Termination Modulo Combinations of Equational Theories

Abstract

Abstract. Rewriting with rules R modulo axioms E is a widely used technique in both rule-based programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativity-commutativity) are known. However, much less seems to be known about termination methods that can be modular in the set E of axioms. In fact, current termination tools and proof methods cannot be applied to commonly occurring combinations of axioms that fall outside their scope. This work proposes a modular termination proof method based on semantics-and termination-preserving transformations that can reduce the proof of termination of rules R modulo E to an equivalent proof of termination of the transformed rules modulo a typically much simpler set B of axioms. Our method is based on the notion of variants of a term recently proposed by Comon and Delaune. We illustrate its practical usefulness by considering the very common case in which E is an arbitrary combination of associativity, commutativity, left-and right-identity axioms for various function symbols.

Folding Variant Narrowing and Optimal Variant Termination

“…Besides yielding in particular a new finitary unification algorithm for FVP equational theories that improves upon the variant algorithm presented in [9], and does not require anymore prior checking of FVP as described in [8], by being applicable to any equational theory modulo under minimal assumptions of confluence, termination, and coherence, many more applications than just cryptographic protocol analysis modulo algebraic properties in the style of the Maude-NPA [6] are opened up. In fact, several such applications, to termination methods modulo axioms [5], and to the most recent Maude CRC and ChC tools modulo axioms (see http://maude.lcc.uma.es/CRChC/), are already exploiting the general power of folding variant narrowing.…”

Strong and Weak Operational Termination of Order-Sorted Rewrite Theories

“…Associativity and commutativity axioms satisfy this requirement, which can even be made to work for identity axioms by perfoming the semantics-preserving transformation described in [3]. Now we can give the main result of this section.…”

“…For a trivial example, consider the single conditional rewrite rule a → b ⇐ a → c. Since the rewrite relation defined by this conditional rule is the empty set, the constant a is trivially irreducible; but the proof tree associated to the normalization of a using the CTRS inference system is infinite [7], and a rewrite engine that tries to evaluate a will loop when trying to satisfy the rule's condition. 3 Therefore, calling a a normal form is a very bad joke, since, intuitively, a term is considered to be a normal form if it is "fully normalized," that is, if it is the result of fully evaluating some input term by rewriting. Our answer to this puzzle is to introduce a precise distinction (fully articulated in the paper) between irreducible terms and normal forms: every normal form is irreducible, but, as the above example shows, not every irreducible term is a normal form.…”

Incremental checking of well-founded recursive specifications modulo axioms

“…The extension of our method to this more general case is accomplished by an automatic theory transformation (Σ, B, R) → (Σ, B 0 , R ∪ ∆) such that: (i) B 0 only involves commutativity and associativity-commutativity axioms; (ii) the theories R ∪ B and B 0 ∪ R ∪ ∆ are semantically equivalent (as inductive theories, see below); and (iii) (Σ, B, R) is confluent, terminating, and sufficiently complete for Ω modulo B iff (Σ, B 0 , R ∪ ∆) has the same properties modulo B 0 . Here we summarize and extend the basic ideas of the transformation and refer to [9] for further details.…”

“…For the first transformation (Σ, B, R) → (Σ, B 1 , R 1 ∪ ∆ 1 ) we are always guaranteed that the set of rules R 1 is finite if R is (see [9]). However, for the second transformation (Σ, B 1 , R 1 ∪ ∆ 1 ) → (Σ, B 0 , R ∪ ∆), which removes associative but not commutative axioms from B 1 , we cannot in general guarantee that (Σ, B 0 , R ∪ ∆) is a finite theory.…”