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Normalization by evaluation for MartinLöf type theory with one universe
 IN 23RD CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, MFPS XXIII, ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2007
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Totality in Applicative Theories
 ANNALS OF PURE AND APPLIED LOGIC
, 1995
"... In this paper we study applicative theories of operations and numbers with (and without) the nonconstructive minimum operator in the context of a total application operation. We determine the prooftheoretic strength of such theories by relating them to wellknown systems like Peano Arithmetic ..."
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Cited by 21 (12 self)
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In this paper we study applicative theories of operations and numbers with (and without) the nonconstructive minimum operator in the context of a total application operation. We determine the prooftheoretic strength of such theories by relating them to wellknown systems like Peano Arithmetic PA and the system (\Pi 0 1 CA) !"0 of second order arithmetic. Essential use will be made of socalled fixedpoint theories with ordinals, certain infinitary term models and Church Rosser properties.
Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory
 Journal of Symbolic Logic
"... In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established. 1 Introduction In classical set theory an ordinal # is called a Mahlo ordinal if it is a regular cardinal and if, ..."
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Cited by 20 (14 self)
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In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established. 1 Introduction In classical set theory an ordinal # is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function f from # to #, there exists a regular cardinal less than # so that {f(#) : # < } # . The statement that there exists a Mahlo ordinal is a powerful set existence axiom going beyond theories like ZFC. It also outgrows the existence of inaccessible cardinals, hyper inaccessibles, hyperhyperinaccessible and the like. There is also an obvious recursive analogue of Mahlo ordinal. Typically, an ordinal # is baptized recursively Mahlo, if it is admissible and reflects every # 2 sentence on a smaller admissible ordinal. The corresponding formal theory KPM has been prooftheoretically analyzed by Rathjen [14, 15]. KPM is a highly impredicative theory, and its proofthe...
Wellordering proofs for MartinLöf Type Theory
 Annals of Pure and Applied Logic
, 1998
"... We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is ..."
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Cited by 18 (11 self)
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We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...
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
Extending the System T_0 of explicit mathematics: the limit and Mahlo axioms
"... In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive def ..."
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Cited by 13 (8 self)
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In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not ne ..."
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Cited by 12 (9 self)
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Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
Typebased uselesscode elimination for functional programs
 Proceedings of the International Workshop on Semantics, Applications, and Implementation of Program Generation (SAIG 2000
, 2000
"... Abstract. In this paper we present a survey of the work on typebased uselesscode elimination for higherorder functional programs. After some historical remarks on the motivations and early approaches we give an informal but complete account of the techniques and results developed at the Computer ..."
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Cited by 10 (0 self)
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Abstract. In this paper we present a survey of the work on typebased uselesscode elimination for higherorder functional programs. After some historical remarks on the motivations and early approaches we give an informal but complete account of the techniques and results developed at the Computer Science Department of the University of Torino. In particular, we focus on the fact that, for each of the typebased techniques developed, there is an optimal program simplification. 1 UselessCode Elimination for HigherOrder Programs: