16,263 research outputs found
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
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
Towards a Maude tool for model checking temporal graph properties
We present our prototypical tool for the verification of graph transformation systems. The major novelty of our tool is that it provides a model checker for temporal graph properties based on counterpart semantics for quantified m-calculi. Our tool can be considered as an instantiation of our approach to counterpart semantics which allows for a neat handling of creation, deletion and merging in systems
with dynamic structure. Our implementation is based on the object-based machinery of Maude, which provides the basics to deal with attributed graphs. Graph transformation
systems are specified with term rewrite rules. The model checker evaluates logical formulae of second-order modal m-calculus in the automatically generated CounterpartModel (a sort of unfolded graph transition system) of the graph transformation system under study. The result of evaluating a formula is a set of assignments for each state, associating node variables to actual nodes
Polytool: polynomial interpretations as a basis for termination analysis of Logic programs
Our goal is to study the feasibility of porting termination analysis
techniques developed for one programming paradigm to another paradigm. In this
paper, we show how to adapt termination analysis techniques based on polynomial
interpretations - very well known in the context of term rewrite systems (TRSs)
- to obtain new (non-transformational) ter- mination analysis techniques for
definite logic programs (LPs). This leads to an approach that can be seen as a
direct generalization of the traditional techniques in termination analysis of
LPs, where linear norms and level mappings are used. Our extension general-
izes these to arbitrary polynomials. We extend a number of standard concepts
and results on termination analysis to the context of polynomial
interpretations. We also propose a constraint-based approach for automatically
generating polynomial interpretations that satisfy the termination conditions.
Based on this approach, we implemented a new tool, called Polytool, for
automatic termination analysis of LPs
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