Developing Modal Tableaux and Resolution Methods via First-Order Resolution

Transcription

1 Developing Modal Tableaux and Resolution Methods via First-Order Resolution Renate Schmidt University of Manchester Reference: Advances in Modal Logic, Vol. 6 (2006)

2 Modal logic: Background Established field, long history in mathematics and philosophy Popular in CS: program specification & verification, NLP, multi-agent systems, description logics, semantic web & ontology reasoning,... Commonly used MLs have many good properties: fragments of FOL, decidable, nice computational complexity language is simple & natural, powerful enough to describe useful structures s 3 e 12 b c d 5 30 s 1 s 2 a transition systems trees Deduktionstreffen, Koblenz, March 2007 p.2

3 Basic modal logic Basic modal logic K (m) = propositional logic plus [a], [b],... Ac = {a, b,...} (index set) Modal formulae φ, ψ p i φ φ ψ [a]φ a φ = def [a] φ Semantics: Kripke model M = (W, {R a a Ac}, ι) M, x = p i iff x ι(p i ) M, x = φ iff M, x = φ M, x = φ ψ iff M, x = φ and M, x = ψ M, x = [a]φ iff for all R a -successors y of x M, y = φ M, x = a φ iff for some R a -successor y of x M, y = φ Deduktionstreffen, Koblenz, March 2007 p.3

4 Extensions of the basic modal logic Traditional MLs: extension of K (m) with extra modal axioms epistemic ML, doxastic ML,... Dynamic MLs: extensions of K (m) with operators on actions dynamic logic PDL = K(m) (, ;,,?) description logics with role operators Reading of [a]φ Notation Logic φ is necessary φ basic modal logic K agent a knows φ K a φ epistemic logic KT 45 (m) agent a believes φ B a φ doxastic logic KD45 (m) program a causes φ [a]φ dynamic logic PDL R a -relatives of only C φ s R a.c φ description logics, ALC family Deduktionstreffen, Koblenz, March 2007 p.4

5 Automating reasoning in modal logic Given: Wanted: What is needed? a modal logic L an automated theorem prover for answering Γ = L ϕ? 1. a deduction calculus Cal for L 2. soundness and completeness results for Cal 3. techniques to ensure termination & efficiency 4. implementation Problem: There are infinitely many modal logics and we do not really want to repeat these steps for every new logic. Deduktionstreffen, Koblenz, March 2007 p.5

6 Automating reasoning in modal logic (cont d) Usual solution: Adapt and extend an existing calculus and theorem prover. Develop calculi for class of MLs and implement in one prover This talk: Develop via first-order resolution approach Systematic method for developing special-purpose calculi, much of which can be automated Which style of deduction? Modal tableau, modal resolution, Rasiowa-Sikorski calculi Deduktionstreffen, Koblenz, March 2007 p.6

7 Overview Tableau Rasiowa-Sikorski Resolution First-order hyperresolution Simulation of tableau by resolution Synthesis of tableau rules / calculus Simulation and reduction Soundness, completeness of synthesised calculi; decidability Synthesising direct resolution calculi & Rasiowa-Sikorski calculi Other consequences Deduktionstreffen, Koblenz, March 2007 p.7

8 Tableau refutation approach, testing (un)satisfiability goal-directed approach rules for each logical operator branching rules derivations are trees ϕ s : ψ 1 ψ 2 s : ψ i s : ψ 1 ψ 2 s : ψ 1 s : ψ 2 s : [α]ψ, (s, u) : α etc u : ψ In this talk: tableau = ground labelled modal tableau!! Deduktionstreffen, Koblenz, March 2007 p.8

9 Rasiowa-Sikorski systems proving approach, testing validity goal-directed approach rules for each logical operator branching rules derivations are trees ϕ s : ψ 1 ψ 2 s : ψ i s : ψ 1 ψ 2 s : ψ 1 s : ψ 2 s : α ψ, (s, u) : α etc u : ψ Deduktionstreffen, Koblenz, March 2007 p.9

10 Resolution refutation approach, testing (un)satisfiability operates on clauses two rules: resolution and factoring no branching rules required derivations are linear N Resolution: Factoring: C A A D C D C A A C A Deduktionstreffen, Koblenz, March 2007 p.10

11 The develop via first-order resolution method Basic idea: define transformation and resolution refinement which simulates behaviour of desired deduction method synthesise modal inference rules read them off from clausal form of transformation Our case study: develop ground semantic calculi Requirements: Π Cls L FOL clause logic Π, Cls efficiently computable, sound and complete, structure preserving transformations range-restriction hyperresolution with splitting Deduktionstreffen, Koblenz, March 2007 p.11

12 Hyperresolution Hyperresolution C 1 A 1... C n A n B 1... B n D (C 1... C n D)σ provided σ = mgu(a 1. = B1,..., A n. = Bn ), and C i A i and D are positive clauses Positive factoring C A B (C A)σ provided σ = mgu(a. = B) and C is positive Splitting N {C D} N {C} N {D} provided C and D are variable-disjoint and positive Theorem: H, H sp are sound and complete for FO clause logic. Deduktionstreffen, Koblenz, March 2007 p.12

