The scope of computation has expanded dramatically beyond the rubric of discrete, deterministic sequential computation under which it has been studied for many decades. That focus, of course, led to a great deal of deep and beautiful theory, but our focus in this special issue of Mathematical Structures in Computer Science is on new directions that have emerged from the study of computational phenomena in other settings, and thus on a celebration of the diversity of ideas, methods, new applications and novel sources of inspiration that have marked the modern era. The papers in this issue come from sources extending far beyond the core of computer science, yet using many of the central ideas that have evolved within computer science and mathematics. The nexus of all this activity has been, on the one hand, the boundary between logic and computation, and, on the other hand, the natural sciences, particularly physics and biology. The papers in this collection are expanded versions of selected papers from the DCM 2010 workshop, which was held in Edinburgh in July 2010. The theme of the workshop was Causality, Computation and Physics.

Indexed monoidal algebras are introduced as an equivalent structure for self-dual compact closed categories, and a coherence theorem is proved for the category of such algebras. Turing automata and Turing graph machines are defined by generalising the classical Turing machine concept so that the collection of such machines becomes an indexed monoidal algebra. On the analogy of the von Neumann data-flow computer architecture, Turing graph machines are proposed as potentially reversible low-level universal computational devices, and a truly reversible molecular size hardware model is presented as an example.

We investigate the computing power of a restricted class of DNA strand displacement structures: those that are made of double strands with nicks (interruptions) in the top strand. To preserve this structural invariant, we impose restrictions on the single strands they interact with: we consider only two-domain single strands consisting of one toehold domain and one recognition domain. We study fork and join signal processing gates based on these structures, and show that these systems are amenable to formalisation and mechanical verification.

A generalised framework of site graphs is introduced in order to provide the first fully semantic definition of the side-effect-free core of the rule-based language Kappa. This formalisation allows the use of types either to confirm that a rule respects a certain invariant or to guide a restricted refinement process that allows us to constrain its run-time applicability.

This paper is concerned with the asymptotic properties of a restricted class of Petri nets equipped with stochastic mass-action semantics. We establish a simple algebraic criterion for the existence of an equilibrium, that is to say, an invariant probability that satisfies the detailed balance condition familiar from the thermodynamics of reaction networks. We also find that when such a probability exists, it can be described by a free energy function that combines an internal energy term and an entropy term. Under strong additional conditions, we show how the entropy term can be deconstructed using the finer-grained individual-token semantics of Petri nets.

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits ‘enough adhesivity’ from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are ‘type safe’ in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discrete and finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parameterised by graphical signatures, which are similar to the monoidal signatures of Joyal and Street, and define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs and more general monoidal theories. Thus, open-graphs give us a tool for mechanised reasoning in monoidal categories.

We give a complete structural characterisation of the map implemented by the positive branch of a one-way pattern. Our approach is based on the phase map decomposition (de Beaudrap et al. 2006; de Beaudrap et al. 2008) and leads to some preliminary results on the connection between the column structure of a given unitary and the angles of measurements in a pattern that implements it. Our characterisation highlights the role of entanglement in the efficiency of the simulation of a one-way pattern, and it is a step forward towards a full characterisation of those unitaries that have an efficient one-way model implementation.

We discuss quantum algorithms based on the Bernstein–Vazirani algorithm for finding which input variables a Boolean function depends on. There are 2n possible linear Boolean functions of n input variables; given a linear Boolean function, the Bernstein–Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. We show how the same quantum algorithm can also be used to learn which input variables any other type of Boolean function} depends on. The success probability of learning that the function depends on a particular input variable depends on} the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure based on another quantum algorithm, the Grover search, to amplify further the success probability. Finally, we discuss quantum algorithms for learning the exact form of certain quadratic and cubic Boolean functions.

In this paper we consider general probabilistic theories pertaining to circuits that satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of space–time employ only mathematical objects associated with that region. We call this formalism locality. It incorporates the idea that space and time should be treated on an equal footing. Formulations that use a foliation of space--time to evolve a state do not have this property, nor do histories-based approaches. An operation has inputs and outputs (through which systems travel), for example,

We study how graph states on fractal lattices can be used to perform measurement-based quantum computation, and investigate which topological features allow this application. We find fractal lattices of arbitrary dimension greater than one that all act as good resources for measurement-based quantum computation, and sets of fractal lattices with dimension greater than one that do not. The difference is put down to other topological factors such as ramification and connectivity. This is in direct analogy to the tendency of lattices to observe criticality in spin systems. We also discuss the analogy between thermodynamics and one-way computation in this context. This work adds confidence to the analogy and highlights new features of what we require for universal resources for measurement-based quantum computation. This paper is an extended version of Markham et al. (2010), which appeared in the proceedings of DCM 2010.

We compare three different characterisations of the globalness of bipartite unitary operations, namely, delocalisation power, entanglement cost and entangling power, to investigate the global properties of unitary operations. We show that the globalness of the same unitary operation depends on whether input states are given by unknown states representing pieces of quantum information or a set of known states for the characterisation. We extend our analysis of delocalisation power in two ways. First we show that the delocalisation power differs according to whether the global operation is applied to one piece or two pieces of quantum information. Then we introduce a new task called LOCC one-piece relocation, and prove that the controlled-unitary operations do not have enough delocalisation power to relocate one of two pieces of quantum information by adding LOCC.

We present a simple stochastic rule-based approach to multi-level modelling for computational systems biology. Populations are modelled using multi-level multisets; these contain both species and agents, with the latter possibly containing further such multisets. Rules are pairs of such multisets, but they may now also include variables (as well as species and agents), together with an associated stochastic rate.

We give two illustrative examples. The first is an extracellular model of virus infection, coupled with an intracellular model of viral reproduction; this model can demonstrate successive waves of infection. The second is a model of cell division in which a repressor protein is diluted in successive generations, so eventually repression no longer occurs. The multi-level multiset approach can also be seen in terms of stochastic term rewriting for the theory of a commutative monoid equipped with extra constants (for the species) and unary operations (for the agents). We further discuss the relationship of this approach with two others: Krivine et al.'s stochastic bigraphs, restricted to Milner's place graphs, and Coppo et al.'s Stochastic Calculus of Wrapped Compartments. These various relationships provide evidence for the fundamental nature of the approach.

Lambda-calculi with vectorial structures have been studied in various ways, but their semantics remain largely uninvestigated. The main contribution of this paper is to provide a categorical framework for the semantics of such algebraic lambda-calculi. We first develop a categorical analysis of a general simply typed lambda-calculus endowed with the structure of a module. We study the problems arising from the addition of a fixed-point combinator and show how to modify the equational theory to solve them. The categorical analysis carries nicely over to the modified language. We provide various concrete models for both the case without fixpoints and for the case with them.