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11
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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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.
A ProofTheoretic Characterization of the Basic Feasible Functionals
 Theoretical Computer Science
, 2002
"... We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27]. ..."
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Cited by 7 (6 self)
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We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].
The provably terminating operations of the subsystem PETJ of explicit mathematics
, 2010
"... In Spescha and Strahm [15], a system PET of explicit mathematics in the style of Feferman [7, 8] is introduced, and in Spescha and Strahm [16] the addition of the join principle to PET is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of PETJ i ..."
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Cited by 4 (1 self)
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In Spescha and Strahm [15], a system PET of explicit mathematics in the style of Feferman [7, 8] is introduced, and in Spescha and Strahm [16] the addition of the join principle to PET is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of PETJ i are the polytime functions on binary words. However, although strongly conjectured, it remained open whether the same holds true for the corresponding theory PETJ with classical logic. This note supplements a proof of this conjecture. Keywords: Explicit mathematics, polytime functions, nonstandard models
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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Cited by 4 (3 self)
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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
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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Cited by 2 (2 self)
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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
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.
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].
Primitive Recursive Selection Functions for Existential Assertions over Abstract Algebras
"... Abstract. We generalize to abstract manysorted algebras the classical prooftheoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a correspond ..."
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Abstract. We generalize to abstract manysorted algebras the classical prooftheoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a corresponding generalization to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalization turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulae in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realizability of existential assertions over such algebras, generalized to realizability of sequents of existential assertions. In this way, the results can be seen to hold for classical proof systems. This investigation may give some insight into the relationship between specifiability and computability for data types such as the reals, where the atomic formulae, i.e., equations between terms of type real, are not computable. Key words and phrases: generalized computability, realizability, selection function2 1
Realisability in weak systems of explicit mathematics
, 2010
"... This paper is a direct successor to Spescha and Strahm [12]. Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in [12] by the socalled join axiom of explicit mathematics. 1 ..."
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This paper is a direct successor to Spescha and Strahm [12]. Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in [12] by the socalled join axiom of explicit mathematics. 1