Congruence Closure Modulo Associativity and Commutativity

Uniform Derivation of Decision Procedures by Superposition

“…For instance, [BT00] focus on abstracting the control of congruence closure algorithms, in order to give a uniform presentation of several known algorithms. A recent extension to deal with equality modulo AC is presented in [BRTV00].…”

Rewrite Rule Inference Using Equality Saturation

“…This approach is generally fast, but complicates support for cyclic or expansive rules, and makes both rewriting performance and output quality dependent on fine-grained rule orderings. Past work has extensively investigated how to mitigate these challenges by scheduling rules [Barendregt et al 1987;Borovanskỳ et al 1998;Dershowitz 1982;Knuth and Bendix 1983], special casing cyclic and expansive rules [Bachmair et al 2000;Dershowitz 1987;Eker 2003;Lucas 2001], and efficiently implementing rewrite rule-based search [Clavel et al 2007;Kirchner 2015;Visser 2001a,b]. Many systems still rely on ad hoc rule orderings and heuristic mitigations developed through trial and error, though recent work [Newcomb et al 2020] has demonstrated how reduction orders [Baader and Nipkow 1998] can be automatically synthesized and then used to effectively guide destructive term rewriting systems.…”

Hacking nondeterminism with induction and coinduction

“…On the other hand, Naive(x, y) returns false as soon as it finds a word which is accepted by one state and not the other. (1) R is empty; todo is empty; (2) insert (x, y) in todo; …”

“…Note that many algorithms were proposed in the literature to compute the congruence closure of a relation (see, e.g., Nelson and Oppen, 18 Shostak, 23 and Bachmair et al 2 ). However, they usually consider uninterpreted symbols or associative and commutative symbols, but not associative, commutative, and idempotent symbols, which is what we need here.…”