23 research outputs found
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
A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing
The motivation for this thesis was to recast quantum self-testing [MY98,MY04]
in operational terms. The result is a category-theoretic framework for
discussing the following general question: How do different implementations of
the same input-output process compare to each other? In the proposed framework,
an input-output process is modelled by a causally structured channel in some
fixed theory, and its implementations are modelled by causally structured
dilations formalising hidden side-computations. These dilations compare through
a pre-order formalising relative strength of side-computations. Chapter 1
reviews a mathematical model for physical theories as semicartesian symmetric
monoidal categories. Many concrete examples are discussed, in particular
quantum and classical information theory. The key feature is that the model
facilitates the notion of dilations. Chapter 2 is devoted to the study of
dilations. It introduces a handful of simple yet potent axioms about dilations,
one of which (resembling the Purification Postulate [CDP10]) entails a duality
theorem encompassing a large number of classic no-go results for quantum
theory. Chapter 3 considers metric structure on physical theories, introducing
in particular a new metric for quantum channels, the purified diamond distance,
which generalises the purified distance [TCR10,Tom12] and relates to the Bures
distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for
causality in terms of '(constructible) causal channels' and 'contractions'. It
simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in
monoidal categories [JSV96]. The formalism allows for the definition of 'causal
dilations' and the establishment of a non-trivial theory of such dilations.
Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus
pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN
978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of
December 2020, defended 11th of February 2021. Keywords: dilations, applied
category theory, quantum foundations, causal structure, quantum self-testing.
242 pages, 1 figure. Comments are welcom
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 Quantum Theory: From Classical Concepts to Operator Algebras
Quantum physics; Mathematical physics; Matrix theory; Algebr
Interacting Hopf Algebras: the theory of linear systems
Scientists in diverse fields use diagrammatic formalisms to reason about various kinds
of networks, or compound systems. Examples include electrical circuits, signal flow graphs,
Penrose and Feynman diagrams, Bayesian networks, Petri nets, Kahn process networks, proof
nets, UML specifications, amongst many others. Graphical languages provide a convenient
abstraction of some underlying mathematical formalism, which gives meaning to diagrams.
For instance, signal flow graphs, foundational structures in control theory, are traditionally
translated into systems of linear equations. This is typical: diagrammatic languages are used
as an interface for more traditional mathematics, but rarely studied per se.
Recent trends in computer science analyse diagrams as first-class objects using formal
methods from programming language semantics. In many such approaches, diagrams are generated
as the arrows of a PROP — a special kind of monoidal category — by a two-dimensional
syntax and equations. The domain of interpretation of diagrams is also formalised as a PROP
and the (compositional) semantics is expressed as a functor preserving the PROP structure.
The first main contribution of this thesis is the characterisation of SVk, the PROP of
linear subspaces over a field k. This is an important domain of interpretation for diagrams
appearing in diverse research areas, like the signal flow graphs mentioned above. We present by
generators and equations the PROP IH of string diagrams whose free model is SVk. The name
IH stands for interacting Hopf algebras: indeed, the equations of IH arise by distributive laws
between Hopf algebras, which we obtain using Lack’s technique for composing PROPs. The
significance of the result is two-fold. On the one hand, it offers a canonical string diagrammatic
syntax for linear algebra: linear maps, kernels, subspaces and the standard linear algebraic
transformations are all faithfully represented in the graphical language. On the other hand,
the equations of IH describe familiar algebraic structures — Hopf algebras and Frobenius
algebras — which are at the heart of graphical formalisms as seemingly diverse as quantum
circuits, signal flow graphs, simple electrical circuits and Petri nets. Our characterisation
enlightens the provenance of these axioms and reveals their linear algebraic nature.
Our second main contribution is an application of IH to the semantics of signal processing
circuits. We develop a formal theory of signal flow graphs, featuring a string diagrammatic
syntax for circuits, a structural operational semantics and a denotational semantics. We
prove soundness and completeness of the equations of IH for denotational equivalence. Also,
we study the full abstraction question: it turns out that the purely operational picture is
too concrete — two graphs that are denotationally equal may exhibit different operational
behaviour. We classify the ways in which this can occur and show that any graph can be
realised — rewritten, using the equations of IH, into an executable form where the operational
behaviour and the denotation coincide. This realisability theorem — which is the culmination
of our developments — suggests a reflection about the role of causality in the semantics of
signal flow graphs and, more generally, of computing devices
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.
Causality in Linear Logic: full completeness and injectivity (unit-free multiplicative-additive fragment)
Commuting conversions of Linear Logic induce a notion of dependencybetween rules inside a proof derivation: a rule depends on a previous rule whenthey cannot be permuted using the conversions. We propose a new interpretation ofproofs of Linear Logic ascausal invariantswhich capturesexactlythis dependency.We represent causal invariants using game semantics based on general eventstructures, carving out, inside the model of [6], a submodel of causal invariants.This submodel supports an interpretation of unit-free Multiplicative AdditiveLinear Logic with MIX (MALL−) which is (1)fully complete: every element ofthe model is the denotation of a proof and (2)injective: equality in the modelcharacterises exactly commuting conversions of MALL−. This improves over thestandard fully complete game semantics model of MALL−