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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...

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The calculus of looping sequences is a formalism for describing evolution of biological systems by means of term rewriting rules. We propose to enrich this calculus with type disciplines to guarantee the soundness of reduction rules with respect to interesting biological properties.

Reversible computation allows computation to proceed not only in the standard, forward direction, but also backward, recovering past states. While reversible computation has attracted interest for its multiple applications, covering areas as different as low-power computing, simulation, robotics and debugging, such applications need to be supported by a clear understanding of the foundations of reversible computation. We report below on many threads of research in the area of foundations of reversible computing, giving particular emphasis to the results obtained in the framework of the European COST Action IC1405, entitled "Reversible Computation-Extending Horizons of Computing", which took place in the years 2015-2019.

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