This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation: Quantum Key Distribution, the quantum Fourier transform, and the teleportation protocol.

This work is about diagrammatic languages, how they can be represented, and what they in turn can be used to represent. More specifically, it focuses on representations and applications of string diagrams. String diagrams are used to represent a collection of processes, depicted as "boxes" with multiple (typed) inputs and outputs, depicted as "wires". If we allow plugging input and output wires together, we can intuitively represent complex compositions of processes, formalised as morphisms in a monoidal category. While string diagrams are very intuitive, existing methods for defining them rigorously rely on topological notions that do not extend naturally to automated computation. The first major contribution of this dissertation is the introduction of a discretised version of a string diagram called a string graph. String graphs form a partial adhesive category, so they can be manipulated using double-pushout graph rewriting. Furthermore, we show how string graphs modulo a rewrite system can be used to construct free symmetric traced and compact closed categories on a monoidal signature. The second contribution is in the application of graphical languages to quantum information theory. We use a mixture of diagrammatic and algebraic techniques to prove a new classification result for strongly complementary observables. Namely, maximal sets of strongly complementary observables of dimension D must be of size no larger than 2, and are in 1-to-1 correspondence with the Abelian groups of order D. We also introduce a graphical language for multipartite entanglement and illustrate a simple graphical axiom that distinguishes the two maximally-entangled tripartite qubit states: GHZ and W. Notably, we illustrate how the algebraic structures induced by these operations correspond to the (partial) arithmetic operations of addition and multiplication on the complex projective line. The third contribution is a description of two software tools developed in part by the author to implement much of the theoretical content described here. The first tool is Quantomatic, a desktop application for building string graphs and graphical theories, as well as performing automated graph rewriting visually. The second is QuantoCoSy, which performs fully automated, model-driven theory creation using a procedure called conjecture synthesis.

Probabilistic model checking is a powerful formal verification method that can ensure the correctness of real-life systems that exhibit stochastic behaviour. The work presented in this thesis aims to solve the scalability challenge of probabilistic model checking, by developing, for the first time, fully-automated compositional verification techniques for probabilistic systems. The contributions are novel approaches for automatically learning probabilistic assumptions for three different compositional verification frameworks. The first framework considers systems that are modelled as Segala probabilistic automata, with assumptions captured by probabilistic safety properties. A fully-automated approach is developed to learn assumptions for various assume-guarantee rules, including an asymmetric rule Asym for two-component systems, an asymmetric rule Asym-N for n-component systems, and a circular rule Circ. This approach uses the L* and NL* algorithms for automata learning. The second framework considers systems where the components are modelled as probabilistic I/O systems (PIOSs), with assumptions represented by Rabin probabilistic automata (RPAs). A new (complete) assume-guarantee rule Asym-Pios is proposed for this framework. In order to develop a fully-automated approach for learning assumptions and performing compositional verification based on the rule Asym-Pios, a (semi-)algorithm to check language inclusion of RPAs and an L*-style learning method for RPAs are also proposed. The third framework considers the compositional verification of discrete-time Markov chains (DTMCs) encoded in Boolean formulae, with assumptions represented as Interval DTMCs (IDTMCs). A new parallel operator for composing an IDTMC and a DTMC is defined, and a new (complete) assume-guarantee rule Asym-Idtmc that uses this operator is proposed. A fully-automated approach is formulated to learn assumptions for rule Asym-Idtmc, using the CDNF learning algorithm and a new symbolic reachability analysis algorithm for IDTMCs. All approaches proposed in this thesis have been implemented as prototype tools and applied to a range of benchmark case studies. Experimental results show that these approaches are helpful for automating the compositional verification of probabilistic systems through learning small assumptions, but may suffer from high computational complexity or even undecidability. The techniques developed in this thesis can assist in developing scalable verification frameworks for probabilistic models.

