Completeness of many-sorted equational logic

Abstract

The rules of deduction which are usually used for many-sorted equational logic in computer science, for example in the study of abstract data types, are not sound. Correcting these rules by introducing explicit quantifiers yields a system which, although it is sound, is not complete; some new rules are needed for the addition and deletion of quantifiers. This note is intended as an informal, but precise, introduction to the main issues and results. It gives an example showing the unsoundness of the usual rules; it also gives a completeness theorem for our new rules, and gives necessary and sufficient conditions for the old rules to agree with the new.

Casl Reference Manual

“…total function symbol, and a predicate to each predicate symbol. Requiring carriers to be non-empty simplifies deduction [21] and will allow axioms in specifications to be implicitly universally quantified, see Sect. 2.4.…”

Recent SRI work in verification

“…Reference [11] exhibits a small but vicious flaw in the usual deductive system for many-sorted equational logic, as used for exampl e in most work on abstract data types ; rather shockingly, this system is not sound . Not only have we given a new axiom system which is Sound and complete, but we have found sufficient conditions for the old deductive system to be valid as well .…”

Fundamentals of Algebraic Specification 1: Equations and Initial Semantics (H. Ehrig and B. Mahr)

“…FAS1 does not support this statement, but instead refers to a report by Loeckx and Mahr [13] which contains an argument based on the claim that Lemma 75 in I14] is false. However, this claim is not supported in [13], and actually it is false, since Lemma 75 and the other results in [5] and [14] hold as stated.…”

“…The treatment ofBirkhoff's theorem in FAS1 follows the precautions for validity that were suggested in [5], but the reader should be aware that in general, as noted on pp. 325 and 326 of [5], varieties are only closed under filtered colimits if the equations quantify over finitely many variables. The statement that "a subalgebra of a free algebra is again a free algebra" in 4.15 of FAS1 is not correct.…”

“…(2) Page 137 states, regarding the work of J.A. Goguen and J. Meseguer on many-sorted equational deduction (e.g., [5], 14]) that "most of their proposals are misleading or incorrect." FAS1 does not support this statement, but instead refers to a report by Loeckx and Mahr [13] which contains an argument based on the claim that Lemma 75 in I14] is false.…”