Generation of Hard Non-Clausal Random Satisfiability Problems

We present the results from experiments with a new family of random formulas for the satisfiability problem. Our proposal is a generalization of the random k-SAT model that introduces non-clausal formulas and exhibits interesting features such as (experimentally observed) sharp phase transition and the easy-hard-easy pattern. The experimental results provide some insights on how the use of different clausal translations can affect the performance of satisfiability solving algorithms. We also expect our model to provide diverse and challenging benchmarks for developers of SAT procedures for non-clausal formulas

A New General Method to Generate Random Modal Formulae for Testing Decision Procedures

The recent emergence of heavily-optimized modal decision procedures has highlighted the key role of empirical testing in this domain. Unfortunately, the introduction of extensive empirical tests for modal logics is recent, and so far none of the proposed test generators is very satisfactory. To cope with this fact, we present a new random generation method that provides benefits over previous methods for generating empirical tests. It fixes and much generalizes one of the best-known methods, the random CNF_[]m test, allowing for generating a much wider variety of problems, covering in principle the whole input space. Our new method produces much more suitable test sets for the current generation of modal decision procedures. We analyze the features of the new method by means of an extensive collection of empirical tests

A New General Method to Generate Random Modal Formulae for Testing Decision Procedures

The recent emergence of heavily-optimized modal decision procedures has highlighted the key role of empirical testing in this domain. Unfortunately, the introduction of extensive empirical tests for modal logics is recent, and so far none of the proposed test generators is very satisfactory. To cope with this fact, we present a new random generation method that provides benefits over previous methods for generating empirical tests. It fixes and much generalizes one of the best-known methods, the random CNF_[]m test, allowing for generating a much wider variety of problems, covering in principle the whole input space. Our new method produces much more suitable test sets for the current generation of modal decision procedures. We analyze the features of the new method by means of an extensive collection of empirical tests

Why solutions can be hard to find: a featural theory of cost for a local search algorithm on random satisfiability instances

The local search algorithm WSat is one of the most successful algorithms for solving the archetypal NP-complete problem of satisfiability (SAT). It is notably effective at solving Random-3-SAT instances near the so-called 'satisfiability threshold', which are thought to be universally hard. However, WSat still shows a peak in search cost near the threshold and large variations in cost over different instances. Why are solutions to the threshold instances so hard to find using WSat? What features characterise threshold instances which make them difficult for WSat to solve? We make a number of significant contributions to the analysis of WSat on these high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are implicates of an instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We undertake a detailed study of the behaviour of the algorithm during search and uncover some interesting patterns. These patterns lead us to introduce a measure of the backbone fragility of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for WSat are those with large backbones which are also backbone-fragile. We suggest that the decay in cost for WSat beyond the satisfiability threshold, which has perplexed a number of researchers, is due to the decreasing backbone fragility. Our hypothesis makes three correct predictions. First, that a measure of the backbone robustness of an instance (the opposite to backbone fragility) is negatively correlated with the WSat cost when other factors are controlled for. Second, that backbone-minimal instances (which are 3-SAT instances altered so as to be more backbone-fragile) are unusually hard for WSat. Third, that the clauses most often unsatisfied during search are those whose deletion has the most effect on the backbone. Our analysis of WSat on random-3-SAT threshold instances can be seen as a featural theory of WSat cost, predicting features of cost behaviour from structural features of SAT instances. In this thesis, we also present some initial studies which investigate whether the scope of this featural theory can be broadened to other kinds of random SAT instance. random-2+p-SAT interpolates between the polynomial-time problem Random-2-SAT when p = 0 and Random-3-SAT when p = 1. At some value p ~ pq ~ 0.41, a dramatic change in the structural nature of instances is predicted by statistical mechanics methods, which may imply the appearance of backbone fragile instances. We tested NovELTY+, a recent variant of WSat, on rand o m- 2 +p-SAT and find some evidence that growth of its median cost changes from polynomial to superpolynomial between p = 0.3 and p = 0.5. We also find evidence that it is the onset of backbone fragility which is the cause of this change in cost scaling: typical instances at p — 0.5 are more backbone-fragile than their counterparts at p — 0.3. Not-All-Equal (NAE) 3-SAT is a variant of the SAT problem which is similar to it in most respects. However, for NAE 3-SAT instances no implicate literals are possible. Hence the backbone for NAE 3-SAT must be redefined. We show that under a redefinition of the backbone, the pattern of factors influencing WSat cost at the NAE Random-3-SAT threshold is much the same as in Random-3-SAT, including the role of backbone fragility

Proof Generation for CDCL Solvers Using Gauss-Jordan Elimination

Traditional Boolean satisfiability (SAT) solvers based on the conflict-driven clause-learning (CDCL) framework fare poorly on formulas involving large numbers of parity constraints. The CryptoMiniSat solver augments CDCL with Gauss-Jordan elimination to greatly improve performance on these formulas. Integrating the TBUDDY proof-generating BDD library into CryptoMiniSat enables it to generate unsatisfiability proofs when using Gauss-Jordan elimination. These proofs are compatible with standard, clausal proof frameworks.Comment: Presented at 2022 Workshop on the Pragmatics of SA

New results on rewrite-based satisfiability procedures

Program analysis and verification require decision procedures to reason on theories of data structures. Many problems can be reduced to the satisfiability of sets of ground literals in theory T. If a sound and complete inference system for first-order logic is guaranteed to terminate on T-satisfiability problems, any theorem-proving strategy with that system and a fair search plan is a T-satisfiability procedure. We prove termination of a rewrite-based first-order engine on the theories of records, integer offsets, integer offsets modulo and lists. We give a modularity theorem stating sufficient conditions for termination on a combinations of theories, given termination on each. The above theories, as well as others, satisfy these conditions. We introduce several sets of benchmarks on these theories and their combinations, including both parametric synthetic benchmarks to test scalability, and real-world problems to test performances on huge sets of literals. We compare the rewrite-based theorem prover E with the validity checkers CVC and CVC Lite. Contrary to the folklore that a general-purpose prover cannot compete with reasoners with built-in theories, the experiments are overall favorable to the theorem prover, showing that not only the rewriting approach is elegant and conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page