Set unification

Abstract

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas-e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes.The algorithms presented are partly drawn from the literature-and properly revisited and analyzed-and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI 1 unification and a new simpler algorithm for general (Ab)(C ℓ) unification.

Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

“…In the case that the top symbol is env, the rules for multi-equations apply, i.e., the rules for env * . Using the properties of the equational theory LC and the considerations in [5]), we see that the rules are complete.…”

“…The following facts about the theory LC can easily be verified: • The equational theory LC has a finitary and decidable unification problem (see [5,4]).…”

“…To model the multi-set property of letrec-environments, i.e., that bindings can be reordered, we use the equational theory left-commutativity (LC) with the following axiom: env(x, env(y, z)) = env(y, env(x, z)) (for the LC-theory and unification modulo LC see [5,4]). The equational theory LC is a congruence relation on the terms, which is denoted as = LC .…”

Graph Mining for Automatic Classification of Logical Proofs

“…In other words, each subgraph is an instance of one generalized subgraph. We can see that the common set unification rules (Dovier et al, 2001) cannot be used here. In this work we focused on generalized subgraphs that consist of three nodes, two parents and their child.…”

javaset

“…. .and (cn) r.eq(s); // equality between sets t.union(r,s); // t = r ∪ s x.eq(y).and(x.eq(3)).and(y.neq(z)) // x = y ∧ x = 3 ∧ y = z Note that solving an equality constraint implies the ability to solve a set unification problem (cf., e.g., [7]). Set unification of two (possibly partially specified) sets s and r means finding an assignment of values to uninitialized variables occurring in them (if any), such that s and r become equal in the underlying set theory.…”