Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications syntax. We also show that while the category of containers has finite limits, it is not locally cartesian closed. 1

Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications syntax. We also show that the category of containers has finite limits, but it is not locally cartesian closed. 1

My thesis is about foundations for formal semantics of name-passing process calculi. These calculi are languages for describing systems of agents that communicate channel names along named channels. This facility provides a natural way of describing the mobility of communication links. (The π-calculus of Milner et al. [1992] is a paradigmatic example of such a language.) The thesis is split into two parts, reflecting the two aspects of the foundations of name-passing calculi that are addressed. • Part I of the thesis is dedicated to operational models for name-passing calculi. Conventional operational models, such as labelled transition systems, are inappropriate for name-passing systems. For this reason I develop and relate two different models of name-passing from the literature: indexed labelled transition systems, based on work of Cattani and Sewell [2004], and a coalgebraic approach introduced by Fiore and Turi [2001]. Connections are made with the History Dependent Automata of Montanari and Pistore [2005], and I introduce a new operational model using the nominal logic of Pitts [2003]. • Part II of the thesis concerns structural operational semantics for name-passing calculi. Various work has been done on the meaning of rule-based transition system specifications, and on

We describe a correspondence between (i) rule-based inductive definitions in the Positive GSOS format, i.e. without negative premises, and (ii) distributive laws in the spirit of Turi and Plotkin (LICS’97), that are suitably monotone. This result can be understood as an isomorphism of lattices, in which the prime elements are the individual rules. We fix a signature, i.e. a set S of operators, each op ∈ S having an arity, ar(op) ∈ N. We also fix a set L of labels. We are concerned with labelled transition relations over the terms of the signature. For an example, we recall a fragment of Milner’s CCS. Fix a set A of actions. There is a binary operation for parallel composition (written x|y), a constant for deadlock, and two unary operations for every action a ∈ A, for prefix and co-action prefix. The set of labels contains actions, co-actions, and silent steps: L = A + A + 1. x a − → x ′ y ā − → y ′ x|y τ − → x ′ |y ′ A labelled transition relation over CCS terms is defined by a transition system specification (TSS). This is a set of rules, including the one in the box on the right. Background on the Positive GSOS rule format li j − → yi j | i ≤ ar(op), j ∈ Ji} {xi lik

Consider a process calculus that allows agents to communicate values. The structural operational semantics involves substitution of values for variables. Existing rule formats, such as the GSOS format, do not allow this kind of explicit substitution in the semantic rules. We investigate how to derive rule formats for languages with substitution, by using categorical logic to interpret the framework of the GSOS format in different categories. The categories in question are categories of ‘substitution actions’. 1 A simple language for value-passing To set the scene, fix a set of channel names, and consider a set V of value-expressions, that includes the channel names. A simple untyped value-passing process language, V-CCS, is given in Figure 1 (c.f. [8]). The precise value expressions of V are not important, but note that since V includes the (static) channel names, V-CCS is a very primitive applied π-calculus without restriction or name generation; c.f. [1]. For the sake of illustration, consider the set Vex of value expressions determined by the following grammar: v:: = n | v + v | (v, v) | π1(v) | π2(v) | c (n is a number, c is a channel name). We will always work with value expressions up-to the evident equations (2 + 3 = 5; π1(v, w) = v; etc.), rather than explicitly evaluating or normalizing them; this is to simplify the presentation. The following transitions are derivable in Vex-CCS. (¯c〈3〉.0) | (c(v).¯c〈2 + v〉.0) τ