Computable concurrent processes

AbstractWe study relative computability for processes and process transformations, in general, and in particular the non-deterministic and concurrent processes which can be specified in terms of various fair merge constructs. The main result is a normal form theorem for these (relatively) computable process functions which implies that although they can be very complex when viewed as classical set-functions, they are all “loosely implementable” in the sense of Park (1980). The precise results are about the player model of concurrency introduced in Moschovakis (1991), which supports both fairness constructs and full recursion

An Intuitionistic Formula Hierarchy Based on High-School Identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism

On disjoint Borel uniformizations

Larman showed that any closed subset of the plane with uncountable vertical cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross-sections which do not have more than aleph_1 disjoint Borel uniformizations, even if the continuum is much larger than aleph_1. This negatively answers some questions of Mauldin. The proof is based on a result of Stern, stating that certain Borel sets cannot be written as a small union of low-level Borel sets. The proof of the latter result uses Steel's method of forcing with tagged trees; a full presentation of this method, written in terms of Baire category rather than forcing, is given here