Results 1 
7 of
7
Total Functional Programming
 Journal of Universal Computer Science
, 2004
"... We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Chur ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Church had developed to investigate this and other problems in the foundations of mathematics: the calculus of lambda conversion. Both the Thesis and the lambda calculus have been of seminal influence on the development of Computing Science. The main subject of this article is the lambda calculus but I will begin with a brief sketch of the emergence of the Thesis. The epistemological status of Church’s Thesis is not immediately clear from the above quotation and remains a matter of debate, as is explored in other papers of this volume. My own view, which I will state but not elaborate here, is that the thesis is empirical because it relies for its significance on a claim about what can be calculated by mechanisms. This becomes clearer in
Programmability of Chemical Reaction Networks
"... Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard c ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and
The ChurchTuring Thesis as an Immature Form of the ZuseFredkin Thesis (More Arguments in Support of the “Universe as a Cellular Automaton” Idea)
"... In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
In [1] we have shown a strong argument in support of the &quot;Universe as a computer &quot; idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only &quot;some kind of computer&quot;, but also a concrete computational model known as a &quot;cellular automaton&quot;.
Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert’s program
"... The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel’s collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophic ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel’s collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and postKantian philosophy, and contains Gödel’s thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their lifelong warm personal relationship. I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Gödel’s Collected Works, pp. 4179. 1 My purpose here is to focus on only one group of interrelated topics from these exchanges, namely the light that it⎯together with assorted published and unpublished articles and lectures by Gödel⎯throws on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert’s program. 2 In that connection, this piece has an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career. 1 The five volumes of Gödel’s Collected Works (19862003) are referred to below, respectively, as CW I,
Some Key Remarks by Turing Bibliography Other Internet Resources Related Entries
, 1997
"... There are various equivalent formulations of the ChurchTuring thesis. A common one is that every effective computation can be carried out by a Turing machine. The ChurchTuring thesis is often misunderstood, particularly in recent writing in the philosophy of mind. ..."
Abstract
 Add to MetaCart
There are various equivalent formulations of the ChurchTuring thesis. A common one is that every effective computation can be carried out by a Turing machine. The ChurchTuring thesis is often misunderstood, particularly in recent writing in the philosophy of mind.
IOS Press Proving as a Computable Procedure
, 2004
"... Abstract. Gödel’s incompleteness theorem states that every finitelypresented, consistent, sound theory which is strong enough to include arithmetic is incomplete. In this paper we present elementary proofs for three axiomatic variants of Gödel’s incompleteness theorem and we use them (a) to illustr ..."
Abstract
 Add to MetaCart
Abstract. Gödel’s incompleteness theorem states that every finitelypresented, consistent, sound theory which is strong enough to include arithmetic is incomplete. In this paper we present elementary proofs for three axiomatic variants of Gödel’s incompleteness theorem and we use them (a) to illustrate the idea that there is more than “complete vs. incomplete”, there are degrees of incompleteness, and (b) to discuss the implications of incompleteness and computerassisted proofs for Hilbert’s Programme. We argue that the impossibility of carrying out Hilbert’s Programme is a thesis and has a similar status to the ChurchTuring thesis. 1.