SAT-solving An old AI technique becomes very popular in modern A.I.

Overview: Basics on SAT Unit propagation SAT-solving Local Search SAT-solving Local Search SAT-solving

3 What is SAT-solving?  Given KR: a set of propositional formula’s  Find a model for KR. More specifically:  Let X 1, X 2, …, X n be all the variables in KR,  Find an assignment I; X i -> {T,F}, for i=1,n, such that all formula’s in KR become true. such that all formula’s in KR become true. Useful ?

Example: 3-queens Representation: X 1,1 X 1,2 X 1,3 X 2,1 X 2,3 X 2,2 X 3,1 X 3,2 X 3,3 represents: there is a queen on i,j represents: there is a queen on i,j X i,j X i,1  X i,2  X i,3 (i=1,3) At least one queen on each row: X i,1  ~X i,2  ~X i,3, X i,2  ~X i,1  ~X i,3, X i,3  ~X i,1  ~X i,2 (i=1,3) At most one queen on each row: Plus: similar formulas for columns and diagonals. 33 formula’s ! Generalisation to q-queens ? Very many formula’s !

Example: personel rostering Representation: - For every employee, i, - For every employee, i, - For every shift in the day, j, - For every shift in the day, j, - For every day in the month, k : - For every day in the month, k : represents: i works on shift j of day k represents: i works on shift j of day k X i,j,k Often this is more elegantly done with assignments to {0,1} and sums instead of  Generalise to higher numbers than just 1 At least one person works on every shift of every day:  X i,j,k (j=1,5, k=1,30) i=1,35 An interpretation assigns to each Xi,j,k: true or false Is a personel assignment !

6 SAT-solving and NP-completeness  SAT-solving: one of the first identified as NP-complete  If you find a P-algorithm for SAT: you get a P- algorithm for all the others.  Also means: most problems can be encoded as SAT-problems and solved using SAT-techniques. Means: all other NP-problems are technically equivalent with this problem

7 But why practically useful?  Although NP(-complete)  Has led to: -Many areas in CS and AI convert problems to SAT -Then use SAT-solvers. Very efficient heuristic approaches exist that work well on certain classes of problems.

Example: Automated Reasoning  Compute a finite grounding of the predicate logic theory.  Marcus example: 2 constants: Marcus and Ceasar. 1. man(Marcus) is ground 2. Pompeian(Marcus) is ground 3.  x Pompeian(x)  Roman(x) Pompeian(Marcus)  Roman(Marcus) Pompeian(Ceasar)  Roman(Ceasar) 4. ruler(Caesar) is ground 5.  x Roman(x)  loyal_to(x,Caesar)  hates(x,Caesar) Roman(Marcus)  loyal_to(Marcus,Caesar)  hates(Marcus,Caesar) Roman(Ceasar)  loyal_to(Ceasar,Caesar)  hates(Ceasar,Caesar) Ect. ….

9 Example: continued  Ground predicate logic is equivalent to propositional ! Example: Pompeian(Marcus) converts to Pompeian_Marcus loyal_to(Marcus,Caesar) converts to loyal_to_Marcus_Caesar  The theorem follows if and only if this propositional KB in unsatisfiable ! KB in unsatisfiable !  Add the propositional version of the negation of the theorem ~loyal_to(Marcus,Caesar) converts to loyal_to_Marcus_Caesar SAT-solving

Conjunctive Normal Form  Every formula is equivalent to a formula of the form: (A1 ...  An)  (B1  …  Bm)  …  (C1  …  Ck)  where all Ai, Bi, …, Ci are either atomic or ~atomic.  Idea: p  q p  q push all ~ as deep as possible apply distributivity of  and  p  q  q  p q  ~p  SAT-solving will work on a collection of disjunctions: X1  …  Xn  ~Y1  … ~Ym

Naive SAT-solving Depth-first left-to-right enumeration of all interpretations

SAT - Standard backtracking ~Y X  ~Y  ~W,, ~X  Y  Z, ~XW ~X  W ~Y T  ~Y  ~W,, F  Y  Z, FW F  W X=TX=TX=TX=T Y=TY=TY=TY=T,TZ,WW,TZ,WWW Z=TZ=TZ=TZ=T, W T, W W=TW=TW=TW=T TF W=FW=FW=FW=F, W F, W Z=FZ=FZ=FZ=F, W T, W Z=TZ=TZ=TZ=T T W=TW=TW=TW=T F W=FW=FW=FW=F, W F, W Z=FZ=FZ=FZ=F ~Y F  ~Y  ~W,, T  Y  Z, TW T  W X=FX=FX=FX=F Y=TY=TY=TY=T F F  ~W, W=TW=TW=TW=T F W=FW=FW=FW=F T,FZ,WW,FZ,WWW Y=FY=FY=FY=F Y=FY=FY=FY=F T T  ~W Success Success Success Success Fail Fail Fail FailFail

