Dealing with Symmetries in Modal Tableaux Carlos Areces and Ezequiel Orbe Universidad Nacional de C´ ordoba, Argentina CONICET, Argentina Frontiers of Combining Systems 2013, Nancy, France
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Introduction A symmetry is a permutation of the variables (literals) of a problem that preserves its structure and its set of solutions. For instance: ϕ = ( ¬ p ∨ r ) ∧ ( q ∨ r ) ∧ � ( ¬ p ∨ q ) has symmetry: ρ = ( ¬ p q )( ¬ q p ) We may improve the performance of theorem proving if: symmetry detection is cheap this information pays off in terms of performance We present a tableau optimization called ”symmetry blocking”.
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Syntax Modal Conjunctive Normal Form: Clausal representation of modal formulas: ( ¬ p ∨ r ) ∧ ( q ∨ r ) ∧ ( r ∨ � ( ¬ p ∨ q ))) → {{¬ p, r } , { q, r } , { r, � {¬ p, q }}} Disregard order and multiplicity: formulas as set of sets. Symmetry: Permutations of literals, ρ : PLIT �→ PLIT ρ is a symmetry of ϕ if ρ ( ϕ ) = ϕ .
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Semantics Models: Kripke model: M = � W, R, V � W is the domain R ⊆ W × W V : W �→ P ( PROP ) Pointed Models: M = � w, W, R, V � , w ∈ W Satisfaction Relation: M | = ϕ iff M | = C for all clauses C ∈ ϕ M | = C iff M | = l for some literal l ∈ C M | = p iff p ∈ V ( w ) for p ∈ PROP = C for all w ′ s.t. wRw ′ � w ′ , W, R, V � | M | = � C iff
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Permutation Sequences In modal logics that have the tree model property , a notion of layer is induced: Formula ϕ Model M ∧ depth = 0 modal depth = 0 Layer 1 p := true ∨ ∨ p q r ¬ � modal depth = 1 depth = 1 ∨ p := true Layer 2 ¬ p q � modal depth = 2 depth = 2 ¬ r Layer 3 r :=true We can consider a different permutation at each layer. Permutation Sequence: ¯ ρ = � ρ 1 , . . . , ρ n � Enables to find more symmetries.
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Detection Create a graph from the formula such that its automorphism group is isomorphic to the symmetry group of the formula. Pass it to graph automorphism tools (eg Saucy, Bliss). The reduction enables the detection of layered symmetries. Node coloring avoids spurious symmetries. ϕ = ¬ � ( ¬ p ∨ � q ∨ � ¬ q ) ∧ ¬ � ( ¬ q ∨ � p ∨ � ¬ p ) A = ¬ � ( ¬ p ∨ � q ∨ � ¬ q ) A B B = ¬ � ( ¬ q ∨ � p ∨ � ¬ p ) C = ¬ p ∨ � q ∨ � ¬ q C D D = ¬ q ∨ � p ∨ � ¬ p E = � q p -p q -q E F G H F = � ¬ q G = � p q -q p -p H = � ¬ p ρ 1 = � ρ Id , ρ Id , ( p ¬ p ) � ¯ Group Generators: ρ 2 = � ρ Id , ρ Id , ( q ¬ q ) � ¯ ρ 3 = � ρ Id , ( p q )( ¬ p ¬ q ) , ( p q )( ¬ p ¬ q ) � ¯
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Detection: Experimental Evaluation Testbed: 378 LWB K + 756 QBFLib + 19000 random. Conclusions: Symmetries do arise in modal formulas. As expected: encoding of the formula drives the existence of symmetries (LWB K). % of symmetries in random instances highly depends on L/N . #Inst #Sym T LWB K 378 208 10.2 100 V20 QBFLib 756 746 16656 V90 V150 80 V210 V300 Class #Inst #Sym AvGen V400 V500 42 42 12 60 k branch %Symm 42 0 0 k d4 42 0 0 k dum 40 42 42 4 k grz 42 1 1 k lin 20 42 42 35 k path 42 39 1 k ph 42 42 18 k poly 0 0 5 10 15 20 25 30 35 42 0 0 k t4p L/N
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions BML Prefixed Tableaux Calculus σ : ϕ σ : C ( ∧ ) for all C i ∈ ϕ ( ∨ ) for all l i ∈ C σ : C i σ : l 1 | . . . | σ : l n σ : ¬ � C σ : � C, σRσ ′ ( ♦ ) 1 ( � ) σ ′ : C σRσ ′ , σ ′ : ∼ C 1 ∼ C : CNF of the negation of C . The prefix σ ′ is new in the tableau. A branch is closed if contains both σ : p and σ : ¬ p , otherwise open . A branch is saturated if no rule can be further applied.
