Satisifiability and Probabilistic Satisifiability

Transcription

1 Satisifiability and Probabilistic Satisifiability Department of Computer Science Instituto de Matemática e Estatística Universidade de São Paulo 2010

2 Topics

3 Next Issue

4 The SAT Problem The Centrality of SAT Satisfiability is a central problem in Computer Science Both theoretical and practical interests

5 The SAT Problem The Centrality of SAT Satisfiability is a central problem in Computer Science Both theoretical and practical interests SAT was the 1st NP-complete problem

6 The SAT Problem The Centrality of SAT Satisfiability is a central problem in Computer Science Both theoretical and practical interests SAT was the 1st NP-complete problem SAT received a lot of attention [1960-now]

7 The SAT Problem The Centrality of SAT Satisfiability is a central problem in Computer Science Both theoretical and practical interests SAT was the 1st NP-complete problem SAT received a lot of attention [1960-now] SAT has very efficient implementations

8 The SAT Problem The Centrality of SAT Satisfiability is a central problem in Computer Science Both theoretical and practical interests SAT was the 1st NP-complete problem SAT received a lot of attention [1960-now] SAT has very efficient implementations SAT has become the assembly language of hard-problems

9 The SAT Problem The Centrality of SAT Satisfiability is a central problem in Computer Science Both theoretical and practical interests SAT was the 1st NP-complete problem SAT received a lot of attention [1960-now] SAT has very efficient implementations SAT has become the assembly language of hard-problems SAT is logic

10 The PSAT Problem Probabilistic satisfiability (PSAT) was proposed by [Boole 1854] On the Laws of Thought Classical probability No assumption of a priori statistical independence Rediscovered several times since Boole PSAT is also NP-complete

11 The PSAT Problem Probabilistic satisfiability (PSAT) was proposed by [Boole 1854] On the Laws of Thought Classical probability No assumption of a priori statistical independence Rediscovered several times since Boole PSAT is also NP-complete But PSAT seems to be more complex than SAT Why?

12 Applications of SAT SAT has good implementations SAT has many applications

13 Applications of SAT SAT has good implementations SAT has many applications Planning: SatPlan Answer Set Programming Integer Programming Map colouring, combinatorial problems Optimisation problems etc NP-complete problems reduced to SAT

14 Potential Applications of PSAT SAT has many potential applications

15 Potential Applications of PSAT SAT has many potential applications Computer models of biological processes Machine learning Fault tolerance. Software design and analysis. Economics, econometrics, etc. But there are no practical algorithms for PSAT

16 Next Issue

17 The Setting: the language Atoms: P = {p 1,...,p n } Literals: p i and p j p = p, p = p A clause is a set of literals. Ex: {p, q,r} or p q r A formula C is a set of clauses

18 The Setting: semantics Valuation for atoms v : P {0,1}

19 The Setting: semantics Valuation for atoms v : P {0,1} An atom p is satisfied if v(p) = 1

20 The Setting: semantics Valuation for atoms v : P {0,1} An atom p is satisfied if v(p) = 1 Valuations are extended to all formulas

21 The Setting: semantics Valuation for atoms v : P {0,1} An atom p is satisfied if v(p) = 1 Valuations are extended to all formulas v( λ) = 1 v(λ) = 0

22 The Setting: semantics Valuation for atoms v : P {0,1} An atom p is satisfied if v(p) = 1 Valuations are extended to all formulas v( λ) = 1 v(λ) = 0 A clause c is satisfied (v(c) = 1) if some literal λ c is satisfied

23 The Setting: semantics Valuation for atoms v : P {0,1} An atom p is satisfied if v(p) = 1 Valuations are extended to all formulas v( λ) = 1 v(λ) = 0 A clause c is satisfied (v(c) = 1) if some literal λ c is satisfied A formula C is satisfied (v(c) = 1) if all clauses in C are satisfied

24 The Problem A formula C is satisfiable if exits v, v(c) = 1. Otherwise, C is unsatisfiable

25 The Problem A formula C is satisfiable if exits v, v(c) = 1. Otherwise, C is unsatisfiable The SAT Problem Given a formula C, decide if C is satisfiable. Witnesses: If C is satisfiable, provide a v such that v(c) = 1.

