Generating Diverse Solutions in SAT: Paper Addendum

This document is an addendum to [1]. We complement Section 4 of [1] in two ways. First, a detailed analysis of the randomized algorithms is provided. Second, an explanation of the behavior of the quality functions of pguide and pbcpguide 100 is proposed. The reader should be familiar with the content of [1] up to and including Section 4. 1 Analyzing Randomized Algorithms This section complements the analysis presented in Section 4 of [1] by analyzing the behavior of three randomized algorithms and comparing them to prand. dpll-based sampling invokes the SAT solver k times to generate k models for the same input formula. The first assignment to a variable is random for each invocation of the SAT solver. dpll-based sampling was mentioned in [2], but we did not find any reference to work introducing it. xor-sample [3] invokes the SAT solver at least k times to generate k models. For each invocation, the initial formula is augmented with random XOR constraints, where an XOR constraint includes variables and, optionally, the constant 1. Adding an XOR constraint means enforcing that an odd number of elements in the constraint are satisfied. In the original definition, a variable is added to the XOR constraint with probability q and 1 is added with probability 1/2. One can also use XOR sampling, where the length and number of XOR constraints are predefined. prandweak [4] (AllSAT-Sampling in [4]) is a compact DiversekSet algorithm. It randomizes the polarity of a new decision variable only when a variable is selected for the first time or for the first time after a model. Let us analyze the behavior of the randomized algorithms for DiversekSet, including prand, on our instances. Table 1 summarizes their behavior for a selected number of models and Fig. 1 presents their behavior as a function of the number of models. In our experiments, we used 4 versions of xor-sample, each one generating either 100 or 10 XOR constraints of length of 100 or 10. We generated the XOR constraints and translated them to clauses using the utilities of [3]. Note that additional variables must be introduced to translate XOR constraints to clauses. Compare the summary of the behavior of xor-sample with that Table 1: Mean quality and mean run-time for randomized DiversekSet algorithms, given 100, 50 and 10 models on 66 benchmarks from semiformal verification of hardware. The algorithms are sorted by the quality obtained when generating 100 models. 100 50 10 Algorithm Quality Time Quality Time Quality Time prand 0.1923 225 0.1915 206 0.1903 186 dpll-based sampling 0.1608 3988 0.1609 1992 0.1617 402 xor-sample-100-100 0.1487 5067 0.1485 2559 0.1482 520 xor-sample-100-10 0.1376 6855 0.1373 3430 0.1371 651 xor-sample-10-100 0.1056 4117 0.1052 2059 0.1062 424 prandweak 0.09991 76 0.09405 55 0.08332 39 xor-sample-10-10 0.08896 4126 0.08894 2076 0.08976 424