Extending DPLL-Based QBF Solvers to Handle Free Variables

Transcription

1 Extending DPLL-Based QBF Solvers to Handle Free Variables Will Klieber, Mikoláš Janota, Joao Marques-Silva, Edmund Clarke July 9,

2 Open QBF Closed QBF: All variables quantified; answer is True or False. Open QBF: Contains free (unquantified) variables. Goal: Find equivalent propositional formula. E.g., given x. x (y z), return y z. 2

3 Open QBF Closed QBF: All variables quantified; answer is True or False. Open QBF: Contains free (unquantified) variables. Goal: Find equivalent propositional formula. E.g., given x. x (y z), return y z. Applications: symbolic MC, synthesis from formal spec, etc. 2

4 Outline Naïve Algorithm Introduce sequents that generalize clauses for open QBF in CNF (without ghost variables) Experimental results Ghost variables: see paper. 3

5 Naïve Algorithm Notation: ite(x, φ 1, φ 2 ) is a formula with an if-then-else: ite(x, φ 1, φ 2 ) = (x φ 1 ) ( x φ 2 ) 4

6 Naïve Algorithm Notation: ite(x, φ 1, φ 2 ) is a formula with an if-then-else: ite(x, φ 1, φ 2 ) = (x φ 1 ) ( x φ 2 ) Recursively Shannon-expand on free variables: Φ = ite(x, Φ x=true, Φ x=false ) 4

7 Naïve Algorithm Notation: ite(x, φ 1, φ 2 ) is a formula with an if-then-else: ite(x, φ 1, φ 2 ) = (x φ 1 ) ( x φ 2 ) Recursively Shannon-expand on free variables: Φ = ite(x, Φ x=true, Φ x=false ) Base case (no more free variables): Give to closed-qbf solver. 4

8 Naïve Algorithm 1. function solve(φ) { 2. if (Φ has no free variables) 3. return closed qbf solve(φ); 7. } 5

9 Naïve Algorithm 1. function solve(φ) { 2. if (Φ has no free variables) 3. return closed qbf solve(φ); 4. x := (a free variable in Φ); 5. return ite(x, solve( Φ x=true ), 6. solve( Φ x=false )); 7. } 5

10 Naïve Algorithm 1. function solve(φ) { 2. if (Φ has no free variables) 3. return closed qbf solve(φ); 4. x := (a free variable in Φ); 5. return ite(x, solve( Φ x=true ), 6. solve( Φ x=false )); 7. } Builds OBDD if: 1. same branch order, 2. formula construction is memoized, and 3. ite(x, φ, φ) is simplified to φ. 5

11 Naïve Algorithm Naïve Algorithm: Similar to DPLL in terms of branching. But lacks many optimizations that make DPLL fast: Non-chronological backtracking Clause learning Our open-qbf technique: Extend existing closed-qbf algorithm to allow free variables. 6

12 Preliminaries Prenex Form: Q 1 x 1...Q n x n. φ where φ has no quantifiers. 7

13 Preliminaries Prenex Form: Q 1 x 1...Q n x n. φ where φ has no quantifiers. In x. y. φ, we say that y is downstream of x. y occurs inside scope of x. 7

14 Preliminaries Prenex Form: Q 1 x 1...Q n x n. φ where φ has no quantifiers. In x. y. φ, we say that y is downstream of x. y occurs inside scope of x. Free variables are upstream of all quantified variables. 7

15 Preliminaries Prenex Form: Q 1 x 1...Q n x n. φ where φ has no quantifiers. In x. y. φ, we say that y is downstream of x. y occurs inside scope of x. Free variables are upstream of all quantified variables. We identify assignment π with the set of literals made true by π. E.g., identify {(e 1, True), (u 2, False)} with {e 1, u 2 }. 7

16 Preliminaries Prenex Form: Q 1 x 1...Q n x n. φ where φ has no quantifiers. In x. y. φ, we say that y is downstream of x. y occurs inside scope of x. Free variables are upstream of all quantified variables. We identify assignment π with the set of literals made true by π. E.g., identify {(e 1, True), (u 2, False)} with {e 1, u 2 }. Substitution: Φ π substitutes assigned variables with values (even if bound by quantifier, which gets deleted). 7

17 QBF as a Game Existential variables are owned by Player. Universal variables are owned by Player. Players assign variables in quantification order. The goal of Player is to make Φ be true. The goal of Player is to make Φ be false. 8

18 Properties of Clauses and Cubes Motivate definition of sequents. If π falsifies all literals in clause C in CNF Φ, then Φ π = False. 9

