Results 1  10
of
25
Higher Type Recursion, Ramification and Polynomial Time
 Annals of Pure and Applied Logic
, 1999
"... It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [9] and later became known as the essential part of Godel's system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion [14] showed that another form of ramification can be used to restrict higher type recursion to PSPACE. However, to characterize the much smaller class of polynomialtime computable functions by higher type recursion, it seems that an additional principle is required. By introducing linear...
Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
Feasible Computation With Higher Types
, 2002
"... We restrict recursion in nite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types ! and terms x r as well as ( and x r. Here we use two sorts of typed variables: complete ones x ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
We restrict recursion in nite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types ! and terms x r as well as ( and x r. Here we use two sorts of typed variables: complete ones x and incomplete ones x . 1.
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
Remarks On Finitism
 Reflections on the Foundations of Mathematics. Essays in Honor of Solomon Feferman, LNL 15. Association for Symbolic Logic
, 2000
"... representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing by the human mind, but possibly to be beyond representablity in the physical world. [2, p. 39]. This di#culty ought to be discussed more adequately then + This paper is based on a talk that I was very pleased to give at the conference Reflections, December 1315, 1998, in honor of Solomon Feferman on his seventieth birthday. The choice of topic is especially appropriate for the conference in view of recent discussions we had had about finitism. I profited from the discussion following my talk and, in particular, from the remarks of Richard Zach. I have since had the advantage of further discussions with Zach and of reading his paper 1998; and I use his scholarshi
Science At the Crossroad Between Randomness and Determinism
, 2000
"... Time and again, man's understanding of Nature is at the crossroad between total worldcomprehension and total randomness. It is suggested that not only are the preferences influenced by the theories and models of today, but also by the very personal subjective inclinations of the people involved ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Time and again, man's understanding of Nature is at the crossroad between total worldcomprehension and total randomness. It is suggested that not only are the preferences influenced by the theories and models of today, but also by the very personal subjective inclinations of the people involved. The second part deals with the principle of selfconsistency and its consequences for totally deterministic systems.
A Quantum Mechanical Look At Time Travel and Free Will
, 2001
"... Consequences of the basic and most evident consistency requirementthat measured events cannot happen and not happen at the same timeare reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, very general bounds on the forecast and control of e ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Consequences of the basic and most evident consistency requirementthat measured events cannot happen and not happen at the same timeare reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, very general bounds on the forecast and control of events within the known laws of physics result. These bounds are of a global, statistical nature and need not aect singular events or groups of events. We also present a quantum mechanical model of time travel and discuss chronology protection schemes. Such models impose restrictions upon certain capacities of event control.
Gödel's Dialectica interpretation and its twoway stretch
 in Computational Logic and Proof Theory (G. Gottlob et al eds.), Lecture Notes in Computer Science 713
, 1997
"... this article has appeared in Computational Logic and Proof Theory (Proc. 3 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
this article has appeared in Computational Logic and Proof Theory (Proc. 3
Syntax and Semantics
"... The year is 2002 and here we are at a symposium on Foundations ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The year is 2002 and here we are at a symposium on Foundations