Finality regained: A coalgebraic study of Scott-sets and multisets

Foundations of Software Science and Computational Structures

2005

Abstract. Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.

“…Coalgebras for B N are directed graphs with N-weighted edges, often referred to as multigraphs [6]. 6.…”

Section: Remarkmentioning

confidence: 99%

Expressivity of Coalgebraic Modal Logic: The Limits and Beyond

Foundations of Software Science and Computational Structures

2005

“…This captures the semantics of graded modalities over multigraphs [15], which are precisely the B-coalgebras. Unlike in the modal case [27], the multigraph semantics does not engender the same notion of satisfiability as the more standard Kripke semantics of graded modalities, as the latter validates all formulas ¬♦ 1 i, i ∈ N. One can however polynomially encode Kripke semantics into multigraph semantics: a graded hybrid formula φ, w.l.o.g.…”

Section: Examplementioning

confidence: 99%

Coalgebraic Hybrid Logic

Myers

Pattinson

Foundations of Software Science and Computational Structures

2009

Abstract. We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators, thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K, a wide variety of logics with non-normal modal operators such as probabilistic, graded, or coalitional modalities and non-monotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.

“…Coalgebra has emerged as the right level of generality for a unified treatment of a wide range of modalities with seemingly disparate semantics beyond the realm of pure relational structures. Examples include monotone modalities [Chellas 1980], probabilistic modalities [Larsen and Skou 1991], graded modalities [Fine 1972;D'Agostino and Visser 2002], coalitional/alternatingtime modalities [Alur et al 2002;Pauly 2002], and various non-monotonic conditionals [Friedman and Halpern 1994;Olivetti et al 2007]. The semantics of coalgebraic logic is parametrized over the choice of an endofunctor on the category of sets, whose coalgebras play the role of frames.…”

Section: Introductionmentioning

confidence: 99%

Flat Coalgebraic Fixed Point Logics

CONCUR 2010 - Concurrency Theory

Venema

2010

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The µ-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 µ-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 µ-calculus including, e.g., flat fragments of the graded µ-calculus and the alternating-time µ-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.