Machines and Recursive Definitions 2.1 Abstract Machines The best-known model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental Church-Turing Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The Church-Turing Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...

From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the "higher infinite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. In this paper, I present a new very general notion of the "unfolding" closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic

machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the well-known definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semi-infinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function 14 f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the number-theoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the correspon...

In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type. 1

This paper is concerned with the equational logic of corecursion, that is of definitions involving final coalgebra maps. The framework for our study is iteration theories (cf. e.g. Bloom and ' Esik [2, 3]), recently re-introduced as models of the FLR 0 fragment of the Formal Language of Recursion [7, 8, 9]. We present a new class of iteration theories derived from final coalgebras. This allows us to reason with a number of types of fixedpoint equations which heretofore seemed to require metric or order-theoretic ideas. All of the work can be done using finality properties and equational reasoning. Having a semantics, we obtain the following completeness result: the equations involving fixed-point terms which are valid for final coalgebra interpretations are exactly those valid in a number of contexts pertaining to recursion. For example, they coincide with the equations valid for least-fixed point recursion on dcpo's. We also present a new version of the proof of the well-...

. We extend the well-known analysis of recursion-removal in first-order program schemes to a higher-order language of finitely typed and polymorphically typed functional programs, the semantics of which is based on call-by-name parameter-passing. We introduce methods for recursion-removal, i.e. for translating higher-order recursive programs into higher-order iterative programs, and determine conditions under which this translation is possible. Just as finitely typed recursive programs are naturally classified by their orders, so are finitely typed iterative programs. This syntactic classification of recursive and iterative programs corresponds to a semantic (or computational) classification: the higher the order of programs, the more functions they can compute. 1 Background and Motivation Although our analysis is entirely theoretical, as it combines methods from typed -calculi, from abstract recursion theory and from denotational semantics, the problems we consider have a strong pra...

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-nitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U 0 (NFA) is equivalent to PA; (ii) U 1 (NFA) is equivalent to RA<! ; (iii) U(NFA) is equivalent to RA< 0 . Thus U(NFA) is proof-theoretically equivalent to predicative analysis.

this paper are related to "program verification" very much like predicate logic and its completeness are related to axiomatic set theory; they are certainly relevant, but not of much help in establishing specific, concrete results. In its most general form, a recursive definition of a function is expressed by a fixpoint equation of the form

In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project -- i.e., the half which tries to expose an initial tension between Cantor's theorem and the Löwenheim-Skolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...

In this thesis, we develop four systems of set theory based on linear logic. All of those systems have the principle of unrestricted comprehension but they are shown to be consistent. The consitency proofs are given by establishing the cut-elimination theorems. Our first system of linear set theory SMALL is formulated in full linear logic, i.e., with exponentials. However we do not allow exponentials to appear inside of set terms. Secondly, we formulate a system of set theory in linear logic with infinitary additive conjunction and disjunction, instead of exponentials. This system is called AS 1 . Thirdly, we present the system of linear set theory LZF which is a conservative extension of Zermelo-Fraenkel set theory without the axiom of regularity or ZF \Gamma . The idea is to build up a linear set theory on top of ZF \Gamma in a style similar to SMALL. We establish a partial cut-elimination result for LZF, and derive from it that LZF is a conservative extension of ZF \Gamma , ...