SAT Encodings of Pseudo-Boolean Constraints with At-Most-One Relations

Savile Row Manual

“…For example, the following is a matrix literal containing four integer literals. [ 1,3,2,4 ] Matrix literals may contain any valid expression in ESSENCE PRIME. For example a matrix literal is allowed to contain other matrix literals to build up a matrix with two or more dimensions.…”

“…Finally a multi-dimensional matrix value can be expressed by nesting matrix literals. Suppose we have the following domain: matrix indexed by [int(-2..0),int (1,2,4)] of int (1..5) One value contained in this domain is the following:…”

“…There is a choice of 5 encodings of pseudo-Boolean and other sum constraints, as described in Section 4.5.6. The MDD, GPW, SWC, and GGT encodings are implemented exactly as described in Bofill et al [4]. Sum-equal constraints are split into sum-≤ and sum-≥ before encoding.…”

“…Four of them are based on Bofill et al [4] (with thanks to Jordi Coll for providing the implementations): Multi-valued Decision Diagrams (MDD); the Global Polynomial Watchdog (GPW); the Sequential Weighted Counter (SWC); and the Generalized Generalized Totalizer (GGT). Each of these is implemented exactly as described in [4]. For these four encodings, any pseudo-Boolean equality constraints are decomposed into ≤ and ≥.…”

SAT Encodings for Pseudo-Boolean Constraints Together With At-Most-One Constraints

“…Consider a non-leaf node O (with children L and R), and digit h. The carry γ h O depends on values of L h and R h . The B 2 : values of the digit O h (represented in MTO with variables o h σ ) depend on L h and R h as well as the carry γ h O (see clause (22)) and as a result the NNF terms corresponding to variables o h σ may not be decomposable. As a concrete example, suppose λ h = 3 and there is no carry-in.…”