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Blocking We write Γ( σ ) = { ψ | σ : � ψ ∈ Θ } for the set of � -formulas at prefix σ in branch Θ . Definition (Symmetry Blocking) Let ¯ ρ be a layered symmetry of ϕ , and Θ a branch in a tableau of ϕ . Rule ( ♦ ) cannot be applied to σ :¯ ρ ( ¬ � ψ ) on Θ if – it has been applied to σ : ¬ � ψ and – Var (¯ ρ ( ¬ � ψ )) ∩ Var (Γ( σ )) = ∅ , Dynamic: Var (Γ( σ )) can change over time.
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Blocking Completeness: Soundness is trivial as we do not modify the set of rules. Completeness requires more work. Theorem The tableau calculus with symmetry blocking for the modal logic BML is sound and complete. Completeness lemma: a saturated open branch and its blocked ¬ � -formulas form a satisfiable set of formulas.
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Blocking: Experimental Evaluation I Implementation on the HTab solver: It only expands a ¬ � -formula if there is no symmetric formula already expanded. Checks blocking condition: iff it gets a saturated open branch. If it holds for all blocked formulas, terminates. Otherwise, reschedules for further expansion. Conclusions: Figure : HTab vs. HTab +SB HTab +SB outperforms HTab . 100 Table : Total Times with SB 10 ab + SB [sec.] 1 Solver #Suc #TO T 1 T 2 HTab +SB 318 636 9657 391167 HT 0.1 HTab 311 643 10634 396434 0.01 0.001 0.001 0.01 0.1 1 10 100 HT ab [sec.]
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Blocking: Experimental Evaluation II Conclusions: Different behavior for sat and unsat instances. Table : Applications of SB. Status #Inst #Trig B 1 B 2 Satisfiable 157 73 6319 6278 Unsatisfiable 163 79 1038 87 100 100 10 10 ab + SB [sec.] ab + SB [sec.] 1 1 HT 0.1 HT 0.1 0.01 0.01 0.001 0.001 0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100 HT ab [sec.] HT ab [sec.] a) Satisfiable formulas b) Unsatisfiable formulas Figure : Performance of HTab vs. HTab +SB.
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Symmetry Blocking: Experimental Evaluation III Conclusions: Effectiveness of SB depends on the problem class. Table : Effect of SB on the LWB K HTab +SB HTab Class n 100 n 600 T n 100 n 600 T 21 21 59.760 13 15 4402.130 k branch p 9 10 7010.200 8 10 7197.000 k branch n 21 21 0.508 21 21 0.276 k grz p 21 21 0.632 21 21 0.380 k grz n 21 21 4.542 21 21 3.812 k path p 21 21 5.348 21 21 3.792 k path n 7 8 8116.900 7 8 8095.48 k ph p 21 21 177.560 21 21 178.579 k ph n 21 21 29.068 21 21 22.949 k poly p 21 21 29.534 21 21 24.229 k poly n
Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions Conclusions Layered permutations enable to find more symmetries. Symmetric blocking performs differently depending on problem class. Not all symmetries are used by Symmetric Blocking (not always happen in ¬− � -formulas). Future work: Symmetry Breaking Predicates for modal logic.
Appendix Resources HTab prover: http://tinyurl.com/orsnu2z Benchmarks: http://tinyurl.com/pq63to7
Appendix Symmetries in Modal Logics: semantic properties Property I Let ϕ be a formula, ρ be a symmetry of ϕ and M a model, then, M | = ϕ iff ρ ( M ) | = ϕ . Symmetries induce a partition in the model set. Static and Dynamic Symmetry Breaking (Symmetry Breaking Predicates). Property II Let ϕ and ψ be formulas and ρ be a symmetry of ϕ then, ϕ | = ψ iff ϕ | = ρ ( ψ ) . This provides a cheap inference mechanism. Symmetric Reasoning (Symmetric Clause Learning). This holds for permutation sequences.
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