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

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 predicates 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.2.1 An established extension of BV with a self-dual quantifier An abstract syntax for predicates, and term-rewriting system for BVQ are defined in Fig 1. The rules can be applied to rewrite a predicate of the form on the left of the long right arrow to the predicate on the right (note rewriting -the direction of proof search -is in the opposite direction to linear implication). The key feature of the calculus of structures is deep inference, with is the ability to apply all rewrite rules in any context, i.e. predicates with a hole of the following form:The term-rewriting system is defined modulo a congruence, where a congruence is an equivalence relation that holds in any context. The congruence, ≡ in Fig. 1, makes par and times commutative and seq non-commutative in general. For the nominal quantifier ∇, the congruence enables: α-conversion for renaming bound names; equivariance which allows names bound by successive Technical report. Publication date: November 2017.Definition 2.2. A proof is a derivation P −→ I from a predicate P to the unit I. When such a derivation exists, we say that P is provable, and write ⊢ P holds.As a basic property of linear implication ⊢ P ⊸ P must hold for any P. Now assume that ⊢ Q ⊸ Q is provable in BVQ (hence, by the above definitions, there exists a derivation Q Q −→ I), and consider a formula of the form ∇x .Q. Using the unify rule, we can construct the following proof.Thus unify is necessary in order to guarantee reflexivity -the most basic property of implication -for an extension of BV with a self-dual nominal quantifier. In the next section, we explain why the unify rule is problematic for modelling proc...

Constructing weak simulations from linear implications for processes with private names

“…A proof in BV1 is a derivation of the form • P. When such a derivation exists, we say that P is provable, and write P. Cut elimination holds for BV1 as a consequence of cut-elimination for MAV1 (Horne et al 2016). A full proof appears in a companion paper (Horne et al 2018).…”

“…No further problematic cases are introduced by quantifiers. As a technical device, we define a form of restricted context called a killing context (Chaudhuri et al 2011) consisting of a hole surrounded by quantifiers ∀ and И. Splitting, proven in a companion paper as the main technical lemma required for proving cut elimination (Horne et al 2018), is formulated as follows for BV1. Traditionally, in the sequent calculus, any connective can be selected and a corresponding rule applied.…”

“…To ensure atomic interaction is always applied in a left context, the above derivation can be transformed to the derivation below. Splitting, proven in a companion paper as the main technical lemma required for proving cut elimination, Horne et al (2018), is formulated as follows for BV1. Traditionally, in the sequent calculus, any connective can be selected and a corresponding rule applied.…”

“…Details of cut elimination for MAV1 appear in a supporting technical report(Horne et al 2018).2 An extensive discussion on this implication appears in a companion paper(Horne et al 2018).…”

“…We also remark that linear implication is strictly finer than any interleaving preorder. † Details of cut elimination for MAV1 appear in a supporting technical report, Horne et al (2018).…”