57 research outputs found
Algebra, coalgebra, and minimization in polynomial differential equations
We consider reasoning and minimization in systems of polynomial ordinary
differential equations (ode's). The ring of multivariate polynomials is
employed as a syntax for denoting system behaviours. We endow this set with a
transition system structure based on the concept of Lie-derivative, thus
inducing a notion of L-bisimulation. We prove that two states (variables) are
L-bisimilar if and only if they correspond to the same solution in the ode's
system. We then characterize L-bisimilarity algebraically, in terms of certain
ideals in the polynomial ring that are invariant under Lie-derivation. This
characterization allows us to develop a complete algorithm, based on building
an ascending chain of ideals, for computing the largest L-bisimulation
containing all valid identities that are instances of a user-specified
template. A specific largest L-bisimulation can be used to build a reduced
system of ode's, equivalent to the original one, but minimal among all those
obtainable by linear aggregation of the original equations. A computationally
less demanding approximate reduction and linearization technique is also
proposed.Comment: 27 pages, extended and revised version of FOSSACS 2017 pape
Algebra and Coalgebra of Stream Products
We study connections among polynomials, differential equations and streams over a field ?, in terms of algebra and coalgebra. We first introduce the class of (F,G)-products on streams, those where the stream derivative of a product can be expressed as a polynomial of the streams themselves and their derivatives. Our first result is that, for every (F,G)-product, there is a canonical way to construct a transition function on polynomials such that the induced unique final coalgebra morphism from polynomials into streams is the (unique) ?-algebra homomorphism - and vice-versa. This implies one can reason algebraically on streams, via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G)-product. As an example of reasoning on streams, we focus on specific products (convolution, shuffle, Hadamard) and show how to obtain closed forms of algebraic generating functions of combinatorial sequences, as well as solutions of nonlinear ordinary differential equations
On the Coalgebra of Partial Differential Equations
We note that the coalgebra of formal power series in commutative variables is final in a certain subclass of coalgebras. Moreover, a system Sigma of polynomial PDEs, under a coherence condition, naturally induces such a coalgebra over differential polynomial expressions. As a result, we obtain a clean coinductive proof of existence and uniqueness of solutions of initial value problems for PDEs. Based on this characterization, we give complete algorithms for checking equivalence of differential polynomial expressions, given Sigma
Coherent states for Hopf algebras
Families of Perelomov coherent states are defined axiomatically in the
context of unitary representations of Hopf algebras possessing a Haar integral.
A global geometric picture involving locally trivial noncommutative fibre
bundles is involved in the construction. A noncommutative resolution of
identity formula is proved in that setup. Examples come from quantum groups.Comment: 19 pages, uses kluwer.cls; the exposition much improved; an example
of deriving the resolution of identity via coherent states for SUq(2) added;
the result differs from the proposals in literatur
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
Enhanced Coalgebraic Bisimulation
International audienceWe present a systematic study of bisimulation-up-to techniques for coalgebras. This enhances the bisimulation proof method for a large class of state based systems, including labelled transition systems but also stream systems and weighted automata. Our approach allows for compositional reasoning about the soundness of enhancements. Applications include the soundness of bisimulation up to bisimilarity, up to equivalence and up to congruence. All in all, this gives a powerful and modular framework for simplified coinductive proofs of equivalence
Languages and models for hybrid automata: A coalgebraic perspective
article in pressWe study hybrid automata from a coalgebraic point of view. We show that such a perspective supports a generic theory of hybrid automata with a rich palette of definitions and results. This includes, among other things, notions of bisimulation and behaviour, state minimisation techniques, and regular expression languages.POCI-01-0145-FEDER-016692. RDF — European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation — COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT — Fundação para a Ciência e a Tecnologia within project POCI-01-0145-FEDER-016692 and by the PT-FLAD Chair on Smart Cities & Smart Governance at Universidade do Minh
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
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