We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than

We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees. 1.

this paper, Slaman was partially supported by National Science Foundation Grant DMS-9500878.

Abstract not found

Abstract not found

We construct a logic-enriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every n-ary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that

Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2-complete. The particular problem of local confluence turns out to be Π 0 2-complete for ground terms, but only Σ 0 1-complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1-complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.