Jingchao Chen
School of Informatics, Donghua University
2999 North Renmin Road, Songjiang District, Shanghai 201620, P. R. China [email protected]
Abstract—StocBCD is a simple stochastic local search (SLS) solver. This solver sets an initial assignment of the input formula according to blocked sets obtained blocked clause decomposition (BCD). Its search is based on probability distributions similar to probSAT [2]. In addition, we speed up the search progress by block flipping heuristic.
I. INTRODUCTION
The StocBCD solver consists mainly of three phases such as initialization, local search, block flipping. The initialization phase sets an initial assignment for the formula to be solved. The local search phase flips a variable randomly selected from unsatisfied clauses according to a probability distribution. The local search is trapped often in a deep local minima. To escape from such a local minima, we set up the block flipping phase.
II. INITIALIZATION TECHNIQUES
Many stochastic local search (SLS) solvers start with a random assignment. However, StocBCD does not like this. It sets up an assignment in the order of blocked clauses. The notion about blocked clauses may be described as follows.
Given a CNF formula F , a clause C, a literal l ∈ C is said to block C w.r.t. F if (i) C is a tautology w.r.t. l, or (ii) for each clause C′ ∈ F with ¯l ∈ C′, C′⊗
lC is a tautology. In
this case, the clause C is called a blocked clause.
Any CNF formula can be decomposed into two blocked subsets such that both can be eliminated into a empty set by blocked clause elimination (BCE). On the other hand, any blocked set is satisfiable. Based on the two properties, we decide to use blocked clause decomposition (BCD) technique to do the initialization. At first, we decompose the formula into two blocked subsets: large subset L and small subset S, which is done by the LessInterfereBCD algorithm [1] recently proposed. Then we append S to L. In the reverse order of clause elimination obtained by BCE, we generate a number for each blocked clause. The number of the clause eliminated before is larger than the number of one eliminated after. The number of a clause in L precedes that of a clause in S. Under such assumption, let L ∧ S = C1∧ C2∧ · · · ∧ Cn, where
Ci (i = 1, 2, . . . , n) is a clause. According to this order of
clauses, an initial assignment (which is assumed to be stored in array value) is set as follows.
Algorithm initialize assignment for i = 1 to n do
for each literal x in clause Ci do
if value[x] is undefined then value[x] ← 1
III. LOCAL SEARCH TECHNIQUES
Here we recall briefly the local search technique of the probSAT solver [3]. ProbSAT is a SLS solver that uses only the break values of a variable. It flips variable v according to probability ∑ f (v,a)
x∈UCf (x,a)
, where a denotes the current assignment, UC is the set of unsatisfiable clauses, and f(x, a) is a probability distribution function, which is defined as an exponential or a polynomial shape as shown below.
f (x, a) = (cb)−break(x,a)
f (x, a) = (ϵ + break(x, a))−cb
where break(x, a) denotes the number of clauses which be- come false by flipping variable x under the current assignment a. StocBCD is also based on probability ∑ f (v,a)
x∈UCf (x,a)
to flip variable v. However, StocBCD uses different f(x, a). In details, its f(x, a) is defined as
f (x, a) = (cb)−break(x,a)× blocking(x)
f (x, a) = (ϵ + break(x, a))−cb× blocking(x)
where
blocking(x) =
1 xis not a blocking literal 0.92 x∈ L and x ∈ S
1.2 x∈ L or x ∈ S
Function blocking(x) is actually a weight function, which is set to larger value if x is a blocking literal in only one blocked subset, and smaller value if x is a blocking literal in both blocked subsets. This may be based on the fact that the blocking literal that occurs in only one blocked subset should be a critical literal, and more important than the other literals.
IV. PARAMETER DYNAMIC SETTINGS
Like probSAT, StocBCD sets ϵ to 0.9. However, parameter cb setting is different. ProbSAT sets up statically parameter
cb, while stocBCD does dynamically it. The following table
s cb[0] cb[t] ϵ 3 2.30 2.3-0.085t 0.9 4 2.95 3.05-0.0625t - 5 3.88 3.95-0.075t - 6 4.831 4.831+0.092t - 7 5.825 6.2-0.205t - ≥ 8 5.4 5.8-0.15t -
where s is the size of the longest clause in CNF formula F . Let #flip denote the total number of flipping. When #flip < 5× 108, parameter c
b is set to cb[0]. Otherwise, cb is set to
cb[t], where t = 2#f lip×108 mod 4. The dynamic change period is
4.
V. BLOCK FLIPPING HEURISTIC
Here we present a block flipping heuristic to escape from local minima. When the total number of unsatisfied clauses is one, we generate n bit patterns every 12 variables. In general, n ≤ 100. Let D(a, k, p) denote the difference between the number of unsatisfied clauses under assignment a and the number of unsatisfied clauses when the value of the (k + i)-th variable become the i-th bit value of p, where 1 ≤ i ≤ 12, and the values of the remaining variables are consistent with assignment a. The algorithm for generating n closest bit patterns may be described as follows.
Algorithm build bit pattern for i = 0 to #var
12 do
for p = 0 to 4096 do
if p < n then pat[i][p] ← p
else if D(a, 12i, p) < max{D(a, 12i, pat[i][j])} then Let D(a, 12i, pat[i][k]) is max
pat[i][k]← p
Using pattern pat, every 3 × 107 flips, we carry out block
flipping according to the following algorithm.
Algorithm block flip(k) for i = 0 to #var
12 do
p← pat[i][k] for j = 0 to 12 do
if value[v(12i+j)]̸= j-th bit of p
then flip variable v(12i+j) VI. CONCLUSIONS
StocBCD adopted a few new heuristics, including BCD- based initialization, BCD-based probability distribution com- putation, block flipping, dynamic parameter setting etc. Based on our experimental observation, these new techniques were efficient. However, it is not clear whether they are suitable for the benchmarks of the SAT competition 2016.
REFERENCES
[1] J.C. Chen: Fast Blocked Clause Decomposition with High Quality, 2015, http://arxiv.org/abs/1507.00459
[2] A. Balint, U. Schning: Choosing Probability Distributions for Stochastic Local Search and the Role of Make versus Break Lecture Notes in Computer Science, 2012, Volume 7317, Theory and Applications of Satisfiability Testing - SAT 2012, pp. 16-29
[3] A. Balint, U. Schning: ProbSAT and pprobSAT, Proceedings of the SAT Competition 2014, pp. 37–38.