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15
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.
Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathemat ..."
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Cited by 8 (5 self)
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This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are prooftheoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and prooftheoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...
Realization of analysis into Explicit Mathematics
 The Journal of Symbolic Logic
, 2000
"... We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard Dinterpretation, which was used before and was shown to interpret only subsystems prooftheoretically weaker than T0 , our interpretation can reach the full strength of T0 . The ..."
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Cited by 4 (2 self)
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We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard Dinterpretation, which was used before and was shown to interpret only subsystems prooftheoretically weaker than T0 , our interpretation can reach the full strength of T0 . The Rinterpretation is an adaptation of Kleene's recursive realizability, and is applicable only to intuitionistic theories. Introduction Systems of Explicit Mathematics were introduced by S. Feferman in the 70es as a logical framework for Bishopstyle constructive mathematics (see [Fef75], [Fef79]). In [Fef79] he gave an embedding of the basic theory T 0 into a subsystem # 1 2 CA+BI of 2nd order arithmetic and conjectured that the converse also holds. In [Ja83] G. Jager carried out a necessary wellordering proof in T 0 , which together with [JP82] completed its prooftheoretical analysis and established prooftheoretic equivalence of the system of Explicit Mathematics T 0 , system o...
Realization of Constructive Set Theory into Explicit Mathematics: a lower bound for impredicative Mahlo universe
 Transactions American Math. Soc
, 2000
"... We define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T0 , thus providing relative lower bounds for the prooftheoretic strength of the latter ..."
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Cited by 4 (2 self)
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We define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T0 , thus providing relative lower bounds for the prooftheoretic strength of the latter. Introduction Several di#erent frameworks have been founded in the 70es aiming to give a foundation for constructive mathematics. The most welldeveloped of them nowadays are MartinLof type theory, Aczel's constructive set theory, and Feferman's explicit mathematics. While constructive set theory was built to have an immediate type interpretation, no theory stronger than # 1 2 CA, which prooftheoretically is still far below the basic system T 0 of Explicit Mathematics, have been shown up to now to be directly embeddable into explicit systems. It also yielded that the only method for establishing lower bounds for T 0 and its extensions remained to be wellordering proofs. This omissi...
A Theory of Explicit Mathematics Equivalent to ID_1
"... We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1. ..."
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Cited by 2 (2 self)
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We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1.
Impredicative Overloading in Explicit Mathematics
, 2000
"... In this article we introduce the system OTN of explicit mathematics based on elementary separation, product, join and weak power types. We present a settheoretical model for OTN, and we develop in OTN a theory of impredicative overloading. Together this yields a solution to the problem of impredica ..."
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Cited by 2 (2 self)
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In this article we introduce the system OTN of explicit mathematics based on elementary separation, product, join and weak power types. We present a settheoretical model for OTN, and we develop in OTN a theory of impredicative overloading. Together this yields a solution to the problem of impredicativity encountered in denotational semantics for overloading and latebinding. Further, our work provides a first example of an application of power types in explicit mathematics. Keywords: Objectoriented constructs, type structure, proof theory. 1 Introduction Overloading is an important concept in objectoriented programming. For example, it occurs when a method is redefined in a subclass or when a class provides several methods with the same name but with di#erent argument types. Theoretically speaking, overloading denotes the possibility that several functions f i with respective types S i # T i may be combined to a new overloaded function f of type {S i # T i } i#I . We then ...
Elementary constructive operational set theory. To appear in: Festschrift for Wolfram Pohlers, Ontos Verlag
"... Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical ..."
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Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has nonextensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1.
Weak theories of truth and explicit mathematics. Submitted for publication. 19
"... We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able t ..."
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We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able to interpret expressive feasible subsystems of explicit mathematics. 1
Universes over Frege Structures
 Annals of Pure and Applied Logic
, 1996
"... We investigate universes axiomatized as sets with natural closure conditions over Frege structures. In the presence of a natural form of induction, we obtain a theory of prooftheoretic strength \Gamma 0 . 1 Introduction Frege structures were introduced by Aczel in [Acz80] as a semantical concept t ..."
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Cited by 1 (0 self)
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We investigate universes axiomatized as sets with natural closure conditions over Frege structures. In the presence of a natural form of induction, we obtain a theory of prooftheoretic strength \Gamma 0 . 1 Introduction Frege structures were introduced by Aczel in [Acz80] as a semantical concept to introduce a notion of sets by means of a partial truth predicate. This approach is closely related to prior work of Scott [Sco75] and was originally developed for questions around MartinLof's type theory. In [Bee85, Ch. XVII] Beeson gave a formalization of Frege structures as a truth theory over applicative theories. Applicative theories go back to Feferman's systems of explicit mathematics introduced in [Fef75, Fef79]. These systems provide a logical basis for functional programming. The basic theory for which Frege structures are defined is the basic theory of operations and numbers TON, introduced and studied in [JS95]. It comprises total combinatorial logic and arithmetic. The notion ...
Formalizing NonTermination of Recursive Programs
 J. of Logic and Algebraic Programming
, 2001
"... In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to dene a least xed point operator. Still, o ..."
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In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to dene a least xed point operator. Still, our theory has a standard recursion theoretic interpretation. 1