26 The Problem A formula C is satisfiable if exits v, v(c) = 1. Otherwise, C is unsatisfiable The SAT Problem Given a formula C, decide if C is satisfiable. Witnesses: If C is satisfiable, provide a v such that v(c) = 1. SAT has small witnesses

27 An NP Algorithm for SAT NP-SAT(C) Input: C, a formula in clausal form Output: v, if v(c) = 1; no, otherwise. 1: Guess a v 2: Show, in polynomial time, that v(c) = 1 3: return v 4: if no such v is guessable then 5: return no 6: end if

28 A Naïve SAT Solver NaiveSAT(C) Input: C, a formula in clausal form Output: v, if v(c) = 1; no, otherwise. 1: for every valuation v over p 1,...,p n do 2: if v(c) = 1 then 3: return v 4: end if 5: end for 6: return no

29 History Next Issue

30 History A Brief History of SAT Solvers [Davis & Putnam, 1960; Davis, Longemann & Loveland, 1962] The DPLL Algorithm, a complete SAT Solver

31 History A Brief History of SAT Solvers [Davis & Putnam, 1960; Davis, Longemann & Loveland, 1962] The DPLL Algorithm, a complete SAT Solver [Tseitin, 1966] DPLL has exponential lower bound

32 History A Brief History of SAT Solvers [Davis & Putnam, 1960; Davis, Longemann & Loveland, 1962] The DPLL Algorithm, a complete SAT Solver [Tseitin, 1966] DPLL has exponential lower bound [Cook 1971] SAT is NP-complete

33 History Incomplete SAT methods Incomplete methods compute valuation if C is SAT; if C is unsat, no answer. [Selman, Levesque & Mitchell, 1992] GSAT, a local search algorithm for SAT

34 History Incomplete SAT methods Incomplete methods compute valuation if C is SAT; if C is unsat, no answer. [Selman, Levesque & Mitchell, 1992] GSAT, a local search algorithm for SAT [Mitchell, Levesque & Selman, 1992] Hard and easy SAT problems

35 History Incomplete SAT methods Incomplete methods compute valuation if C is SAT; if C is unsat, no answer. [Selman, Levesque & Mitchell, 1992] GSAT, a local search algorithm for SAT [Mitchell, Levesque & Selman, 1992] Hard and easy SAT problems [Kautz & Selman, 1992] SAT planning

36 History Incomplete SAT methods Incomplete methods compute valuation if C is SAT; if C is unsat, no answer. [Selman, Levesque & Mitchell, 1992] GSAT, a local search algorithm for SAT [Mitchell, Levesque & Selman, 1992] Hard and easy SAT problems [Kautz & Selman, 1992] SAT planning [Kautz & Selman, 1993] WalkSAT Algorithm

37 History Incomplete SAT methods Incomplete methods compute valuation if C is SAT; if C is unsat, no answer. [Selman, Levesque & Mitchell, 1992] GSAT, a local search algorithm for SAT [Mitchell, Levesque & Selman, 1992] Hard and easy SAT problems [Kautz & Selman, 1992] SAT planning [Kautz & Selman, 1993] WalkSAT Algorithm [Gent & Walsh, 1994] SAT phase transition

38 History Incomplete SAT methods Incomplete methods compute valuation if C is SAT; if C is unsat, no answer. [Selman, Levesque & Mitchell, 1992] GSAT, a local search algorithm for SAT [Mitchell, Levesque & Selman, 1992] Hard and easy SAT problems [Kautz & Selman, 1992] SAT planning [Kautz & Selman, 1993] WalkSAT Algorithm [Gent & Walsh, 1994] SAT phase transition [Shang & Wah, 1998] Discrete Lagrangian Method (DLM)

39 History DPLL: Second Generation Second Generation of DPLL SAT Solvers: Posit [1995], SATO [1997], GRASP [1999]. Heuristics but no learning.

40 History DPLL: Second Generation Second Generation of DPLL SAT Solvers: Posit [1995], SATO [1997], GRASP [1999]. Heuristics but no learning. SAT competitions since 2002:

41 History DPLL: Second Generation Second Generation of DPLL SAT Solvers: Posit [1995], SATO [1997], GRASP [1999]. Heuristics but no learning. SAT competitions since 2002: Aggregation of several techniques to SAT, such as learning, unlearning, backjumping, watched literal, special heuristics.

42 History DPLL: Second Generation Second Generation of DPLL SAT Solvers: Posit [1995], SATO [1997], GRASP [1999]. Heuristics but no learning. SAT competitions since 2002: Aggregation of several techniques to SAT, such as learning, unlearning, backjumping, watched literal, special heuristics. Very competitive SAT solvers: Chaff [2001], BerkMin [2002],zChaff [2004].