13 Naïve SAT-solving algorithm NaiveSAT( i) For Truth = T, F do X i := Truth; X i := Truth; Remove all Dj containing T from S; Remove all Dj containing T from S; Remove all  F and F  from all Dj; Remove all  F and F  from all Dj; If no Dj in S is equal to F then If no Dj in S is equal to F then Else i := i + 1; NaiveSAT( i ); i := i - 1; If S= {} then return( X 1, X 2, …, X n ); return( X 1, X 2, …, X n ); Form is a CNF-formula with variables X 1, X 2,...X n S:= { Dj | Dj is a disjunction in Form} Call: NaiveSAT(1) End-For

Davis-Putman (1960): unit propagation The basis for VERY efficient SAT-solvers An early form of Dynamic Search Rearrangement

15 1. Dealing with pure symbols Example: S = { } S = { X  ~Y, ~Y  ~Z, Z  X } X is pure: only occurs positive Y is pure: only occurs negative Z is not pure.  A variable Xi is pure if it only appears with one sign in all disjunctions.  Assign a pure variable the value that makes it true everywhere (don’t consider the other assignment).

SAT + treating pure symbols ~Y X  ~Y  ~W,, ~X  Y  Z, ~XW ~X  W Success Z=TZ=TZ=TZ=T ~Y X  ~Y  ~W,, ~X  Y  T, ~XW ~X  W Y=FY=FY=FY=F T X  T  ~W, ~XW ~X  W X=FX=FX=FX=F T T  W Z is pure Y is pure X is pure

17 The order of choosing variables has become dynamic. Effect of dealing with pure symbols: Yet: we do not loose completeness: If another solution exist, then the current assignment is ok too. + If unsatisfiable: this will fail too. But not first-fail based. We get a _much_ smaller search space ! But we are no longer considering all assignments.

18 1. Unit propagation Example: S = { } S = { X  ~Y, ~Y, X } ~Y and X are units ~Y and X are units  A disjunction is unit if it only contains one variable Xi.  Assign a unit the value that makes it true (don’t consider the other assignment).

SAT – with unit propagation ~Y X  ~Y  ~W,, ~X  Y  Z, ~XW ~X  W ~Y T  ~Y  ~W,, F  Y  Z, FW F  W X=TX=TX=TX=T ~Y F  ~Y  ~W,, T  Y  Z, TW T  W X=FX=FX=FX=F Y=TY=TY=TY=T F F  ~W Y=FY=FY=FY=F T T  ~W Success Success Success Success,YZ,TT,YZ,TTT W=TW=TW=TW=T Y=TY=TY=TY=T,TZ,,TZ, FZFZ Y=FY=FY=FY=F Z=TZ=TZ=TZ=T T W=FW=FW=FW=F T unit !

20 Many more optimzations in real SAT-solvers! Effect of unit propagation Obviously: combine both optimizations! - Components analysis. - More Dynamic Search Rearrangement -Eg.: take most frequent variable first - Intelligent backtracking, Indexing, … Smaller search space again. But example is not well suited for unit propagation.

Marcus as SAT-solving: man_Marcus, ruler_Caesar, try_assassinate_Marcus_Ceasar, loyal_to_Marcus_Caesar, ~man_Marcus  ~ruler_Ceasar  ~try_assassinate_Marcus_Ceasar  ~local_to_Marcus_Ceasar,...  ~local_to_Marcus_Ceasar,...  Part of the grounding of Marcus-example  plus the negation of the theorem: 4 unit propagations: try_assassinate_Marcus_Ceasar = T man_Marcus = T ruler_Caesar = T loyal_to_Marcus_Caesar = T T, T, T, T, F  F  F  F,... Fails ! Theorem proved.

SAT-solving by Local Search WalkSAT algorithm

Local Search representation: X i, i=1,n propositional variables D j, j=1,m the disjunctions in the CNF States are n-tuples: (T,F,F,T, …,F) Objective function: the number of D j ‘s that evaluate to F = an interpretation for the X i ‘s Find the global minimum Neighbors: flip one truth value of an X i in failing D j To avoid local minima: probabilistically do Not take the best flip

24 The WalkSAT algorithm: For i= 1 to Max_flip do IF all Dj’s are true in State Then return State; IF all Dj’s are true in State Then return State; Else Else Disj:= random Dj that is false under State; Disj:= random Dj that is false under State; With probability P flip random Xi in Dj; With probability P flip random Xi in Dj; Else Else flip the Xi of Dj that minimizes false Dj’s; flip the Xi of Dj that minimizes false Dj’s; Max_flip:= some number; P:= some probability; S:= { Dj | Dj disjunction}; State:= some interpretation for X 1, X 2,...X n ; End-For Report Failure

Evaluation:  Unclear whether optimized SAT-solving or Local Search is better.  but today more people are using local search  Efficiency of methods depends on underconstraint versus overconstraintness of problems  see Norvig & Russel for a discussion

26 What about Disjunctive Normal From?  DNF: a dual representation for propositional formulas:  Satisfiability of DNF can be checked in linear time!  Find 1 conjunction that does not contain a variable and the negation of that variable. But: conversion to DNF requires exponential time and exponential space !  CNF and DNF are dual: so conversion to CNF is also exponential !!! (A1 ...  An)  (B1  …  Bm)  …  (C1  …  Ck) So why do we still prefer CNF ?? Think about it.