text: Constraints over finite sequences of variables are ubiquitous in sequencing and timetabling. Moreover, the wide variety of such constraints in practical applications led to general modelling techniques and generic propagation algorithms, often based on deterministic finite automata (DFA) and their extensions. We consider counter-DFAs (cDFA), which provide concise models for regular counting constraints, that is constraints over the number of times a regular-language pattern occurs in a sequence. We show how to enforce domain consistency in polynomial time for atmost and atleast regular counting constraints based on the frequent case of a cDFA with only accepting states and a single counter that can be incremented by transitions. We also prove that the satisfaction of exact regular counting constraints is NP-hard and indicate that an incomplete algorithm for exact regular counting constraints is faster and provides more pruning than the existing propagator from [3]. Regular counting constraints are closely related to the CostRegular constraint but contribute both a natural abstraction and some computational advantages.
  @cite_9: We revisit the SeqBin constraint. This meta-constraint subsumes a number of important global constraints like Change, Smooth and IncreasingNValue. We show that the previously proposed filtering algorithm for SeqBin has two drawbacks even under strong restrictions: it does not detect bounds disentailment and it is not idempotent. We identify the cause for these problems, and propose a new propagator that overcomes both issues. Our algorithm is based on a connection to the problem of finding a path of a given cost in a restricted @math -partite graph. Our propagator enforces domain consistency in O(nd^2) and, for special cases of SeqBin that include Change, Smooth and IncreasingNValue, in O(nd) time.
  @cite_1: This paper introduces the SEQ BIN meta-constraint with a polytime algorithm achieving generalized arc-consistency. SEQ BIN can be used for encoding counting constraints such as CHANGE, SMOOTH, or INCREASING NVALUE. For all of them the time and space complexity is linear in the sum of domain sizes, which improves or equals the best known results of the literature.
summary: The @math constraint @cite_1 @cite_9 can be represented by a regular counting constraint, but it would require non-unary signature constraints.