43 History DPLL: Second Generation Second Generation of DPLL SAT Solvers: Posit [1995], SATO [1997], GRASP [1999]. Heuristics but no learning. SAT competitions since 2002: Aggregation of several techniques to SAT, such as learning, unlearning, backjumping, watched literal, special heuristics. Very competitive SAT solvers: Chaff [2001], BerkMin [2002],zChaff [2004]. Applications to planning, microprocessor test and verification, software design and verification, AI search, games, etc.

44 History DPLL: Second Generation Second Generation of DPLL SAT Solvers: Posit [1995], SATO [1997], GRASP [1999]. Heuristics but no learning. SAT competitions since 2002: Aggregation of several techniques to SAT, such as learning, unlearning, backjumping, watched literal, special heuristics. Very competitive SAT solvers: Chaff [2001], BerkMin [2002],zChaff [2004]. Applications to planning, microprocessor test and verification, software design and verification, AI search, games, etc. Some non-dpll SAT solvers incorporate all those techniques: [Dixon 2004]

45 Next Issue

46 The SAT Phase Transition

47 The Phase Transition Diagram 3-SAT, N is fixed Higher N, more abrupt transition M/N: low (SAT); high (UNSAT) Phase transition point: M/N 4.3, 50%SAT [Toby & Walsh 1994] Invariant with N Invariant with algorithm!

48 The Phase Transition Diagram 3-SAT, N is fixed Higher N, more abrupt transition M/N: low (SAT); high (UNSAT) Phase transition point: M/N 4.3, 50%SAT [Toby & Walsh 1994] Invariant with N Invariant with algorithm! No theoretical explanation There is another phase-transition for SAT based on Impurity [Lozinskii 2006]

49 Next Issue

50 DPLL Through Examples p q p q p t s p t s p s p s a

51 Initial Simplifications Delete all clauses that contain λ, if λ does not occur. p q p q p t s p t s p s p s a ////////////////

52 Construction of a Partial Valuation Choose a literal: s. V = {s} Propagate choice: Delete clauses containing s. Delete s from other clauses. p q p q p t s ////////////// p t s //////////////// p s /////

53 Unit Propagation Enlarge the partial valuation with unit clauses. V = {s, p} Propagate unit clauses as before. ////q p //// q p p //// Another propagation step leads to V = {s, p,q, q}

54 Backtracking Unit propagation may lead to contradictory valuation: V = {s, p,q, q} Backtrack to the previous choice, and propagate: V = { s} p q p q p t s /// p t s /// p s ///////////

55 New Choice When propagation finishes, a new choice is made: p. V = { s,p}. This leads to an inconsistent valuation: V = { s,p,t, t} Backtrack to last choice: V = { s, p} ////q p //// q p p t ///////// p t /////////// Propagation leads to another contradiction: V = { s, p, q, q}

56 The Formula is UnSAT There is nowhere to backtrack to now! The formula is unsatisfiable, with a proof sketched below. s p ( p s) q (p q) q (p q) s p t ( p t s) t ( p t s) p q (p q) q (p q)

57 Resolution Next Issue

58 Resolution The Resolution Inference For Clauses Usual Resolution C λ λ D C D Clauses as Sets Γ {λ} { λ} Γ Note that, as clauses are sets Γ {µ,λ} { λ,µ} Γ {µ}

59 Resolution DPLL Proofs and Resolution s p ( p s) q (p q) q (p q) s p t ( p t s) t ( p t s) p q (p q) q (p q)

60 Resolution DPLL Proofs and Resolution s p ( p s) p p q p q s p t ( p t s) t ( p t s) p q (p q) q (p q)

61 Resolution DPLL Proofs and Resolution s p p q p q p s s p t ( p t s) t ( p t s) p q (p q) q (p q)

62 Resolution DPLL Proofs and Resolution s p p q p q p s s p s p t s p t s p q (p q) q (p q)

63 Resolution DPLL Proofs and Resolution s p p q p q p s s p s p t s p t s p p q p q

64 Resolution DPLL Proofs and Resolution s p p q p q p s s p s p t s p t s p p q p q

65 Resolution Conclusion DPLL is isomorphic to (a restricted form of) resolution

66 Resolution Conclusion DPLL is isomorphic to (a restricted form of) resolution DPLL inherits all properties of this (restricted form of resolution

