent systems/calculi – Logics: C ; J ; S 5 (varying, constant) ; K, K 4 , D, D 4 , T, S 4 Converting Non-Classical Matrix Proofs . . . 1 CADE-13, 1996

Non-classical Matrix Proofs D -formula F ≡ ✷ ∃ x. ✷ C ( x ) ∧ ✸ B ⇒ ✸ ( B ∧ ✸ ∃ x. C ( x )) b 0 : ⇒ 0 Substitutions a 1 : ∧ 1 a 8 : ✸ 0 σ M (¯ a 3 )= σ M (¯ a 9 )= a 7 σ M (¯ a 12 )=¯ a 5 a 2 : ✷ 1 a 6 : ✸ 1 a 9 : ∧ 0 ¯ ❁ M = { ( a 7 , ¯ a 3 ) , ( a 7 , ¯ a 9 ) } a 3 : ∃ 1 a 7 : B 1 a 10 : B 0 a 11 : ✸ 0 ¯ σ Q (¯ a 13 )= a 4 a 4 : ✷ 1 a 12 : ∃ 0 ¯ ❁ Q = { ( a 4 , ¯ a 13 ) } a 5 : C ( a 4 ) 1 a 13 ) 0 ¯ a 13 : C (¯ ¯ Irreflexive ordering ≡ ≪ ∪ ❁ Q ∪ ❁ M ✁     a 9 : B 0 b 0 ¯ � �     a 5 : C ( a 4 ) 1 b 0 a 7 : B 1 b 0 ¯ a 3 ¯         a 13 ) 0     b 0 ¯ a 9 ¯ a 12 : C (¯     Converting Non-Classical Matrix Proofs . . . 2 CADE-13, 1996

Unified Matrix Characterizations M A formula F is L -valid if there is a multiplicity µ , a combined substitution σ = � σ Q , σ M � and a set of σ - complementary connections such that each path through F 0 contains at least one connection. (Wallen, 1990) - σ Q quantifier substitution (induces ❁ Q ) - σ m modal substitution defined on prefixes (induces ❁ M ) ❀ depends on accessibility relation pR 0 q on prefixes - σ -complementarity: � σ Q , σ M � unifies connected atoms & admissibility (i) Irreflexive reduction ordering ✁ ≡ ≪ ∪ ❁ Q ∪ ❁ M (ii) σ m respects R 0 and interacts with σ Q wrt. modal domain conditions Converting Non-Classical Matrix Proofs . . . 3 CADE-13, 1996

Prefixed Sequent Systems L P D -formula: F ≡ ✷ ∃ x. ✷ C ( x ) ∧ ✸ B ⇒ ✸ ( B ∧ ✸ ∃ x. C ( x )) b 0 a 7 ¯ a 5 : C ( a 4 ) , b 0 a 7 : B ⊢ b 0 a 7 ¯ a 5 : C ( a 4 ) a 5 : ∃ x.C ( x ) ∃ –r ( ¯ a 12 , ¯ a 13 ) a 5 : C ( a 4 ) , b 0 a 7 : B ⊢ b 0 a 7 ¯ b 0 a 7 ¯ a 5 : C ( a 4 ) , b 0 a 7 : B ⊢ b 0 a 7 : ✸ ∃ x.C ( x ) ✸ –r ( a 11 ) b 0 a 7 ¯ a 5 : C ( a 4 ) , b 0 a 7 : B ⊢ b 0 a 7 : B b 0 a 7 ¯ ∧ –r ( ¯ a 9 , a 10 ) b 0 a 7 ¯ a 5 : C ( a 4 ) , b 0 a 7 : B ⊢ b 0 a 7 : B ∧ ✸ ∃ x.C ( x ) a 5 : C ( a 4 ) , b 0 a 7 : B ⊢ b 0 : ✸ ( B ∧ ✸ ∃ x.C ( x )) ✸ –r ( a 8 ) b 0 a 7 ¯ b 0 a 7 : ∃ x. ✷ C ( x ) , b 0 a 7 : B ⊢ b 0 : ✸ ( B ∧ ✸ ∃ x.C ( x )) ∃ –l ( ¯ a 3 ) , ✷ –l ( a 4 , ¯ a 5 ) b 0 : ✷ ∃ x. ✷ C ( x ) , b 0 a 7 : B ⊢ b 0 : ✸ ( B ∧ ✸ ∃ x.C ( x )) ✷ –l ( a 2 ) b 0 : ✷ ∃ x. ✷ C ( x ) , b 0 : ✸ B ⊢ b 0 : ✸ ( B ∧ ✸ ∃ x.C ( x )) ✸ –l ( a 6 , a 7 ) ⇒ –l ( b 0 ) , ∧ –l ( a 1 ) ⊢ b 0 : ✷ ∃ x. ✷ C ( x ) ∧ ✸ B ⇒ ✸ ( B ∧ ✸ ∃ x.C ( x )) Prefix construction: respect pR 0 q , new ( π, ν ), used ( ν ) – Reflects conditions on prefix unification in matrix-proof M ⇒ Conversion ≡ traverse reduction ordering ✁ using � σ Q , σ M � – Position ⇒ main-operator & polarity ⇒ unique L P -rule Converting Non-Classical Matrix Proofs . . . 4 CADE-13, 1996

