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20
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 ..."
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Cited by 204 (4 self)
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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.
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 ..."
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Cited by 28 (4 self)
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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...
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 ..."
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Cited by 20 (4 self)
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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
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 ..."
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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 ..."
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Cited by 7 (3 self)
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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 ..."
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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 ..."
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Cited by 4 (4 self)
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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.
Demuth’s path to randomness
 In Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond, WTCS’12
, 2012
"... Abstract. Osvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic random ..."
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Abstract. Osvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later. In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth’s work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth’s independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truthtable reducibility, algorithmic randomness, and semigenericity in Demuth’s work. §1. Introducing Demuth. The mathematician Osvald Demuth worked primarily on constructive analysis in the Russian style, which was initiated by Markov, Šanin, Cĕıtin, and others in the 1950s. Born in 1936 in Prague, Demuth
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 ..."
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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 ..."
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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.