135 research outputs found
Completeness of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
Completeness for Flat Modal Fixpoint Logics
This paper exhibits a general and uniform method to prove completeness for
certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form
\gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the
language L\sharp (\Gamma) is obtained by adding to the language of polymodal
logic a connective \sharp\_\gamma for each \gamma \epsilon. The term
\sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the
least fixed point of the functional interpretation of the term \gamma(x,
\varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma,
construct an axiom system which is sound and complete with respect to the
concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We
prove two results that solve this problem. First, let K\sharp (\Gamma) be the
logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint
axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma.
Provided that each indexing formula \gamma satisfies the syntactic criterion of
being untied in x, we prove this axiom system to be complete. Second,
addressing the general case, we prove the soundness and completeness of an
extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an
effective procedure that, given an indexing formula \gamma as input, returns a
finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded
by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever
\Gamma is finite
Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics
We establish a generic upper bound ExpTime for reasoning with global
assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier
results of this kind, our bound does not require a tractable set of tableau
rules for the instance logics, so that the result applies to wider classes of
logics. Examples are Presburger modal logic, which extends graded modal logic
with linear inequalities over numbers of successors, and probabilistic modal
logic with polynomial inequalities over probabilities. We establish the
theoretical upper bound using a type elimination algorithm. We also provide a
global caching algorithm that potentially avoids building the entire
exponential-sized space of candidate states, and thus offers a basis for
practical reasoning. This algorithm still involves frequent fixpoint
computations; we show how these can be handled efficiently in a concrete
algorithm modelled on Liu and Smolka's linear-time fixpoint algorithm. Finally,
we show that the upper complexity bound is preserved under adding nominals to
the logic, i.e. in coalgebraic hybrid logic.Comment: Extended version of conference paper in FCT 201
Coalgebraic Methods for Object-Oriented Specification
This thesis is about coalgebraic methods in software specification and verification. It extends known techniques of coalgebraic specification to a more general level to pave the way for real world applications of software verification. There are two main contributions of the present thesis: 1. Chapter 3 proposes a generalisation of the familiar notion of coalgebra such that classes containing methods with arbitrary types (including binary methods) can be modelled with these generalised coalgebras. 2. Chapter 4 presents the specification language CCSL (short for Coalgebraic Class Specification Language), its syntax, its semantics, and a prototype compiler that translates CCSL into higher-order logic.Die Dissertation beschreibt coalgebraische Mittel und Methoden zur Softwarespezifikation und -verifikation. Die Ergebnisse dieser Dissertation vereinfachen die Anwendung coalgebraischer Spezifikations- und Verifikationstechniken und erweitern deren Anwendbarkeit. Damit werden Softwareverifikation im Allgemeinen und im Besonderen coalgebraische Methoden zur Softwareverifikation der praktischen Anwendbarkeit ein Stück nähergebracht. Diese Dissertation enthält zwei wesentliche Beiträge: 1. Im Kapitel 3 wird eine Erweiterung des klassischen Begriffs der Coalgebra vorgestellt. Diese Erweiterung erlaubt die coalgebraische Modellierung von Klassenschnittstellen mit beliebigen Methodentypen (insbesondere mit binären Methoden). 2. Im Kapitel 4 wird die coalgebraische Spezifikationssprache CCSL (Coalgebraic Class Specification Language) vorgestellt. Die Bescheibung umfasst Syntax, Semantik und einen Prototypcompiler, der CCSL Spezifikationen in Logik höherer Ordnung (passend für die Theorembeweiser PVS und Isabelle/HOL) übersetzt
Precongruences and Parametrized Coinduction for Logics for Behavioral Equivalence
We present a new proof system for equality of terms which present elements of the final coalgebra of a finitary set functor. This is most important when the functor is finitary, and we improve on logical systems which have already been proposed in several papers. Our contributions here are (1) a new logical rule which makes for proofs which are somewhat easier to find, and (2) a soundness/completeness theorem which works for all finitary functors, in particular removing a weak pullback preservation requirement that had been used previously. Our work is based on properties of precongruence relations and also on a new parametrized coinduction principle
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