hypothetical reasoning in propositional satisfiability sat02, may, 2002 joao marques-silva technical...

Hypothetical Reasoning inHypothetical Reasoning inPropositional SatisfiabilityPropositional Satisfiability

SAT’02, May, 2002

Joao Marques-Silva

Technical University of Lisbon,

IST/INESC, CEL

Lisbon, Portugal

Hypothetical ReasoningHypothetical Reasoning

Preliminary ongoing research work– Not yet published

• Main ideas available as a (preliminary) technical report:– I. Lynce and J. Marques-Silva, “Hypothetical Reasoning in Propos

itional Satisfiability,” Technical Report 1/2002, INESC-ID, March 2001

– http://sat.inesc.pt/~jpms/research/tech-reports/RT-01-2002.pdf

– Some of the concepts still evolving

Feedback welcome !

Joint work with Ines Lynce

MotivationMotivation

SAT solvers have been the subject of significant improvements in recent years

The utilization of SAT is increasing in industry– More challenging problem instances

Improvements to current key techniques unlikely(?)– Better (non-chronological) backtracking?– Better data structures?– Newer (more competitive) strategies?

How to improve SAT solvers?– Devise new paradigms…– Integrate already used techniques

OutlineOutline

Notation & Definitions Evolution of SAT Solvers

– Overview established approaches

Next challenges in SAT Other promising approaches Our proposed approach Hypothetical reasoning (HR)

– The overall approach– Applying reasoning conditions

• Relation with other existing techniques

Preliminary experimental results

Notation & DefinitionsNotation & Definitions

CNF Formula, clauses, literals:– A CNF formula () is a conjunction of clauses– A clause () is a disjunction of literals– A literal (l) is a propositional variable or its complement

Assignments: x, 0 denotes the assignment of value 0 to variable x– Can also use x = 0 to denote an assignment

= (a + c)(b + c)(d + c)(¬a + ¬b + ¬c)

= (a c)(b c)(d c)(¬a ¬b ¬c)

Notation & Definitions (Cont’d)Notation & Definitions (Cont’d)

Unit-clause rule:– If clause is unit (has a single free literal l), then the free

literal l must be satisfied for the clause to be satisfied – Iterated application of the unit-clause rule is referred to as

unit propagation (UP) or boolean constraint propagation (BCP)

– BCP(x, vx): denotes the set of implied variable assignments obtained by applying BCP as the result of the triggering assignment x, vx

• If BCP( x, vx) unsatisfies one or more clauses, then we say that BCP( x, vx)

A Taxonomy of SAT AlgorithmsA Taxonomy of SAT Algorithms

Backtrack search (DPLL)

Resolution (original DP)

Stalmarck’s method (SM)

Recursive learning (RL)

BDDs

...

Local search (hill climbing)

Continuous formulations

Genetic algorithms

Simulated annealing

...

Tabu search

SAT Algorithms

Complete Incomplete

Can prove unsatisfiability Cannot prove unsatisfiability

The Most Effective SAT SolversThe Most Effective SAT Solvers

Backtrack search Boolean constraint propagation “Reasonable” branching heuristic Clause recording

– Non-chronological backtracking

Search strategies– Restarts– Random backtracking

Efficient data structures– E.g. head/tail lists; watched literals; literal sifting

Examples: BerkMin; Chaff; SATO; rel_sat; GRASP

Other Effective SAT SolversOther Effective SAT Solvers

Backtrack search Unit propagation Chronological backtracking Fine-tuned branching

heuristics Probing & reasoning

techniques– Lookahead (variable probing)

– Equivalency reasoning Search strategies

– Restarts Efficient data structures

– E.g. head/tail lists

Examples:– EQSATZ

• Built on top of SATZ• Uses equivalency

reasoninig– RAND-SATZ

• Built on top of SATZ• Branching randomization• Search restarts

– SATZ• No search restarts• No equivalency reasoning• Forms of look-ahead

probing

Other Dedicated SAT SolversOther Dedicated SAT Solvers

Local search for dedicated classes of instances– Incomplete class of algorithms– Useful if instances known to be satisfiable

Solvers with domain-specific information– Incremental SAT– SAT on Boolean networks– …

Challenging Problem InstancesChallenging Problem Instances

SAT is being applied in industrial settings– Electronic design automation

– Formal verification of hardware/software systems

– …

SAT solvers are expected to handle problem instances:– that have hundred thousand / few million variables

– that have tens of million clauses

– that may be unsatisfiable

• SAT solvers must be capable of proving unsatisfiability– completeness is a key issue !

