149 research outputs found
Bialgebraic Semantics for Logic Programming
Bialgebrae provide an abstract framework encompassing the semantics of
different kinds of computational models. In this paper we propose a bialgebraic
approach to the semantics of logic programming. Our methodology is to study
logic programs as reactive systems and exploit abstract techniques developed in
that setting. First we use saturation to model the operational semantics of
logic programs as coalgebrae on presheaves. Then, we make explicit the
underlying algebraic structure by using bialgebrae on presheaves. The resulting
semantics turns out to be compositional with respect to conjunction and term
substitution. Also, it encodes a parallel model of computation, whose soundness
is guaranteed by a built-in notion of synchronisation between different
threads
Interacting Hopf Algebras
We introduce the theory IH of interacting Hopf algebras, parametrised over a
principal ideal domain R. The axioms of IH are derived using Lack's approach to
composing PROPs: they feature two Hopf algebra and two Frobenius algebra
structures on four different monoid-comonoid pairs. This construction is
instrumental in showing that IH is isomorphic to the PROP of linear relations
(i.e. subspaces) over the field of fractions of R
The Power of Convex Algebras
Probabilistic automata (PA) combine probability and nondeterminism. They can
be given different semantics, like strong bisimilarity, convex bisimilarity, or
(more recently) distribution bisimilarity. The latter is based on the view of
PA as transformers of probability distributions, also called belief states, and
promotes distributions to first-class citizens.
We give a coalgebraic account of the latter semantics, and explain the
genesis of the belief-state transformer from a PA. To do so, we make explicit
the convex algebraic structure present in PA and identify belief-state
transformers as transition systems with state space that carries a convex
algebra. As a consequence of our abstract approach, we can give a sound proof
technique which we call bisimulation up-to convex hull.Comment: Full (extended) version of a CONCUR 2017 paper, to be submitted to
LMC
Coalgebraic Behavioral Metrics
We study different behavioral metrics, such as those arising from both
branching and linear-time semantics, in a coalgebraic setting. Given a
coalgebra for a functor , we define a framework for deriving pseudometrics on which
measure the behavioral distance of states.
A crucial step is the lifting of the functor on to a
functor on the category of pseudometric spaces.
We present two different approaches which can be viewed as generalizations of
the Kantorovich and Wasserstein pseudometrics for probability measures. We show
that the pseudometrics provided by the two approaches coincide on several
natural examples, but in general they differ.
If has a final coalgebra, every lifting yields in a
canonical way a behavioral distance which is usually branching-time, i.e., it
generalizes bisimilarity. In order to model linear-time metrics (generalizing
trace equivalences), we show sufficient conditions for lifting distributive
laws and monads. These results enable us to employ the generalized powerset
construction
Towards Trace Metrics via Functor Lifting
We investigate the possibility of deriving metric trace semantics in a
coalgebraic framework. First, we generalize a technique for systematically
lifting functors from the category Set of sets to the category PMet of
pseudometric spaces, showing under which conditions also natural
transformations, monads and distributive laws can be lifted. By exploiting some
recent work on an abstract determinization, these results enable the derivation
of trace metrics starting from coalgebras in Set. More precisely, for a
coalgebra on Set we determinize it, thus obtaining a coalgebra in the
Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we
can equip the final coalgebra with a behavioral distance. The trace distance
between two states of the original coalgebra is the distance between their
images in the determinized coalgebra through the unit of the monad. We show how
our framework applies to nondeterministic automata and probabilistic automata
Combining Semilattices and Semimodules
We describe the canonical weak distributive law of the powerset monad over
the -left-semimodule monad , for a class of semirings . We
show that the composition of with by means of such
yields almost the monad of convex subsets previously introduced by
Jacobs: the only difference consists in the absence in Jacobs's monad of the
empty convex set. We provide a handy characterisation of the canonical weak
lifting of to as well as an algebraic
theory for the resulting composed monad. Finally, we restrict the composed
monad to finitely generated convex subsets and we show that it is presented by
an algebraic theory combining semimodules and semilattices with bottom, which
are the algebras for the finite powerset monad
Up-To Techniques for Behavioural Metrics via Fibrations
Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras and we provide abstract results to prove their soundness in a compositional way.
In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale, for which pseudo-metric spaces are an example. To illustrate our approach we provide an example on distances between regular languages
Bisimilarity of Open Terms in Stream GSOS
Stream GSOS is a specification format for operations and calculi on infinite
sequences. The notion of bisimilarity provides a canonical proof technique for
equivalence of closed terms in such specifications. In this paper, we focus on
open terms, which may contain variables, and which are equivalent whenever they
denote the same stream for every possible instantiation of the variables. Our
main contribution is to capture equivalence of open terms as bisimilarity on
certain Mealy machines, providing a concrete proof technique. Moreover, we
introduce an enhancement of this technique, called bisimulation up-to
substitutions, and show how to combine it with other up-to techniques to obtain
a powerful method for proving equivalence of open terms
The Axiom of Choice in Cartesian Bicategories
We argue that cartesian bicategories, often used as a general categorical algebra of relations, are also a natural setting for the study of the axiom of choice (AC). In this setting, AC manifests itself as an inequation asserting that every total relation contains a map. The generality of cartesian bicategories allows us to separate this formulation from other set-theoretically equivalent properties, for instance that epimorphisms split. Moreover, via a classification result, we show that cartesian bicategories satisfying choice tend to be those that arise from bicategories of spans
Behavioral Metrics via Functor Lifting
We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra alpha : X -> FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states.
A first crucial step is the lifting of the functor F on Set to a functor /F in the category PMet of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ.
Then a final coalgebra for F in Set can be endowed with a behavioral distance resulting as the smallest solution of a fixed-point equation, yielding the final /F-coalgebra in PMet. The same technique, applied to an arbitrary coalgebra alpha : X -> FX in Set, provides the behavioral distance on X. Under some constraints we can prove that two states are at distance 0 if and only if they are behaviorally equivalent
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