4,271 research outputs found
Equational reasoning with context-free families of string diagrams
String diagrams provide an intuitive language for expressing networks of
interacting processes graphically. A discrete representation of string
diagrams, called string graphs, allows for mechanised equational reasoning by
double-pushout rewriting. However, one often wishes to express not just single
equations, but entire families of equations between diagrams of arbitrary size.
To do this we define a class of context-free grammars, called B-ESG grammars,
that are suitable for defining entire families of string graphs, and crucially,
of string graph rewrite rules. We show that the language-membership and
match-enumeration problems are decidable for these grammars, and hence that
there is an algorithm for rewriting string graphs according to B-ESG rewrite
patterns. We also show that it is possible to reason at the level of grammars
by providing a simple method for transforming a grammar by string graph
rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The
final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-21145-9_
!-Graphs with Trivial Overlap are Context-Free
String diagrams are a powerful tool for reasoning about composite structures
in symmetric monoidal categories. By representing string diagrams as graphs,
equational reasoning can be done automatically by double-pushout rewriting.
!-graphs give us the means of expressing and proving properties about whole
families of these graphs simultaneously. While !-graphs provide elegant proofs
of surprisingly powerful theorems, little is known about the formal properties
of the graph languages they define. This paper takes the first step in
characterising these languages by showing that an important subclass of
!-graphs--those whose repeated structures only overlap trivially--can be
encoded using a (context-free) vertex replacement grammar.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Towards 3-Dimensional Rewriting Theory
String rewriting systems have proved very useful to study monoids. In good
cases, they give finite presentations of monoids, allowing computations on
those and their manipulation by a computer. Even better, when the presentation
is confluent and terminating, they provide one with a notion of canonical
representative of the elements of the presented monoid. Polygraphs are a
higher-dimensional generalization of this notion of presentation, from the
setting of monoids to the much more general setting of n-categories. One of the
main purposes of this article is to give a progressive introduction to the
notion of higher-dimensional rewriting system provided by polygraphs, and
describe its links with classical rewriting theory, string and term rewriting
systems in particular. After introducing the general setting, we will be
interested in proving local confluence for polygraphs presenting 2-categories
and introduce a framework in which a finite 3-dimensional rewriting system
admits a finite number of critical pairs
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
Encoding !-tensors as !-graphs with neighbourhood orders
Diagrammatic reasoning using string diagrams provides an intuitive language
for reasoning about morphisms in a symmetric monoidal category. To allow
working with infinite families of string diagrams, !-graphs were introduced as
a method to mark repeated structure inside a diagram. This led to !-graphs
being implemented in the diagrammatic proof assistant Quantomatic. Having a
partially automated program for rewriting diagrams has proven very useful, but
being based on !-graphs, only commutative theories are allowed. An enriched
abstract tensor notation, called !-tensors, has been used to formalise the
notion of !-boxes in non-commutative structures. This work-in-progress paper
presents a method to encode !-tensors as !-graphs with some additional
structure. This will allow us to leverage the existing code from Quantomatic
and quickly provide various tools for non-commutative diagrammatic reasoning.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Tensors, !-graphs, and non-commutative quantum structures
Categorical quantum mechanics (CQM) and the theory of quantum groups rely
heavily on the use of structures that have both an algebraic and co-algebraic
component, making them well-suited for manipulation using diagrammatic
techniques. Diagrams allow us to easily form complex compositions of
(co)algebraic structures, and prove their equality via graph rewriting. One of
the biggest challenges in going beyond simple rewriting-based proofs is
designing a graphical language that is expressive enough to prove interesting
properties (e.g. normal form results) about not just single diagrams, but
entire families of diagrams. One candidate is the language of !-graphs, which
consist of graphs with certain subgraphs marked with boxes (called !-boxes)
that can be repeated any number of times. New !-graph equations can then be
proved using a powerful technique called !-box induction. However, previously
this technique only applied to commutative (or cocommutative) algebraic
structures, severely limiting its applications in some parts of CQM and
(especially) quantum groups. In this paper, we fix this shortcoming by offering
a new semantics for non-commutative !-graphs using an enriched version of
Penrose's abstract tensor notation.Comment: In Proceedings QPL 2014, arXiv:1412.810
Linear lambda terms as invariants of rooted trivalent maps
The main aim of the article is to give a simple and conceptual account for
the correspondence (originally described by Bodini, Gardy, and Jacquot) between
-equivalence classes of closed linear lambda terms and isomorphism
classes of rooted trivalent maps on compact oriented surfaces without boundary,
as an instance of a more general correspondence between linear lambda terms
with a context of free variables and rooted trivalent maps with a boundary of
free edges. We begin by recalling a familiar diagrammatic representation for
linear lambda terms, while at the same time explaining how such diagrams may be
read formally as a notation for endomorphisms of a reflexive object in a
symmetric monoidal closed (bi)category. From there, the "easy" direction of the
correspondence is a simple forgetful operation which erases annotations on the
diagram of a linear lambda term to produce a rooted trivalent map. The other
direction views linear lambda terms as complete invariants of their underlying
rooted trivalent maps, reconstructing the missing information through a
Tutte-style topological recurrence on maps with free edges. As an application
in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent
maps as linear lambda terms containing no closed proper subterms, and conclude
by giving a natural reformulation of the Four Color Theorem as a statement
about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio
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