51 research outputs found
A short overview of Hidden Logic
In this paper we review a hidden (sorted) generalization of k-deductive systems - hidden k-logics. They encompass deductive systems as
well as hidden equational logics and inequational logics. The special case of
hidden equational logics has been used to specify and to verify properties in
program development of behavioral systems within the dichotomy visible vs.
hidden data. We recall one of the main applications of this work - the study
of behavioral equivalence. Related results are obtained through combinatorial
properties of the Leibniz congruence relation.
In addition we obtain a few new developments concerning hidden equational
logic, namely we present a new characterization of the behavioral consequences of a theory
Weak Similarity in Higher-Order Mathematical Operational Semantics
Higher-order abstract GSOS is a recent extension of Turi and Plotkin's
framework of Mathematical Operational Semantics to higher-order languages. The
fundamental well-behavedness property of all specifications within the
framework is that coalgebraic strong (bi)similarity on their operational model
is a congruence. In the present work, we establish a corresponding congruence
theorem for weak similarity, which is shown to instantiate to well-known
concepts such as Abramsky's applicative similarity for the lambda-calculus. On
the way, we develop several techniques of independent interest at the level of
abstract categories, including relation liftings of mixed-variance bifunctors
and higher-order GSOS laws, as well as Howe's method
Behavioral equivalence of hidden k-logics: an abstract algebraic approach
This work advances a research agenda which has as its main aim the application
of Abstract Algebraic Logic (AAL) methods and tools to the specification and
verification of software systems. It uses a generalization of the notion of an abstract
deductive system to handle multi-sorted deductive systems which differentiate
visible and hidden sorts. Two main results of the paper are obtained by generalizing
properties of the Leibniz congruence — the central notion in AAL.
In this paper we discuss a question we posed in [1] about the relationship between
the behavioral equivalences of equivalent hidden logics. We also present a necessary
and sufficient intrinsic condition for two hidden logics to be equivalent
A general account of coinduction up-to
Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras. In the second part of the paper, we provide a new categorical treatment of weak bisimilarity on labeled transition systems and we prove the soundness of up-to context for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. The weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend the categorical framework developed in the first part to an ordered setting
Composition and Recursion for Causal Structures
Causality appears in various contexts as a property where present behaviour can only depend on past events, but not on future events. In this paper, we compare three different notions of causality that capture the idea of causality in the form of restrictions on morphisms between coinductively defined structures, such as final coalgebras and chains, in fairly general categories. We then focus on one presentation and show that it gives rise to a traced symmetric monoidal category of causal morphisms. This shows that causal morphisms are closed under sequential and parallel composition and, crucially, under recursion
Codensity Lifting of Monads and its Dual
We introduce a method to lift monads on the base category of a fibration to
its total category. This method, which we call codensity lifting, is applicable
to various fibrations which were not supported by its precursor, categorical
TT-lifting. After introducing the codensity lifting, we illustrate some
examples of codensity liftings of monads along the fibrations from the category
of preorders, topological spaces and extended pseudometric spaces to the
category of sets, and also the fibration from the category of binary relations
between measurable spaces. We also introduce the dual method called density
lifting of comonads. We next study the liftings of algebraic operations to the
codensity liftings of monads. We also give a characterisation of the class of
liftings of monads along posetal fibrations with fibred small meets as a limit
of a certain large diagram.Comment: Extended version of the paper presented at CALCO 2015, accepted for
publication in LMC
Extensions of Functors From Set to V-cat
We show that for a commutative quantale V every functor from Set to V-cat has an enriched left-Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric
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