The Expressive Power of Ad-Hoc Constraints for Modelling CSPs

Abstract

Ad-hoc constraints (also called generic constraints) are important for modelling Constraint Satisfaction Problems (CSPs). Many representations have been proposed to define ad-hoc constraints, such as tables, decision diagrams, binary constraint trees, automata and context-free grammars. However, prior works mainly focus on efficient Generalized Arc Consistency (GAC) propagators of ad-hoc constraints using the representations. In this paper, we ask a more fundamental question which bears on modelling constraints in a CSP as ad-hoc constraints, how the choice of constraints and operations affect tractability. Rather than ad-hoc constraints and their GAC propagators, our focus is on their expressive power in terms of succinctness (polysize) and cost of operations/queries (polytime). We use a large set of constraint families to investigate the expressive power of 14 existing ad-hoc constraints. We show a complete map of the succinctness of the ad-hoc constraints. We also present results on the tractability of applying various operations and queries on the ad-hoc constraints. Finally, we give case studies illustrating how our results can be useful for questions in the modelling of CSPs.

Encoding Constraints as Binary Constraint Networks Satisfying BTP

“…Except for the smart table and segmented table constraints, the CFG constraint is more succinct than the other 11 ad-hoc constraints discussed in (Wang and Yap 2023). Therefore, BTPN is strictly more succinct than all 14 ad-hoc constraints discussed in (Wang and Yap 2023).…”

“…Then we can construct a CN P T = (X, C T ) such that sol(P ) = sol(P T ) and C T = {c B |B is a bag in T } where scp(c B ) = B and rel(c B ) = sol(P ) [B]. (Dechter and Pearl 1989;Wang and Yap 2023;Kučera 2023) showed that P T can be encoded as polysize BCT, thus, we cannot directly define a strictly more succinct constraint representation with existing tractable structural classes.…”

“…Table 1 shows that BTPN is strictly more succinct than the DNNF constraint and all 14 ad-hoc constraints discussed in (Wang and Yap 2023), namely the table (Wang et al 2016;Demeulenaere et al 2016), regular (Pesant 2004), nondeterministic finite automaton (NFA) (Quimper and Walsh 2006), context free grammar (CFG) (Quimper and Walsh 2006), c-table (Katsirelos and Walsh 2007), MDD (Cheng and Yap 2010), short table (Jefferson and Nightingale 2013), sliced table (Gharbi et al 2014) In addition, some special purpose global constraints are incomparable to the BTPN constraint, such as the permutation, AllDifferent (Régin 1994), GCC (Régin 1996), NValue (Pachet and Roy 1999), channelling (Cheng et al 1999), cycle, circuit (Beldiceanu and Contejean 1994), linear (Yuanlin and Yap 2000) and knapsack (Trick 2003) constraints. Theorem 2.…”

“…In the rest of this section, we give the proof of Theorem 2. The context free grammar, smart table and segmented table constraints are incomparable to each other (Wang and Yap 2023), thus, we only need to prove BTPN is more succinct than them. In addition, for the special purpose global constraints, if they cannot be encoded as polysize BTPN, then BTPN is incomparable to them, as they cannot be used to encode arbitrary constraint relations.…”

“…To be specific, we prove that the consistency checker (Bessiere et al 2009) of the BTPN constraint can be computed by polysize monotone circuit, thus, some global constraints cannot be encoded as polysize BTPN, such as Circuit (Beldiceanu and Contejean 1994), Channelling (Cheng et al 1999) and Linear (Yuanlin and Yap 2000) constraints. Meanwhile, we introduce a binary encoding to transform the DNNF constraint (Gange and Stuckey 2012) into polysize BTPN, and show that BTPN is strictly more succinct than the DNNF constraint and all 14 ad-hoc constraints discussed in (Wang and Yap 2023), such as the context-free grammar (Quimper and Walsh 2006), BCT and smart table (Mairy, Deville, and Lecoutre 2015) constraints. Moreover, we prove that it is NP-hard to minimizing the size of the BTPN encoding a constraint.…”