Section: Research Program

General overview

There are two broad approaches for computational specifications. In the computation as model approach, computations are encoded as mathematical structures containing nodes, transitions, and state. Logic is used to describe these structures, that is, the 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.

The computation as deduction approach, in contrast, expresses computations logically, using formulas, terms, types, and proofs as computational elements. Unlike the model approach, general logical apparatus such as cut-elimination or automated deduction becomes directly applicable as tools for defining, analyzing, and animating computations. Indeed, we can identify two main aspects of logical specifications that have been very fruitful:

• Proof normalization, which treats the state of a computation as a proof term and computation as normalization of the proof terms. General reduction principles such as $\beta$-reduction or cut-elimination are merely particular forms of proof normalization. Functional programming is based on normalization  [64] , and normalization in different logics can justify the design of new and different functional programming languages  [38] .

• Proof search, which views the state of a computation as a a structured collection of formulas, known as a sequent, and proof search in a suitable sequent calculus as encoding the dynamics of the computation. Logic programming is based on proof search  [70] , and different proof search strategies can be used to justify the design of new and different logic programming languages  [68] .

While the distinction between these two aspects is somewhat informal, it helps to identify and classify different concerns that arise in computational semantics. For instance, confluence and termination of reductions are crucial considerations for normalization, while unification and strategies are important for search. A key challenge of computational logic is to find means of uniting or reorganizing these apparently disjoint concerns.

An important organizational principle is structural proof theory, that is, the study of proofs as syntactic, algebraic and combinatorial objects. Formal proofs often have equivalences in their syntactic representations, leading to an important research question about canonicity in proofs – when are two proofs “essentially the same?” The syntactic equivalences can be used to derive normal forms for proofs that illuminate not only the proofs of a given formula, but also its entire proof search space. The celebrated focusing theorem of Andreoli  [39] identifies one such normal form for derivations in the sequent calculus that has many important consequences both for search and for computation. The combinatorial structure of proofs can be further explored with the use of deep inference; in particular, deep inference allows access to simple and manifestly correct cut-elimination procedures with precise complexity bounds.

Type theory is another important organizational principle, but most popular type systems are generally designed for either search or for normalization. To give some examples, the Coq system  [76] that implements the Calculus of Inductive Constructions (CIC) is designed to facilitate the expression of computational features of proofs directly as executable functional programs, but general proof search techniques for Coq are rather primitive. In contrast, the Twelf system  [72] that is based on the LF type theory (a subsystem of the CIC), is based on relational specifications in canonical form (i.e., without redexes) for which there are sophisticated automated reasoning systems such as meta-theoretic analysis tools, logic programming engines, and inductive theorem provers. In recent years, there has been a push towards combining search and normalization in the same type-theoretic framework. The Beluga system  [73] , for example, is an extension of the LF type theory with a purely computational meta-framework where operations on inductively defined LF objects can be expressed as functional programs.

The Parsifal team investigates both the search and the normalization aspects of computational specifications using the concepts, results, and insights from proof theory and type theory.