Polluted Resolution and other Combined Proof Search Methods for Propositional Modal Logics Cláudia Nalon nalon@unb.br University of Brasília WOLLI, 2015 C. Nalon WOLLI, 2015 – 1 / 22
Polluted Resolution and other Combined Proof Search Methods for Propositional Modal Logics A Modal-Layered Resolution Calculus for K - Tableaux 2015 Cláudia Nalon nalon@unb.br University of Brasília Ullrich Hustadt Clare Dixon U.Hustadt@liverpool.ac.uk C.Dixon@liverpool.ac.uk University of Liverpool WOLLI, 2015 C. Nalon WOLLI, 2015 – 1 / 22
Motivation ⊲ Motivation K n , the smallest multi-modal normal logic, extends propositional logic � Reasoning Tasks with a fixed, finite set of modal operators. Complexity Proof Methods Formally, the set of well-formed formulae , WFF K n , is the least set such � Implementation that: Example Previous work The main idea p ∈ P = { p, q, p ′ , q ′ , p 1 , q 1 , . . . } and true are in WFF K n ; – The Normal Form a ϕ for if ϕ and ψ are in WFF K n , then so are ¬ ϕ , ( ϕ ∧ ψ ) , and � – Clauses Transformation Rules each a ∈ A n = { 1 , . . . , n } . Inference Rules Inference Rules Formulae are interpreted, as usual, with respect to Kripke structures: Inference Rules � Inference Rules Example Negative Resolution �W , w 0 , R 1 , . . . , R n , π � Ordered Resolution LWB – K_T4P where QBF Conclusion and a ϕ if, and only if, for all w ′ , w R a w ′ implies �M , w ′ � | Future Work = � �M , w � | = ϕ . Abbreviations: false = ¬ true, ( ϕ ∨ ψ ) = ¬ ( ¬ ϕ ∧ ¬ ψ ) , � a ¬ ϕ . ( ϕ → ψ ) = ( ¬ ϕ ∨ ψ ) , and ♦ a ϕ = ¬ � C. Nalon WOLLI, 2015 – 2 / 22
Reasoning Tasks Motivation ⊲ Reasoning Tasks �W , w 0 , R 1 , . . . , R n , π � Complexity Proof Methods Implementation For local satisfiability, formulae are interpreted with respect to the root � Example of M , that is, w 0 . A formula ϕ is locally satisfied in M , denoted by Previous work The main idea M | = L ϕ , if �M , w 0 � | = ϕ . The Normal Form Clauses The formula ϕ is locally satisfiable if there is a model M such that � Transformation Rules �M , w 0 � | = ϕ . Inference Rules Inference Rules A formula ϕ is globally satisfied in M , if for all w ∈ W , �M , w � | = ϕ . � Inference Rules A formula ϕ is globally satisfiable if there is a model M such that M Inference Rules � Example globally satisfies ϕ , denoted by M | = G ϕ . Negative Resolution Ordered Resolution Given a set of formulae Γ and a formula ϕ , the local satisfiability of ϕ � LWB – K_T4P under the global constraints Γ consists of showing that there is a model QBF Conclusion and that globally satisfies the formulae in Γ and that there is a world in this Future Work model that satisfies ϕ . C. Nalon WOLLI, 2015 – 3 / 22
Complexity Motivation Local satisfiability: PSPACE-complete; � Reasoning Tasks ⊲ Complexity Global satisfiability: EXPTIME-complete; � Proof Methods Local satisfiability under global constraints: EXPTIME-complete. � Implementation Example Previous work The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work C. Nalon WOLLI, 2015 – 4 / 22
Proof Methods Motivation Translation into first-order logic; � Reasoning Tasks Sequent calculus; � Complexity ⊲ Proof Methods Tableaux; � Implementation Example Inverse method; � Previous work BDD; � The main idea The Normal Form SAT; � Clauses Resolution; Transformation Rules � Inference Rules . . . � Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work C. Nalon WOLLI, 2015 – 5 / 22
Implementation $./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds C. Nalon WOLLI, 2015 – 6 / 22
Implementation $./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds C. Nalon WOLLI, 2015 – 6 / 22
