An Intuitionistic Formula Hierarchy Based on High-School Identities

TL;DR

This paper introduces an exponential polynomial framework for intuitionistic logic, leading to a compact proof system and a hierarchy of formulas that preserves isomorphism, offering new insights into proof representations and classifications.

Contribution

It develops a novel exponential polynomial approach to intuitionistic logic, resulting in a compact proof system and a hierarchy of formulas that maintains isomorphism.

Findings

Invertible proof rules correspond to high-school identities

A compact proof system with non-invertible rules and normalization

An intuitionistic hierarchy of formulas analogous to the arithmetical hierarchy

Abstract

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variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure.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.