195 research outputs found
A Correspondence between Maximal Abelian Sub-Algebras and Linear Logic Fragments
We show a correspondence between a classification of maximal abelian
sub-algebras (MASAs) proposed by Jacques Dixmier and fragments of linear logic.
We expose for this purpose a modified construction of Girard's hyperfinite
geometry of interaction which interprets proofs as operators in a von Neumann
algebra. The expressivity of the logic soundly interpreted in this model is
dependent on properties of a MASA which is a parameter of the interpretation.
We also unveil the essential role played by MASAs in previous geometry of
interaction constructions
The algebra of entanglement and the geometry of composition
String diagrams turn algebraic equations into topological moves that have
recurring shapes, involving the sliding of one diagram past another. We
individuate, at the root of this fact, the dual nature of polygraphs as
presentations of higher algebraic theories, and as combinatorial descriptions
of "directed spaces". Operations of polygraphs modelled on operations of
topological spaces are used as the foundation of a compositional universal
algebra, where sliding moves arise from tensor products of polygraphs. We
reconstruct several higher algebraic theories in this framework.
In this regard, the standard formalism of polygraphs has some technical
problems. We propose a notion of regular polygraph, barring cell boundaries
that are not homeomorphic to a disk of the appropriate dimension. We define a
category of non-degenerate shapes, and show how to calculate their tensor
products. Then, we introduce a notion of weak unit to recover weakly degenerate
boundaries in low dimensions, and prove that the existence of weak units is
equivalent to a representability property.
We then turn to applications of diagrammatic algebra to quantum theory. We
re-evaluate the category of Hilbert spaces from the perspective of categorical
universal algebra, which leads to a bicategorical refinement. Then, we focus on
the axiomatics of fragments of quantum theory, and present the ZW calculus, the
first complete diagrammatic axiomatisation of the theory of qubits.
The ZW calculus has several advantages over ZX calculi, including a
computationally meaningful normal form, and a fragment whose diagrams can be
read as setups of fermionic oscillators. Moreover, its generators reflect an
operational classification of entangled states of 3 qubits. We conclude with
generalisations of the ZW calculus to higher-dimensional systems, including the
definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures;
University of Oxford doctoral thesi
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Arithmetic-arboreal residue structures induced by Prufer extensions : An axiomatic approach
We present an axiomatic framework for the residue structures induced by
Prufer extensions with a stress upon the intimate connection between their
arithmetic and arboreal theoretic properties. The main result of the paper
provides an adjunction relationship between two naturally defined functors
relating Prufer extensions and superrigid directed commutative regular
quasi-semirings.Comment: 56 page
Memoization for Unary Logic Programming: Characterizing PTIME
We give a characterization of deterministic polynomial time computation based
on an algebraic structure called the resolution semiring, whose elements can be
understood as logic programs or sets of rewriting rules over first-order terms.
More precisely, we study the restriction of this framework to terms (and logic
programs, rewriting rules) using only unary symbols. We prove it is complete
for polynomial time computation, using an encoding of pushdown automata. We
then introduce an algebraic counterpart of the memoization technique in order
to show its PTIME soundness. We finally relate our approach and complexity
results to complexity of logic programming. As an application of our
techniques, we show a PTIME-completeness result for a class of logic
programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201
Order, algebra, and structure: lattice-ordered groups and beyond
This thesis describes and examines some remarkable relationships existing between seemingly quite different properties (algebraic, order-theoretic, and structural) of ordered groups. On the one hand, it revisits the foundational aspects of the structure theory of lattice-ordered groups, contributing a novel systematization of its relationship with the theory of orderable groups. One of the main contributions in this direction is a connection between validity in varieties of lattice-ordered groups, and orders on groups; a framework is also provided that allows for a systematic account of the relationship between orders and preorders on groups, and the structure theory of lattice-ordered groups. On the other hand, it branches off in new directions, probing the frontiers of several different areas of current research. More specifically, one of the main goals of this thesis is to suitably extend results that are proper to the theory of lattice-ordered groups to the realm of more general, related algebraic structures; namely, distributive lattice-ordered monoids and residuated lattices. The theory of lattice-ordered groups provides themain source of inspiration for this thesis’ contributions on these topics
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