67 Resolution Conclusion DPLL is isomorphic to (a restricted form of) resolution DPLL inherits all properties of this (restricted form of resolution In particular, DPLL inherits the exponential lower bounds

68 Resolution Enhancing DPLL For the reasons discussed, DPLL needs to be improved to achieve better efficiency. Several techniques have been applied: Learning Unlearning Backjumping Watched literals Heuristics for choosing literals

69 Resolution Enhancing DPLL For the reasons discussed, DPLL needs to be improved to achieve better efficiency. Several techniques have been applied: Learning Unlearning Backjumping Watched literals Heuristics for choosing literals

70 Next Issue

71 WatchLit Next Issue

72 WatchLit Watched Literals

73 WatchLit The Cost of Unit Propagation Empirical measures show that 80% of time DPLL is doing Unit Propagation Propagation is the main target for optimization Chaff introduced the technique of Watched Literals Unit Propagation speed up No need to delete literals or clauses No need to watch all literals in a clause Constant time backtracking (very fast)

74 WatchLit DPLL and 3-valued Logic DPLL underlying logic is 3-valued Given a partial valuation V = {λ 1,...,λ k } Let λ be any literal. 1(true) if λ V V(λ) = 0(false) if λ V (undefined) otherwise

75 WatchLit The Watched Literal Data Structure Every clause c has two selected literals: λ c1,λ c2 For each c, λ c1,λ c2 are dynamically chosen and varies with time λ c1,λ c2 are properly watched under partial valuation V if: they are both undefined; or at least one of them is true

76 WatchLit Dynamics of Watched Literals Initially, V = A pair of watched literals is chosen for each clause. It is proper. Literal choice and unit propagation expand V One or both watched literals may be falsified If λ c1,λ c2 become improper then The falsified watched literal is changed if no proper pair of watched literals can be found, two things may occur to alter V Unit propagation (V is expanded) Backtracking (V is reduced)

77 WatchLit Example clause λ c1 λ c2 p q r p = q = p q s p = q = p r s p = r = Initially V = A pair of literals was elected for each clause All are undefined, all pairs are proper

78 WatchLit p is chosen V = { p} All watched literals become (0, ), improper New literals are chosen to be watched clause λ c1 λ c2 p q r r = q = p q s s = q = p r s s = r =

79 WatchLit r is chosen V = { p, r} WL in clauses 1,3 become improper No other *- or 1-literal to be chosen Unit propagation: q, s become true clause λ c1 λ c2 p q r r = 0 q = / 1 p q s s = q = p r s s = / 1 r = 0

80 WatchLit Unit propagation leads to backtracking V = { p, r,q, s} WL in clause 2 becomes improper No other *- or 1-literal to be chosen No unit propagation is possible: clause 2 is false clause λ c1 λ c2 p q r r = 0 q = 1 p q s s = 0 q = 0 p r s s = 1 r = 0

81 WatchLit Fast Backtracking V is contracted to last choice point V = { p, r,q, s} ////////////// { p,r} clause λ c1 λ c2 p q r r = 1 q = p q s s = q = p r s s = r = 1 Only affected WLs had to be recomputed No need to reestablish previous context from a stack of contexts Very quick backtracking

82 Further Techniques Next Issue

83 Further Techniques Improvements Techniques from Chaff Learning new clauses VSDIS Heuristics Random restarts Backjumping

84 Further Techniques Learning

85 Further Techniques When do we learn? Sages learn from their bad choices

86 Further Techniques When do we learn? Sages learn from their bad choices Every time a contradiction is derived (closed branch) we can lament our choice...

87 Further Techniques When do we learn? Sages learn from their bad choices Every time a contradiction is derived (closed branch) we can lament our choice or learn from our mistakes:

88 Further Techniques When do we learn? Sages learn from their bad choices Every time a contradiction is derived (closed branch) we can lament our choice or learn from our mistakes: We learn that a choice of literals lead to contradiction

89 Further Techniques When do we learn? Sages learn from their bad choices Every time a contradiction is derived (closed branch) we can lament our choice or learn from our mistakes: We learn that a choice of literals lead to contradiction We learn that the clauses involved have enough information to avoid the mistake

90 Further Techniques When do we learn? Sages learn from their bad choices Every time a contradiction is derived (closed branch) we can lament our choice or learn from our mistakes: We learn that a choice of literals lead to contradiction We learn that the clauses involved have enough information to avoid the mistake We can added learned information to the problem

91 Further Techniques When do we learn? Sages learn from their bad choices Every time a contradiction is derived (closed branch) we can lament our choice or learn from our mistakes: We learn that a choice of literals lead to contradiction We learn that the clauses involved have enough information to avoid the mistake We can added learned information to the problem Learning means adding new clauses, without adding new variables

92 Further Techniques Learning 1: Bad Choices Suppose we had choices which after propagation led to V = { p, r} V = { p, r,q, s,s} That is, in that context we learned that we cannot have both p and r.

