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19
Semantics vs. Syntax vs. Computations  Machine Models For Type2 . . .
 JOURNAL OF COMPUTER AND SYSTEM SCIENCE
, 1997
"... This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their bas ..."
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This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic polynomialtime functionals. We show that if P = NP, these polynomialtime effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions. We also consider a weaker notion of polynomialtime effective operation where the machines computing these functionals have access to the computations of their procedural parameter, but not to its program text. For this version of polynomialtime effective operations, the analog of the KreiselLacombeShoenfield is shown to holdtheir power matches that of the basic feasible functionals on R.
Equational Theories for Inductive Types
 Annals of Pure and Applied Logic
, 1997
"... This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain synt ..."
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This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model; a version of the KreiselLacombeShoenfield theorem allows us to transfer the result from the domain model to the per model. 0 Introduction This paper concerns a typed calculus with inductive types which correspond semantically to initial algebras of (covariant) functors; the calculus lies between Godel's T and Girard's F in prooftheoretic strength. The goal of the paper is to analyse the structure of the model of this calculus given by the category PER of partial equivalence relations over the natural numbers. We shall show that ...
What is a Universal HigherOrder Programming Language?
 In Proc. International Conference on Automata, Languages, and Programming. Lecture Notes in Computer Science
, 1993
"... . In this paper, we develop a theory of higherorder computability suitable for comparing the expressiveness of sequential, deterministic programming languages. The theory is based on the construction of a new universal domain T and corresponding universal language KL. The domain T is universal for ..."
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. In this paper, we develop a theory of higherorder computability suitable for comparing the expressiveness of sequential, deterministic programming languages. The theory is based on the construction of a new universal domain T and corresponding universal language KL. The domain T is universal for observably sequential domains; KL can define all the computable elements of T, including the elements corresponding to computable observably sequential functions. In addition, domain embeddings in T preserve the maximality of finite elementspreserving the termination behavior of programs over the embedded domains. 1 Background and Motivation Classic recursion theory [7, 13, 18] asserts that all conventional programming languages are equally expressive because they can define all partial recursive functions over the natural numbers. This statement, however, is misleading because real programming languages support and enforce a more abstract view of data than bitstrings. In particular, mo...
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.
Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics, MartinLöf formalization, in Agda notation, of part of the paper with the same title
 University of Birmingham, UK, http://www.cs.bham.ac.uk/~mhe/papers/ omniscient/AnInfiniteOmniscientSet.html, September 2011. SETS IN CONSTRUCTIVE MATHEMATICS 21
"... Abstract. We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the onepoint compactification of the natural numbers, also k ..."
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Abstract. We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the onepoint compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot’s and Ishihara’s Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a wellfoundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below ɛ0, and richer type systems allow us to get higher. §1. Introduction. We show that there are plenty of infinite sets X that satisfy the omniscience principle for every function p: X → 2, ∃x ∈ X(p(x) = 0) ∨ ∀x ∈ X(p(x) = 1). For X finite this is trivial, and for X = N, this is LPO, the limited principle of omniscience, which of course is and will remain a taboo in any variety of
2007. Computability of simple games: A characterization and application to the core
"... It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that comp ..."
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It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted.
Metric Spaces in Synthetic Topology
, 2010
"... We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and me ..."
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We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree. 1
Kleene’s Amazing Second Recursion Theorem (Extended Abstract)
"... This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number ..."
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This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every nary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric