An overview of a proof theoretical approach to reasoning about computation

Typing rules and structural operational semantics are often given via inference rules: that is, the justification of a typing or an evaluation is actually a proof. Thus it is not surprising that proof theory can be used to benefit the specification of and the reasoning about computation. An additional advantage of using proof theory is that it can support such “intensional” aspects of computation as resources (say, via linear logic) and bindings (say, via term-level and proof-level bindings). In this talk, I will overview recent work on designing a proof theoretic framework for reasoning about both the static and dynamic semantics of specifications languages and programming languages. A synthesis of the following topics will be provided: λ-tree syntax, mobility of binders, ∇-quantification, two-level logic architecture, induction and coinduction, and focusing proof systems. An annotated, partial bibliography The technical material for this overview is contained in a number of papers, some of which are briefly described in the following annotated bibliography. λ-tree syntax. An earlier illustration of the kinds of computations that are possible in λProlog by directly manipulating λ-terms is contained in [MN87]. Later this style of manipulation was called higher-order abstract syntax, but since that term came to mean different things to different communities, the term λ-tree syntax was introduced in [Mil00] to denote the original form of λ-term manipulation. [MN87] D. Miller and G. Nadathur. A logic programming approach to manipulating formulas and programs. In Seif Haridi, editor, Sym. on Logic Programming, pp. 379–388, 1987. [Mil00] D. Miller. Abstract syntax for variable binders: An overview. In John Lloyd and et. al., editors, Computational Logic CL 2000, LNAI 1861, pp. 239–253. Springer, 2000. Definitions and induction. In order to reason about what can and cannot be proved from a given specification, it is necessary to be able to think of a logic specification as being closed or defined. Induction can then also be described on such definitions. A suitable proof theory presentation of these ideas can be applied 1 INRIA Saclay Île-de-France & LIX/École Polytechnique, 91128 Palaiseau, France. Support from INRIA through the “Equipes Associées” Slimmer is gratefully acknowledged. c ©2008 Published by Elsevier Science B. V.