93 Further Techniques Learning 1: Bad Choices Suppose we had choices which after propagation led to V = { p, r} V = { p, r,q, s,s} That is, in that context we learned that we cannot have both p and r. Add the new clause: p r

94 Further Techniques Learning 1: Properties From a closed branch with choices λ 1,...,λ k, learn λ 1... λ k

95 Further Techniques Learning 1: Properties From a closed branch with choices λ 1,...,λ k, learn λ 1... λ k In general, this form of learning does not improve efficiency, for

96 Further Techniques Learning 1: Properties From a closed branch with choices λ 1,...,λ k, learn λ 1... λ k In general, this form of learning does not improve efficiency, for it will only be used if k 1 literals occur in another branch

97 Further Techniques Learning 1: Properties From a closed branch with choices λ 1,...,λ k, learn λ 1... λ k In general, this form of learning does not improve efficiency, for it will only be used if k 1 literals occur in another branch not very likely

98 Further Techniques Learning 1: Properties From a closed branch with choices λ 1,...,λ k, learn λ 1... λ k In general, this form of learning does not improve efficiency, for it will only be used if k 1 literals occur in another branch not very likely Useful in the presence of forgetting (random restarts)

99 Further Techniques Learning 1: Properties From a closed branch with choices λ 1,...,λ k, learn λ 1... λ k In general, this form of learning does not improve efficiency, for it will only be used if k 1 literals occur in another branch not very likely Useful in the presence of forgetting (random restarts) Other learning techniques may be applied

100 Further Techniques Learning 2: Relevant information Clauses involved in a contradiction present useful information

101 Further Techniques Learning 2: Relevant information Clauses involved in a contradiction present useful information In particular, some clauses leading to a contradiction can be resolved

102 Further Techniques Learning 2: Relevant information Clauses involved in a contradiction present useful information In particular, some clauses leading to a contradiction can be resolved We learn the resolved clause and add it

103 Further Techniques Learning 2: Relevant information Clauses involved in a contradiction present useful information In particular, some clauses leading to a contradiction can be resolved We learn the resolved clause and add it For that, we have to store extra information in a partial valuation:

104 Further Techniques Learning 2: Relevant information Clauses involved in a contradiction present useful information In particular, some clauses leading to a contradiction can be resolved We learn the resolved clause and add it For that, we have to store extra information in a partial valuation: For each unit clause, its original clause

105 Further Techniques Learning 2: Relevant information Clauses involved in a contradiction present useful information In particular, some clauses leading to a contradiction can be resolved We learn the resolved clause and add it For that, we have to store extra information in a partial valuation: For each unit clause, its original clause Choice literals are associted to

106 Further Techniques Learning 2: An Example In the previous example, the choices are represented as: V = {( p, ),( r, )}

107 Further Techniques Learning 2: An Example In the previous example, the choices are represented as: V = {( p, ),( r, )} After linear propagation, the contradiction V = {( p, ),( r, ),(q,p q r),( s,p r s),(s,p q s)}

108 Further Techniques Learning 2: An Example In the previous example, the choices are represented as: V = {( p, ),( r, )} After linear propagation, the contradiction V = {( p, ),( r, ),(q,p q r),( s,p r s),(s,p q s)} We learn the resolution of contradição: p r s e p q s

109 Further Techniques Learning 2: An Example In the previous example, the choices are represented as: V = {( p, ),( r, )} After linear propagation, the contradiction V = {( p, ),( r, ),(q,p q r),( s,p r s),(s,p q s)} We learn the resolution of contradição: p r s e p q s That is, we add the clause: p r q

110 Further Techniques Learning 2: An Example In the previous example, the choices are represented as: V = {( p, ),( r, )} After linear propagation, the contradiction V = {( p, ),( r, ),(q,p q r),( s,p r s),(s,p q s)} We learn the resolution of contradição: p r s e p q s That is, we add the clause: p r q This form of learning has better effects to proof efficiency

