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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 162 (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.
On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the rea ..."
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Cited by 34 (1 self)
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We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.
A proof of strong normalisation using domain theory
 In LICS’06
, 2006
"... U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show tha ..."
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Cited by 13 (1 self)
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U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and MartinLöf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for MartinLöf dependent type theory extended with bar recursion, and with some form of proofirrelevance. 1
Recursion on the partial continuous functionals
 Logic Colloquium ’05
, 2006
"... We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the wellknown abbreviation ..."
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Cited by 7 (5 self)
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We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the wellknown abbreviation
Continuous semantics for strong normalization
 In CiE’05, volume 3526 of LNCS
, 2005
"... Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for ..."
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Cited by 4 (2 self)
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Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for which strong normalization was hitherto unknown.
1962 Spector Dialectica Interpretation of PA2 1998 Berardi,Bezem, Realizability Interpretation of
, 2005
"... These interpretations use strong computation principles with difficult termination proofs. 2 The problem Given a strongly normalizing type theory T and a system of rewrite rules R, show that the combined system T + R is strongly normalizing. We are mainly interested in higherorder rewrite rules tha ..."
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These interpretations use strong computation principles with difficult termination proofs. 2 The problem Given a strongly normalizing type theory T and a system of rewrite rules R, show that the combined system T + R is strongly normalizing. We are mainly interested in higherorder rewrite rules that go beyond structural recursion. Example T = simple types over the booleans and the natural numbers R = Gödel primitive recursion + modified bar recursion:
considered for publication in Logical Methods in Computer Science STRONG NORMALISATION FOR APPLIED LAMBDA CALCULI
, 2005
"... Abstract. We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domaintheoretic model such that any term not denoting ⊥ is strongly normalising provided all its ‘stratified approximations ’ are. From this we derive a general normalisation theore ..."
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Abstract. We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domaintheoretic model such that any term not denoting ⊥ is strongly normalising provided all its ‘stratified approximations ’ are. From this we derive a general normalisation theorem for applied typed λcalculi: If all constants have a total value, then all typeable terms are strongly normalising. We apply this result to extensions of Gödel’s system T and system F extended by various forms of bar recursion for which strong normalisation was hitherto unknown. 1.