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A mathematical modeling of pure, recursive algorithms
 Logic at Botik ’89
, 1989
"... This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are ..."
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This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are the soundness of the reduction calculus of FLR (which models faithful, algorithm–preserving compilation) for this semantics, and the robustness of the class of algorithms assigned to a structure under algorithm adjunction. This is the second in a sequence of papers begun with [16] in which we develop a foundation for the theory of computation based on a mathematical modeling of recursive algorithms. The general features, aims and methodological assumptions of this program were discussed and illustrated by examples in the preliminary, expository report [15]. In [16] we studied the formal language of recursion FLR which is the main technical tool for this work, we developed several alternative denotational semantics for it and we established a key unique termination theorem for a reduction calculus which models faithful (algorithm–preserving) compilation. Here we will define the intensional semantics of FLR for structures with given (pure) recursors, the set–theoretic objects we use to model pure (side–effect–free) algorithms: the intension of a term t on each structure A is a recursor which models the algorithm expressed by t on A. Basic results of the paper During the preparation of this paper the author was partially supported by an NSF
Elementary constructive operational set theory. To appear in: Festschrift for Wolfram Pohlers, Ontos Verlag
"... Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical ..."
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Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has nonextensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1.
ERECURSIVE INTUITIONS
"... Abstract. An informal sketch (with intermittent details) of parts of ERecursion theory, mostly old, some new, that stresses intuition. The lack of effective unbounded search is balanced by the availability of divergence witnesses. A set is Eclosed iff it is transitive and closed under the applicat ..."
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Abstract. An informal sketch (with intermittent details) of parts of ERecursion theory, mostly old, some new, that stresses intuition. The lack of effective unbounded search is balanced by the availability of divergence witnesses. A set is Eclosed iff it is transitive and closed under the application of partial Erecursive functions. Some finite injury, forcing, and model theoretic constructions can be adapted to Eclosed sets that are not Σ1 admissible.