17 research outputs found
Rule Algebras for Adhesive Categories
We demonstrate that the most well-known approach to rewriting graphical
structures, the Double-Pushout (DPO) approach, possesses a notion of sequential
compositions of rules along an overlap that is associative in a natural sense.
Notably, our results hold in the general setting of -adhesive
categories. This observation complements the classical Concurrency Theorem of
DPO rewriting. We then proceed to define rule algebras in both settings, where
the most general categories permissible are the finitary (or finitary
restrictions of) -adhesive categories with -effective
unions. If in addition a given such category possess an -initial
object, the resulting rule algebra is unital (in addition to being
associative). We demonstrate that in this setting a canonical representation of
the rule algebras is obtainable, which opens the possibility of applying the
concept to define and compute the evolution of statistical moments of
observables in stochastic DPO rewriting systems
Rule Algebras for Adhesive Categories
We demonstrate that the most well-known approach to rewriting graphical
structures, the Double-Pushout (DPO) approach, possesses a notion of sequential
compositions of rules along an overlap that is associative in a natural sense.
Notably, our results hold in the general setting of -adhesive
categories. This observation complements the classical Concurrency Theorem of
DPO rewriting. We then proceed to define rule algebras in both settings, where
the most general categories permissible are the finitary (or finitary
restrictions of) -adhesive categories with -effective
unions. If in addition a given such category possess an -initial
object, the resulting rule algebra is unital (in addition to being
associative). We demonstrate that in this setting a canonical representation of
the rule algebras is obtainable, which opens the possibility of applying the
concept to define and compute the evolution of statistical moments of
observables in stochastic DPO rewriting systems
Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)
Tracelets are the intrinsic carriers of causal information in categorical
rewriting systems. In this work, we assemble tracelets into a symmetric
monoidal decomposition space, inducing a cocommutative Hopf algebra of
tracelets. This Hopf algebra captures important combinatorial and algebraic
aspects of rewriting theory, and is motivated by applications of its
representation theory to stochastic rewriting systems such as chemical reaction
networks.Comment: In Proceedings ACT 2021, arXiv:2211.0110
Convolution Products on Double Categories and Categorification of Rule Algebras
Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product
On The Axioms Of -Adhesive Categories
Adhesive and quasiadhesive categories provide a general framework for the
study of algebraic graph rewriting systems. In a quasiadhesive category any two
regular subobjects have a join which is again a regular subobject. Vice versa,
if regular monos are adhesive, then the existence of a regular join for any
pair of regular subobjects entails quasiadhesivity. It is also known
(quasi)adhesive categories can be embedded in a Grothendieck topos via a
functor preserving pullbacks and pushouts along (regular) monomorphisms. In
this paper we extend these results to -adhesive
categories, a concept recently introduced to generalize the notion of
(quasi)adhesivity. We introduce the notion of -adhesive morphism,
which allows us to express -adhesivity as a condition
on the subobjects's posets. Moreover, -adhesive morphisms allows
us to show how an -adhesive category can be embedded
into a Grothendieck topos, preserving pullbacks and -pushouts
Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems
We develop a novel method to analyze the dynamics of stochastic rewriting
systems evolving over finitary adhesive, extensive categories. Our formalism is
based on the so-called rule algebra framework and exhibits an intimate
relationship between the combinatorics of the rewriting rules (as encoded in
the rule algebra) and the dynamics which these rules generate on observables
(as encoded in the stochastic mechanics formalism). We introduce the concept of
combinatorial conversion, whereby under certain technical conditions the
evolution equation for (the exponential generating function of) the statistical
moments of observables can be expressed as the action of certain differential
operators on formal power series. This permits us to formulate the novel
concept of moment-bisimulation, whereby two dynamical systems are compared in
terms of their evolution of sets of observables that are in bijection. In
particular, we exhibit non-trivial examples of graphical rewriting systems that
are moment-bisimilar to certain discrete rewriting systems (such as branching
processes or the larger class of stochastic chemical reaction systems). Our
results point towards applications of a vast number of existing
well-established exact and approximate analysis techniques developed for
chemical reaction systems to the far richer class of general stochastic
rewriting systems