Section: Scientific Foundations

General Overview

In the specification of computational systems, logics are generally used in one of two approaches. In the computation-as-model approach, computations are encoded as mathematical structures, containing such items as nodes, transitions, and state. Logic is used in an external sense to make statements about those structures. That is, computations are used as models for logical expressions. Intensional operators, such as the modals of temporal and dynamic logics or the triples of Hoare logic, are often employed to express propositions about the change in state. This use of logic to represent and reason about computation is probably the oldest and most broadly successful use of logic in computation.

The computation-as-deduction approach, uses directly pieces of logic's syntax (such as formulas, terms, types, and proofs) as elements of the specified computation. In this much more rarefied setting, there are two rather different approaches to how computation is modeled.

The proof normalization approach views the state of a computation as a proof term and the process of computing as normalization (know variously as -reduction or cut-elimination). Functional programming can be explained using proof-normalization as its theoretical basis [50] and has been used to justify the design of new functional programming languages [22] .

The proof search approach views the state of a computation as a sequent (a structured collection of formulas) and the process of computing as the process of searching for a proof of a sequent: the changes that take place in sequents capture the dynamics of computation. Logic programming can be explained using proof search as its theoretical basis [54] and has been used to justify the design of new logic programming languages [53] .

The divisions proposed above are informal and suggestive: such a classification is helpful in pointing out different sets of concerns represented by these two broad approaches (reductions, confluence, etc, versus unification, backtracking search, etc). Of course, a real advance in computation logic might allow us merge or reorganize this classification.

Although type theory has been essentially designed to fill the gap between these two kinds of approaches, it appears that each system implementing type theory up to now only follows one of the approaches. For example, the Coq system implementing the Calculus of Inductive Constructions (CIC) uses proof normalization while the Twelf system [57] , implementing the Edinburgh Logical Framework (LF, a sub-system of CIC), follows the proof search approach (normalization appears in LF, but it is much weaker than in, say, CIC).

The Parsifal team works on both the proof normalization and proof search approaches to the specification of computation.