Solving Non-clausal Formulas with DPLL search

Christian Thiffault Fahiem BacchusUniversity of Toronto

Toby WalshUNSW

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CNF• The classical DPLL algorithm (most modern SAT

solvers) work with Conjunctive Normal Form (CNF)• However, CNF is not the most natural representation.• Hence, to use modern SAT solver technology problems

must be converted to CNF• In this work we demonstrate that such a conversion is

unnecessary. • For a number of reasons such a conversion is also

undesirable.– We demonstrate that original structure lost in conversion can

be utilized to significantly improve the performance of SAT solvers.

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Tseitin Encodings• The most commonly used CNF encoding is

the Tseitin encoding [Tseitin 1970].• This encoding converts a propositional

formula recursively by adding a new variable for every subformula.

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Tseitin EncodingsA → (C & D)– V1 ≡ (C & D)

• (~V1, C), (~V1, D), (~C,~D,V1)

1. (~V1, C)

2. (~V1, D)

3. (~C,~D,V1)

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Tseitin EncodingsA → (C & D)– V1 ≡ (C & D)

• (~V1, C), (~V1, D), (~C,~D,V1)

– V2 ≡ (A → V1)• (~V2, ~A,V1), (A,V2), (~V1,V2)

1. (~V1, C)

2. (~V1, D)

3. (~C,~D,V1)

4. (~V2, ~A,V1)

5. (A,V2)

6. (~V1,V2)

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Tseitin EncodingsA → (C & D)– V1 ≡ (C & D)

• (~V1, C), (~V1, D), (~C,~D,V1)

– V2 ≡ (A → V1)• (~V2, ~A,V1), (A,V2), (V1,V2)

– The formula must be true: (V2)

• (~V1, C)• (~V1, D)• (~C,~D,V1)

4. (~V2, ~A,V1)

5. (A,V2)

6. (V1,V2)

7. (V2)

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Converting to CNF is Undesirable • There are two obvious problems arising

from this conversion.

1. Structural information is lost– The fact that a particular group of clauses were

generated from the same subformula is lost.– The global interconnections between the

various subformulas are lost.

2. Additional variables are added. This potentially increases the size of the DPLL search.

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Structural Information • Some of this information could be recovered

from the CNF encoding, but not all of it [Lang & Marquis, 1989].

– There is good empirical evidence that recovering structural information can yield considerable benefits in solving performance. [EqSatZ, LSAT].

– But why lose this information in the first place? – We will show conversion to CNF is not necessary.– We also develop techniques which confirm the

benefits of retaining the original structure.

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Extra Variables• Various works have noted the potential

difficulty of introducing new variables. • Some have suggested restricting the DPLL

search so that it cannot branch on any on the newly introduced “subformula” variables.

• However, it is not difficult to show that restricting branching in this manner causes an exponential slowdown on some problems [Jarvisalo et al. 2004]

• Solvers that attempt to restrict their branching in this manner can perform very poorly on many problems.

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Extra Variables• The alternative is unrestricted branching.• However, with unrestricted branching a

CNF solver can waste a lot of time branching on variables that have dynamically become irrelevant.

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Irrelevant Variables

A → (C & D) A=False.

Formula satisfied

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Irrelevant Variables CNFA → (C & D)– V1 ≡ (C & D)– V2 ≡ (A → V1)

• (~V1, C)• (~V1, D)• (~C,~D,V1)

4. (~V2, ~A,V1)

5. (A,V2)

6. (V1,V2)

7. (V2)8. (~A)

Solver must still determine that the remaining clauses are SAT

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Converting to CNF is Unnecessary • DPLL search can be performed on the

original formula.– This has been noted in previous work on

circuit based solvers [Ganai et al. 2002]

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DPLL on formulas • Convert formula to a dag.

A → (C & D) \/ B → (C & D)

\/

→ →

A B

&

C D

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DPLL on formulas • Associate a truth value (True/ False/ Unassigned)

with every node of the DAG.• Now perform DPLL search by splitting on the

truth value of an unassigned node.• Use the logic of the boolean operators to

propagate truth values to neighboring nodes.• A solution is found when all nodes have been

labeled with consistent truth values (i.e., the parent and child values are consistent with the logic of the parent’s operator.)