19 Properties of Clauses and Cubes Motivate definition of sequents. If π falsifies all literals in clause C in CNF Φ, then Φ π = False. If π falsifies all existential literals in clause C in CNF Φ and doesn t satisfy any universal literals in C, then Φ π = False. 9

20 Properties of Clauses and Cubes Motivate definition of sequents. If π falsifies all literals in clause C in CNF Φ, then Φ π = False. If π falsifies all existential literals in clause C in CNF Φ and doesn t satisfy any universal literals in C, then Φ π = False. If π satisfies all universal literals in a cube C in a DNF Φ and doesn t falsify any existential literals in C, then Φ π = True. 9

21 Properties of Clauses and Cubes Motivate definition of sequents. If π falsifies all literals in clause C in CNF Φ, then Φ π = False. If π falsifies all existential literals in clause C in CNF Φ and doesn t satisfy any universal literals in C, then Φ π = False. If π satisfies all universal literals in a cube C in a DNF Φ and doesn t falsify any existential literals in C, then Φ π = True. Tautological clauses learned via long-distance resolution? (Assuming -reduction is done only on-the-fly, during unit prop.) 9

22 L now, L fut Sequents Definition. A game-state specifier is a pair L now, L fut consisting of two sets of literals, L now and L fut. Definition. We say that L now, L fut matches assignment π iff: 1. for every literal l in L now, l π = True, and 2. for every literal l in L fut, either l π = True or l vars(π). 10

23 L now, L fut Sequents Definition. A game-state specifier is a pair L now, L fut consisting of two sets of literals, L now and L fut. Definition. We say that L now, L fut matches assignment π iff: 1. for every literal l in L now, l π = True, and 2. for every literal l in L fut, either l π = True or l vars(π). E.g., {e}, {u} matches {e} and {e, u}, 10

24 L now, L fut Sequents Definition. A game-state specifier is a pair L now, L fut consisting of two sets of literals, L now and L fut. Definition. We say that L now, L fut matches assignment π iff: 1. for every literal l in L now, l π = True, and 2. for every literal l in L fut, either l π = True or l vars(π). E.g., {e}, {u} matches {e} and {e, u}, but does not match {} or {e, u}. 10

25 L now, L fut Sequents Definition. A game-state specifier is a pair L now, L fut consisting of two sets of literals, L now and L fut. Definition. We say that L now, L fut matches assignment π iff: 1. for every literal l in L now, l π = True, and 2. for every literal l in L fut, either l π = True or l vars(π). E.g., {e}, {u} matches {e} and {e, u}, but does not match {} or {e, u}. L now, {l, l} matches π only if π doesn t assign l. 10

26 L now, L fut Sequents Definition. A game-state specifier is a pair L now, L fut consisting of two sets of literals, L now and L fut. Definition. We say that L now, L fut matches assignment π iff: 1. for every literal l in L now, l π = True, and 2. for every literal l in L fut, either l π = True or l vars(π). Definition. L now, L fut = (Φ ψ) means for all assignments π that match L now, L fut, Φ π is logically equivalent to ψ π unless π is a don t-care assignment. 11

27 L now, L fut Sequents Definition. A game-state specifier is a pair L now, L fut consisting of two sets of literals, L now and L fut. Definition. We say that L now, L fut matches assignment π iff: 1. for every literal l in L now, l π = True, and 2. for every literal l in L fut, either l π = True or l vars(π). Definition. L now, L fut = (Φ ψ) means for all assignments π that match L now, L fut, Φ π is logically equivalent to ψ π unless π is a don t-care assignment. Without ghost literals: No assignments are don t-care. With ghost literals: Some assignments are don t-care. 11

28 Correspondence of Sequents to Clauses and Cubes Consider a QBF with existential literals e 1... e n and universal literals u 1... u m. Clause (e 1... e n u 1... u m ) in CNF Φ in corresponds to sequent { e 1,..., e n }, { u 1,..., u m } = (Φ in False). 12

29 Correspondence of Sequents to Clauses and Cubes Consider a QBF with existential literals e 1... e n and universal literals u 1... u m. Clause (e 1... e n u 1... u m ) in CNF Φ in corresponds to sequent { e 1,..., e n }, { u 1,..., u m } = (Φ in False). Cube (e 1... e n u 1... u m ) in DNF Φ in corresponds to sequent {u 1,..., u m }, {e 1,..., e n } = (Φ in True). 12

