Results 1  10
of
16
Computational foundations of basic recursive function theory
 In Third IEEE Symposium on Logic in Computer Science
, 1988
"... ..."
(Show Context)
Transitive closure and the mechanization of mathematics
 Thirtyfive Years of Automating Mathematics
, 2003
"... Abstract. We argue that the concept of transitive closure is the key for understanding nitary inductive denitions and reasoning, and we provide evidence for the thesis that logics which are based on it (in which induction is a logical rule) are the right logical framework for the formalization and ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
Abstract. We argue that the concept of transitive closure is the key for understanding nitary inductive denitions and reasoning, and we provide evidence for the thesis that logics which are based on it (in which induction is a logical rule) are the right logical framework for the formalization and mechanization of Mathematics. We investigate the expressive power of languages with the most basic transitive closure operation TC. We show that with TC one can dene all recursive predicates and functions from 0, the successor function and addition, yet with TC alone addition is not denable from 0 and the successor function. However, in the presence of a pairing function, TC does suÆce for having all types of nitary inductive denitions of relations and functions. This result is used for presenting a simple version of Feferman's framework FS 0, demonstrating that TClogics provide in general an excellent framework for mechanizing formal systems. An interesting side eect of these results is a simple characterization of recursive enumerability and a new, concise version of Church thesis. We end with a use of TC for a formalization of Set Theory which is based on purely syntactical considerations, and re
ects real mathematical practice. 1.
On Founding the Theory of Algorithms
, 1998
"... machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to comp ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semiinfinite 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 numbertheoretic 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...
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 ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
(Show Context)
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
What Is an Algorithm
 SOFSEM, Lecture Notes in
"... We present a twopart exposition on the notion of algorithm and foundational analyses of computation. The first part is below, and the second is here: ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
We present a twopart exposition on the notion of algorithm and foundational analyses of computation. The first part is below, and the second is here:
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.
A Scheme for Defining Partial HigherOrder Functions by Recursion
"... This paper describes a scheme for defining partial higherorder functions as the least fixed points of monotone functionals. The scheme can be used to define both single functions by recursion and systems of functions by mutual recursion. The scheme is implemented in the IMPS Interactive Mathematica ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
This paper describes a scheme for defining partial higherorder functions as the least fixed points of monotone functionals. The scheme can be used to define both single functions by recursion and systems of functions by mutual recursion. The scheme is implemented in the IMPS Interactive Mathematical Proof System. The IMPS implementation includes an automatic syntactic check for monotonicity that succeeds for many common recursive definitions. 1
Elementary algorithms and their implementations
"... In the sequence of articles [3, 5, 4, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a settheoretic “definition ” of algorithms, much like the “definition ” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
In the sequence of articles [3, 5, 4, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a settheoretic “definition ” of algorithms, much like the “definition ” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a probability space. The aim is to provide a traditional foundation for the theory of algorithms, a development of it within axiomatic set theory in the same way as analysis and probability theory are rigorously developed within set theory on the basis of the set theoretic modeling of their basic notions. A characteristic feature of this approach is the adoption of a very abstract notion of algorithm which takes recursion as a primitive operation, and is so wide as to admit “nonimplementable ” algorithms: implementations are special, restricted algorithms (which include the familiar models of computation, e.g., Turing and random access machines), and an algorithm is implementable if it is reducible to an implementation. Our main aim here is to investigate the important relation between an
An Intensional Investigation of Parallelism
, 1994
"... Denotational semantics is usually extensional in that it deals only with input/output properties of programs by making the meaning of a program a function. Intensional semantics maps a program into an algorithm, thus enabling one to reason about complexity, order of evaluation, degree of parallelism ..."
Abstract
 Add to MetaCart
Denotational semantics is usually extensional in that it deals only with input/output properties of programs by making the meaning of a program a function. Intensional semantics maps a program into an algorithm, thus enabling one to reason about complexity, order of evaluation, degree of parallelism, efficiencyimproving program transformations, etc. I propose to develop intensional models for a number of parallel programming languages. The semantics will be implemented, resulting in a programming language of parallel algorithms, called CDSP. Applications of CDSP will be developed to determine its suitability for actual use. The thesis will hopefully make both theoretical and practical contributions: as a foundational study of parallelism by looking at the expressive power of various constructs, and with the design, implementation, and applications of an intensional parallel programming language. 1 Introduction Denotational semantics has now been around for about 25 years, which makes...