A coalgebraic perspective on linear weighted automata

Weighted automata are a generalization of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterize weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterize weighted language equivalence. Relying on the second characterization, we show three different procedures for computing weighted language equivalence. The first one consists in a generalizion of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages

Coalgebraic Aspects of Bidirectional Computation

We have previously (Bx, 2014; MPC, 2015) shown that several statebased bx formalisms can be captured using monadic functional programming, using the state monad together with possibly other monadic effects, giving rise to structures we have called monadic bx (mbx). In this paper, we develop a coalgebraic theory of state-based bx, and relate the resulting coalgebraic structures (cbx) to mbx. We show that cbx support a notion of composition coherent with, but conceptually simpler than, our previous mbx definition. Coalgebraic bisimulation yields a natural notion of behavioural equivalence on cbx, which respects composition, and essentially includes symmetric lens equivalence as a special case. Finally, we speculate on the applications of this coalgebraic perspective to other bx constructions and formalisms

Coend calculus

The book formerly known as "This is the (co)end, my only (co)friend".Comment: This is the version ready for submissio

Why H Z-algebra Spectra are Differential Graded Algebras?

In homological algebra, to understand commutative rings R, one studies R-modules, chain complexes of R-modules and their monoids, the differential graded R-algebras. The category of R-modules has a rich structure, but too rigid to efficiently work with homological invariants and homotopy invariant properties. It appears more appropriate to operate in the derived category D(R), which is the homotopy category of differential graded R-modules. Algebra of symmetric spectra offers a generalization of homological algebra. In this frame, spectra are objects that take the place of abelian groups; in particular, the analogue of the initial ring Z is the sphere spectrum S. Tensoring over S endows the category of spectra with a symmetric monoidal smash product, analogous to the tensor product of abelian groups. Thus, spectra are S-modules, and ring spectra, which extend the notion of rings, are the S-algebras. To any discrete ring R, one can associate the Eilenberg-Mac Lane ring spectrum HR, which is commutative if R is

Abstract Copower

We give a common generalization of two earlier constructions in [2], that yielded coalgebraic type functors for weighted, resp. fuzzy transition systems. Transition labels for these systems were drawn from a commutative monoid M or a complete semilattice L, with the transition structure interacting with the algebraic structure on the labels. Here, we show that those earlier signature functors are in fact instances of a more general construction, provided by the so-called copower functor. Exemplarily, we instantiate this functor in categories given by varieties V of algebras. In particular, for the variety S of all semigroups, or the variety M of all (not necessarily commutative) monoids, and with M any monoid, we find that the resulting copower functors MS[−] (resp MM[−]) weakly preserve pullbacks if and only if M is equidivisible (resp. conical and equidivisible). Finally, we show that copower functors are universal in the sense that every Setfunctor can be seen as an instance of an appropreiate copower functor.