111 Further Techniques Learning 2: An Example In the previous example, the choices are represented as: V = {( p, ),( r, )} After linear propagation, the contradiction V = {( p, ),( r, ),(q,p q r),( s,p r s),(s,p q s)} We learn the resolution of contradição: p r s e p q s That is, we add the clause: p r q This form of learning has better effects to proof efficiency Theorem: proofs with DPLL + Learning2 can polynomially simulate full resolution proofs

112 Further Techniques Heuristics for Choosing Literals

113 Further Techniques Choosing Literals A DPLL step starts when linear propagation ends without contradiction

114 Further Techniques Choosing Literals A DPLL step starts when linear propagation ends without contradiction A each step, a new literal has to be chosen to start a new cycle propagation/backtrack

115 Further Techniques Choosing Literals A DPLL step starts when linear propagation ends without contradiction A each step, a new literal has to be chosen to start a new cycle propagation/backtrack Several heuristics can be used to choose that literal

116 Further Techniques Choosing Literals A DPLL step starts when linear propagation ends without contradiction A each step, a new literal has to be chosen to start a new cycle propagation/backtrack Several heuristics can be used to choose that literal Heuristics are rules that help the choice of one among several possibilities

117 Further Techniques Choosing Literals A DPLL step starts when linear propagation ends without contradiction A each step, a new literal has to be chosen to start a new cycle propagation/backtrack Several heuristics can be used to choose that literal Heuristics are rules that help the choice of one among several possibilities Some heuristics are clearly more efficient than others

118 Further Techniques Heuristics without Learning Do not take in consideration the learning of new clauses

119 Further Techniques Heuristics without Learning Do not take in consideration the learning of new clauses MOM Heuristics Maximum number of literal Ocorrences with Minimum length Easy to implement: select the literal that is present in the largest number of clauses of minimum size. Increases the probability of backtracking

120 Further Techniques Heuristics without Learning Do not take in consideration the learning of new clauses MOM Heuristics Maximum number of literal Ocorrences with Minimum length Easy to implement: select the literal that is present in the largest number of clauses of minimum size. Increases the probability of backtracking SATO Heuristics is a variation of MOM. Let f(p) be 1 plus the number of clauses of smallest size which contain p. Choose p that maximizes f(p) f( p). Choose p if f(p) > f( p); p otherwise.

121 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum

122 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration

123 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal

124 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal Initialize each with n. of occurrences of each literal

125 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal Initialize each with n. of occurrences of each literal Increment counter when a clause containing that literal is added

126 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal Initialize each with n. of occurrences of each literal Increment counter when a clause containing that literal is added Choose literal with highest count

127 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal Initialize each with n. of occurrences of each literal Increment counter when a clause containing that literal is added Choose literal with highest count Periodically, counters are divided by a constant

128 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal Initialize each with n. of occurrences of each literal Increment counter when a clause containing that literal is added Choose literal with highest count Periodically, counters are divided by a constant Highest priority to literals in clauses recently learned

129 Further Techniques Heuristics VSIDS Variable State Independent Decaying Sum Takes the learning of clauses in consideration One counter for each literal Initialize each with n. of occurrences of each literal Increment counter when a clause containing that literal is added Choose literal with highest count Periodically, counters are divided by a constant Highest priority to literals in clauses recently learned Low overheads: counters updated only during learning

130 Further Techniques Random Restarts

131 Further Techniques Forgetting, or Random Restarts Suppose a formula is SAT, but bad initial choices

132 Further Techniques Forgetting, or Random Restarts Suppose a formula is SAT, but bad initial choices A valuation will be found only after exhausting the consequences of those bad choices

133 Further Techniques Forgetting, or Random Restarts Suppose a formula is SAT, but bad initial choices A valuation will be found only after exhausting the consequences of those bad choices Idea: restart periodically the search for a valuation

134 Further Techniques Forgetting, or Random Restarts Suppose a formula is SAT, but bad initial choices A valuation will be found only after exhausting the consequences of those bad choices Idea: restart periodically the search for a valuation Problem: if the formula is UNSAT, this may lead to great inefficiency

135 Further Techniques Forgetting, or Random Restarts Suppose a formula is SAT, but bad initial choices A valuation will be found only after exhausting the consequences of those bad choices Idea: restart periodically the search for a valuation Problem: if the formula is UNSAT, this may lead to great inefficiency This problem is avoided if learned formulas are kept