Challenges to SAT Solvers Challenges to SAT Solvers

Dramatic improvements to backtrack search SAT solvers unlikely– Can utilize equivalency reasoning

• Hard to interact with clause recording and non-chronological backtracking

– Can apply lookahead techniques• Hard to interact with clause recording and non-

chronological backtracking– Can devise new search strategies

• Search restarts, random backtracking, … ?

Other ApproachesOther Approaches

Resolution– Unlikely to be a practical proof procedure

Variable probing (branch-merge rule) Clause probing (recursive learning)

– Not (yet) extensively evaluated

Additional mechanisms for identifying necessary assignments and inferring new clauses– Integrated solution still lacking

Resolution (original DP)Resolution (original DP)

Iteratively apply resolution (consensus) to eliminate one variable each time– i.e., resolution between all pairs of clauses containing x and ¬x– formula satisfiability is preserved

Stop applying resolution when,– Either empty clause is derived instance is unsatisfiable– Or only clauses satisfied or with pure literals are obtained

instance is satisfiable

= (a + c)(b + c)(d + c)(¬a + ¬b + ¬c) Eliminate variable c

1 = (a + ¬a + ¬b)(b + ¬a + ¬b )(d + ¬a + ¬b )= (d + ¬a + ¬b ) Instance is SAT !

Stalmarck’s Method (SM) Stalmarck’s Method (SM) in CNFin CNF

Recursive application of the branch-merge rule to each variable with the goal of identifying common conclusions

Try a = 0: (a = 0) (b = 1) (d = 1)

Try a = 1: (a = 1) (c = 1) (d = 1)

C(a = 0) = {a = 0, b = 1, d = 1}

C(a = 1) = {a = 1, c = 1, d = 1}

C(a = 0) C(a = 1) = {d = 1} Any assignment to variable a implies d = 1.Hence, d = 1 is a necessary assignment !

Recursion can be of arbitrary depth

= (a + b)(¬a + c) (¬b + d)(¬c + d) = (a + b)(¬a + c) (¬b + d)(¬c + d) = (a + b)(¬a + c) (¬b + d)(¬c + d) = (a + b)(¬a + c) (¬b + d)(¬c + d)

An Alternative Explanation for SMAn Alternative Explanation for SM

= (a + b)(¬a + c) (¬b + d)(¬c + d)

(b + c)

resolution

(c + d)resolution

Comment: SM provides amechanism for identifyingsuitable resolution operations

(d)resolution

Sequence of resolutionoperations for findingnecessary assignments

Recursion can be of arbitrary depth

Recursive Learning (RL) Recursive Learning (RL) in CNFin CNF

Recursive evaluation of clause satisfiability requirements for identifying common assignments

Try a = 1:

= (a + b)(¬a + d) (¬b + d)

(a = 1) (d = 1)

Try b = 1: (b = 1) (d = 1)

C(a = 1) = {a = 1, d = 1}

C(b = 1) = {b = 1, d = 1}

C(a = 1) C(b = 1) = {d = 1} Every way of satisfying (a + b) implies d = 1. Hence, d = 1 is a necessary assignment !

= (a + b)(¬a + d) (¬b + d) = (a + b)(¬a + d) (¬b + d) = (a + b)(¬a + d) (¬b + d)

An Alternative Explanation for RLAn Alternative Explanation for RL

= (a + b)(¬a + d) (¬b + d)

(b + d)

resolution

(d)

resolutionSequence of resolutionoperations for findingnecessary assignments

Comment: RL provides yet another mechanism for identifyingsuitable resolution operations

SM vs. RLSM vs. RL

Both complete procedures for SAT Stalmarck’s method (in CNF):

