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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 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,