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Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
The Expressive Power of Indeterminate Primitives in Asynchronous Computation
 Proceedings of the Fifteenth Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes In Computer Science
, 1995
"... It has long been realized that the exigencies of systems programming require primitives that behave indeterminately. The bestknown dataflow primitive is the so called fair merge which abstracts aspects of fair resource allocation. It has been known for about two deacdes that fair primitives lead t ..."
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It has long been realized that the exigencies of systems programming require primitives that behave indeterminately. The bestknown dataflow primitive is the so called fair merge which abstracts aspects of fair resource allocation. It has been known for about two deacdes that fair primitives lead to unbounded indeterminacy. Around seven years ago E. W. Stark, Vasant Shanbhogue and I discovered that various variants of fair merge primitives, all manifesting unbounded indeterminacy, were provably different. These differences are based on simple monotonicity properties. In the present paper I review these results and discuss some related phenomena involving a fair stack. I then describe results about fair splitting. These results are based on topological properties rather than simple ordertheoretic properties. This gives some basic insight into what can and cannot be described by oracles and the relative power of various oracles. Finally I describe a result, implicitly due to Jim Russel...
A stable programming language
 I&C
"... It is wellknown that stable models (as dIdomains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cann ..."
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It is wellknown that stable models (as dIdomains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cannot be defined by programs ofPCF. In this paper, a paradigmatic programming language namedStPCF is proposed, which extends the languagePCF with two additional operators. The operational description of the extended language is presented in an effective way, although the evaluation of one of the new operators cannot be formalized in a PCFlike rewrite system. SinceStPCF can define all finite cliques of coherence spaces the above gap with stable models is filled, consequently stable models are fully abstract for the extended language. 1
Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for PCF and PCF+
 LOGICAL METHODS IN COMPUTER SCIENCE 3(3:7), 1–50 (2007)
, 2007
"... A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the sevent ..."
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A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given levelbylevel in the finite type hierarchy. To prove full abstraction and nondcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are nonomegacomplete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omegacompleteness), “naturally” continuous (with respect to existing directed “pointwise”, or “natural” lubs) and also “naturally” omegaalgebraic and “naturally” bounded complete—appropriate generalisation of the ordinary notions of domain theory to the case of nondcpos.
Hypergraphs and degrees of parallelism: A completeness result, in: I. Walukiewicz (Ed
 Proceedings of the 7th International Conference of Foundations of Software Science and Computation Structures – FOSSACS 2004
, 2004
"... Abstract. In order to study relative PCFdenability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [2, 4]). We introduce the notion of timed hypergraph morphism and show that it is: { Sound: if there exists a timed morphism from Hf to Hg then f is PCFde nab ..."
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Abstract. In order to study relative PCFdenability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [2, 4]). We introduce the notion of timed hypergraph morphism and show that it is: { Sound: if there exists a timed morphism from Hf to Hg then f is PCFde nable relatively to g. { Complete for subsequential functions: if f is PCFdenable relatively to g, and g is subsequential, then there exists a timed morphism from Hf to Hg. 1
INVESTIGATIONS ON RELATIVE DEFINABILITY IN PCF by
, 2005
"... The focus of this thesis is the study of relative definability of firstorder boolean functions with respect to the language PCF, a paradigmatic typed, higherorder language based on the simplytyped λcalculus. The basic core language is sequential. We study the effect of adding construct that embo ..."
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The focus of this thesis is the study of relative definability of firstorder boolean functions with respect to the language PCF, a paradigmatic typed, higherorder language based on the simplytyped λcalculus. The basic core language is sequential. We study the effect of adding construct that embody various notions of parallel execution. The resulting set of equivalence classes with respect to relative definability forms a supsemilattice analoguous to the lattice of degrees in recursion theory. Recent results of Bucciarelli show that the lattice of degrees of parallelism has both infinite chains and infinite antichains. By considering a very simple subset of Sieber’s sequentiality relations, we identify levels in the lattice and derive inexpressiblity results concerning functions on different levels. This allows us to explore further the structure of the lattice of degrees of parallelism and show the existence of new infinite hierarchies. We also identify four subsemilattices of this structure, all characterized by a simple property. ii Résumé Dans ce mémoire nous nous concentrons sur l’étude de la definition relative de fonctions
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"... We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has ..."
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We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (ChainAntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey’s Theorem for pairs (RT2 2) splits into a stable version (SRT22) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases these versions are strictly weaker (which is not known to be the case for RT 2 2 and SRT2 2). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2, showing for instance that WKL0 is incomparable with all of the systems we study; and prove computabilitytheoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman,
A “Book ” proof that parallel convergence tester cannot implement parallel or
, 2009
"... I give a short and elementary proof that paralle convergence tester cannot implement parallel or. Parallel convergence tester is a twoargument function c: O → O × O with the following graph: c(⊥, ⊥) = ⊥ c(⊥, ⊤) = ⊤ c(⊤, ⊥) = ⊤ c(⊤, ⊤) = ⊤. Parallel or is the wellknown function p: B × B → B de ..."
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I give a short and elementary proof that paralle convergence tester cannot implement parallel or. Parallel convergence tester is a twoargument function c: O → O × O with the following graph: c(⊥, ⊥) = ⊥ c(⊥, ⊤) = ⊤ c(⊤, ⊥) = ⊤ c(⊤, ⊤) = ⊤. Parallel or is the wellknown function p: B × B → B defined on the domain of booleans B with the following graph: p(ff,ff) = ff p(tt, ⊥) = tt p(⊥,ff) = ⊥ p(⊥,tt) = tt p(ff, ⊥) = ⊥ with all other values being determined by monotonicity. These functions arise in the discussion of full abstraction of PCF [Plo77] and the lazy λcalculus [AO93]. It is now wellknown that the lattice of degrees of parallelism is very rich and infinite in two directions [Buc97, PP01]: the fact that one cannot implement p with c is a tiny part of these results. There is, however, a very simple proof that PCF with c cannot implement p assuming that PCF by itself cannot implement p. Suppose that such an implementation exists so that there is some pure PCF context C[·] with C[c] = p. The functional λx.C[x] is monotone. Therefore the pure PCF 1 term C[λu.⊤] is extensionally above p, but p is maximal so the pure PCF term C[λu.⊤] = p, a contradiction. In fact this argument applies to any function with return type O even if it is horribly nonrecursive.