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Operational semantics in a natural deduction setting

from Representing Formal Systems

Published online by Cambridge University Press:  04 August 2010

By
Rod Burstall  and
Furio Honsell
Edited by
Gerard Huet  and
G. Plotkin
Rod Burstall
Affiliation:
University of Edinburgh
Furio Honsell
Affiliation:
University of Udine
Gerard Huet
Affiliation:
Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt
G. Plotkin
Affiliation:
University of Edinburgh
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Summary

Abstract

We show how Natural Deduction extended with two replacement operators can provide a framework for defining programming languages, a framework which is more expressive than the usual Operational Semantics presentation in that it permits hypothetical premises. This allows us to do without an explicit environment and store. Instead we use the hypothetical premises to make assumptions about the values of variables. We define the extended Natural Deduction logic using the Edinburgh Logical Framework.

Introduction

The Edinburgh Logical Framework (ELF) provides a formalism for defining Natural Deduction style logics. Natural Deduction is rather more powerful than the notation which is commonly used to define programming languages in “inference-style” Operational Semantics, following Plotkin and others, for example Kahn. So one may ask

“Can a Natural Deduction style be used with advantage to define programming languages?”.

We show here that, with a slight extension, it can, and hence that the ELF can be used as a formal meta-language for defining programming languages. However ELF employs the “judgements as types” paradigm and takes the form of a typed lambda calculus with dependent types. We do not need all this power here, and in this paper we present a slight extension of Natural Deduction as a semantic notation for programming language definition. This extension can itself be defined in ELF.

The inspiration for using a meta-logic for Natural Deduction proofs comes from Martin-Löf.

Type
Chapter
Information
Logical Frameworks , pp. 185 - 214
Publisher: Cambridge University Press
Print publication year: 1991

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