Uniform Conversion M �− → L P • Divide logical calculi into invariant and variant parts wrt. logic L Matrix characterization M Prefixed sequent systems L P invariant variant invariant variant connection method prefix unification σ m , R 0 sequent rules prefix construction π , ν , R 0 • Represent variant parts in tables ; define mappings betweeen (in)variants of calculi • Algorithm structure ≡ traverse ✁ ; table access ≡ using � σ Q , σ M � wrt. L function convert ( L , ✁ , σ ) : L P -rules while ¬ proven ( ✁ ) do select open position a ∈ ≪ , not blocked by ❁ Q ∪ ❁ M ; update ( a, y ) ∈ ❁ Q ∪ ❁ M ; case Ptype ( a ) = atom : proven ( ✁ ) if axiom -rule is possible ; Ptype ( a ) ∈ { α, γ, δ } : apply L P -rule using σ Q and tables wrt. L (for γ, δ ) ; Ptype ( a ) ∈ { π, ν } : apply L P -rule using σ m and tables wrt. L ; Ptype ( a ) = β : apply β -rule ; [ ✁ 1 , ✁ 2 ] := β -split ( ✁ , a ) ; convert ( L , ✁ 1 , σ ) ; convert ( L , ✁ 2 , σ ) ; proven ( ✁ ) . Converting Non-Classical Matrix Proofs . . . 5 CADE-13, 1996

Conventional (Prefix-free) Sequent Calculi L MC D -formula: F ≡ ✷ ∃ x. ✷ C ( x ) ∧ ✸ B ⇒ ✸ ( B ∧ ✸ ∃ x. C ( x )) C ( a 4 ) ⊢ C ( a 4 ) C ( a 4 ) ⊢ ∃ x.C ( x ) ∃ –r ( ¯ a 12 , ¯ a 13 ) ✷ C ( a 4 ) , B ⊢ ✸ ∃ x.C ( x ) ✸ –r ( a 11 ) ✷ C ( a 4 ) , B ⊢ B ∧ –r ( ¯ a 9 , a 10 ) ✷ C ( a 4 ) , B ⊢ B ∧ ✸ ∃ x.C ( x ) ∃ x. ✷ C ( x ) , B ⊢ B ∧ ✸ ∃ x.C ( x ) ∃ –l ( ¯ a 3 ) ✷ ∃ x. ✷ C ( x ) , ✸ B ⊢ ✸ ( B ∧ ✸ ∃ x.C ( x )) ✸ –l ( a 6 , a 7 ) ⊢ ✷ ∃ x. ✷ C ( x ) ∧ ✸ B ⇒ ✸ ( B ∧ ✸ ∃ x.C ( x )) ⇒ –l ( b 0 ) , ∧ –l ( a 1 ) • Semantics of L encoded by sequent rules of L MC – ν / π -rules cause deletion of sequent formulae & “macro”-reductions • Reduction ordering ✁ not “complete” wrt. rule permutabilities ❀ Add dynamic restrictions: wait 2 -labels depending on traversing order ⇒ Delete irrelevant subrelations of ✁ after β -splits – correctness & completeness Converting Non-Classical Matrix Proofs . . . 6 CADE-13, 1996

β -splits and Non-normalform Reductions Θ -node a ≡ at least two sucessors in ≪ and Ptype ( a ) � = β β -purity at c Θ -purity at c k k k k β Θ Θ pre ( β ) c c Θ -reduction: c 1 is β -free. k k k k Θ Θ Θ Θ c 2 c 2 c 1 c 1 Converting Non-Classical Matrix Proofs . . . 7 CADE-13, 1996

Extended Conversion M �− → L MC Modified variant/invariant parts of sequent calculi L MC wrt. logic L Multiple-conclusioned sequent calculi L MC invariant variant sequent rules of type { α, β, δ, γ } sequent rules of type { π, ν } (cause “macro”-reductions and formulae deletion) ❀ New table system for variant parts of L MC : conditions on ν / π -rules wrt. L ❀ Extend table system : conditions on wait 2 -labels ❀ complete rule permutabilities ❀ Extend algorithm : dynamic computation of wait 2 -labels & macro-reductions for ν / π ❀ Extend algorithm : integrate β –split with non-normal form reductions function split ( ✁ , a ) : [ ✁ 1 , ✁ 2 ]; [ ✁ 1 , ✁ 2 ] := β –split ( ✁ , a ) ; for i = 1 , 2 do ✁ i := Θ –reduction ( ✁ i ) ; ✁ i := ( β, Θ) –purity ( ✁ i ) . Converting Non-Classical Matrix Proofs . . . 8 CADE-13, 1996

Conclusion & Further Work Results 1. Uniform representation of calculi M , L P , L MC via tables 2. Uniform conversion procedure • M �− → L P : C ; J ; S 5 (varying, constant) ; K, K 4 , D, D 4 , T, S 4 (cumulative, varying, constant) • M �− → L MC : C ; J ; D, D 4 , T, S 4 (cumulative, varying) 3. No search during the transformation process ❀ polynomial time 4. Guidance of a constructive proof-development system Future work • Build uniform “prover �− → conversion”-procedure (Connection Method) • Extend conversion approach to Resolution • Develop “prover �− → conversion”-component for induction Converting Non-Classical Matrix Proofs . . . 9 CADE-13, 1996