7,002 research outputs found
Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC
We describe a new approach to the Word Problem for Artin-Tits groups and,
more generally, for the enveloping group U(M) of a monoid M in which any two
elements admit a greatest common divisor. The method relies on a rewrite system
R(M) that extends free reduction for free groups. Here we show that, if M
satisfies what we call the 3-Ore condition about common multiples, what
corresponds to type FC in the case of Artin-Tits monoids, then the system R(M)
is convergent. Under this assumption, we obtain a unique representation result
for the elements of U(M), extending Ore's theorem for groups of fractions and
leading to a solution of the Word Problem of a new type. We also show that
there exist universal shapes for the van Kampen diagrams of the words
representing 1.Comment: 29 pages ; v2 : cross-references updated ; v3 : typos corrected;
final version due to appear in Journal of Combinatorial Algebr
Rewriting Modulo \beta in the \lambda\Pi-Calculus Modulo
The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with
dependent types where beta-conversion is extended with user-defined rewrite
rules. It is an expressive logical framework and has been used to encode logics
and type systems in a shallow way. Basic properties such as subject reduction
or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo.
However, they hold if the rewrite system generated by the rewrite rules
together with beta-reduction is confluent. But this is too restrictive. To
handle the case where non confluence comes from the interference between the
beta-reduction and rewrite rules with lambda-abstraction on their left-hand
side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus
Modulo. We prove that confluence of rewriting modulo beta is enough to ensure
subject reduction and uniqueness of types. We achieve our goal by encoding the
lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a
consequence, we also make the confluence results for HRSs available for the
lambda-Pi-calculus Modulo.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
A general conservative extension theorem in process algebras with inequalities
We prove a general conservative extension theorem for transition system based process theories with easy-to-check and reasonable conditions. The core of this result is another general theorem which gives sufficient conditions for a system of operational rules and an extension of it in order to ensure conservativity, that is, provable transitions from an original term in the extension are the same as in the original system. As a simple corollary of the conservative extension theorem we prove a completeness theorem. We also prove a general theorem giving sufficient conditions to reduce the question of ground confluence modulo some equations for a large term rewriting system associated with an equational process theory to a small term rewriting system under the condition that the large system is a conservative extension of the small one. We provide many applications to show that our results are useful. The applications include (but are not limited to) various real and discrete time settings in ACP, ATP, and CCS and the notions projection, renaming, stage operator, priority, recursion, the silent step, autonomous actions, the empty process, divergence, etc
Linear-Logic Based Analysis of Constraint Handling Rules with Disjunction
Constraint Handling Rules (CHR) is a declarative committed-choice programming
language with a strong relationship to linear logic. Its generalization CHR
with Disjunction (CHRv) is a multi-paradigm declarative programming language
that allows the embedding of horn programs. We analyse the assets and the
limitations of the classical declarative semantics of CHR before we motivate
and develop a linear-logic declarative semantics for CHR and CHRv. We show how
to apply the linear-logic semantics to decide program properties and to prove
operational equivalence of CHRv programs across the boundaries of language
paradigms
Using groups for investigating rewrite systems
We describe several technical tools that prove to be efficient for
investigating the rewrite systems associated with a family of algebraic laws,
and might be useful for more general rewrite systems. These tools consist in
introducing a monoid of partial operators, listing the monoid relations
expressing the possible local confluence of the rewrite system, then
introducing the group presented by these relations, and finally replacing the
initial rewrite system with a internal process entirely sitting in the latter
group. When the approach can be completed, one typically obtains a practical
method for constructing algebras satisfying prescribed laws and for solving the
associated word problem
Multifraction reduction II: Conjectures for Artin-Tits groups
Multifraction reduction is a new approach to the word problem for Artin-Tits
groups and, more generally, for the enveloping group of a monoid in which any
two elements admit a greatest common divisor. This approach is based on a
rewrite system ("reduction") that extends free group reduction. In this paper,
we show that assuming that reduction satisfies a weak form of convergence
called semi-convergence is sufficient for solving the word problem for the
enveloping group, and we connect semi-convergence with other conditions
involving reduction. We conjecture that these properties are valid for all
Artin-Tits monoids, and provide partial results and numerical evidence
supporting such conjectures.Comment: 41 pages , v2 : cross-references updated , v3 : exposition improved,
typos corrected, final version due tu appear in Journal of Combinatorial
Algebr
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
- …