Semantic models for a logic of partial functions

2012

Specifications of programs frequently involve operators and functions that are not defined over all of their (syntactic) domains. Proofs about specifications–and those to discharge proof obligations that arise in justifying steps of design–must be based on formal rules. Since classical logic deals only with defined values, some extra thought is required. There are several ways of handling terms that can fail to denote a value—this paper provides a semantically based comparison of three of the best known approaches.

2012

Abstract It has been pointed out by a number of authors that partial terms (ie terms that can fail to denote a value) arise frequently in the specification and development of programs. Furthermore, earlier papers describe and argue for the use of a nonclassical logic (the" Logic of Partial Functions") to facilitate sound and convenient reasoning about such terms. This paper addresses some of the issues that arise in trying to provide (semi-) decision procedures-such as resolution-for such a logic.

Information Processing Letters, 1995

Science of Computer Programming, 1999

Partiality abounds in specifications and programs. We present a three-valued typed logic for reasoning equationally about programming in the presence of partial functions. The logic in essence is a combination of the equational logic E and typed LPF. Of course, there are already many logics in which some classical theorems acquire the status of neither-true-nor-false. What is distinctive here is that we preserve the equational reasoning style of E, as well as most of its main theorems. The principal losses among the theorems are the law of the excluded middle, the anti-symmetry of implication, a small complication in the trading law for existential quantification, and the requirement to show delinedness when using instantiation. The main loss among proof methods is proof by mutual implication; we present some new proof strategies that make up for this loss. Some proofs are longer than in E, but the heuristics commonly used in the proof methodology of E remain valid. We present a Hilbert-style axiomatisation of the logic in which modus ponens and generalisation are the only inference rules. The axiomatisation is easily modified to yield a classical axiomatisation of E itself. We suggest that the logic may be readily extended to a many-valued logic, and that this will have its uses.

2006

Partial functions and operators are used extensively in the formal development of programs and thus development methods have to clarify how to reason about them. There are a number of approaches which cover up the fact that “First Order Predicate Calculus” does not handle undefined logical values. There is also at least one specific “Logic of Partial Functions”(LPF) which tackles the issue at its root by using a weaker logic. Recently, we have come to realise that LPF fits a particular way of developing programs.

In this paper we analyse total correctness operation refinementon a partial relation semantics for specification. In particular weshow that three theories: a relational completion approach, a prooftheoreticapproach and a functional models approach, are all equivalent. This result holds whether or not preconditions are taken to be minimalor fixed conditions for establishing the postcondition. Keyword: Specification Language; Specification Logic; Refinement; 1

1989

This dissertation addresses the problem of incorporating into lazy higher-order functional programming the relational programming capability of Horn logic. The language design is based on set abstraction, a feature whose denotational semantics has until now not been rigorously defined. A novel approach is taken in constructing an operational semantics directly from the denotational description. The main results of this dissertation are: (i) Relative set abstraction can combine lazy higher-order functional programming with not only first-order Horn logic, but also with a useful subset of higherorder Horn logic. Sets, as well as functions, can be treated as first-class objects. (ii) Angelic powerdomains provide the semantic foundation for relative set abstraction. (iii) The computation rule appropriate for this language is a modified paralleloutermost, rather than the more familiar left-most rule. (iv) Optimizations incorporating ideas from narrowing and resolution greatly improve the...

Journal of Symbolic Logic, Vol. 37, 159-169, 1972

Introduction. In what follows there is presented a unified semantic treatment of certain "paradox-free" systems of entailment, including Church's weak theory of implication (Church [7D and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6D.1 We shall refer to these systems generally as relevant logics.

University of Essex, technical report …, 2002

Notre Dame Journal of Formal Logic

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality, and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.