De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic

TL;DR

This paper introduces MAV1, a new first-order logic system with dual nominal quantifiers 'new' and 'wen' that enable modeling private names in processes, supported by a proof theory establishing cut elimination and normalization.

Contribution

The paper presents MAV1, a novel logic with polarized nominal quantifiers, and develops a proof theory including cut elimination for modeling private names in process calculus.

Findings

MAV1 successfully models private names in process calculus.

Cut elimination is established for MAV1, ensuring consistency.

Novel techniques for proof normalization are introduced.

Abstract

This paper explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a de Morgan dual pair of nominal quantifiers. These nominal quantifiers called `new' and `wen' are distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of these nominal quantifiers is they are polarised in the sense that `new' distributes over positive operators while `wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as formulae in MAV1. The technical challenge is to establish a cut elimination result, from which essential properties including the transitivity of implication follow. Since the system is defined using the calculus of structures, a generalisation of the sequent calculus, novel techniques are employed. The proof relies on an intricately designed multiset-based measure of the size of a proof, which is used to guide a normalisation technique called splitting. The presence of equivariance, which swaps successive quantifiers, induces complex inter-dependencies between nominal quantifiers, additive conjunction and multiplicative operators in the proof of splitting. Every rule is justified by an example demonstrating why the rule is necessary for soundly embedding processes and ensuring that cut elimination holds.