Exploiting Circuit Reconvergence through Static Learning in CNF SAT Solvers

Most contemporary SAT solvers use a Conjunctive- Normal-Form (CNF) representation for logic functions due to the availability of efficient algorithms for this form, such as deduction through unit propagation and conflict driven learning using clause resolution. The use of CNF gen- erally entails transformation to this form from other rep- resentations such as logic circuits [23]. However, this transformation results in loss of information such as di- rection of signal flow and observability of signals at cir- cuit outputs [12][13]. This has prompted the development of various circuit-based solvers [2][14][17][22], hybrid CNF+circuit-based solvers[13], as well as augmented CNF solvers [12]. Having the circuit available provides for addi- tional capabilities at a cost, and thus requires careful anal- ysis to determine the viability of each approach. This paper highlights one specific capability provided by a circuit: the ability to consider reconvergent paths in unit propagation. Unit propagation is the workhorse of contemporary SAT solvers, thus any improvement to this has significant practical potential. We first demonstrate that the Tseitin circuit-to-CNF transformation limits back- ward unit propagation and how additional implications can be derived when unit propagation across multiple paths is considered. Next, we show how these implications can be exploited by statically learning clauses during circuit pre-processing. The results of the practical implementa- tion of these algorithms show that the static learning can provide significant speed-up on several classes of bench- mark circuits. Finally, we discuss how this work com- pares with other circuit-based approaches, especially those arising from the automatic-test-pattern-generation (ATPG) community (e.g. recursive learning) and circuit and non- circuit based pre-processors.