Implementation $./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.14 seconds C. Nalon WOLLI, 2015 – 6 / 22
Implementation $./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.14 seconds $./prover -i benchmarks/lwb/k_branch_p.03.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.49 seconds C. Nalon WOLLI, 2015 – 6 / 22
Implementation $./prover -i benchmarks/lwb/k_branch_p.01.ksp -fsub -ires Unsatisfiable. 0.02 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires ^C 363.98 seconds $./prover -i benchmarks/lwb/k_branch_p.02.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.14 seconds $./prover -i benchmarks/lwb/k_branch_p.03.ksp -fsub -ires -bnfsimp -bsub -unit -ple Unsatisfiable. 0.49 seconds $./prover -i benchmarks/lwb/k_branch_p.04.ksp -fsub -ires -bnfsimp -bsub -unit -ple ^C 118.26 seconds C. Nalon WOLLI, 2015 – 6 / 22
Example Motivation ♦♦ p ∧ � ¬ p Reasoning Tasks Complexity Proof Methods Implementation 1 . start → t 0 ⊲ Example t 0 → ♦ t 1 Previous work 2 . The main idea t 1 → ♦ p 3 . The Normal Form Clauses t 0 → � ¬ p 4 . Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work C. Nalon WOLLI, 2015 – 7 / 22
Previous work Motivation Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based � Reasoning Tasks heuristics in modal theorem proving. In: Proc. of ECAI 2000. pp. Complexity Proof Methods 199-203. IOS Press (2000). Implementation Example ⊲ Previous work ♦♦ p ∧ � ¬ p = ⇒ ♦♦ p 2 ∧ � ¬ p 1 The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work C. Nalon WOLLI, 2015 – 8 / 22
Previous work Motivation Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based � Reasoning Tasks heuristics in modal theorem proving. In: Proc. of ECAI 2000. pp. Complexity Proof Methods 199-203. IOS Press (2000). Implementation Example ⊲ Previous work ♦♦ p ∧ � ¬ p = ⇒ ♦♦ p 2 ∧ � ¬ p 1 The main idea The Normal Form Clauses Transformation Rules p ∧ � ¬ p = ⇒ p 0 ∧ � ¬ p 1 Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work C. Nalon WOLLI, 2015 – 8 / 22
Previous work Motivation Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based � Reasoning Tasks heuristics in modal theorem proving. In: Proc. of ECAI 2000. pp. Complexity Proof Methods 199-203. IOS Press (2000). Implementation Example ⊲ Previous work ♦♦ p ∧ � ¬ p = ⇒ ♦♦ p 2 ∧ � ¬ p 1 The main idea The Normal Form Clauses Transformation Rules p ∧ � ¬ p = ⇒ p 0 ∧ � ¬ p 1 Inference Rules Inference Rules Inference Rules Areces, C., de Nivelle, H., de Rijke, M.: Prefixed Resolution: A � Inference Rules Resolution Method for Modal and Description Logics. In: Ganzinger, H. Example Negative Resolution (ed.) Proc. CADE-16. LNAI, vol. 1632, pp. 187-201. Springer, Berlin Ordered Resolution LWB – K_T4P (Jul 7-10 1999). QBF Conclusion and Formulae labelled by either constants or pair of constants. – Future Work The inference rule for ♦ generates new labels. – The inference rule for � corresponds to propagation. – C. Nalon WOLLI, 2015 – 8 / 22
The main idea Motivation The calculus should allow for both local and modal reasoning. � Reasoning Tasks A formula to be tested for (un)satisfiability is translated into a normal � Complexity Proof Methods form, where labels refer to the modal level they occur. Implementation Example Inference rules are then applied by modal level. � Previous work ⊲ The main idea The Normal Form Clauses Transformation Rules Inference Rules Inference Rules Inference Rules Inference Rules Example Negative Resolution Ordered Resolution LWB – K_T4P QBF Conclusion and Future Work C. Nalon WOLLI, 2015 – 9 / 22
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