30 Correspondence of Sequents to Clauses and Cubes Consider a QBF with existential literals e 1... e n and universal literals u 1... u m. Clause (e 1... e n u 1... u m ) in CNF Φ in corresponds to sequent { e 1,..., e n }, { u 1,..., u m } = (Φ in False). Cube (e 1... e n u 1... u m ) in DNF Φ in corresponds to sequent {u 1,..., u m }, {e 1,..., e n } = (Φ in True). Sequents generalize clauses/cubes because L now, L fut = (Φ ψ) can have ψ be a formula in terms of free variables. 12

31 Alternate Sequent Notation L now, L fut = ( loses Φ) means L now, L fut = (Φ False). L now, L fut = ( loses Φ) means L now, L fut = (Φ True). 13

32 Resolution rule for free variable Literal r is free L now 1 {r}, L fut 1 = (Φ in ψ 1 ) L now 2 { r}, L fut 2 = (Φ in ψ 2 ) L now 1 L now 2, L fut 1 Lfut 2 {r, r} = (Φ in ite(r, ψ 1, ψ 2 )) 14

33 Top-level algorithm 1. initialize_sequent_database(); 2. π cur := ; Propagate(); 3. while (true) { 12. } 15

34 Top-level algorithm 1. initialize_sequent_database(); 2. π cur := ; Propagate(); 3. while (true) { 4. while (π cur doesn t match any database sequent) { 5. DecideLit(); 6. Propagate(); 7. } 12. } 15

35 Top-level algorithm 1. initialize_sequent_database(); 2. π cur := ; Propagate(); 3. while (true) { 4. while (π cur doesn t match any database sequent) { 5. DecideLit(); 6. Propagate(); 7. } 8. Learn(); 9. if (learned seq has form, L fut = (Φ in ψ)) return ψ; 10. Backtrack(); 11. Propagate(); 12. } 15

36 Propagation Let seq be a sequent L now, L fut = (Φ in ψ) in database. If there is a literal l L now such that 1. π cur {l} matches seq, and 2. l is not downstream of any unassigned literals in L fut, then l is forced; it is added to the current assignment π cur. 16

37 Propagation Let seq be a sequent L now, L fut = (Φ in ψ) in database. If there is a literal l L now such that 1. π cur {l} matches seq, and 2. l is not downstream of any unassigned literals in L fut, then l is forced; it is added to the current assignment π cur. Propagation ensures that the solver never re-explores areas of the search space for which it already knows the answer. 16

38 Learning func Learn() { 1. seq := (the database sequent that matches π cur ); 2. while (true) { } } 17

39 Learning func Learn() { 1. seq := (the database sequent that matches π cur ); 2. while (true) { 3. r := (the most recently assigned literal in seq.lnow) 4. seq := Resolve(seq, antecedent[r]); } } 17

40 Learning func Learn() { 1. seq := (the database sequent that matches π cur ); 2. while (true) { 3. r := (the most recently assigned literal in seq.lnow) 4. seq := Resolve(seq, antecedent[r]); 5. if (seq.lnow = or has good UIP(seq)) 6. return seq; 7. } } 17

41 Resolution rule for quantified variable (case 1) The quantifier type of r in Φ is Q L now 1 {r}, L fut 1 = (Q loses Φ in) L now 2 { r}, L fut 2 = (Q loses Φ in) Opponent of Q owns all literals in L fut 1 r is not downstream of any l such that l L fut 1 and l (Lfut 1 Lfut 2 ) L now 1 L now 2, L fut 1 Lfut 2 = (Q loses Φ in) 18

42 Resolution rule for quantified variable (case 2) The quantifier type of r in Φ is Q L now 1 {r}, L fut 1 = (Q loses Φ in) L now 2 { r}, L fut 2 = (Φ in ψ) Opponent of Q owns all literals in L fut 1 r is not downstream of any l such that l L fut 1 and l (Lfut 1 Lfut 2 ) L now 1 L now 2, L fut 1 Lfut 2 { r} = (Φ in ψ) 19

43 Experimental Comparison Our solver: GhostQ. Compared to computational-learning solver from: B. Becker, R. Ehlers, M. Lewis, and P. Marin, ALLQBF solving by computational learning (ATVA 2012). Benchmarks (from same paper): synthesis from formal specifications. 20

44 Cactus Plot CPU time (s) learner learner-d learner-c GQ instances 21

45 Formula Size GQ learner-c 22

46 Conclusion DPLL-based solver for open QBF. Sequents generalize clauses and cubes. Generates proof certificates. Our solver produces unordered BDDs. Unordered because of unit propagation. In our experience, often larger than OBDDs. More details: preprint of CP 2013 paper on Will Klieber s website. 23