On the Expressive Power of First-Order Boolean Functions in PCF

TL;DR

This paper investigates the structure of the semilattice of degrees of parallelism of first-order boolean functions in PCF, revealing infinite hierarchies and inexpressibility results related to levels of sequentiality.

Contribution

It introduces a level-based analysis of the semilattice, identifying new hierarchies and characterizing semilattices with simple level properties.

Findings

Existence of infinite chains and antichains in the semilattice.

Identification of levels in the semilattice using Sieber's sequentiality relations.

Discovery of natural infinite hierarchies related to these levels.

Abstract

Recent results of Bucciarelli show that the semilattice of degrees of parallelism of first-order boolean functions in PCF has both infinite chains and infinite antichains. By considering a simple subclass of Sieber's sequentiality relations, we identify levels in the semilattice and derive inexpressibility results concerning functions on different levels. This allows us to further explore the structure of the semilattice of degrees of parallelism: we identify semilattices characterized by simple level properties, and show the existence of new infinite hierarchies which are in a certain sense natural with respect to the levels.