Alternating timed automata are a powerful extension of classical Alur-Dill timed automata that are closed under all Boolean operations. They have played a key role, among others, in providing verification algorithms for prominent specification formalisms such as Metric Temporal Logic. Unfortunately, when interpreted over an infinite dense time domain (such as the reals), alternating timed automata have an undecidable language emptiness problem. In this thesis we consider restrictions on this model that restore the decidability of the language emptiness problem. We consider the restricted class of safety alternating timed automata, which can encode a corresponding Safety fragment of Metric Temporal Logic. This thesis connects these two formalisms with insertion channel machines, a model of faulty communication, and demonstrates that the three formalisms are interreducible. We thus prove a non-elementary lower bound for the language emptiness problem for 1-clock safety alternating timed automata and further obtain a new proof of decidability for this problem. Complementing the restriction to safety properties, we consider interpreting the automata over bounded dense time domains. We prove that the time-bounded language emptiness problem is decidable but non-elementary for unrestricted alternating timed automata. The language emptiness problem for alternating timed automata is a special case of a much more general and abstract logical problem: Church's synthesis problem. Given a logical specification S(I,O), Church's problem is to determine whether there exists an operator F that implements the specification in the sense that S(I,F(I)) holds for all inputs I. It is a classical result that the synthesis problem is decidable in the case that the specification and implementation are given in monadic second-order logic over the naturals. We prove that this decidability extends to MSO over the reals with order and furthermore to MSO over every fixed bounded interval of the reals with order and the +1 relation.

The aim of this thesis is to present an extension to the string graphs of Dixon, Duncan and Kissinger that allows the finite representation of certain infinite families of graphs and graph rewrite rules, and to demonstrate that a logic can be built on this to allow the formalisation of inductive proofs in the string diagrams of compact closed and traced symmetric monoidal categories. String diagrams provide an intuitive method for reasoning about monoidal categories. However, this does not negate the ability for those using them to make mistakes in proofs. To this end, there is a project (Quantomatic) to build a proof assistant for string diagrams, at least for those based on categories with a notion of trace. The development of string graphs has provided a combinatorial formalisation of string diagrams, laying the foundations for this project. The prevalence of commutative Frobenius algebras (CFAs) in quantum information theory, a major application area of these diagrams, has led to the use of variable-arity nodes as a shorthand for normalised networks of Frobenius algebra morphisms, so-called "spider notation". This notation greatly eases reasoning with CFAs, but string graphs are inadequate to properly encode this reasoning. This dissertation firstly extends string graphs to allow for variable-arity nodes to be represented at all, and then introduces !-box notation – and structures to encode it – to represent string graph equations containing repeated subgraphs, where the number of repetitions is abitrary. This can be used to represent, for example, the "spider law" of CFAs, allowing two spiders to be merged, as well as the much more complex generalised bialgebra law that can arise from two interacting CFAs. This work then demonstrates how we can reason directly about !-graphs, viewed as (typically infinite) families of string graphs. Of particular note is the presentation of a form of graph-based induction, allowing the formal encoding of proofs that previously could only be represented as a mix of string diagrams and explanatory text.

