Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae

Transcription

1 1/15 Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Sylvie Coste-Marquis Daniel Le Berre Florian Letombe Pierre Marquis CRIL, CNRS FRE 2499 Lens, Université d Artois, France Monday, July 11, 2005

2 2/15 Introduction The qbf problem Canonical PSPACE-complete problem Can be used in many AI areas: planning, nonmonotonic reasoning, paraconsistent inference, abduction, etc High complexity, both in theory and in practice A possible solution: tractable classes Instances of those tractable classes hard for current qbf solvers (e.g. (renamable) Horn benchmarks)

3 3/15 Introduction Outline QBF Target fragments Negation normal form Other propositional fragments Complexity results Complexity landscape A glimpse at some proofs A polynomial case Conclusion and perspectives

4 4/15 QBF QBF: formal definition Definition (QBF) A QBF Π is an expression of the form Q 1 X 1... Q n X n Φ, (n 0) X 1... X n sets of propositional variables Φ a propositional formula on those variables Q i (0 i n) an existential or universal quantifier

5 5/15 QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ) Example x y 1, y 2 [(y 1 y 2 ) ( y 2 x) ( y 1 y 2 ) (y 2 x)]

6 5/15 QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ) Example x y 1, y 2 [(y 1 y 2 )( y /////////////// 2 x) ( y 1 y 2 ) (y 2 //////)] x x y 2 y 1

7 5/15 QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ) Example x y 1, y 2 [(y 1 y 2 ) ( y 2 ////) x ( y 1 y 2 )(y ///////////////] 2 x) x y 2 y 2 y 1 y 1

8 5/15 QBF Validity of a QBF Existence of a winning strategy in a game against nature ( ) Example x y 1, y 2 [(y 1 y 2 ) ( y 2 x) ( y 1 y 2 ) (y 2 x)] y 1 x y 2 [(y 1 y 2 ) ( y 2 x) ( y 1 y 2 ) (y 2 x)] x y 2 y 2 y 1 y 1 x y 1 y 2

9 6/15 Target fragments Negation normal form Definition (NNF [Darwiche 1999]) A formula in NNF PS is a rooted DAG where: each leaf node is labeled with true, false, x or x, x PS each internal node is labeled with or and can have arbitrarily many children Example a b b a c d d c

10 7/15 Target fragments Other propositional fragments Properties [Darwiche 1999] Decomposability Determinism Smoothness Decision Ordering

11 8/15 Target fragments Other propositional fragments Fragments of NNF PS : examples Example a b b a c d d c

12 8/15 Target fragments Other propositional fragments Fragments of NNF PS : examples Decomposability: if C 1,..., C n are the children of and-node C, then Var(C i ) Var(C j ) = for i j Example Decomposability a b b a c d d c

13 8/15 Target fragments Other propositional fragments Fragments of NNF PS : examples Determinism: if C 1,..., C n are the children of or-node C, then C i C j = false for i j Example Determinism a b b a c d d c

14 8/15 Target fragments Other propositional fragments Fragments of NNF PS : examples Smoothness: if C 1,..., C n are the children of or-node C, then Var(C i ) = Var(C j ) Example Smoothness a b b a c d d c

15 9/15 Target fragments Other propositional fragments Fragments of NNF PS : definitions Definition (Propositional fragments [Darwiche & Marquis 2001]) DNNF: NNF PS + decomposability. d-dnnf: NNF PS + decomposability and determinism. FBDD: NNF PS + decomposability and decision. OBDD < : NNF PS + decomposability, decision and ordering. MODS: DNF d-dnnf + smoothness.

16 10/15 Complexity results Complexity landscape Complexity results for qbf Fragment PROP PS (general case) CNF DNF d-dnnf DNNF FBDD OBDD < OBDD < (compatible prefix) PI IP MODS Complexity PSPACE-c PSPACE-c PSPACE-c PSPACE-c PSPACE-c PSPACE-c PSPACE-c P PSPACE-c PSPACE-c P

17 11/15 Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-nnf s-nnf DNNF f-nnf BDD d-dnnf FBDD sd-dnnf DNF CNF OBDD OBDD < MODS IP PI

18 11/15 Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-nnf s-nnf DNNF f-nnf BDD d-dnnf FBDD sd-dnnf DNF CNF OBDD OBDD < MODS IP PI

19 11/15 Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-nnf s-nnf DNNF f-nnf BDD d-dnnf FBDD sd-dnnf DNF CNF OBDD OBDD < MODS IP PI

20 11/15 Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-nnf s-nnf DNNF f-nnf BDD d-dnnf FBDD sd-dnnf DNF CNF OBDD OBDD < MODS IP PI

21 11/15 Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-nnf s-nnf DNNF f-nnf BDD d-dnnf FBDD sd-dnnf DNF CNF OBDD OBDD < MODS IP PI

22 11/15 Complexity results A glimpse at some proofs Inclusion of fragments [Darwiche & Marquis 2001] NNF d-nnf s-nnf DNNF f-nnf BDD d-dnnf FBDD sd-dnnf DNF CNF OBDD OBDD < MODS IP PI

23 12/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Prefix compatible: < extension of the variable ordering induced by the prefix of the QBF Eliminating quantifiers from the innermost to the outermost Eliminating existential quantifiers Eliminating universal quantifiers ( x x ) Reduce the OBDD< at each elimination step Remark: Negation in constant time in OBDD <

24 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = x y z φ φ (x y) z

25 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = x y z φ φ (x y) z x φ = y z

26 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = x y z φ φ (x y) z x z φ = y

27 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = x y z φ φ (x y) z x z φ = y

28 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = x y z φ φ (x y) z y z φ = x

29 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case y }{{} z φ = y Σ = x y z φ φ (x y) z x

30 13/15 Complexity results A polynomial case OBDD < with compatible prefix: a polynomial case Σ = x y z φ φ (x y) z x y z φ = Σ is valid

31 14/15 Conclusion and perspectives Conclusion Presentation of propositional fragments Numerous intractable fragments Some tractability results (under very restrictive conditions) Perspectives Study of other (polynomial) complete or incomplete fragments Integrate specialized algorithms into our solver A sharper analysis of the prefix in order to improve our solver

32 15/15 That s all folks! Any questions?