Cartesian closed 2-categories and permutation equivalence in higher-order rewriting

We propose a semantics for permutation equivalence in higher-order rewriting. This semantics takes place in cartesian closed 2-categories, and is proved sound and complete

Full abstraction for fair testing in CCS (expanded version)

In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent form of presheaf semantics and as a concurrent form of game semantics. We define in this setting an analogue of fair testing equivalence, which we prove fully abstract w.r.t. standard fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the `rule of the game'. From any playground, we derive two languages equipped with labelled transition systems, as well as a strong, functional bisimulation between them.Comment: 80 page

Shapely monads and analytic functors

In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that "any two operations of the same shape agree". An important part of this work is the study of analytic functors between presheaf categories, which are a common generalisation of Joyal's analytic endofunctors on sets and of the parametric right adjoint functors on presheaf categories introduced by Diers and studied by Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among the analytic endofunctors, and may be characterised as the submonads of a universal analytic monad with "exactly one operation of each shape". In fact, shapeliness also gives a way to define the data and axioms of a structure directly from its graphical calculus, by generating a free shapely monad on the basic operations of the calculus. In this paper we do this for some of the examples listed above; in future work, we intend to do so for graphical calculi such as Milner's bigraphs, Lafont's interaction nets, or Girard's multiplicative proof nets, thereby obtaining canonical notions of denotational model

Graphical Presentations of Symmetric Monoidal Closed Theories

We define a notion of symmetric monoidal closed (SMC) theory, consisting of a SMC signature augmented with equations, and describe the classifying categories of such theories in terms of proof nets.Comment: Uses Paul Taylor's diagram

Rigid Mixin Modules

International audienceMixin modules are a notion of modules that allows cross-module recursion and late binding, two features missing in ML-style modules. They have been well defined in a call-by-name setting, but in a call-by-value setting, they tend to conflict with the usual static restrictions on recursive definitions. Moreover, the semantics of instantiation has to specify an order of evaluation, which involves a difficult design choice. Previous proposals rely on the dependencies between components to compute a valid order of evaluation. In such systems, mixin module types must carry some information on the dependencies between their components, which makes them verbose. In this paper, we propose a new, simpler design for mixin modules in a call-by-value setting, which avoids this problem

Modules over monads and operational semantics

This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as lambda-bar-mu-calculus, pi-calculus, Positive GSOS specifications, differential lambda-calculus, and the big-step, simply-typed, call-by-value lambda-calculus. Moreover, we design a suitable notion of signature for transition monads

Justified sequences in string diagrams: A comparison between two approaches to concurrent game semantics

We compare two approaches to concurrent game semantics, one by Tsukada and Ong for a simply-typed λ-calculus and the other by the authors and collaborators for CCS and the π-calculus. The two approaches are obviously related, as they both define strategies as sheaves for the Grothendieck topology induced by embedding ‘views’ into ‘plays’. However, despite this superficial similarity, the notions of views and plays differ significantly: the former is based on standard justified sequences, the latter uses string diagrams. In this paper, we relate both approaches at the level of plays. Specifically, we design a notion of play (resp. view) for the simply-typed λ-calculus, based on string diagrams as in our previous work, into which we fully embed Tsukada and Ong's plays (resp. views). We further provide a categorical explanation of why both notions yield essentially the same model, thus demonstrating that the difference is a matter of presentation. In passing, we introduce an abstract framework for producing sheaf models based on string diagrams, which unifies our present and previous models

Modules over Monads and Operational Semantics

This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as ???-calculus, ?-calculus, Positive GSOS specifications, differential ?-calculus, and the big-step, simply-typed, call-by-value ?-calculus. Finally, we design a suitable notion of signature for transition monads