SLIDE 1 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
SLIDE 2 Converting Non-Classical Matrix Proofs . . . 1 CADE-13, 1996
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; S5 (varying, constant); K, K4, D, D4, T, S4
SLIDE 3 Converting Non-Classical Matrix Proofs . . . 2 CADE-13, 1996
Non-classical Matrix Proofs D-formula F ≡ ✷∃x. ✷C(x) ∧✸B ⇒ ✸(B ∧✸∃x. C(x))
¯ a12 : ∃0 b0 : ⇒ 0 a1 : ∧ 1 a8 : ✸0 a2 : ✷1 a6 : ✸1 a7 : B1 ¯ a3 : ∃1 a4 : ✷1 ¯ a5 : C(a4)1 ¯ a13 : C(¯ a13)0 a11 : ✸0 a10 : B0 ¯ a9 : ∧ 0
a3¯ a5 : C(a4)1 b0a7 : B1
b0¯ a9 : B0 b0¯ a9¯ a12 : C(¯ a13)0
Substitutions σM(¯ a3)=σM(¯ a9)=a7 σM(¯ a12)=¯ a5 ❁M ={(a7, ¯ a3), (a7, ¯ a9)} σQ(¯ a13)=a4 ❁Q ={(a4, ¯ a13)} Irreflexive ordering ✁ ≡ ≪ ∪ ❁Q ∪ ❁M
SLIDE 4 Converting Non-Classical Matrix Proofs . . . 3 CADE-13, 1996
Unified Matrix Characterizations M
A formula F is L-valid if there is a multiplicity µ, a combined substitution σ = σQ, σM and a set of σ-complementary connections such that each path through F 0 contains at least one connection.
(Wallen, 1990)
- σQ quantifier substitution (induces ❁Q)
- σm modal substitution defined on prefixes (induces ❁M)
❀ depends on accessibility relation pR0q on prefixes
- σ-complementarity: σQ, σM unifies connected atoms & admissibility
(i) Irreflexive reduction ordering ✁ ≡ ≪ ∪ ❁Q ∪ ❁M (ii) σm respects R0 and interacts with σQ wrt. modal domain conditions
SLIDE 5 Converting Non-Classical Matrix Proofs . . . 4 CADE-13, 1996
Prefixed Sequent Systems LP
D-formula: F ≡ ✷∃x. ✷C(x) ∧✸B ⇒ ✸(B ∧✸∃x. C(x))
b0a7¯ a5 : C(a4), b0a7 : B ⊢ b0a7 : B b0a7¯ a5 : C(a4), b0a7 : B ⊢ b0a7¯ a5 : C(a4) b0a7¯ a5 : C(a4), b0a7 : B ⊢ b0a7¯
a5 : ∃x.C(x) ∃–r (¯
a12, ¯ a13) b0a7¯ a5 : C(a4), b0a7 : B ⊢ b0a7 : ✸∃x.C(x) ✸–r (a11) b0a7¯ a5 : C(a4), b0a7 : B ⊢ b0a7 : B ∧✸∃x.C(x)
∧–r (¯
a9, a10)
b0a7¯ a5 : C(a4), b0a7 : B ⊢ b0 : ✸(B ∧✸∃x.C(x)) ✸–r (a8) b0a7 : ∃x.✷C(x), b0a7 : B ⊢ b0 : ✸(B ∧✸∃x.C(x)) ∃–l (¯
a3), ✷–l (a4, ¯ a5) b0 : ✷∃x.✷C(x), b0a7 : B ⊢ b0 : ✸(B ∧✸∃x.C(x)) ✷–l (a2) b0 : ✷∃x.✷C(x), b0 : ✸B ⊢ b0 : ✸(B ∧✸∃x.C(x)) ✸–l (a6, a7) ⊢ b0 : ✷∃x.✷C(x) ∧✸B ⇒ ✸(B ∧✸∃x.C(x)) ⇒ –l (b0),
∧–l (a1)
Prefix construction: respect pR0q, new (π, ν), used (ν) – Reflects conditions on prefix unification in matrix-proof M ⇒ Conversion ≡ traverse reduction ordering ✁ using σQ, σM – Position ⇒ main-operator & polarity ⇒ unique LP-rule
SLIDE 6 Converting Non-Classical Matrix Proofs . . . 5 CADE-13, 1996
Uniform Conversion M − → LP
- Divide logical calculi into invariant and variant parts wrt. logic L
Matrix characterization M Prefixed sequent systems LP invariant variant invariant variant connection method prefix unification σm,R0 sequent rules prefix construction π,ν,R0
- Represent variant parts in tables; define mappings betweeen (in)variants of calculi
- Algorithm structure ≡ traverse ✁; table access ≡ using σQ, σM wrt. L
function convert (L, ✁, σ) : LP-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 LP-rule using σQ and tables wrt. L (for γ, δ); Ptype (a) ∈ {π, ν} : apply LP-rule using σm and tables wrt. L; Ptype (a) = β : apply β-rule; [✁1, ✁2] := β-split(✁, a); convert (L, ✁1, σ); convert (L, ✁2, σ); proven (✁).
