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TERM EXTRACTION AND RAMSEY’S THEOREM FOR PAIRS
"... Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This ..."
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Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL0 + Π0 1CP + COH is Π0 2conservative over PRA. Recent work of the first author showed that Π0 1CP + COH is equivalent to a weak variant of the BolzanoWeierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey’s theorem for pairs and two colors (RT2 2) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA0 + Σ0 2IA+RT2 2 are in T1. This is the fragment of Gödel’s system T containing only type 1 recursion — roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T1bounds from proofs that use RT2 2. Moreover, this yields a new proof of the fact that WKL0 + Σ0 2IA + RT2 2 is Π0 3conservative over RCA0 + Σ0 2IA. The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel’s functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π0 1comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard’s ordinal analysis of bar recursion (for RT2 2). 1.
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
A Constructive Model of Uniform Continuity
"... Abstract. We construct a continuous model of Gödel’s system T and its logic HA ω in which all functions from the Cantor space 2 N to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in partic ..."
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Abstract. We construct a continuous model of Gödel’s system T and its logic HA ω in which all functions from the Cantor space 2 N to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in particular, we can compute moduli of uniform continuity of Tdefinable functions 2 N → N. Moreover, the model has a continuous Fan functional of type (2 N → N) → N that calculates moduli of uniform continuity. We work with sheaves, and with a full subcategory of concrete sheaves that can be presented as sets with structure, which can be regarded as spaces, and whose natural transformations can be regarded as continuous maps.
Limit Spaces and Transfinite Types
, 1998
"... We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types. ..."
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We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.
Domain representations of spaces of compact subsets
, 2010
"... We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domai ..."
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We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.
cfl2004 Published by Elsevier
"... 2 Abstract. Synthetic topology as conceived in this monograph has three fundamental aspects: 1. to explain what has been done in classical topology in conceptual terms, 2. to provide oneline, enlightening proofs of the theorems that constitute the core of the theory, and 3. to smoothly export topol ..."
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2 Abstract. Synthetic topology as conceived in this monograph has three fundamental aspects: 1. to explain what has been done in classical topology in conceptual terms, 2. to provide oneline, enlightening proofs of the theorems that constitute the core of the theory, and 3. to smoothly export topological concepts and theorems to unintended situations, keeping the synthetic proofs unmodified.
12345efghi UNIVERSITY OF WALES SWANSEA REPORT SERIES
"... Computability on topological spaces via domain representations by V StoltenbergHansen and J V Tucker Report # CSR 22007Computability on topological spaces via domain representations ..."
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Computability on topological spaces via domain representations by V StoltenbergHansen and J V Tucker Report # CSR 22007Computability on topological spaces via domain representations
On the Expressive Power of Existential Quantification in PolynomialTime Computability
"... this paper to study the expressive power of bounded existential quantification in polynomialtime computability. Our goal was to characterize nondeterministic polynomialtime computations in a machineindependent way. The following considerations are intended to make our idea clear. Let # be the fin ..."
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this paper to study the expressive power of bounded existential quantification in polynomialtime computability. Our goal was to characterize nondeterministic polynomialtime computations in a machineindependent way. The following considerations are intended to make our idea clear. Let # be the finite alphabet