110 research outputs found
Termination orders for 3-polygraphs
This note presents the first known class of termination orders for
3-polygraphs, together with an application.Comment: 4 pages, 12 figure
Termination orders for 3-dimensional rewriting
This paper studies 3-polygraphs as a framework for rewriting on
two-dimensional words. A translation of term rewriting systems into
3-polygraphs with explicit resource management is given, and the respective
computational properties of each system are studied. Finally, a convergent
3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In
order to prove these results, it is explained how to craft a class of
termination orders for 3-polygraphs.Comment: 30 pages, 35 figure
Soft linear logic and polynomial time
AbstractWe present a subsystem of second-order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice versa
Two polygraphic presentations of Petri nets
This document gives an algebraic and two polygraphic translations of Petri
nets, all three providing an easier way to describe reductions and to identify
some of them. The first one sees places as generators of a commutative monoid
and transitions as rewriting rules on it: this setting is totally equivalent to
Petri nets, but lacks any graphical intuition. The second one considers places
as 1-dimensional cells and transitions as 2-dimensional ones: this translation
recovers a graphical meaning but raises many difficulties since it uses
explicit permutations. Finally, the third translation sees places as
degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is
a setting equivalent to Petri nets, equipped with a graphical interpretation.Comment: 28 pages, 24 figure
Intensional properties of polygraphs
We present polygraphic programs, a subclass of Albert Burroni's polygraphs,
as a computational model, showing how these objects can be seen as first-order
functional programs. We prove that the model is Turing complete. We use
polygraphic interpretations, a termination proof method introduced by the
second author, to characterize polygraphic programs that compute in polynomial
time. We conclude with a characterization of polynomial time functions and
non-deterministic polynomial time functions.Comment: Proceedings of TERMGRAPH 2007, Electronic Notes in Computer Science
(to appear), 12 pages, minor changes from previous versio
A folk model structure on omega-cat
We establish a model structure on the category of strict omega-categories.
The constructions leading to the model structure in question are expressed
entirely within the scope of omega-categories, building on a set of generating
cofibrations and a class of weak equivalences as basic items. All object are
fibrant while cofibrant objects are exactly the free ones. Our model structure
transfers to n-categories along right-adjoints, for each n, thus recovering the
known cases n = 1 and n = 2.Comment: 33 pages, expanded version of the original 17 pages synopsis, new
sections adde
Orientals as free algebras
The aim of this paper is to give an alternative construction of Street's
cosimplicial object of orientals, based on an idea of Burroni that orientals
are free algebras for some algebraic structure on strict -categories.
More precisely, following Burroni, we define the notion of an expansion on an
-category and we show that the forgetful functor from strict
-categories endowed with an expansion to strict -categories is
monadic. By iterating this monad starting from the empty -category, we
get a cosimplicial object in strict -categories. Our main contribution
is to show that this cosimplicial object is the cosimplicial objects of
orientals. To do so, we prove, using Steiner's theory of augmented directed
chain complexes, a general result for comparing polygraphs having same
generators and same linearized sources and targets.Comment: 28 page
Higher-dimensional normalisation strategies for acyclicity
We introduce acyclic polygraphs, a notion of complete categorical cellular
model for (small) categories, containing generators, relations and
higher-dimensional globular syzygies. We give a rewriting method to construct
explicit acyclic polygraphs from convergent presentations. For that, we
introduce higher-dimensional normalisation strategies, defined as homotopically
coherent ways to relate each cell of a polygraph to its normal form, then we
prove that acyclicity is equivalent to the existence of a normalisation
strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical
finiteness condition for higher categories which extends Squier's finite
derivation type for monoids. We relate this homotopical property to a new
homological finiteness condition that we introduce here.Comment: Final versio
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