SLIDE 7 Converting Non-Classical Matrix Proofs . . . 6 CADE-13, 1996
Conventional (Prefix-free) Sequent Calculi LMC
D-formula: F ≡ ✷∃x. ✷C(x) ∧✸B ⇒ ✸(B ∧✸∃x. C(x))
✷C(a4), B ⊢ B C(a4) ⊢ C(a4) C(a4) ⊢ ∃x.C(x) ∃–r (¯ a12, ¯ a13) ✷C(a4), B ⊢ ✸∃x.C(x) ✸–r (a11) ✷C(a4), B ⊢ B ∧✸∃x.C(x)
∧–r (¯
a9, a10) ∃x.✷C(x), B ⊢ B ∧✸∃x.C(x) ∃–l (¯ a3) ✷∃x.✷C(x), ✸B ⊢ ✸(B ∧✸∃x.C(x)) ✸–l (a6, a7) ⊢ ✷∃x.✷C(x) ∧✸B ⇒ ✸(B ∧✸∃x.C(x)) ⇒ –l (b0),
∧–l (a1)
- Semantics of L encoded by sequent rules of LMC
– ν/π-rules cause deletion of sequent formulae & “macro”-reductions
- Reduction ordering ✁ not “complete” wrt. rule permutabilities
❀ Add dynamic restrictions: wait2-labels depending on traversing order ⇒ Delete irrelevant subrelations of ✁ after β-splits – correctness & completeness
SLIDE 8 Converting Non-Classical Matrix Proofs . . . 7 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 pre(β) k Θ k c Θ β k c Θ-reduction: c1 is β-free. k Θ k Θ c1 c2 k Θ k Θ c1 c2
SLIDE 9 Converting Non-Classical Matrix Proofs . . . 8 CADE-13, 1996
Extended Conversion M − → LMC
Modified variant/invariant parts of sequent calculi LMC wrt. logic L Multiple-conclusioned sequent calculi LMC 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 LMC: conditions on ν/π-rules wrt. L ❀ Extend table system: conditions on wait2-labels ❀ complete rule permutabilities ❀ Extend algorithm: dynamic computation of wait2-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).
SLIDE 10 Converting Non-Classical Matrix Proofs . . . 9 CADE-13, 1996
Conclusion & Further Work Results
- 1. Uniform representation of calculi M, LP, LMC via tables
- 2. Uniform conversion procedure
- M −
→ LP : C; J; S5 (varying, constant); K, K4, D, D4, T, S4 (cumulative, varying, constant)
→ LMC : C; J; D, D4, T, S4 (cumulative, varying)
- 3. No search during the transformation process ❀ polynomial time
- 4. Guidance of a constructive proof-development system
Future work
→ conversion”-procedure (Connection Method)
- Extend conversion approach to Resolution
- Develop “prover −
→ conversion”-component for induction