– hypothetical reasoning based on variables

– use variable assignment conditions to probe assignments

• variable probing

Recursive learning (in CNF):– hypothetical reasoning based on clauses

– use clause satisfiability conditions to probe assignments

• clause probing

Both can be viewed as the process of identifying selective resolution operations

Both can be integrated into backtrack search algorithms

The Objectives of HRThe Objectives of HR

Integrate variable probing and clause probing– Complete proof procedure

for SAT Devise conditions for a priori

identification of new clauses– That entail most of existing

clause inference procedures

Evolve from identification of necessary assignments to generalized clause reasoning

Applications:– Complete proof procedure

for SAT

– Preprocessing engine to existing SAT solvers

• With polynomial effort

– Replace unit propagation with HR with backtrack search solvers

• With polynomial effort

– Cooperate with backtrack search solvers

• In parallel solutions for SAT

The Organization of HRThe Organization of HR

Recursive procedure that:– Extends variable probing

• To incorporate clause probing • Ensures completeness

– Establishes general clause inference conditions • That cover (most) existing clause inference conditions

– Readily implements a number of additional techniques• 2-var equivalence; hyper resolution (with binary clauses);

equivalency reasoning; binary clause inference conditions; …

Can be integrated into backtrack search

How to Implement HR ?How to Implement HR ?

Independent probing, given conditions on variables and on clauses, may not be practical– O(L2+ L N) = O(L2) at each step

• L: number of literals• N: number of variables

Construct assignment & trigger tables, for implementing variable and clause probing– O(L N) at each step

• In practice, worst-case complexity is extremely unlikely

– OBS: unrestricted clause inference conditions are computationally hard to implement

Assignment TableAssignment Table

Captures the result of applying BCP to each variable assignment

Create a (2N x 2N) matrix:– Each row is associated with an assignment x, vx

– 1-valued entries denote assignments y, vy implied by BCP due to trigger assignment x, vx, i.e. BCP(x, vx)

OBS: In practice can use a sparse matrix representation !

Assignment Table (Example)Assignment Table (Example)

b, 0 implies (with BCP) the assignments b, 0, c, 0 and d, 0

a=0 a=1 b=0 b=1 c=0 c=1 d=0 d=1a=0 1 1a=1 1 1b=0 1 1 1b=1 1c=0 1c=1 1d=0 1d=1 1 1

Trigger TableTrigger Table

Captures which variable assignments directly imply (w/ BCP) each variable assignment

Create a (2N x 2N) matrix:– Each row is associated with an assignment x, vx

– 1-valued entries denote assignments y, vy that imply, with BCP, the assignment x, vx

OBS: The trigger table is the transpose of the assignment table ! – Required to create trigger table if using a sparse matrix

representation of the assignment table

Trigger Table (Example)Trigger Table (Example)

b, 1 is implied (due to BCP) by the assignments a, 0, a, 1 and b, 1

a=0 a=1 b=0 b=1 c=0 c=1 d=0 d=1a=0 1a=1 1b=0 1b=1 1 1 1c=0 1 1 1c=1 1d=0 1 1d=1 1

Utilizations of Assignment TablesUtilizations of Assignment Tables

Necessary assignments from variable assignment conditions — variable probing

For both assignments to a, a, 0 and a, 1 , we obtain b, 1. b, 1 is a necessary assignment



Necessary assignments from clause satisfiability conditions — clause probing– Assuming existence of clause (b d)

Every assignment thatsatisfies (b d), also implies c, 0. c, 0 is a necessary assignment



Clause inference from variable assignment conditions

a=0 a=1 b=0 b=1 c=0 c=1 d=0 d=1a=0 1 1a=1 1 1b=0 1b=1 1 1c=0 1c=1 1 1d=0 1d=1 1

One of these assignments must hold (because of a) create clause (b c)


Clause inference from clause satisfiability conditionsa=0 a=1 b=0 b=1 c=0 c=1 d=0 d=1

a=0 1 1a=1 1 1b=0 1b=1 1 1c=0 1c=1 1 1d=0 1d=1 1

Assume clause = (a b c) exists. Each assignment that satisfies implies either c, 0 or d, 0 create clause (c d)

