3,477 research outputs found
Cyclic rewriting and conjugacy problems
Cyclic words are equivalence classes of cyclic permutations of ordinary
words. When a group is given by a rewriting relation, a rewriting system on
cyclic words is induced, which is used to construct algorithms to find minimal
length elements of conjugacy classes in the group. These techniques are applied
to the universal groups of Stallings pregroups and in particular to free
products with amalgamation, HNN-extensions and virtually free groups, to yield
simple and intuitive algorithms and proofs of conjugacy criteria.Comment: 37 pages, 1 figure, submitted. Changes to introductio
The Tutte-Grothendieck group of a convergent alphabetic rewriting system
The two operations, deletion and contraction of an edge, on multigraphs
directly lead to the Tutte polynomial which satisfies a universal problem. As
observed by Brylawski in terms of order relations, these operations may be
interpreted as a particular instance of a general theory which involves
universal invariants like the Tutte polynomial, and a universal group, called
the Tutte-Grothendieck group. In this contribution, Brylawski's theory is
extended in two ways: first of all, the order relation is replaced by a string
rewriting system, and secondly, commutativity by partial commutations (that
permits a kind of interpolation between non commutativity and full
commutativity). This allows us to clarify the relations between the semigroup
subject to rewriting and the Tutte-Grothendieck group: the later is actually
the Grothendieck group completion of the former, up to the free adjunction of a
unit (this was even not mention by Brylawski), and normal forms may be seen as
universal invariants. Moreover we prove that such universal constructions are
also possible in case of a non convergent rewriting system, outside the scope
of Brylawski's work.Comment: 17 page
Coherent Presentations of Monoidal Categories
Presentations of categories are a well-known algebraic tool to provide
descriptions of categories by means of generators, for objects and morphisms,
and relations on morphisms. We generalize here this notion, in order to
consider situations where the objects are considered modulo an equivalence
relation, which is described by equational generators. When those form a
convergent (abstract) rewriting system on objects, there are three very natural
constructions that can be used to define the category which is described by the
presentation: one consists in turning equational generators into identities
(i.e. considering a quotient category), one consists in formally adding
inverses to equational generators (i.e. localizing the category), and one
consists in restricting to objects which are normal forms. We show that, under
suitable coherence conditions on the presentation, the three constructions
coincide, thus generalizing celebrated results on presentations of groups, and
we extend those conditions to presentations of monoidal categories
Canonical Abstract Syntax Trees
This paper presents Gom, a language for describing abstract syntax trees and
generating a Java implementation for those trees. Gom includes features
allowing the user to specify and modify the interface of the data structure.
These features provide in particular the capability to maintain the internal
representation of data in canonical form with respect to a rewrite system. This
explicitly guarantees that the client program only manipulates normal forms for
this rewrite system, a feature which is only implicitly used in many
implementations
Topological finiteness properties of monoids. Part 1: Foundations
We initiate the study of higher dimensional topological finiteness properties
of monoids. This is done by developing the theory of monoids acting on CW
complexes. For this we establish the foundations of -equivariant homotopy
theory where is a discrete monoid. For projective -CW complexes we prove
several fundamental results such as the homotopy extension and lifting
property, which we use to prove the -equivariant Whitehead theorems. We
define a left equivariant classifying space as a contractible projective -CW
complex. We prove that such a space is unique up to -homotopy equivalence
and give a canonical model for such a space via the nerve of the right Cayley
graph category of the monoid. The topological finiteness conditions
left- and left geometric dimension are then defined for monoids
in terms of existence of a left equivariant classifying space satisfying
appropriate finiteness properties. We also introduce the bilateral notion of
-equivariant classifying space, proving uniqueness and giving a canonical
model via the nerve of the two-sided Cayley graph category, and we define the
associated finiteness properties bi- and geometric dimension. We
explore the connections between all of the these topological finiteness
properties and several well-studied homological finiteness properties of
monoids which are important in the theory of string rewriting systems,
including , cohomological dimension, and Hochschild
cohomological dimension. We also develop the corresponding theory of
-equivariant collapsing schemes (that is, -equivariant discrete Morse
theory), and among other things apply it to give topological proofs of results
of Anick, Squier and Kobayashi that monoids which admit presentations by
complete rewriting systems are left-, right- and bi-.Comment: 59 pages, 1 figur
On insertion-deletion systems over relational words
We introduce a new notion of a relational word as a finite totally ordered
set of positions endowed with three binary relations that describe which
positions are labeled by equal data, by unequal data and those having an
undefined relation between their labels. We define the operations of insertion
and deletion on relational words generalizing corresponding operations on
strings. We prove that the transitive and reflexive closure of these operations
has a decidable membership problem for the case of short insertion-deletion
rules (of size two/three and three/two). At the same time, we show that in the
general case such systems can produce a coding of any recursively enumerable
language leading to undecidabilty of reachability questions.Comment: 24 pages, 8 figure
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
Geodesic rewriting systems and pregroups
In this paper we study rewriting systems for groups and monoids, focusing on
situations where finite convergent systems may be difficult to find or do not
exist. We consider systems which have no length increasing rules and are
confluent and then systems in which the length reducing rules lead to
geodesics. Combining these properties we arrive at our main object of study
which we call geodesically perfect rewriting systems. We show that these are
well-behaved and convenient to use, and give several examples of classes of
groups for which they can be constructed from natural presentations. We
describe a Knuth-Bendix completion process to construct such systems, show how
they may be found with the help of Stallings' pregroups and conversely may be
used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory,
Dortmund and Carleton Conferences". Series: Trends in Mathematics.
Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009,
Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause
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