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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 PCFdefinability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [3, 5]). 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 PCFdefin ..."
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Abstract. In order to study relative PCFdefinability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [3, 5]). 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 PCFdefinable relatively to g. – Complete for subsequential functions: if f is PCFdefinable relatively to g, and g is subsequential, then there exists a timed morphism from Hf to Hg. We show that the problem of deciding the existence of a timed morphism between two given hypergraphs is NPcomplete. 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