13 Tableau derivation for (p q) p in K 1. a : (p q) p 2. a : (p q) 1, ( ) 1 3. a : p 1, ( ) 2 4. (a, b) : R 2, ( ) 1 5. b : p q 2, ( ) 2 6. b : p 3, 4, () 7. b : p 5, ( ) 8. b : q 5, ( ) 9. 6, 7, ( ) Deduktionstreffen, Koblenz, March 2007 p.13

14 Transformation to clausal form Simplified structural transformation of ϕ = (p q) p: 1. Q ϕ (a) 2. Q ϕ (x) Q (p q) (x) 3. Q ϕ (x) Q p (x) 4. Q (p q) (x) R(x, f (x)) 5. Q (p q) (x) Q p q (f (x)) 6. Q p q (x) Q p (x) Q q (x) 7. Q p (x) R(x, y) Q p (y) 8. Q p (x) Q p (x) ϕ satisfiable in K iff {1,..., 8} satisfiable in FOL Deduktionstreffen, Koblenz, March 2007 p.14

15 Transformation to clausal form Simplified structural transformation of ϕ = (p q) p: 1. Q ϕ (a) 2. Q ϕ (x) Q (p q) (x) x = ϕ.(x = (p q)) 3. Q ϕ (x) Q p (x) (x = p) 4. Q (p q) (x) R(x, f (x)) 5. Q (p q) (x) Q p q (f (x)) 6. Q p q (x) Q p (x) Q q (x) 7. Q p (x) R(x, y) Q p (y) 8. Q p (x) Q p (x) ϕ satisfiable in K iff {1,..., 8} satisfiable in FOL Deduktionstreffen, Koblenz, March 2007 p.14

16 Hyperresolution derivation 1. Q ϕ (a) a : ϕ 2. Q ϕ (x) Q (p q) (x) 3. Q ϕ (x) Q p (x) 4. Q (p q) (x) R(x, f (x)) 5. Q (p q) (x) Q p q (f (x)) 6. Q p q (x) Q p (x) Q q (x) 7. Q p (x) R(x, y) Q p (y) 8. Q p (x) Q p (x) 9. Q (p q) (a) (1, 2) a : (p q) 10. Q p (a) (1, 3) a : p 11. R(a, f (a)) (9, 4) (a, b) : R 12. Q p q (f (a)) (9, 5) b : p q Deduktionstreffen, Koblenz, March 2007 p.15

17 Hyperresolution derivation (cont d) 1. Q ϕ (a) 6. Q p q (x) Q p (x) Q q (x) 7. Q p (x) R(x, y) Q p (y) 8. Q p (x) Q p (x) 9. Q (p q) (a) (1, 2) 10. Q p (a) (1, 3) 11. R(a, f (a)) (9, 4) 12. Q p q (f (a)) (9, 5) 13. Q p (f (a)) (10, 11, 7) 14. Q p (f (a)) Q q (f (a)) (12, 6) 15. Q p (f (a)) 16. Q q (f (a)) (14, Sp) 17. (13, 15, 8) Deduktionstreffen, Koblenz, March 2007 p.16

18 Hyperresolution derivation & tableau derivation 1. Q ϕ (a) a : ϕ 6. Q p q (x) Q p (x) Q q (x) 7. Q p (x) R(x, y) Q p (y) 8. Q p (x) Q p (x) 9. Q (p q) (a) (1, 2) a : (p q) 10. Q p (a) (1, 3) a : p 11. R(a, f (a)) (9, 4) (a, b) : R 12. Q p q (f (a)) (9, 5) b : p q 13. Q p (f (a)) (10, 11, 7) b : p 14. Q p (f (a)) Q q (f (a)) (12, 6) 15. Q p (f (a)) b : p 16. Q q (f (a)) (14, Sp) b : q 17. (13, 15, 8) Deduktionstreffen, Koblenz, March 2007 p.16

19 Synthesising tableau rules 1. Q ψ1 ψ 2 (s) 2. Q ψ1 ψ 2 (x) Q ψ1 (x) 3. Q ψ1 (s) hyp.res, 1, 2 s : ψ 1 ψ 2 s : ψ 1 1. Q [α]ψ (s) 2. R α (s, u) 3. Q [α]ψ (x) R α (x, y) Q ψ (y) 4. Q ψ (u) hyp.res, 1, 2, 3 s : [α]ψ, (s, u) : α u : ψ 1. Q ψ1 ψ 2 (s) 2. Q ψ1 ψ 2 (x) Q ψ1 (x) Q ψ2 (x) 3. Q ψ1 (s) Q ψ2 (s) hyp.res, 1, 2 4. Q ψ1 (s) 5.Q ψ2 (s) split, 3 s : ψ 1 ψ 2 s : ψ 1 s : ψ 2 Deduktionstreffen, Koblenz, March 2007 p.17

