Fibred Coalgebraic Logic and Quantum Protocols

Motivated by applications in modelling quantum systems using coalgebraic techniques, we introduce a fibred coalgebraic logic. Our approach extends the conventional predicate lifting semantics with additional modalities relating conditions on different fibres. As this fibred setting will typically involve multiple signature functors, the logic incorporates a calculus of modalities enabling the construction of new modalities using various composition operations. We extend the semantics of coalgebraic logic to this setting, and prove that this extension respects behavioural equivalence. We show how properties of the semantics of modalities are preserved under composition operations, and then apply the calculational aspect of our logic to produce an expressive set of modalities for reasoning about quantum systems, building these modalities up from simpler components. We then demonstrate how these modalities can describe some standard quantum protocols. The novel features of our logic are shown to allow for a uniform description of unitary evolution, and support local reasoning such as "Alice's qubit satisfies condition" as is common when discussing quantum protocols.Comment: In Proceedings QPL 2013, arXiv:1412.791

Big Toy Models: Representing Physical Systems As Chu Spaces

We pursue a model-oriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we aim for a `big toy model', in which both quantum and classical systems can be faithfully represented - as well as, possibly, more exotic kinds of systems. To this end, we show how Chu spaces can be used to represent physical systems of various kinds. In particular, we show how quantum systems can be represented as Chu spaces over the unit interval in such a way that the Chu morphisms correspond exactly to the physically meaningful symmetries of the systems - the unitaries and antiunitaries. In this way we obtain a full and faithful functor from the groupoid of Hilbert spaces and their symmetries to Chu spaces. We also consider whether it is possible to use a finite value set rather than the unit interval; we show that three values suffice, while the two standard possibilistic reductions to two values both fail to preserve fullness.Comment: 24 pages. Accepted for Synthese 16th April 2010. Published online 20th April 201

The Post‐Modern Transcendental of Language in Science and Philosophy

In this chapter I discuss the deep mutations occurring today in our society and in our culture, the natural and mathematical sciences included, from the standpoint of the “transcendental of language”, and of the primacy of language over knowledge. That is, from the standpoint of the “completion of the linguistic turn” in the foundations of logic and mathematics using Peirce’s algebra of relations. This evolved during the last century till the development of the Category Theory as universal language for mathematics, in many senses wider than set theory. Therefore, starting from the fundamental M. Stone’s representation theorem for Boolean algebras, computer scientists developed a coalgebraic first-order semantics defined on Stone’s spaces, for Boolean algebras, till arriving to the definition of a non-Turing paradigm of coalgebraic universality in computation. Independently, theoretical physicists developed a coalgebraic modelling of dissipative quantum systems in quantum field theory, interpreted as a thermo-field dynamics. The deep connection between these two coalgebraic constructions is the fact that the topologies of Stone spaces in computer science are the same of the C*-algebras of quantum physics. This allows the development of a new class of quantum computers based on coalgebras. This suggests also an intriguing explanation of why one of the most successful experimental applications of this coalgebraic modelling of dissipative quantum systems is just in cognitive neuroscience

Bialgebraic foundations for the operational semantics of string diagrams

Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their semantics as a distributive law for that monad. As a proof of concept, we provide bialgebraic semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory)

Towards a Uniform Theory of Effectful State Machines

Using recent developments in coalgebraic and monad-based semantics, we present a uniform study of various notions of machines, e.g. finite state machines, multi-stack machines, Turing machines, valence automata, and weighted automata. They are instances of Jacobs' notion of a T-automaton, where T is a monad. We show that the generic language semantics for T-automata correctly instantiates the usual language semantics for a number of known classes of machines/languages, including regular, context-free, recursively-enumerable and various subclasses of context free languages (e.g. deterministic and real-time ones). Moreover, our approach provides new generic techniques for studying the expressivity power of various machine-based models.Comment: final version accepted by TOC

Old and New Minimalism: a Hopf algebra comparison

In this paper we compare some old formulations of Minimalism, in particular Stabler's computational minimalism, and Chomsky's new formulation of Merge and Minimalism, from the point of view of their mathematical description in terms of Hopf algebras. We show that the newer formulation has a clear advantage purely in terms of the underlying mathematical structure. More precisely, in the case of Stabler's computational minimalism, External Merge can be described in terms of a partially defined operated algebra with binary operation, while Internal Merge determines a system of right-ideal coideals of the Loday-Ronco Hopf algebra and corresponding right-module coalgebra quotients. This mathematical structure shows that Internal and External Merge have significantly different roles in the old formulations of Minimalism, and they are more difficult to reconcile as facets of a single algebraic operation, as desirable linguistically. On the other hand, we show that the newer formulation of Minimalism naturally carries a Hopf algebra structure where Internal and External Merge directly arise from the same operation. We also compare, at the level of algebraic properties, the externalization model of the new Minimalism with proposals for assignments of planar embeddings based on heads of trees.Comment: 27 pages, LaTeX, 3 figure

Bialgebraic Semantics for String Diagrams

Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their bialgebraic semantics in terms of a distributive law over that monad. As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, \`a la Hoare and \`a la Milner.Comment: Accepted for publications in the proceedings of the 30th International Conference on Concurrency Theory (CONCUR 2019

Higher prequantum geometry

This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey examples and problems of interest. In section 3 we survey the abstract cohesive homotopy theory that serves to make all this precise and tractable.Comment: expanded version of my contribution to Catren, Anel (eds.) "New Spaces in Mathematics and Physics" (ercpqg-espace.sciencesconf.org