De Morgan dual nominal quantifiers modelling private names in non-commutative logic

Ross Horne, Alwen Tiu, Bogdan Aman, Gabriel Ciobanu

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

This article 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.

Original languageEnglish
Article number4
Pages (from-to)1-44
Number of pages44
JournalACM Transactions on Computational Logic
Volume20
Issue number4
DOIs
Publication statusPublished - 16 Jul 2019

Keywords

  • calculus of structures
  • nominal logic
  • non-commutative logic
  • theory of computation
  • models of computation

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