20 Obtained ground tableau calculus Tab for K (m) s : ψ φ s : ψ φ ( ) 1 s : ψ ( ) 2 s : φ ( ) s : (ψ φ) s : ψ s : φ ( [ ]) 1 s : [α]ψ (s, t) : α ( [ ]) 2 s : [α]ψ t : ψ ([ ]) s : [α]ψ, (s, u) : α u : ψ ( ) s : ψ, s : ψ (c) s : (ψ ψ) s : ψ t = term uniquely associated with s : [α]ψ Deduktionstreffen, Koblenz, March 2007 p.18

21 Simulation Calculus C 2 step-wise simulates calculus C 1 (wrt. to Π) iff n such that inference step in C 1 there are n inference steps in C 2 which derive the corresponding conclusion.... More C2 -inferences may be possible s : ψ... H sp step-wise simulates the Q ψ (s) generated Tab (wrt. structural transformation). Theorem 1 Suppose C 2 step-wise simulates C 1. C 1 sim C 2 (i) If C 1 is (refutationally) complete then C 2 is (ref.) complete. (ii) If C 2 is sound then C 1 is sound. Deduktionstreffen, Koblenz, March 2007 p.19

22 Reduction Calculus C 1 is a reduct of C 2 iff n such that the inference steps in any C 2 derivation can be uniquely and exhaustively grouped into macro inference steps (of size n) corresponding to inference steps in C All C2 inferences are used s : ψ... Tab is a reduct of H sp (wrt. Q ψ (s) structural transformation). Theorem 2 Suppose C 1 is a reduct of C 2. C 1 C 2 red (i) If C 2 is (ref.) complete then C 1 is (ref.) complete. (ii) If C 1 is sound then C 2 is sound.

23 Simulation & reduction Corollary 3 sim Suppose C 1 C 2 Then red (i) If C 2 is sound and complete, then C 1 is sound and complete. (ii) If C 2 is a decision procedure, then C 1 is a decision procedure. Deduktionstreffen, Koblenz, March 2007 p.21

24 Tab: Soundness, completeness and decidability Theorem 4 (i) H sp on structural clause form Tab step-wise simulates Tab. (ii) Tab is a reduct of H sp on structural clause form. sim red H sp Theorem 5 (JIGPL 2000) H sp, H decide a clausal class subsuming the structural clausal form associated with K (m). Corollary 6 (i) Tab w/wo contraction is sound, complete. (ii) Tab provides a decision procedure for K (m). Deduktionstreffen, Koblenz, March 2007 p.22

25 Case studies for different logics Tableau synthesis via first-order resolution: Dynamic modal logic K(m) (,,, ) [AiML 2006] Description logic with modal operators ALCM [Mostafavi 2007] Common traditional modal logics [CADE 2003, TOCL] Linear simulation by first-order resolution modal ground labelled tableau, single-step prefix tableau [FTP98, JIGPL 2000, SAT 2000] tableau for ALC with non-empty TBoxes [IJCAI 1999] Deduktionstreffen, Koblenz, March 2007 p.23

26 Variations of the method Varying the definition of resolution calculus [AiML 2006] Variation Res Generated calculi H sp tableau calculi omit splitting H modal resolution calculi add ordering OH ordered modal resolution calculi dualise DH sp Rasiowa-Sikorski calculi DH DOH dual modal resolution dual ordered modal resolution Deduktionstreffen, Koblenz, March 2007 p.24

27 Variations of the method (cont d) Varying the translation to FOL [TOCL] Translation Res Generated calculi std relational H sp tableaux using structural rules functional H sp prefix tableau calculi axiomatic H sp tableaux using propagation rules Deduktionstreffen, Koblenz, March 2007 p.25

28 Develop via f.o. resolution method: Contribution Non-standard application of first-order resolution Method for synthesis of semantic ground calculi (tableau, dual tableau, direct resolution) Automatic soundness and completeness Other transferring properties: - decidability - finite model building - complexity Techniques carrying over: - ordering restrictions - different notions of redundancy Deduktionstreffen, Koblenz, March 2007 p.26

29 Other consequences Easy back-translation of resolution derivations to modal logic derivations Demo First-order resolution provers can be used as special-purpose provers free provers / implemented decision procedures fast prototyping Uniform framework for comparing different calculi Deduktionstreffen, Koblenz, March 2007 p.27

30 Analytical comparison of different calculi Tab sim red H sp d-sim d-red d-sim d-red RS sim red DH sp Res sim red H ORes sim red OH d-sim d-red d-sim d-red d-sim d-red d-sim d-red DRes sim red DH DORes sim red DOH Deduktionstreffen, Koblenz, March 2007 p.28

31 Empirical comparison with TABSPASS Selection refinement with splitting Selection refinement without splitting L/N L/N [SAT 2000] Deduktionstreffen, Koblenz, March 2007 p.29

32 Further work Methodology extends to: other logics other styles of deduction calculi Implementing a tableau calculus generator and generator of other styles of deduction calculi... Deduktionstreffen, Koblenz, March 2007 p.30