Complete Propagation Rules for Lexicographic Order Constraints over Arbitrary Domains

Abstract

Abstract. We give an efficiently executable specification of the global constraint of lexicographic order in the Constraint Handling Rules (CHR) language. In contrast to previous approaches, the implementation is short and concise without giving up on the best known worst case time complexity. It is incremental and concurrent by nature of CHR. It is provably correct and confluent. It is independent of the underlying constraint system, and therefore not restricted to finite domains. We have found a direct recursive decomposition of the problem. We also show completeness of constraint propagation, i.e. that all possible logical consequences of the constraint are generated by the implementation. Finally, we report about some practical implementation experiments.

As time goes by: Constraint Handling Rules

“…Lexicographic order. Frühwirth (2006a) presented a constraint solver for a lexicographic order constraint in terms of inequality constraints offered by the underlying solver. The approach is general in that it can be used for any constraint domain offering inequality (less-than) constraints between problem variables.…”

Constructing Rule-Based Solvers for Intentionally-Defined Constraints

“…Consider the lexicographic order constraint [2][3][4]. Given two sequences L 1 and L 2 of variables of the same length, then lex holds if L 1 is lexicographically smaller than or equal to L 2 .…”

“…By comparison we find that the backward reasoning rules are not subsumed by our combined lex solver rendering the solver incomplete. In the case of the lex solver of [4], rules (8), (7), (10), and (9) are applied in that order and further constrain the domains of the variables to: …”