Utilizations of Trigger TablesUtilizations of Trigger Tables

Clause inference from variable assignment conditionsa=0 a=1 b=0 b=1 c=0 c=1 d=0 d=1

a=0 1 1a=1 1b=0 1 1b=1 1 1c=0 1c=1 1d=0 1d=1 1

The assignments a, 0 and b, 1 imply the assignments c, 0 and c, 1; are disallowed create clause (a b)

Utilizations of Trigger TablesUtilizations of Trigger Tables

Clause inference from clause unsatisfiability conditionsa=0 a=1 b=0 b=1 c=0 c=1 d=0 d=1

a=0 1 1a=1 1b=0 1 1b=1 1 1c=0 1c=1 1d=0 1d=1 1

Assume clause = (c d) exists. The assignments a, 0 or b, 0 unsatisfy . create clause (a b)

Reasoning Conditions SummaryReasoning Conditions Summary

Necessary assignments:– From variable assignment conditions (variable probing)– From clause satisfiability conditions (clause probing)

Inferred clauses:– Satisfiability conditions

• Variable assignments• Clause satisfiability

– Unsatisfiability conditions• Variable assignments• Clause satisfiability

Inference of ClausesInference of Clauses

HR reasoning conditions can only infer binary clauses ? – No. Can infer arbitrary clauses !

– Clause satisfiability conditions:

• For each clause = (t1 t2 tm) of formula , all clauses of the form (s1 s2 sm), such that s1, s2,,sm BCP(t1, 1) … BCP(tm, 1), are implicates of

– Clearly subsumption can potentially be applied

– Clause unsatisfiability conditions:

• For each set of assignments A = { t1, 0, t2, 0, , tm, 0 }, such that BCP(t1, 0) BCP(t2, 0) BCP(tm, 0), then clause = (t1 t2 tm) is an implicate of

Reasoning Conditions ComplexityReasoning Conditions Complexity

O(L N) for constructing assignment & trigger tables and implementing variable and clause probing

Why ? – BCP for filling each row is O(L) – For the 2N rows, construction of table is O(L N)

– Each set intersection can trivially be accomplished in O(N) ! – All intersections can be done in O((N + L) N) = O(L N)

• Corresponding to variable and clause probing– Total time complexity is O(L N)

– OBS: In practice worst-case complexity extremely unlikely– OBS: unrestricted clause inference conditions are computationally hard to

implement; must use restrictions

The HR AlgorithmThe HR Algorithm

Basic HR algorithm (with depth d, target variables V)– return if depth d 0– For each variable v in set of target variables V

• For each assignment to variable v– L1: Apply unit propagation (BCP)

• Apply (tabular) reasoning conditions

• Recur HR with depth (d-1)

• Re-apply (tabular) reasoning conditions

– [Optional] Loop from L1 if more assignments

O(L N)

Can loop O(N) times

Polynomial time if depth is constant !

Relation with Other TechniquesRelation with Other Techniques

Assignment & Trigger tables naturally capture:– Variable probing (branch-merge rule)

• Lookahead techniques– Clause probing (recursive learning)– New clause inference conditions

Assignment & Trigger tables allow capturing:– Failed-literal rule– Two-variable equivalence– Closure of binary clause implication graph– Literal dropping– Equivalency-reasoning / Inference of binary clauses– Hyper resolution (with binary clause inference)– … ?


Failed literal rule: – If an assignment x, 0 yields an unsatisfied clause, then x, 1

is a necessary assignment

– In the construction of the assignment table,• If BCP(x, 0), then x, 1 is a necessary assignment


2-variable equivalence: – First form:

• If both (x y) and (y x) exist in formula, then x y – Second form:

• Utilize binary clause implication graph• Identify strongly connected components (SCCs)

– If x, 0 and y, 0 in the same SCC, then x y

– If, from construction of the assignment table, y, 0 BCP(x, 0) and y, 1 BCP(x, 1), then x y

• Captures all SCCs in binary clause implication graph• Can identify additional 2-variable equivalences !


Closure of binary clause implication graph:– If l1, 1 l2, 1 and l2, 1 l3, 1, then l1, 1 l3, 1 and infer

clause (l1 l3)

– From construction of the assignment table, if l2, 1 BCP(l1, 1), then create clause (l1 l2)

• Captures the identification of the transitive closure of the implication graph

• Can identify additional implications (and clauses) !


