Order-sorted Equality Enrichments Modulo Axioms

Gutiérrez, Raúl; Meseguer, José; Rocha, Camilo

Permalink

https://hdl.handle.net/2142/28597 Copy

Description

Title

Order-sorted Equality Enrichments Modulo Axioms

Author(s)

• Gutiérrez, Raúl
• Meseguer, José
• Rocha, Camilo

Issue Date

2011-12-22

Keyword(s)

• equality predicate
• rewriting logic
• equality enrichment
• Maude
• abstract data types

Abstract

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to \emph{effectively} define a theory transformation $\mathcal{E} \mapsto \mathcal{E}^{\simeq}$ that extends an algebraic specification $\mathcal{E}$ to a specification $\mathcal{E}^{\simeq}$ where equationally-defined equality predicates have been added? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications $\mathcal{E}$ that are sort-decreasing, ground confluent, and operationally terminating modulo axioms $B$ and have subsignature of constructors. The axioms $B$ can consist of associativity, or commutativity, or associativity-commutativity axioms, so that the constructors are \emph{free modulo} $B$. We prove that the transformation $\mathcal{E} \mapsto \mathcal{E}^{\simeq}$ preserves all the just-mentioned properties of $\mathcal{E}$. The transformation has been automated in Maude using reflection and it is used in Maude formal tools.

Type of Resource

text

Language

en

Permalink

http://hdl.handle.net/2142/28597

Owning Collections