This paper axiomatises the structure of bigraphs, and proves that the resulting theory is complete. Bigraphs are graphs with double structure, representing locality and connectivity. They have been shown to represent dynamic theories for the $\pi$-calculus, mobile ambients and Petri nets in a way that is faithful to each of those models of discrete behaviour. While the main purpose of bigraphs is to understand mobile systems, a prerequisite for this understanding is a well-behaved theory of the structure of states in such systems. The algebra of bigraph structure is surprisingly simple, as this paper demonstrates; this is because bigraphs treat locality and connectivity orthogonally.

The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource that is rich enough, for example, to form the logical basis for ‘pointer logic’ and ‘separation logic’ semantics for programs that manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels for BI's tableaux to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, $\bot$, the challenge consists in dealing with BI's Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. Then, from these results, we can define a new resource semantics of BI, based on partially defined monoids, and prove that this semantics is complete. Such a semantics, based on partiality, is closely related to the semantics of BI's (intuitionistic) pointer and separation logics. Returning to the tableaux calculus, we propose a new version with liberalised rules for which the countermodels are closely related to the topological Kripke semantics of BI. As consequences of the relationships between semantics of BI and resource tableaux, we prove two new strong results for propositional BI: its decidability and the finite model property with respect to topological semantics.

We present a modular correspondence between various categorical structures and their internal languages in terms of extensional dependent type theories à la Martin-Löf. Starting from lex categories, through regular ones, we provide internal languages of pretopoi and topoi and some variations of them, such as, for example, Heyting pretopoi.

With respect to the internal languages already known for some of these categories, such as topoi, the novelty of these calculi is that formulas corresponding to subobjects can be regained as particular types that are equipped with proof-terms according to the isomorphism ‘propositions as mono types’, which was invisible in previously described internal languages.

We develop a multidimensional syntax for cut-free proofs of Multiplicative Linear Logic. This syntax is essentially equivalent to the traditional formalism of proof-nets; the interest of the multi-dimensional formalism consists in its explicit relationship with the formalism of bordisms. Bordisms are compact manifolds with boundary, which are treated as morphisms between the ‘incoming’ and ‘outgoing’ boundary components (composition is given by glueing bordisms along matching boundaries). The category of bordisms has recently become important in contemporary mathematics, in particular, because of developments in topological quantum theory and quantum gravity. A semantics of MLL underlying the multi-dimensional syntax is based on a certain category of bordisms, which we call ‘coherent space-times’. The resulting model has an extremely intuitive geometric description. The dual multiplicative connectives $\otimes$ and $\smash{\raise5.99pt\hbox{\rotatebox{-180}{\&}}}$ correspond simply to disjoint unions and connected sums of bordisms. Following ideas from topological quantum field theory, we also discover deep relationships between this new model and the author's coherent phase spaces model (Slavnov 2003), which is based on the context of symplectic geometry.