Extracting Minimum Unsatisfiable Cores with a Greedy Genetic Algorithm Jianmin Zhang, Sikun Li, and Shengyu Shen School of Computer Science, National University of Defense Technology Australasian Joint Conference on Artificial Intelligence AI 06

Abstract Problem: finding minimum unsatisfiable core Method: Efficient greedy genetic algorithm takes advantage of the relationship between maximal satisfiability and minimum unsatisfiability generate all minimal UCs then select the smallest one experiment: SAT benchmarks from Automotive Product Configuration (size of 22 instances from 4k~8k clauses and 1.6k ~1.9k variables) compared with the branch-and-bound algorithm Conclusion : Efficient than branch-and-bound algorithm

Introduction Explaining the causes of unsatisfiability of Boolean formulas is a key requirement in formal verification, artificial intelligence, and electronic design. checking minimal unsatisfiability is hard ? D p completeD p complete a smallest sub-formula that is also unsatisfiable (minimum unsatisfiable core). How to find minimun unsat core ? Using relationship between maximal satisfiability and minimal unsatisfiability derive the complements of maximal satisfiable subsets. extract a minimum unsatisfiable core by computing the minimum hitting set of those complement sets

Related Work Bruni and A. Sassano. Restoring Satisfiability or Maintaining Unsatisfiability by Finding Small Unsatisfiable Subformulae (SAT’01) a method of adaptive core search L. Zhang and S. Malik. Extracting small unsatisfiable cores from unsatisfiable Boolean formula (SAT’03) use SAT solvers to produce resolution refutations. AMUSE: a minimally-unsatisfiable subformula extractor.(DAC 04) MUP: A minimal unsatisfiability prover.(DAC 05)  : small unsatisfiable subformula no guarantee minimal subformula

Related Work (con’t) I. Lynce and J. P. Marques-Silva. On computing minimum unsatisfiable cores.(SAT’04) algorithm enumerates all possible subsets of a formula and checks which are unsatisfiable A branch and bound algorithm for extracting smallest minimal unsatisfiable formulas. (SAT’05) utilizes iterative maximal satisfiability solutions to generate lower and upper bounds on smallest minimal UC branch on specific subformulas to find it beter than previous our approach compared with this on the same benchmarks

Preliminaries Definition 1. (Unsatisfiable Core). Given a formula ϕ, ψ is an unsatisfiable core for ϕ iff ψ is an unsatisfiable formula and ψ ⊆ ϕ. Definition 2. (Minimal Unsatisfiable Core). Given an unsatisfiable core ψ for a formula ϕ, ψ is a minimal unsatisfiable core iff removing any clause ω ∈ ψ from ψ implies that ψ − {ω} is satisfiable. Definition 3. (Minimum Unsatisfiable Core). Consider the set of all unsatisfiable cores for a formula ϕ : {ψ 1,..., ψ j }. Then, ψ k ∈ {ψ 1,..., ψ j } is a minimum unsatisfiable core iff ∀ ψ i ∈ {ψ 1,..., ψ j }, 1 ≤ i ≤ j : |ψ k | ≤ |ψ i |.

Preliminaries (con’t) Definition 4. (Satisfiable Subformula). Consider a formula ϕ, φ is a satisfiable subformula for ϕ iff φ is a satisfiable formula and φ ⊆ ϕ. Definition 5. (Maximal Satisfiable Subformula). Given a satisfiable subformula φ for a formula ϕ, φ is a maximal satisfiable subformula iff adding any clause ω ∈ { ϕ − φ} into φ implies that φ ∪ {ω} is unsatisfiable Definition 6. (Minimal Hitting Set). Suppose C = {S1,..., Sn} is a collection of sets. Then H is a hitting set of C iff H ⊆ 1≤i≤n ∪ Si, and H ∩ Si != ∅ for each i with 1 ≤ i ≤ n. H is a minimal hitting set iff ∀ R ⊂ H : R is not a hitting set. C={ {M1, M2, A1}, {M1, A1, A2, M3}}. The minimal hitting sets are {M1}, {A1}, {M2, A2}, {M2, M3}.

Preliminaries (con’t) Lemma 1. A minimal unsatisfiable core for a formula ϕ is a minimal hitting set of the complements of maximal satisfiable subformulas for ϕ. Definition 7. (Minimum Hitting Set). Given a collection C of sets and the set of all hitting sets of C : {H 1,..., Hj}, then Hk ∈ {H 1,..., Hj} is a minimum hitting set iff ∀ Hi ∈ {H 1,..., Hj}, 1 ≤ i ≤ j : |Hk| ≤ |Hi|. Lemma 2. A minimum unsatisfiable core for a formula ϕ is a minimum hitting set of the comp- lements of maximal satisfiable subformulas for ϕ.

Example CNF formula ϕ = (x 1 ) ∧ (~x 2 ) ∧ (~x 1 ∨ x 2 ) ∧ (~x 2 ∨ x 3 ) ∧ (~x 3 ) The SAT and UNSAT Solutions of ϕ Maximal satisfiable subformulas {2, 3, 4, 5} {1, 2, 4, 5} {1, 3, 5} {1, 3, 4} Complements of MSSes {1} {3} {2, 4} {2, 5} Minimal unsatisfiable cores {1, 2, 3} {1, 3, 4, 5} Minimum unsatisfiable core {1, 2, 3}

Extracting Minimum UC generate all minimal UCs then select the smallest one

GA Main idea How it work? populations of candidate solution natural selection evolve to a better solution

GA GA definitions&operators Genetic encoding.: chromosome code of CoMSSes. For example, CoMSSes is S = {1, 3, 4, 7}, then the corresponding binary genes are Fitness function.: f(S) = H S /|S|; (H S :number of sets hit by some of the elements in S, and |S| represents the cardinality of the set S.) Crossover operator.: S1 : {x1, x2, xk.,xm} ; S2:{y1, y2,yk.,ym}; the crossover offspring : S3 : {x1,.,xk,yk+1,.ym},S4 : {y1,.,yk,xk+1,.xm} Mutation operator.: S1 = {x1, x2,..., xm}, S2 = {si | if i = k, si = ￢ xi, else si = xi}. is the mutation offspring of S1 Inversion operator.: S1 = {x1,., xk, xk+1,., xl−1,xl,.., xm}, then S2= {x1,.., xl, xl−1,., xk+1, xk,..., xm}.is the inversion offspring of S1 Selection operator.: use fitness function. We choose hit more sets and have smaller cardinality than others

Greedy Genetic Algorithm algorithm proceeds as follows: 1. Find all complements of maximal satisfiable subformulas for a formula ϕ ; 2. Recode the clause number in CoMSSes to make the sets more compressed; 3. Adopt a greedy algorithm to derive two good near minimum unsatisfiable cores as the initial parents. We incrementally solve this problem, removing solutions as they are found to continue the search until all have been found; 4. Encode the sets of CoMSSes as binary vector and obtain the binary genes; 5. Apply the genetic operators, including crossover, mutation and inversion, to the selected parents, in order to give birth to the offspring; 6. Recombine the offspring and current parents to form a new population, and employ the selection operator to choose the best solutions as the new parents; 7. If the upper limit of iterations is reached, or the minimum unsatisfiable cores remain the same size for 10 successive offspring, stop and report the best solution found. Otherwise go to the step 5.

More example

(DP-complete problems are both NP-hard and coNP-hard). (DP-complete problems are both NP-hard and coNP-hard).