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Universality results for models in locally Boolean domains
 IN COMPUTER SCIENCE LOGIC
, 2006
"... In [6] J. Laird has shown that an infinitary sequential extension of PCF has a fully abstract model in his category of locally boolean domains (introduced in [8]). In this paper we introduce an extension SPCF ∞ of his language by recursive types and show that it is universal for its model in locall ..."
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In [6] J. Laird has shown that an infinitary sequential extension of PCF has a fully abstract model in his category of locally boolean domains (introduced in [8]). In this paper we introduce an extension SPCF ∞ of his language by recursive types and show that it is universal for its model in locally boolean domains. Finally we consider an infinitary target language CPS ∞ for (the) CPS translation (of [16]) and show that it is universal for a model in locally boolean domains which is constructed like Dana Scott’s D ∞ where D = 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
Functional Reachability
"... Abstract—What is reachability in higherorder functional programs? We formulate reachability as a decision problem in the setting of the prototypical functional language PCF, and show that even in the recursionfree fragment generated from a finite base type, several versions of the reachability pro ..."
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Abstract—What is reachability in higherorder functional programs? We formulate reachability as a decision problem in the setting of the prototypical functional language PCF, and show that even in the recursionfree fragment generated from a finite base type, several versions of the reachability problem are undecidable from order 4 onwards, and several other versions are reducible to each other. We characterise a version of the reachability problem in terms of a new class of tree automata introduced by Stirling at FoSSaCS 2009, called Alternating Dependency Tree Automata (ADTA). As a corollary, we prove that the ADTA nonemptiness problem is undecidable, thus resolving an open problem raised by Stirling. However, by restricting to contexts constructible from a finite set of variable names, we show that the corresponding solution set of a given instance of the reachability problem is regular. Hence the relativised reachability problem is decidable. I.
An
, 2012
"... Under consideration for publication in Math. Struct. in Comp. Science ..."
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The Primitive Recursive Functions are Recursively Enumerable
"... Abstract. Metaoperations on primitive recursive functions sit at the brink of what is computationally possible: the semantic equality of primitive recursive programs is undecidable, and yet this paper shows that the whole class of p.r. functions can be enumerated without semantic duplicates. More g ..."
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Abstract. Metaoperations on primitive recursive functions sit at the brink of what is computationally possible: the semantic equality of primitive recursive programs is undecidable, and yet this paper shows that the whole class of p.r. functions can be enumerated without semantic duplicates. More generally, the construction shows that for any equivalence relation≈on natural numbers, N/ ≈ is r.e. if≈is cosemidecidable.
Natural nondcpo Domains and fSpaces Abstract
"... hereditarilysequential functionals is not ωcomplete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially (universal) model for PCF + = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natura ..."
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hereditarilysequential functionals is not ωcomplete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially (universal) model for PCF + = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natural ’ domains which, although being nondcpos, allow considering ‘naturally ’ continuous functions (with respect to existing directed ‘pointwise’, or ‘natural ’ least upper bounds). There is also an appropriate version of ‘naturally ’ algebraic and ‘naturally ’ bounded complete ‘natural’ domains which serves as the nondcpo analogue of the wellknown concept of Scott domains, or equivalently, the complete fspaces of Ershov. It is shown that this special version of ‘natural ’ domains, if considered under ‘natural ’ Scott topology, exactly corresponds to the class of fspaces, not necessarily complete. Key words: domain theory, dcpo and nondcpo domains, Scott topology,
An
, 2013
"... Under consideration for publication in Math. Struct. in Comp. Science ..."
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