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68
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 190 (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.
Local stability of ergodic averages
 Transactions of the American Mathematical Society
"... We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages An ..."
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Cited by 46 (10 self)
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We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0,1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space, and any element f, it is possible to compute a bound on the rate of convergence of (Anf) from T, f, and the norm ‖f ∗ ‖ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2(X), then there is a computable bound on the rate of convergence of the sequence (Anf). The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every ε> 0, there is an n with the property that the ergodic averages Amf are stable to within ε on the interval [n, K(n)]. Even in situations where the sequence (Anf) does not have a computable limit, one can give explicit bounds on such n in terms of K and ‖f‖/ε. This tells us how far one has to search to find an n so that the ergodic averages are “locally stable ” on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general prooftheoretic methods falling under the heading of “proof mining.” 1
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Ramsey's theorem and the pigeonhole principle in intuitionistic mathematics
, 1992
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Relative Constructivity
 J. Symbolic Logic
, 1996
"... This paper essentially contains material from chapter 8 of the author's Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at ClermontFerrand (see [7]). 1 As is wellknown (cf. the discussion at the end of x3 of [10]), the use of classical logic (on which the ..."
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Cited by 14 (8 self)
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This paper essentially contains material from chapter 8 of the author's Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at ClermontFerrand (see [7]). 1 As is wellknown (cf. the discussion at the end of x3 of [10]), the use of classical logic (on which the systems GnA ! are based) has the consequence that the extractability of an effective (and for n = 2 polynomial) bound from a proof of an 89Asentence is (in general) guaranteed only if A is quantifier free (or purely existential). In the present paper we study proofs which may use mathematically strong nonconstructive analytical principles as e.g. 1) Attainment of the maximum of f 2 C([0; 1] d ; IR) 2) Mean value theorem for integrals 3) CauchyPeano existence theorem 4) Brouwer's fixed point theorem for continuous functions f : [0; 1] d ! [0; 1] d 5) A generalization WKL 2 seq of the binary Konig's lemma WKL 6) Comprehension for negated formulas: CA ae : : 9\Phi 0ae x ae :1 0 8y ae \Gamma \Phiy = 0 0 $ :A(y) \Delta ; where A is arbitrary: as well as the nonintuitionistic logical principles 7) The `double negation shift' DNS : 8x ae ::A ! ::8x ae A for arbitrary types ae and formulas A 8) The `lesser limited principle of omniscience' LLPO : 8x 1 ; y 1 9k 0 1([k = 0 ! x IR y] [k = 1 ! y IR x]) 9) The independence of premise principle for negated formulas IP: : (:A ! 9y ae B) ! 9y ae (:A ! B); where y is not free in A, plus the schema AC of full choice but apply these principles only in the context of the intuitionistic versions (E)GnA ! i of the theories (E)G n A ! . The restriction to intuitionistic logic guarantees the extractability of (uniform) effective bounds for arbitrary 89Asentences (see theorem 4.1 below). Indeed w...
Intuitionism As Generalization
 Philosophia Math
, 1990
"... ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But an ..."
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Cited by 11 (1 self)
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ms that hold only in computational models. For example, Brouwer proved that every totally defined function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But any theorem in constructive mathematics is a theorem in classical mathematics, so in this case it is not a question of choosing between models, but of deciding whether it is worthwhile to talk about computational models in addition to the classical model. By comparing mathematical realism with intuitionism from an informal axiomatic point of view, we can steer clear of most of the metaphysical problems involved in analyzing these notions from the ground up, and concentrate on what may be termed the purely mathematical aspects. In particular we won't have to consider whether intuitionists hold that a theorem "isn't true until it's known to be true", as Blais 1 <F
Computable versions of Baire’s category theorem
 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2001
, 2001
"... We study different computable versions of Baire’s Category Theorem in computable analysis. Similarly, as in constructive analysis, different logical forms of this theorem lead to different computational interpretations. We demonstrate that, analogously to the classical theorem, one of the computab ..."
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Cited by 10 (1 self)
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We study different computable versions of Baire’s Category Theorem in computable analysis. Similarly, as in constructive analysis, different logical forms of this theorem lead to different computational interpretations. We demonstrate that, analogously to the classical theorem, one of the computable versions of the theorem can be used to construct interesting counterexamples, such as a computable but nowhere differentiable function.
MetaPRL  A Modular Logical Environment
, 2003
"... MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive ..."
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Cited by 8 (2 self)
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MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive LCFstyle tacticbased proof assistant, a logical framework, a logical programming environment, and a formal methods programming toolkit. MetaPRL is distributed under an opensource license and can be downloaded from http://metaprl.org/. This paper provides an overview of the system focusing on the features that did not exist in the previous generations of PRL systems.
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 (2 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.
Markov’s principle for propositional type theory
 Computer Science Logic, Proceedings of the 10 th Annual Conference of the EACSL
, 2001
"... Abstract. In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of class ..."
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Abstract. In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of classical reasoning we get more powerful resulting system which remains constructive and valid in the standard constructive semantics of a type theory. We also show that this principle can be formulated and used in a propositional fragment of a type theory.