20 research outputs found

    The independence of control structures in abstract programming systems

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    AbstractAn instance of a control structure is a mapping which takes one or more programs into a new program whose behavior is based on that of the original programs. An instance of a control structure is effective iff it is effectively computable. In order to study the interrelationships of control structures, . we consider abstract programming systems (numberings of the partial recursive functions) in which some control structures, effective or otherwise, are present, but others are not. This paper uses the techniques of recursive function theory, including recursion theorems and priority arguments to prove the independence of certain control structures in abstract programming systems. For example, we have obtained the following results. In effective numberings of the partial recursive functions, the one-one effective Kleene recursion theorem and the one-one effective (partial) if-then-else control structure are independent, but together, they yield all effective control structures. In any effective numbering, the effective Kleene form of the double recursion theorem yields all effective control structures

    An arithmetic theory of consistency enforcement

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    Consistency enforcement starts from a given program specification S and a static invariant I and aims to replace S by a slightly modified program specification SI that is provably consistent with respect to I. One formalization which suggests itself is to define SI as the greatest consistent specialization of S with respect to I, where specialization is a partial order on semantic equivalence classes of program specifications. In this paper we present such a theory on the basis of arithmetic logic. We show that with mild technical restrictions and mild restrictions concerning recursive program specifications it is possible to obtain the greatest consistent specialization gradually and independently from the order of given invariants as well as by replacing basic commands by their respective greatest consistent specialization. Furthermore, this approach allows to discuss computability and decidability aspects for the first time

    On the depth of G\"{o}del's incompleteness theorem

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    In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of G\"{o}del's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with arXiv:2009.0488

    On the Semantics of Intensionality and Intensional Recursion

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    Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

    Hyperarithmetical properties of relations on abelian p-groups and orderings

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    Realizability, Covers, and Sheaves I. Application to the Simply-Typed Lambda-Calculus

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    We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part I of this paper applies the above approach to the simply-typed λ-calculus (with types →, ×, +, and ⊥). Part II of this paper deals with the second-order (polymorphic) λ-calculus (with types → and ∀)

    Acta Cybernetica : Volume 21. Number 3.

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