Lincoln A. Wallen

On the intuitionistic force of classical search

The combinatorics of classical propositional logic lies at the heart of both local and global methods of proof-search enabling the achievement of least-commitment search. Extension of such methods to the predicate calculus, or to non-classical systems, presents us with the problem of recovering this least-commitment principle in the context of non-invertible rules. One successful approach is to view the non-classical logic as a perturbation on search in classical logic and characterize when a least-commitment (classical) search yields sufficient evidence for provability in the (non-classical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a type-theoretic view of this approach for the case in which the non-classical logic is intuitionistic. We develop a system of realizers (proof-objects) for sequents in classical propositional logic (the types) by extending Parigots λμ-calculus, a system of realizers for classical free deduction (cf. natural deduction). Our treatment of disjunction exploits directly the multiple-conclusioned form of LK as opposed to the single-conclusioned form of LJ. Consequently, it requires the addition of another binding operator, called ν, to λμ. This choice is motivated by our concern to reflect the properties of classical proof-search in the system of realizers. Using this framework, we illustrate the sense in which intuitionistic search can be viewed as a perturbation on classical search. As an application, we develop a proof procedure based on the natural extension of the notion of uniform proof to the multiple-conclusioned classical sequent calculus Harrop fragment of intuitionistic logic. This paper develops the proof-theoretic aspects of the approach.

Tableaux for Intuitionistic Logics

Despite the fact that for many years intuitionistic logic has served its function primarily in relation to foundational questions in mathematics, there has been a significant revival of interest over the last couple of decades stimulated by the application of intuitionistic formalisms in computer science (1982) . It is beyond the scope of this chapter to comment on these applications in detail which, broadly speaking, either exploit formalisations of the intuitionistic meaning of general mathematical abstractions as programming logics [Martin-Lof, 1984; Martin-Lof, 1996; Constable et al., 1986] , or exploit the similarity of systems of formal intuitionistic proofs under cut-elimination to systems of typed lambda terms under various forms of reduction (e.g. [Howard, 1980; Girard, 1989; Coquand, 1990]) .1 Both types of application rely on the rich proof theory possessed by intuitionistic formalisms in comparison with their classical counterparts.2

Investigations into proof-search in a system of first-order dependent function types

We present a series of proof systems for λII-calculus: a theory of first-order dependent function types. The systems are complete for the judgement of interest but differ substantially as bases for algorithmic proof-search. Each calculus in the series induces a search space that is properly contained within that of its predecessor. The λII-calculus is a candidate general logic in that it provides a metalanguage suitable for the encoding of logical systems and mathematics. Proof procedures formulated for the metalanguage extend to suitably encoded object logics, thus removing the need to develop procedures for each logic independently. This work is also an exploration of a systematic approach to the design of proof procedures. It is our contention that the task of designing a computationally efficient proof procedure for a given logic can be approached by formulating a series of calculi that possess specific proof-theoretic properties. These properties indicate that standard computational techniques such as unification are applicable, sometimes in novel ways. The study below is an application of this design method to an intuitionistic type theory.

Logic Programming Via Proof-Valued Computations

We argue that the computation of a logic program can be usefully divided into two distinct phases: the first being a proof-valued computation or proof-search; the second a residual computation, or answer extraction. Extension of extraction techniques to various theories then permits more extensive languages and proof procedures to be employed for the computational solution of problems.

Proof-terms for classical and intuitionistic resolution

We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resolution rules. The techniques use the λμe-calculus, a development of Parigots λμ-calculus.

On the Intuitionistic Force of Classical Search (Extended Abstract)

The combinatorics of proof-search in classical propositional logic lies at the heart of most efficient proof procedures because the logic admits least-commitment search. The key to extending such methods to quantifiers and non-classical connectives is the problem of recovering this least-commitment principle in the context of the non-classical/non-propositional logic; i.e., characterizing when a least-commitment (classical) search yields sufficient evidence for provability in the (non-classical) logic.

On form, formalism and equivalence

Sight is perhaps our most sensitive physical sense and can be used to reduce conceptual complexity. This potential is exploited within mathematics when complex ideas are expressed using geometrical relationships between symbols. By conscious manipulation of visual form we achieve unconscious manipulation of conceptual content. For example, the formal rules of Calculus help us to apply complex theorems of Analysis accurately. Moreover, the rules can be used without detailed knowledge of the mathematics that justifies them.