This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...

We show that the monadic second-order theory of an infinite tree recognized by a higher-order pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higher-order grammars of level n. Our decidability result extends the result of Courcelle on algebraic (pushdown of level 1) trees and our own result on trees of level 2.

We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Set-functors straightforwardly gives rise to a transformation of coalgebras for the respective functors. This latter transformation preserves homomorphisms and thus bisimulations. For comparison of probabilistic system types we also need reflection of bisimulation. We build the hierarchy of probabilistic systems by exploiting the new result that the transformation also reflects bisimulation in case the natural transformation is componentwise injective and the first functor preserves weak pullbacks. Additionally, we illustrate the correspondence of concrete and coalgebraic bisimulation in the case of general Segala-type systems.

We consider observational specifications of state-based systems which incorporate the declaration of a distinguished set of observer operations. These observers determine an indistinguishability relation for states which is called "observational equality". An important requirement for the non-observer operations is the compatibility with the observational equality. In the CafeOBJ language (and in extended hidden algebra) this property is called "behavioural coherence". In this presentation we introduce the notion of an "observer complete definition" and we show that any (non-observer) operation which is defined using this pattern is behaviourally coherent. We also discuss some consequences of this result for relating observational logic and extended hidden algebra semantics and for proving the correctness of observational implementations.

By paying more attention to semantics-based tool generation, programming language semantics can significantly increase its impact. Ultimately, this may lead to "Language Design Assistants" incorporating substantial amounts of semantic knowledge. 1991 ACM Computing Classification System: D.2.2, D.3.1, D.3.4, F.3.2 Keywords and Phrases: semantics of programming languages, tool generation, language development system, language design assistant, domain-specific language, compiler toolkit, software renovation tool Note: Submitted to ACM SIGPLAN Notices. This research was supported in part by the Telematica Instituut under the Domain-Specific Languages project. 1 The Role of Programming Language Semantics Programming language semantics has lost touch with large groups of potential users [39]. Among the reasons for this unfortunate state of a#airs, one stands out. Semantic results are rarely incorporated in practical systems that would help language designers to implement and test a ...

This paper presents some techniques of this kind in the area called hidden algebra, clustered around the central notion of coinduction. We believe hidden algebra is the natural next step in the evolution of algebraic semantics and its first order proof technology. Hidden algebra originated in [7], and was developed further in [8, 10, 3, 12, 5] among other places; the most comprehensive survey currently available is [12]

Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.

. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and co-induction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...

We survey various notions of probabilistic automata and probabilistic bisimulation, accumulating in an expressiveness hierarchy of probabilistic system types. The aim of this paper is twofold: On the one hand it provides an overview of existing types of probabilistic systems and, on the other hand, it explains the relationship between these models.

This paper explains the details of the memory model underlying the verification of sequential Java programs in the "LOOP" project ([14, 20]). The building blocks of this memory are cells, which are untyped in the sense that they can store the contents of the fields of an arbitrary Java object. The main memory is modeled as three infinite series of such cells, one for storing instance variables on a heap, one for local variables and parameters on a stack, and and one for static (or class) variables. Verification on the basis of this memory model is illustrated both in PVS and in Isabelle/HOL, via several examples of Java programs, involving various subtleties of the language (wrt. memory storage).