Proving Ground Confluence of Equational Specifications Modulo Axioms

Abstract

Abstract. Terminating functional programs should be deterministic, i.e., should evaluate to a unique result, regardless of the evaluation order. For equational functional programs such determinism is exactly captured by the ground confluence property. For terminating equations this is equivalent to ground local confluence, which follows from local confluence. Checking local confluence by computing critical pairs is the standard way to check ground confluence. The problem is that some perfectly reasonable equational programs are not locally confluent and it can be very hard or even impossible to make them so by adding more equations. We propose a three-step strategy to prove that an equational program as is is ground confluent: First: apply the strategy proposed in [8] to use non-joinable critical pairs as completion hints to either achieve local confluence or reduce the number of critical pairs. Second: use the inductive inference system proposed in this paper to prove the remaining critical pairs ground joinable. Third: to show ground confluence of the original specification, prove also ground joinable the equations added. These methods apply to order-sorted and possibly conditional equational programs modulo axioms such as, e.g., Maude functional modules.

Programming and symbolic computation in Maude

“…Rather than doing so ourselves, we encourage the reader, particularly if not yet acquainted with Maude, to do it him/herself in order to have some fun playing with Maude and get a taste for how one can specify finitary set theory operations in Maude just as one would like to do it out of a textbook. Also, for a taste of how one can specify finitary set theory (the so-called hereditarily finite sets) in a single Maude functional module we refer the reader to [51].…”

“…We can check that it is FVP by computing the variants of its function symbols. For example, the generation of variants for the following terms all stop with a finite number of variants: As shown in [51], this FVP example (borrowed from [51]) can be easily extended to an even richer FVP example INT-FVP where all the above functions (except monus, which is superseded by actual integer difference using unary minus and +) are extended to the integers, and an absolute value function on integers is added.…”