136 Further Techniques Random Restart Consider ǫ, 0 < ǫ 1

137 Further Techniques Random Restart Consider ǫ, 0 < ǫ 1 When a branch is closed

138 Further Techniques Random Restart Consider ǫ, 0 < ǫ 1 When a branch is closed With probability (1 ǫ), perform usual backtracking

139 Further Techniques Random Restart Consider ǫ, 0 < ǫ 1 When a branch is closed With probability (1 ǫ), perform usual backtracking With probability ǫ, restart the search process with V =

140 Further Techniques Random Restart Consider ǫ, 0 < ǫ 1 When a branch is closed With probability (1 ǫ), perform usual backtracking With probability ǫ, restart the search process with V = Empirically checked: this brings efficiency gains

141 Further Techniques Backjumping

142 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s

143 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s The chose literal a does not occur in clauses associated with subsequent unit propagations

144 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s The chose literal a does not occur in clauses associated with subsequent unit propagations That is, the choice of a is useless

145 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s The chose literal a does not occur in clauses associated with subsequent unit propagations That is, the choice of a is useless Backtracking would lead to useless repetition V = {( p, ),( r, ),(ā, ),(q,p q r),( s,p r s),(s,p q s

146 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s The chose literal a does not occur in clauses associated with subsequent unit propagations That is, the choice of a is useless Backtracking would lead to useless repetition V = {( p, ),( r, ),(ā, ),(q,p q r),( s,p r s),(s,p q s Backtracking should jump back over a and ignore it, instead of generating V = {( p, ),( r, ),(ā, ), gera V = {( p, ),(r, )}

147 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s The chose literal a does not occur in clauses associated with subsequent unit propagations That is, the choice of a is useless Backtracking would lead to useless repetition V = {( p, ),( r, ),(ā, ),(q,p q r),( s,p r s),(s,p q s Backtracking should jump back over a and ignore it, instead of generating V = {( p, ),( r, ),(ā, ), gera This is backjumping V = {( p, ),(r, )}

148 Further Techniques Backjumping via Examples Suppose we have the partial valuation V = {( p, ),( r, ),(a, ),(q,p q r),( s,p r s),(s,p q s The chose literal a does not occur in clauses associated with subsequent unit propagations That is, the choice of a is useless Backtracking would lead to useless repetition V = {( p, ),( r, ),(ā, ),(q,p q r),( s,p r s),(s,p q s Backtracking should jump back over a and ignore it, instead of generating V = {( p, ),( r, ),(ā, ), gera V = {( p, ),(r, )} This is backjumping Backjumping always brings efficiency gains

149 Next Issue

150 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers

151 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers Use of smart techniques have improved DPLL s performance: N = 15 N =

152 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers Use of smart techniques have improved DPLL s performance: N = 15 N = There are still very hard formulas that make DPLL exponential, e.g. PHP

153 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers Use of smart techniques have improved DPLL s performance: N = 15 N = There are still very hard formulas that make DPLL exponential, e.g. PHP Experiments show that these formulas do occur in practice

154 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers Use of smart techniques have improved DPLL s performance: N = 15 N = There are still very hard formulas that make DPLL exponential, e.g. PHP Experiments show that these formulas do occur in practice The future of SAT solvers may be in non-dpll, non-clausal methods

155 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers Use of smart techniques have improved DPLL s performance: N = 15 N = There are still very hard formulas that make DPLL exponential, e.g. PHP Experiments show that these formulas do occur in practice The future of SAT solvers may be in non-dpll, non-clausal methods But the techniques learned from DPLL are incorporated in new techniques

156 Conclusion of the Talk DPLL is > 40 years old, but still the most used strategy for SAT solvers Use of smart techniques have improved DPLL s performance: N = 15 N = There are still very hard formulas that make DPLL exponential, e.g. PHP Experiments show that these formulas do occur in practice The future of SAT solvers may be in non-dpll, non-clausal methods But the techniques learned from DPLL are incorporated in new techniques SAT can also be enhanced: SAT Modulo Theories

157 SAT is not a Panacea Reduction to SAT may be ok for some NPc problems, but some NPc problems with no known polynomial SAT-reduction. E.g. Answer Set Programming... some NPc problems with no efficient polynomial SAT-reduction. E.g. Probabilistic SAT