Dealing with Symmetries in Modal Tableaux Carlos Areces and Ezequiel - - PowerPoint PPT Presentation

Dealing with Symmetries in Modal Tableaux

Carlos Areces and Ezequiel Orbe

Universidad Nacional de C´

• rdoba, 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 M | = C iff w′, W, R, V | = C for all w′ s.t. wRw′

Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions

Permutation Sequences

In modal logics that have the tree model property, a notion of layer is induced:

∧ ¬ q r ∨ ∨ ∨ ¬p

• q

¬r Layer 1 Layer 2 Layer 3 Model M Formula ϕ p := true p := true r :=true p

modal depth = 0 modal depth = 1 modal depth = 2 depth = 0 depth = 1 depth = 2

We can consider a different permutation at each layer. Permutation Sequence: ¯ ρ = ρ1, . . . , ρn 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)

• p

C A E p F

• q

q

• q

D B G q H

• p

p

A = ¬(¬p ∨ q ∨ ¬q) B = ¬(¬q ∨ p ∨ ¬p) C = ¬p ∨ q ∨ ¬q D = ¬q ∨ p ∨ ¬p E = q F = ¬q G = p H = ¬p Group Generators: ¯ ρ1 = ρId, ρId, (p ¬p) ¯ ρ2 = ρId, ρId, (q ¬q) ¯ ρ3 = ρId, (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 QBFLib 756 746 16656 Class #Inst #Sym AvGen k branch 42 42 12 k d4 42 k dum 42 k grz 42 42 4 k lin 42 1 1 k path 42 42 35 k ph 42 39 1 k poly 42 42 18 k t4p 42

20 40 60 80 100 5 10 15 20 25 30 35 %Symm L/N V20 V90 V150 V210 V300 V400 V500

Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions

BML Prefixed Tableaux Calculus

σ:ϕ (∧) σ:Ci for all Ci ∈ ϕ σ:C (∨) σ:l1 | . . . | σ:ln for all li ∈ C σ:¬C (♦)1 σRσ′, σ′:∼C 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

• pen.

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: HTab+SB outperforms HTab.

Table : Total Times with SB

Solver #Suc #TO T1 T2 HTab+SB 318 636 9657 391167 HTab 311 643 10634 396434

Figure : HTab vs. HTab+SB

0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100 HT ab + SB [sec.] 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 B1 B2 Satisfiable 157 73 6319 6278 Unsatisfiable 163 79 1038 87

0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100 HT ab + SB [sec.] HT ab [sec.]

a) Satisfiable formulas

0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100 HT ab + SB [sec.] HT ab [sec.]

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

Class HTab+SB HTab n100 n600 T n100 n600 T k branch p 21 21 59.760 13 15 4402.130 k branch n 9 10 7010.200 8 10 7197.000 k grz p 21 21 0.508 21 21 0.276 k grz n 21 21 0.632 21 21 0.380 k path p 21 21 4.542 21 21 3.812 k path n 21 21 5.348 21 21 3.792 k ph p 7 8 8116.900 7 8 8095.48 k ph n 21 21 177.560 21 21 178.579 k poly p 21 21 29.068 21 21 22.949 k poly n 21 21 29.534 21 21 24.229

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.