3,638 research outputs found
Coalgebraic Infinite Traces and Kleisli Simulations
Kleisli simulation is a categorical notion introduced by Hasuo to verify
finite trace inclusion. They allow us to give definitions of forward and
backward simulation for various types of systems. A generic categorical theory
behind Kleisli simulation has been developed and it guarantees the soundness of
those simulations with respect to finite trace semantics. Moreover, those
simulations can be aided by forward partial execution (FPE)---a categorical
transformation of systems previously introduced by the authors.
In this paper, we give Kleisli simulation a theoretical foundation that
assures its soundness also with respect to infinitary traces. There, following
Jacobs' work, infinitary trace semantics is characterized as the "largest
homomorphism." It turns out that soundness of forward simulations is rather
straightforward; that of backward simulation holds too, although it requires
certain additional conditions and its proof is more involved. We also show that
FPE can be successfully employed in the infinitary trace setting to enhance the
applicability of Kleisli simulations as witnesses of trace inclusion. Our
framework is parameterized in the monad for branching as well as in the functor
for linear-time behaviors; for the former we mainly use the powerset monad (for
nondeterminism), the sub-Giry monad (for probability), and the lift monad (for
exception).Comment: 39 pages, 1 figur
Expressive Logics for Coinductive Predicates
The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata
Fair Simulation for Nondeterministic and Probabilistic Buechi Automata: a Coalgebraic Perspective
Notions of simulation, among other uses, provide a computationally tractable
and sound (but not necessarily complete) proof method for language inclusion.
They have been comprehensively studied by Lynch and Vaandrager for
nondeterministic and timed systems; for B\"{u}chi automata the notion of fair
simulation has been introduced by Henzinger, Kupferman and Rajamani. We
contribute to a generalization of fair simulation in two different directions:
one for nondeterministic tree automata previously studied by Bomhard; and the
other for probabilistic word automata with finite state spaces, both under the
B\"{u}chi acceptance condition. The former nondeterministic definition is
formulated in terms of systems of fixed-point equations, hence is readily
translated to parity games and is then amenable to Jurdzi\'{n}ski's algorithm;
the latter probabilistic definition bears a strong ranking-function flavor.
These two different-looking definitions are derived from one source, namely our
coalgebraic modeling of B\"{u}chi automata. Based on these coalgebraic
observations, we also prove their soundness: a simulation indeed witnesses
language inclusion
Embedding locales and formal topologies into positive topologies
A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure
of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a re\ufb02ective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales
On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos
In the effective topos there exists a chain-complete distributive lattice
with a monotone and progressive endomap which does not have a fixed point.
Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for
chain-complete lattices do not have constructive (topos-valid) proofs
Some problems of infinite regress in social-choice models: a category theory solution
An analysis of the infinite regress that appears in the statement of Gauthier's bargaining approach to social choice. The author shows how category theory provides the tools for constructing the appropriate bargaining models by furnishing a setting for the concepts of continuity, limits, and fixed points.Game theory
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