Literal dropping [Dubois & Dequen, IJCAI’01]: – Given a clause = (l1 l2 lk), if exists a proper subset of literals {s1,

s2, …, sj } of , such that

BCP(s1, 0 s2, 0 sj, 0), then create a new clause (s1 s2 sj), that subsumes

– Using the assignment table, if exists a set of assignments A = { t1, 0, t2, 0, , tm, 0 }, such that BCP(t1, 0) BCP(t2, 0) BCP(tm, 0), then create the clause: = (t1 t2 tm)

– Two techniques similar, but not comparable• Literal dropping less general (starts from existing clauses), but more

accurate (considers BCP of set of assignments)


Equivalency reasoning [Li, AAAI’00]: – Shown to be covered with:

• Unit propagation; 2-variable equivalence; conditions for inferring binary clauses

Binary clause inference conditions [MS, CP’00]: – Inference from pattern 2B/1T:

• Given (l1 x) (l2 x) (l1 l2 y), infer (x y)

• From the assignment table:– If x, 0 y, 1, then infer the clause (x y)


Binary clause inference conditions [MS, CP’00]: – Inference from pattern 0B/4T:

• Given (l1 l2 x) (l1 l2 x) (l1 l2 y) (l1 l2 y), infer clause (x y)

– From the assignment table:

• Assume l1 = 0 (depth 1)

– Can infer (x y) (depth 2)

• From (l2 x) (l2 y),

• Assume l1 = 1 (depth 1)

– Can infer (x y) (depth 2)

• From ( l2 x) (l2 y),

infer (x y)– But HR with depth 2 required !


Hyper resolution (w/ binary clauses) [Bacchus, SAT’02]: – Allows inference of binary clause

– Given (l1 x) (l2 x) (lk x) (l1 l2 lk y), infer (x y)

– From the assignment table:• If x, 0 y, 1, then infer the clause (x y)

Preliminary ResultsPreliminary Results

Implemented reasoning conditions on top of JQuest– Assigment tables– Trigger tables– Necessary assignments

• Probing due to variables and clauses (binary and ternary)– Clause inference conditions

• Simplified version: only binary clauses can be inferred

Results for reasoning conditions on example problem instances– #Vars: number of variables; #Cls: number of clauses– #NA: necessary of assignments; #IBC: inferred binary clauses

Experimental ResultsExperimental Results

Instance #Vars #Cls #NA #IBC

sss1.0/dlx2_cc_bug11 1513 12800 26 2663

sss1.0a/dlx2_cc_a_bug40 2654 19516 237 2079

sss-sat-1.0/2dlx_..._bug007 4824 48229 511 3667

bmc/barrel9 8903 36606 909 13150

bmc/queueinvar20 2435 20671 15 3460

cec/c2670 2703 6756 79 5432

cec/c7552 7652 20423 168 17556

dimacs/bf1355-638 2177 6768 638 5746

dimacs/ssa2670-141 986 2315 19 2464

satplan/unsat/bw_large.c 2729 45368 726 3023

satplan/sat/bw_large.c 3016 50457 726 3089

Implementing/Completing HRImplementing/Completing HR

Implement (efficient) recursive wrapper– Incrementally define set of variables in recursive step

• Reduce significantly the number of row updates in assignment and trigger tables

Instead of BCP-based reasoning conditions, evolve to clause-based reasoning conditions

How to use HR?– Standalone complete proof procedure ?– Integrated within backtrack search SAT solver ?

• Hard to interact with clause recording and non-chronological backtracking

– Used as a preprocessing engine to backtrack search SAT solvers ?

ConclusionsConclusions

Proposed the Hypothetical Reasoning algorithm– Integrates variable probing (branch-merge rule) and clause probing

(recursive learning)– Implements a number of additional techniques

• That allow inferring new clauses• That entail most existing clause inference conditions• That entail a significant number of simplification techniques

Preliminary results for practical problem instances:– By applying reasoning conditions,

• a significant number of necessary assignments can be identified and a significant number of new clauses can be inferred

hypothetical reasoning in propositional satisfiability sat02, may, 2002 joao marques-silva technical...

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