Results

**11 - 19**of**19**### Fundamentals of Computing I

- Logic, Problem Solving, Programs, & Computers
, 1992

"... on topological spaces via domain representations ..."

### On the Expressive Power of Existential Quantification in Polynomial-Time Computability

"... this paper to study the expressive power of bounded existential quantification in polynomial-time computability. Our goal was to characterize nondeterministic polynomial-time computations in a machine-independent 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 polynomial-time computability. Our goal was to characterize nondeterministic polynomial-time computations in a machine-independent way. The following considerations are intended to make our idea clear. Let # be the finite alphabet

### Domain-Theoretic Methods for Program Synthesis

"... formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group ..."

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formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group at the University of Munich (7). As a former member of this group I was mainly involved in the theoretical background steering the implementation of the system. The system also exploits the so-called proofs-as-programs paradigm as a logical approach to correct software development: from a formal proof that a certain specication has a solution one fully automatically extracts a program that provably meets the specication. We carried out a number of extended case studies extracting programs from proofs in areas such as arithmetic (6), graph theory (7), innitary combinatorics (7), and lambda calculus (1,2). Special emphasis has been put on an ecient implemen

### 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.

### TERM EXTRACTION AND RAMSEY’S THEOREM FOR PAIRS

"... Abstract. In this paper we study with proof-theoretic 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 1-CP are primitive recursive. This ..."

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Abstract. In this paper we study with proof-theoretic 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 1-CP 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 1-CP + COH is Π0 2-conservative over PRA. Recent work of the first author showed that Π0 1-CP + COH is equivalent to a weak variant of the Bolzano-Weierstraß 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 2-IA+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 T1-bounds from proofs that use RT2 2. Moreover, this yields a new proof of the fact that WKL0 + Σ0 2-IA + RT2 2 is Π0 3-conservative over RCA0 + Σ0 2-IA. 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 1-comprehension. 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.

### 12345efghi UNIVERSITY OF WALES SWANSEA REPORT SERIES

"... Computability on topological spaces via domain representations by V Stoltenberg-Hansen and J V Tucker Report # CSR 2-2007Computability on topological spaces via domain representations ..."

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Computability on topological spaces via domain representations by V Stoltenberg-Hansen and J V Tucker Report # CSR 2-2007Computability on topological spaces via domain representations