140 research outputs found
Observation and abstract behaviour in specification and implementation of state-based systems
Classical algebraic specification is an accepted framework for specification. A criticism which applies is the
fact that it is functional, not based on a notion of state as most software development and implementation languages
are. We formalise the idea of a state-based object or abstract machine using algebraic means. In contrast to similar approaches we consider dynamic logic instead of equational logic as the framework for specification and implementation. The advantage is a more expressive language allowing us to specify safety and liveness conditions. It also allows a clearer distinction of functional and state-based parts which require different treatment in order to achieve behavioural abstraction when necessary. We shall in particular focus on abstract behaviour and observation. A behavioural notion of satisfaction for state-elements is needed in order to abstract from irrelevant details of the state realisation
(Ω, Ξ)-Logic: On the Algebraic Extension of Coalgebraic Specifications
We present an extension of standard coalgebraic specification techniques for statebased systems which allows us to integrate constants and n-ary operations in a smooth way and, moreover, leads to a simplification of the coalgebraic structure of the models of a specification. The framework of (Ω,Ξ)-logic can be considered as the result of a translation of concepts of observational logic (cf. [9]) into the coalgebraic world. As a particular outcome we obtain the notion of an (Ω, Ξ)- structure and a sound and complete proof system for (first-order) observational properties of specifications
Towards a Uniform Theory of Effectful State Machines
Using recent developments in coalgebraic and monad-based semantics, we
present a uniform study of various notions of machines, e.g. finite state
machines, multi-stack machines, Turing machines, valence automata, and weighted
automata. They are instances of Jacobs' notion of a T-automaton, where T is a
monad. We show that the generic language semantics for T-automata correctly
instantiates the usual language semantics for a number of known classes of
machines/languages, including regular, context-free, recursively-enumerable and
various subclasses of context free languages (e.g. deterministic and real-time
ones). Moreover, our approach provides new generic techniques for studying the
expressivity power of various machine-based models.Comment: final version accepted by TOC
A modal proof theory for final polynomial coalgebras
AbstractAn infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal” sets of formulas that have natural syntactic closure properties.The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if the deducibility relation is generated by countably many inference rules.A counter-example to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable
Coalgebras and Their Logics
Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics
Probabilistic call by push value
We introduce a probabilistic extension of Levy's Call-By-Push-Value. This
extension consists simply in adding a " flipping coin " boolean closed atomic
expression. This language can be understood as a major generalization of
Scott's PCF encompassing both call-by-name and call-by-value and featuring
recursive (possibly lazy) data types. We interpret the language in the
previously introduced denotational model of probabilistic coherence spaces, a
categorical model of full classical Linear Logic, interpreting data types as
coalgebras for the resource comonad. We prove adequacy and full abstraction,
generalizing earlier results to a much more realistic and powerful programming
language
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