Relative Hilbert-Post Completeness for Exceptions

Abstract

A theory is complete if it does not contain a contradiction, while all of its proper extensions do. In this paper, first we introduce a relative notion of syntactic completeness; then we prove that adding exceptions to a programming language can be done in such a way that the completeness of the language is not made worse. These proofs are formalized in a logical system which is close to the usual syntax for exceptions, and they have been checked with the proof assistant Coq.

IMP with exceptions over decorated logic

“…This paper builds upon several papers by Domínguez and Duval (2010), Dumas et al (2014a), Dumas et al (2015), Dumas et al (2014b), Dumas et al (2012) and Dumas et al (2014c) . The novel points presented here can be itemized as follows:…”

“…Similar to the ones of the logic L st , the rules of the logic L exc also designed to be sound with respect to a categorical model which is detailed in (Ekici, 2015, §6.2, §6.3, §6.4, §6.5). In (Dumas et al (2015)), we prove that this set of rules is complete with respect to the notion of relative Hilbert-Post completeness.…”

“…However, their syntactic completeness is not immediate. Dumas et al (2015) defines a new syntactic completeness property, subsuming a consistency check, called the relative Hilbert-Post completeness. In (Ekici, 2015, §5.4), it is proven that this set of rules is complete with due respect.…”

“…In a way, it is meant to make sure that we are not using too many axioms to construct a denotational semantics for the IMP+Exc language using the logic L st+exc as the target language. This syntactic completeness property is called relative Hilbert-Post Completeness (rHPC) and elaborately defined by Dumas et al (2015). Briefly, given two logics L 0 and L such that L 0 ⊆ L (L 0 is a sub-logic of L) and a theory T of L. T is relatively Hilbert-Post complete with respect to L 0 if (1) at least one sentence is unprovable in T (not the maximal theory ensuring consistency), and (2) every theory containing T can be generated from T and some sentences from L 0 .…”