Substructural Simple Type Theories for Separation and In-place Update

This thesis studies two substructural simple type theories, extending the "separation" and "number-of-uses" readings of the basic substructural simply typed lambda-calculus with exchange. The first calculus, lambda_sep, extends the alpha lambda-calculus of O'Hearn and Pym by directly considering the representation of separation in a type system. We define type contexts with separation relations and introduce new type constructors of separated products and separated functions. We describe the basic metatheory of the calculus, including a sound and complete type-checking algorithm. We then give new categorical structure for interpreting the type judgements, and prove that it coherently, soundly and completely interprets the type theory. To show how the structure models separation we extend Day's construction of closed symmetric monoidal structure on functor categories to our categorical structure, and describe two instances dealing with the global and local separation. The second system, lambda_inplc, is a re-presentation of substructural calculus for in-place update with linear and non-linear values, based on Wadler's Linear typed system with non-linear types and Hofmann's LFPL. We identify some problems with the metatheory of the calculus, in particular the failure of the substitution rule to hold due to the call-by-value interpretation inherent in the type rules. To resolve this issue, we turn to categorical models of call-by-value computation, namely Moggi's Computational Monads and Power and Robinson's Freyd-Categories. We extend both of these to include additional information about the current state of the computation, defining Parameterised Freyd-categories and Parameterised Strong Monads. These definitions are equivalent in the closed case. We prove that by adding a commutativity condition they are a sound class of models for lambda_inplc. To obtain a complete class of models for lambda_inplc we refine the structure to better match the syntax. We also give a direct syntactic presentation of Parameterised Freyd-categories and prove that it is soundly and completely modelled by the syntax. We give a concrete model based on Day's construction, demonstrating how the categorical structure can be used to model call-by-value computation with in-place update and bounded heaps

Components as coalgebras

In the tradition of mathematical modelling in physics and chemistry, constructive formal specification methods are based on the notion of a software model, understood as a state-based abstract machine which persists and evolves in time, according to a behavioural model capturing, for example, partiality or (different degrees of) nondeterminism. This can be identified with the more prosaic notion of a software component advocated by the software industry as ‘building block’ of large, often distributed, systems. Such a component typically encapsulates a number of services through a public interface which provides a limited access to a private state space, paying tribute to the nowadays widespread object-oriented programming principles. The tradition of communicating systems formal design, by contrast, has developed the notion of a process as an abstraction of the behavioural patterns of a computing system, deliberately ignoring the data and state aspects of software systems. Both processes and components are among the broad group of computing phenomena which are hardly definable (or simply not definable) algebraically, i.e., in terms of a complete set of constructors. Their semantics is essentially observational, in the sense that all that can be traced of their evolution is their interaction with the environment. Therefore, coalgebras, whose theory has recently witnessed remarkable developments, appear as a suitable modelling tool. The basic observation of category theory that universal constructions always come in pairs, has motivated research on the duality between algebras and coalgebras, which provides a bridge between models of static (constructive, data-oriented) and dynamical (observational, behaviour-oriented) systems. At the programming level, the intuitive symmetry between data and behaviour provides evidence of such a duality, in its canonical initial-final specialisation. This line of thought entails both definitional and proof principles, i.e., a basis for the development of program calculi directly based on (actually driven by) type specifications. Moreover, such properties can be expressed in terms of generic programming combinators which are used, not only to calculate programs, but also to program with. Framed in this context, this thesis addresses the following main themes: The investigation of a semantic model for (state-based) software components. These are regarded as concrete coalgebras for some Set endofunctors, with specified initial conditions, and organise themselves in a bicategorical setting. The model is able to capture both behavioural issues, which are usually left implicit in state-based specification methods, and interaction through structured data, which is usually a minor concern on process calculi. Two basic cases are considered entailing, respectively, a ‘functional’ and an ‘object-oriented’ shape for components. Both cases are parametrized by a model of behaviour, introduced as a strong (usually commutative) monad. The development of corresponding component calculi, also parametric on the behaviour model, which adds to the genericity of the approach. The study of processes and the ‘reconstruction’ of classical (CCS-like) process calculi on top of their representation as inhabitants of (the carriers of) final coalgebras, in an essentially pointfree, calculational style. An overall concern for genericity, in the sense that models and calculi for both components and processes are parametric on the behaviour model and the interaction discipline, respectively. The animation of both processes and components in CHARITY, a functional programming language entirely based on inductive and coinductive categorical data types. In particular this leads to the development of a process calculi interpreter parametric on the interaction discipline.PRAXIS XXI - Projecto LOGCAMP; POO11/IC-PME/II/S -Projecto KARMA; Fundação para a Ciência e Tecnologia; ALGORITMI Research Center

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Categorical Quantum Dynamics

We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford University, Michaelmas Term 2016 (273 pages

Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing

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. [...] 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. [...] 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. [...] 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.Comment: PhD Thesis. Passed examination. Minor corrections made and one theorem added at the end of Chapter 5. 182 pages, ~300 figures. See full text for unabridged abstrac

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

A functional quantum programming language

This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms into superoperators, via the operational semantics, made precise by the category Q. Extensional equality for QML programs is also presented, via a mapping from FQC morphisms into the category Q

A functional quantum programming language

This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms into superoperators, via the operational semantics, made precise by the category Q. Extensional equality for QML programs is also presented, via a mapping from FQC morphisms into the category Q