• A contradiction occurs when True and False are propagated to the same node.

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Example

\/

→ →

A B

&

C D

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Example

\/

→ →

A B

&

C D

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Example

\/

→ →

A B

&

C D

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Comparison • Assigning a truth value to a node in the

DAG is equivalent to assigning a truth value to the corresponding variable in the CNF encoding.

• Propagation in the DAG is equivalent to unit propagation in the CNF encoding.

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Comparison • The rule that propagated a truth value can

be recorded. Once a contradiction is detected a conflict clause can be learned (a set of impossible node assignments). This clause can be made equivalent to the 1-UIP clauses learned by CNF solvers.

• The learned clauses can be stored and used to unit propagate node truth values in the rest of the search.

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Comparison • Complex gates, e.g., n-ary XOR can be

more efficiently propagated in the DAG representation (these gates require a number of clauses exponential in the number of their inputs).

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Efficient Propagation • Efficient Unit Propagation via lazy data

structures is a key element of modern SAT solvers.

• One new feature of our circuit based solver over previous circuit solvers is that it utilizes similar lazy data structures to make its propagation as efficient as CNF solvers.

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Efficient Propagation • E.g., an AND gate becomes true only when

all of its children become true. – Assign one child as a true watch, and don’t

check for true propagating to the AND gate node unless its true watch becomes true.

– Some benchmarks have AND gates with thousands of children.

– Previous “table-lookup” circuit solvers required some computation each time a child became true.

• Using these techniques there is no intrinsic loss of efficiency in using the DAG over CNF.

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Structure Based Optimizations• Since no penalty is being paid for using the

original formula, we can turn to exploiting the extra structural information it provides.

• We implemented two structure based optimizations.– Don’t care propagation to deal with irrelevant

variables.– Conflict Clause reduction.

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Don’t Care Propagation• We propagate a third “truth” value through

the DAG: don’t cares.• A node C is don’t care wrt a particular

parent P – if its truth value can no longer affect the truth

value of P nor any of its P siblings.– or P is don’t care.

• A node C is don’t care if it is don’t care wrt to all of its parents.

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Don’t Care Propagation• Assign a don’t care watch parent for each

node.• C becoming don’t care wrt to its watch

parent P can be detected when truth values are propagated to P.

• If C becomes don’t care wrt to its don’t care watch we look for another watch.

• If we can’t find one we know that C has become don’t care.

• Finally, we stop the search from branching on don’t care nodes. This eliminates branching of irrelevant values.

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Conflict Clause Reductions• If one learns a conflict clause (L1,L2,...) and one has L1

→ L2, then we can reduce the conflict clause by a resolution step:– (-L1,L2) (L1,L2,...) (L2,...)– The resolvant subsumes the original conflict clause.

• In a CNF encoding one would have to search the clauses to detect this situation—probably not going to be effective.

• In the DAG one can examine the neighbors of each node assignment in the conflict clause to see if any of the other assignments in the clause are implied by it.

• If so we can remove the implying node assignment.• If this is done with discrimination it can yield a

significant decrease in the size of the learned clauses with little overhead.

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Empirical Results.• We compared with Zchaff.• Tried to make the two solvers as close as

possible, same magic numbers (e.g., clause database cleanup criteria, restart intervals etc.), same branching heuristics.

• Hence, we tried to isolate the impact of the non-clausal representation and the structure based optimizations.

• We believe that the same improvements could be obtained with others CNF solvers via this technique.

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Empirical Results caveats • Our experiments were hampered by a lack

of non-clausal benchmarks.• The performance of our solver was also

limited by the fact that the benchmarks we did obtain has already been transformed into simpler formulas, e.g., no complex XOR of IFF gates were present in the benchmarks.

