Results 1  10
of
15
Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
Abstract

Cited by 162 (4 self)
 Add to MetaCart
Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
The fundamental theorem of algebra: a constructive development without choice
 Pacific Journal of Mathematics
, 2000
"... Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice come ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice comes into play andhow to proceedin its absence. Along the way, a notion of a complete metric space, suitable for a choiceless environment, is developed. By constructive mathematics I mean, essentially, mathematics that is developed along the lines proposed by Errett Bishop [1]. More precisely, I mean mathematics that is done in the context of intuitionistic logic — without the lawof excluded middle. My reasons for identifying these notions are discussed in [9] and [10], the basic contention being that constructive mathematics has the same subject matter as classical mathematics. Ruitenburg [11] treated the fundamental theorem of algebra in a choiceless
Computability on the Probability Measures on the Borel Sets of the Unit Interval
 THEORETICAL COMPUTER SCIENCE
, 1996
"... Scientists apply digital computers to perform computations on natural numbers, finite strings, real numbers and more general objects like sets, functions and measures. While computability theory on many countable sets is well established and for computability on the real numbers several (unfortunate ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Scientists apply digital computers to perform computations on natural numbers, finite strings, real numbers and more general objects like sets, functions and measures. While computability theory on many countable sets is well established and for computability on the real numbers several (unfortunately mutually nonequivalent) definitions are being studied, in particular for measures no computability concept at all has been available until now. In this contribution we introduce a natural computability concept on the set M of probability measures on the Borel subsets of the unit interval [0; 1]. As background we consider TTE, Type 2 Theory of Effectivity, [KW84, KW85], where computability is defined on finite and infinite sequences of symbols explicitly by Turing machines and on other sets by means of notations and representations. A standard representation ffi m :` \Sigma ! \Gamma! M is introduced via some natural information structure [Wei95a] (M; oe; ), where oe is a subbase of so...
Exact Exploration and Hanging Algorithms ⋆
"... Abstract. Recent analysis of sequential algorithms resulted in their axiomatization and in a representation theorem stating that, for any sequential algorithm, there is an abstract state machine (ASM) with the same states, initial states and state transitions. That analysis, however, abstracted from ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
Abstract. Recent analysis of sequential algorithms resulted in their axiomatization and in a representation theorem stating that, for any sequential algorithm, there is an abstract state machine (ASM) with the same states, initial states and state transitions. That analysis, however, abstracted from details of intrastep computation, and the ASM, produced in the proof of the representation theorem, may and often does explore parts of the state unexplored by the algorithm. We refine the analysis, the axiomatization and the representation theorem. Emulating a step of the given algorithm, the ASM, produced in the proof of the new representation theorem, explores exactly the part of the state explored by the algorithm. That frugality pays off when state exploration is costly. The algorithm may be a highlevel specification, and a simple function call on the abstraction level of the algorithm may hide expensive interaction with the environment. Furthermore, the original analysis presumed that state functions are total. Now we allow state functions, including equality, to be partial so that a function call may cause the algorithm as well as the ASM to hang. Since the emulating ASM does not make any superfluous function calls, it hangs only if the algorithm does. [T]he monotony of equality can only lead us to boredom. —Francis Picabia 1
Constructive Mathematics, in Theory and Programming Practice
, 1997
"... The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the pap ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on MartinLof's theory of types as a formal system for BISH.
Physicallyrelativized ChurchTuring Hypotheses. Applied Mathematics and Computation 215, 4
 in the School of Mathematics at the University of Leeds, U.K. © 2012 ACM 00010782/12/03 $10.00 march 2012  vol. 55  no. 3  communications of the acm 83
"... Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), sug ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory—rather than vaguely referring to “nature ” in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability
A Weak Countable Choice Principle
 Proc. Amer. Math. Soc
, 1998
"... A weak choice principle is introduced that is implied both by countable choice and by the law of excluded middle. This principle suffices to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact field, and to prove the fundamental th ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
A weak choice principle is introduced that is implied both by countable choice and by the law of excluded middle. This principle suffices to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact field, and to prove the fundamental theorem of algebra.
F.: Propositional fuzzy logics: Decidable for some (algebraic) operators; undecidable for more complicated ones
 Inter. Jour. Intel. Systems
, 1999
"... If we view fuzzy logic as a logic, i.e., as a particular case of a multivalued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
If we view fuzzy logic as a logic, i.e., as a particular case of a multivalued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these two formulas always have the same truth value. It is known that the simplest fuzzy logic, in which & = min and ∨ = max, is decidable. In this paper, we prove a more general result: that all propositional fuzzy logics with algebraic operations are decidable, We also show that this result cannot be generalized further: e.g., no deciding algorithm is possible for logics in which operations are algebraic with constructive (nonalgebraic) coefficients.
Sets, Complements and Boundaries
"... The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible.
Series Preproceedings of the Workshop “Physics and Computation ” 2008
, 2008
"... In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between ..."
Abstract
 Add to MetaCart
In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model — which can only process information transformed and coded in binary — wouldn’t be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942–52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCullochPitts ’ digital model