Nogood learning is a powerful approach to reducing search in Constraint Programming (CP) solvers. The current state of the art, called Lazy Clause Generation (LCG), uses resolution to derive nogoods expressing the reasons for each search failure. Such nogoods can prune other parts of the search tree, producing exponential speedups on a wide variety of problems. Nogood learning solvers can be seen as resolution proof systems. The stronger the proof system, the faster it can solve a CP problem. It has recently been shown that the proof system used in LCG is at least as strong as general resolution. However, stronger proof systems such as \emph{extended resolution} exist. Extended resolution allows for literals expressing arbitrary logical concepts over existing variables to be introduced and can allow exponentially smaller proofs than general resolution. The primary problem in using extended resolution is to figure out exactly which literals are useful to introduce. In this paper, we show that we can use the structural information contained in a CP model in order to introduce useful literals, and that this can translate into significant speedups on a range of problems.

Abstract We tackle the problem of enumerating set-theoretic solutions to the Yang-Baxter equation. This equation originates from statistical and quantum mechanics, but also has applications in knot theory, cryptography, quantum computation and group theory. Non-degenerate, involutive solutions have been enumerated for sets up to size 10 using constraint programming with partial static symmetry breaking [1]; for general non-involutive solutions, a similar approach was used to enumerate solutions for sets up to size 8. In this paper, we use and extend the SAT Modulo Symmetries framework (SMS), to expand the boundaries for which solutions are known. The SMS framework relies on a minimality check ; we present two solutions to this, one that stays close to the original one designed for enumerating graphs and a new incremental, SAT-based approach. With our new method, we can reproduce previously known results much faster and also report on results for sizes that have remained out of reach so far.