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FVP-UNSAT-2.0 (Velev) Time Problem #Vars Time Imp/Sec

Zchaff NoClause Zchaff NoClause 4pipe 5,237 188.89 9.87 467,001 509,433 4pipe_1 4,647 26.55 35.52 512,108 327,098 4pipe_2 4,941 49.76 36.5 482,896 327,298 4pipe_3 5,233 144.34 62.03 424,551 316,049 4pipe_4 5,525 93.83 42.26 470,936 326,186 5pipe 9,471 54.68 33.34 526,457 409,154 5pipe_1 8,441 126.11 116.18 425,921 280,758 5pipe_2 8,851 138.62 177.24 437,166 279,298 5pipe_3 9,267 137.7 134.08 441,319 295,976 5pipe_4 9,764 873.81 284.62 370,906 270,234 5pipe_5 10,113 249.11 137.09 456,400 298,903 6pipe 15,800 4,550.92 297.13 322,039 288,855 6pipe_6 17,064 1,406.18 1,056.56 402,301 267,207 7pipe 23,910 12,717.00 1,657.70 306,433 244,343 7pipe_bug 24,065 128.9 0.29 266,901 403,148

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FVP-UNSAT-2.0 Decisions Problem #Vars Zchaff NoClause 4pipe 5,237 541,195 41,637 4pipe_1 4,647 131,223 114,512 4pipe_2 4,941 210,169 112,720 4pipe_3 5,233 392,564 169,117 4pipe_4 5,525 295,841 122,497 5pipe 9,471 334,761 102,077 5pipe_1 8,441 381,921 255,894 5pipe_2 8,851 397,550 362,840 5pipe_3 9,267 385,239 292,802 5pipe_4 9,764 1,393,529 503,128 5pipe_5 10,113 578,432 283,554 6pipe 15,800 5,232,321 435,781 6pipe_6 17,064 2,153,346 1,326,371 7pipe 23,910 12,437,654 1,276,763 7pipe_bug 24,065 1,075,907 481

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FVP-UNSAT-2.0 Don’t CaresProblem DC Time No DC DC Decision No DC 4pipe 9.87 57.68 41,637 198,828 4pipe_1 35.52 62.65 114,512 159,049 4pipe_2 36.5 94.46 112,720 212,986 4pipe_3 62.03 213.27 169,117 365,007 4pipe_4 42.26 318.64 122,497 525,623 5pipe 33.34 246.93 102,077 650,312 5pipe_1 116.18 300.59 255,894 489,825 5pipe_2 177.24 360.67 362,840 585,133 5pipe_3 134.08 387.65 292,802 593,815 5pipe_4 284.62 2097.31 503,128 1,842,074 5pipe_5 137.09 379.19 283,554 543,535 6pipe 297.13 10,241.64 435,781 4,726,470 6pipe_6 1,056.56 3,455.35 1,326,371 2,615,479 7pipe 1,657.70 12,685.59 1,276,763 6,687,186 7pipe_bug 0.29 1 481 2,006

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FVP-UNSAT-2.0 Clause ReductionProblem Red. Time No Red. Red. Ave Cls. SizeNo Red 4pipe 9.87 30.5 40 148 4pipe_1 35.52 45.13 77 132 4pipe_2 36.5 54.97 84 132 4pipe_3 62.03 69.7 108 162 4pipe_4 42.26 80.13 112 157 5pipe 33.34 16.29 93 113 5pipe_1 116.18 195.81 140 204 5pipe_2 177.24 159.45 165 199 5pipe_3 134.08 154.02 165 218 5pipe_4 284.62 504.27 208 264 5pipe_5 137.09 216.11 172 237 6pipe 297.13 647.78 232 540 6pipe_6 1,056.56 1,421.42 309 380 7pipe 1,657.70 2,053.92 336 761 7pipe_bug 0.29 0.29 10 10

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Other Series

Series (#probs) ZchaffTime Dec. Imp/Sec Cls Size

sss-sat-1.0 (100) 128 2,970,794 728,144 70 vliw-sat-1.1 (100) 3,284 154,742,779 302,302 82 fvp-unsat-1.0 (4) 245 3,620,014 322,587 326 fvp-unsat-2.0 (22) 20,903 26,113,810 327,590 651

Series (#probs) NoClauseTime Dec. Imp/Sec Cls Size

sss-sat-1.0 (100) 225 1,532,843 616,705 39 vliw-sat-1.1 (100) 1,033 4,455,378 260,779 55 fvp-unsat-1.0 (4) 172 554,100 402,621 100 fvp-unsat-2.0 (22) 4,104 5,537,711 267,858 240

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Conclusions• No intrinsic reason to convert to CNF.• Many other structure based optimizations

remain to be investigated, e.g.– better branching– non-clausal representation of conflicts– more complex gates.