In this dissertation, we study the verification of concurrent programs written in the programming language Erlang using infinite-state model-checking. Erlang is a widely used, higher order, dynamically typed, call-by-value functional language with algebraic data types and pattern-matching. It is further augmented with support for actor concurrency, i.e. asynchronous message passing and dynamic process creation. With decidable model-checking in mind, we identify actor communicating systems (ACS) as a suitable target model for an abstract interpretation of Erlang. ACS model a dynamic network of finite-state processes that communicate over a fixed, finite number of unordered, unbounded channels. Thanks to being equivalent to Petri nets, ACS enjoy good algorithmic properties. We develop a verification procedure that extracts a sound abstract model, in the form of an ACS, from a given Erlang program; the resulting ACS simulates the operational semantics of the input. Using this abstract model, we can conservatively verify coverability properties of the input program, i.e. a weak form of safety properties, with a Petri net model-checker. We have implemented this procedure in our tool Soter, which is the first sound verification tool for Erlang programs using infinite-state model-checking. In our experiments, we find that Soter is accurate enough to verify a range of interesting and non-trivial benchmarks. Even though ACS coverability is Expspace-complete, Soter's analysis of these verification problems is surprisingly quick. In order to improve the precision of our verification procedure with respect to recursion, we investigate an extension of ACS that allows pushdown processes: asynchronously communicating pushdown systems (ACPS). ACPS that satisfy the empty-stack constraint (a pushdown process may receive only when its stack is empty) are a popular subclass of ACPS with good decision and complexity properties. In the context of Erlang, the empty stack constraint is unfortunately not realistic. We introduce a relaxation of the empty-stack constraint for ACPS called the shaped stack constraint. Stacks that fit the shape constraint may reach arbitrary heights. Further, a process may execute any communication action (be it process creation, message send or retrieval) whether or not its stack is empty. We prove that coverability for shaped ACPS, i.e. ACPS that satisfy the shaped constraint, reduces to the decidable coverability problem for well-structured transition systems (WSTS). Thus, shaped ACPS enable the modelling and verification of a larger class of message passing programs. We establish a close connection between shaped ACPS and a novel extension of Petri nets: nets with nested coloured tokens (NNCT). Tokens in NNCT are of two types: simple and complex. Complex tokens carry an arbitrary number of coloured tokens. The rules of a NNCT can synchronise complex and simple tokens, inject coloured tokens into a complex token, and eject all tokens of a specified set of active colours to predefined places. We show that the coverability problem for NNCT is Tower-complete, a new complexity class for non-elementary decision problems introduced by Schmitz. To prove Tower-membership, we devise a geometrically inspired version of the Rackoff technique, and we obtain Tower-hardness by adapting Stockmeyer's ruler construction to NNCT. To our knowledge, NNCT is the first extension of Petri nets (belonging to the class of nets with an infinite set of token types) that is proven to have primitive recursive coverability. This result implies Tower-completeness of coverability for ACPS that satisfy the shaped stack constraint.

Model checking is a powerful technique for analyzing reach ability and temporal properties of finite state systems. Model-checking finite state systems has been well-studied and there are well known efficient algorithms for this problem. However these algorithms may not terminate when applied directly to in finite state systems. Counter systems are a class of in fininite state systems where the domain of counter values is possibly in finite. Many practical systems like cache coherence protocols, broadcast protocols etc, can naturally be modeled as counter systems. In this thesis we identify a class of counter systems, and propose a new technique to check whether a system from this class satires’ a given CTL formula. The key novelty of our approach is a way to use existing reach ability analysis techniques to answer both \until" and \global" properties; also our technique for \global" properties is different from previous techniques that work on other classes of counter systems, as well as other classes of in finite state systems. We also provide some results by applying our approach to several natural examples, which illustrates the scope of our approach.

The Protein Data Bank (PDB) is the definitive electronic repository for experimentally-derived protein structures, composed mainly of those determined by X-ray crystallography. Approximately 200 new structures are added weekly to the PDB, and at the time of writing, it contains approximately 97,000 structures. This represents an expanding wealth of high-quality information but there seem to be few bioinformatics tools that consider and analyse these data as an ensemble. This thesis explores the development of three efficient, fast algorithms and software implementations to study protein structure using the entire PDB. The first project is a crystal-form matching tool that takes a unit cell and quickly (< 1 second) retrieves the most related matches from the PDB. The unit cell matches are combined with sequence alignments using a novel Family Clustering Algorithm to display the results in a user-friendly way. The software tool, Nearest-cell, has been incorporated into the X-ray data collection pipeline at the Diamond Light Source, and is also available as a public web service. The bulk of the thesis is devoted to the study and prediction of protein disorder. Initially, trying to update and extend an existing predictor, RONN, the limitations of the method were exposed and a novel predictor (called MoreRONN) was developed that incorporates a novel sequence-based clustering approach to disorder data inferred from the PDB and DisProt. MoreRONN is now clearly the best-in-class disorder predictor and will soon be offered as a public web service. The third project explores the development of a clustering algorithm for protein structural fragments that can work on the scale of the whole PDB. While protein structures have long been clustered into loose families, there has to date been no comprehensive analytical clustering of short (~6 residue) fragments. A novel fragment clustering tool was built that is now leading to a public database of fragment families and representative structural fragments that should prove extremely helpful for both basic understanding and experimentation. Together, these three projects exemplify how cutting-edge computational approaches applied to extensive protein structure libraries can provide user-friendly tools that address critical everyday issues for structural biologists.