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Infinite sets that admit fast exhaustive search
 In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science
, 2007
"... Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive sea ..."
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Cited by 14 (8 self)
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Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive search over infinite sets be performed? Keywords. Highertype computability and complexity, Kleene–Kreisel functionals, PCF, Haskell, topology. 1.
Exhaustible sets in highertype computation
 Logical Methods in Computer Science
"... Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no examp ..."
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Cited by 13 (12 self)
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Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications. 1.
An operational domaintheoretic treatment of recursive types
 in: TwentySecond Mathematical Foundations of Programming Semantics
, 2006
"... We develop a domain theory for treating recursive types with respect to contextual equivalence. The principal approach taken here deviates from classical domain theory in that we do not produce the recursive types via the usual inverse limits constructions we have it for free by working directly wi ..."
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Cited by 2 (2 self)
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We develop a domain theory for treating recursive types with respect to contextual equivalence. The principal approach taken here deviates from classical domain theory in that we do not produce the recursive types via the usual inverse limits constructions we have it for free by working directly with the operational semantics. By extending type expressions to endofunctors on a ‘syntactic ’ category, we establish algebraic compactness. To do this, we rely on an operational version of the minimal invariance property. In addition, we apply techniques developed herein to reason about FPC programs. Key words: Operational domain theory, recursive types, FPC, realisable functor, algebraic compactness, generic approximation lemma, denotational semantics 1
A HofmannMislove theorem for bitopological spaces
, 2007
"... We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that ar ..."
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We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated HofmannMislove theorem for sober spaces. We link the characterisation to Taylor’s and Escardó’s reading of the HofmannMislove theorem as continuous quantification over a subspace.
niques]: functional programming
, 2010
"... This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a highertype functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a compu ..."
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This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a highertype functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the DoubleNegation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation. Therefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here). D.1.1 [Programming tech
External Examiner
, 2006
"... The results reported in Part III consist of joint work with Martín Escardó [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of th ..."
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The results reported in Part III consist of joint work with Martín Escardó [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of the thesis, produced on October 31, 2006, is the result of completing all the minor modifications as suggested by both the examiners in the viva report (Ref: CLM/AC/497773). We develop an operational domain theory to reason about programs in sequential functional languages. The central idea is to export domaintheoretic techniques of the Scott denotational semantics directly to the study of contextual preorder and equivalence. We investigate to what extent this can be done for two deterministic functional programming languages: PCF (Programminglanguage for Computable Functionals) and FPC (Fixed Point Calculus).
in a highertype setting
"... We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic prog ..."
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We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semidecidability of similar tests under the simultaneous presence of nondeterministic and probabilistic choice. Keywords: Nondeterministic and probabilistic computation, highertype computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains. 1
EXHAUSTIBLE SETS IN HIGHERTYPE COMPUTATION
, 808
"... Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no examp ..."
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Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications. 1.