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17
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 n ..."
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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.
Weak theories of operations and types
"... This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywor ..."
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This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1
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.
Systems of explicit mathematics with nonconstructive µoperator and join
 ANNALS OF PURE AND APPLIED LOGIC
, 1996
"... The aim of this article is to give the prooftheoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the nonconstructive µoperator and join. We make use of standard prooftheoretic techniques such as cutelimination of appropriate semiformal sy ..."
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The aim of this article is to give the prooftheoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the nonconstructive µoperator and join. We make use of standard prooftheoretic techniques such as cutelimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.
Unfolding feasible arithmetic and weak truth
, 2012
"... In this paper we continue Feferman’s unfolding program initiated in [11] which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried thro ..."
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In this paper we continue Feferman’s unfolding program initiated in [11] which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried through for a schematic system of nonfinitist arithmetic NFA in Feferman and Strahm [13] and for a system FA (with and without Bar rule) in Feferman and Strahm [14]. The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apart from the operational unfolding U0(FEA) of FEA, we study two full unfolding notions, namely the predicate unfolding U(FEA) and a more general truth unfolding UT(FEA) of FEA, the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth TPT over combinatory logic, which has recently been introduced in Eberhard and Strahm [7] and Eberhard [6] and whose involved prooftheoretic analysis is due to Eberhard [6]. The results of this paper were first announced in [8].
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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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 ...
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
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A feasible theory of truth over combinatory algebra
"... We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the ..."
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We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low prooftheoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic. The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on descriptions of realisers instead of realisers themselves. In this way, we can express and manipulate realisation information more efficiently.
Weak theories of truth and explicit mathematics
, 2011
"... 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.