298 research outputs found
On Single-Pushout Rewriting of Partial Algebras
We introduce Single-Pushout Rewriting for arbitrary partial algebras. Thus, we give up the usual restriction to graph structures, which are algebraic categories with unary operators only. By this generalisation, we obtain an integrated and straightforward treatment of graphical structures (objects) and attributes (data). We lose co-completeness of the underlying category. Therefore, a rule is no longer applicable at any match. We characterise the new application condition and make constructive use of it in some practical examples
Transformation of Attributed Structures with Cloning (Long Version)
Copying, or cloning, is a basic operation used in the specification of many
applications in computer science. However, when dealing with complex
structures, like graphs, cloning is not a straightforward operation since a
copy of a single vertex may involve (implicitly)copying many edges. Therefore,
most graph transformation approaches forbid the possibility of cloning. We
tackle this problem by providing a framework for graph transformations with
cloning. We use attributed graphs and allow rules to change attributes. These
two features (cloning/changing attributes) together give rise to a powerful
formal specification approach. In order to handle different kinds of graphs and
attributes, we first define the notion of attributed structures in an abstract
way. Then we generalise the sesqui-pushout approach of graph transformation in
the proposed general framework and give appropriate conditions under which
attributed structures can be transformed. Finally, we instantiate our general
framework with different examples, showing that many structures can be handled
and that the proposed framework allows one to specify complex operations in a
natural way
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
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
Algebraic transformation of unary partial algebras II: Single-pushout approach
AbstractThe single-pushout approach to graph transformation is extended to the algebraic transformation of partial many-sorted unary algebras. Such a generalization has been motivated by the need to model the transformation of structures which are richer and more complex than graphs and hypergraphs.The main result presented in this article is an algebraic characterization of the single-pushout transformation in the categories of all conformisms, all closed quomorphisms, and all closed-domain closed quomorphisms of unary partial algebras over a given signature, together with a corresponding operational characterization that may serve as a basis for implementation.Moreover, all three categories are shown to satisfy all of the HLR (high-level replacement) conditions for parallelism, taking as occurrences the total morphisms in each category. Another important result presented in this article is the definition of HLR conditions for amalgamation, which are also satisfied by the categories of partial homomorphisms considered here, taking again the corresponding total morphisms as occurrences
String diagram rewrite theory II: Rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras
String diagram rewrite theory II: rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras
Graph Transformation with Symbolic Attributes via Monadic Coalgebra Homomorphisms
We show how a coalgebraic approach leads to more natural representations of many kinds of graph structures that in the algebraic approach are frequently dealt with using ad-hoc constructions. For the case of symbolically attributed graphs, we demonstrate how using substituting coalgebra homomorphisms in double-pushout rewriting steps yields a powerful and easily understandable transformation mechanism
Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)
Tracelets are the intrinsic carriers of causal information in categorical
rewriting systems. In this work, we assemble tracelets into a symmetric
monoidal decomposition space, inducing a cocommutative Hopf algebra of
tracelets. This Hopf algebra captures important combinatorial and algebraic
aspects of rewriting theory, and is motivated by applications of its
representation theory to stochastic rewriting systems such as chemical reaction
networks.Comment: In Proceedings ACT 2021, arXiv:2211.0110
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