ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...

. Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (non-temporal) object-oriented systems). The monoid action captures the influence of time on the state space, where the monoids that we consider are the natural numbers monoid (N; 0; +) of discrete time, and the positive reals monoid (R0 ; 0; +) of real time. Based on this semantics we develop a hybrid specification formalism with timed method applications: it involves expressions like s:meth@ff, with the following meaning: in state s let the state evolve for ff units of time (according to the monoid action), and then apply the (coalgebraic) method meth. In this formalism we specify various (elementary) hybrid systems, investigate their correctness, and display their behaviour in simulations. We furthe...

The context of this article is the program to develop monoidal bicategories with a feedback operation as an algebra of processes, with applications to concurrency theory. The objective here is to study reachability, minimization and minimal realization in these bicategories. In this setting the automata are 1-cells in contrast with previous studies where they appeared as objects. As a consequence we are able to study the relation of minimization and minimal realization to serial composition of automata using (co)lax (co)monads. We are led to define suitable behaviour categories and prove minimal realization theorems which extend classical results. This work has been supported by NSERC Canada, Italian MURST and the Australian Research Council 1 Introduction Katis, Sabadini, Walters, and Weld have described bicategories equipped with operations of serial and parallel composition, and feedback modelled as, respectively, composition of 1cells, a tensor product and an operation called...

Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly languages that allowed symbolic codes for operations and addresses. fortran began the spread of "high level" programming languages, though at the time it was strongly opposed by many assembly programmers; important features that developed later include blocks, recursive procedures, flexible types, classes, inheritance, modules, and genericity. Without going into the philosophical problems raised by abstraction (which in view of the discussion of realism in Section 4 may be considerable), it seems clear that the mathematics used to describe programming concepts should in general get more abstract as the programming concepts get more abstract. Nevertheless, there has been great resistance to u...

There is a growing interest in models for probabilistic systems. This fact is motivated by engineering applications, namely in problems concerning the evaluation of the performance of systems. It is of

An early result of Goguen [4, 5] describes the fundamental adjunction between categories of deterministic automata and their behaviours. Our first step is to redefine (morphisms in) these categories of automata and behaviours so that a restriction in Goguen's approach can be avoided. Subsequently we give a coalgebraic analysis of this behaviour-realization adjunction; it yields a second generalization to other types of (not only deterministic) automata (and their behaviours). We further show that our (redefined) categories of automata and behaviours support elementary process combinators like renaming, restriction, parallel composition, replication and feedback (some of which also occur, for example, in the -calculus). One of the main contributions is that replication !P is defined for an automaton P such that !P is the terminal coalgebra !P = ! Pk!P of the functor Pk(\Gamma) "compose with P ". The behaviour functor from automata to their behaviours preserves these process combinato...

Abstract. Computations on trees form a classical topic in computing. These computations can be described in terms of machines (typically called tree transducers), or in terms of functions. This paper focuses on three flavors of bottom-up computations, of increasing generality. It brings categorical clarity by identifying a category of tree transducers together with two different behavior functors. The first sends a tree transducer to a coKleisli or biKleisli map (describing the contribution of each local node in an input tree to the global transformation) and the second to a tree function (the global tree transformation). The first behavior functor has an adjoint realization functor, like in Goguen’s early work on automata. Further categorical structure, in the form of Hughes’s Arrows, appears in properly parameterized versions of these structures. 1

Contents 1 Necessary definitions from categorical automata theory and from topos theory 3 1.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Necessary definitions from categorical automata theory : : : : : : 4 1.2.1 Definition of an automaton in a category : : : : : : : : : : 4 1.2.2 Input varietors, output varietors : : : : : : : : : : : : : : : 4 1.2.3 Reachable morphisms, observable morphisms, reaction morphisms, implementations : : : : : : : : : : : : : : : : : : : 6 1.2.4 Pair kernels : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2.5 Coequalizers : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2.6 The category Aut C (X; Y ) : : : : : : : : : : : : : : : : : : : 6 1.3 Necessary definitions from topos theory : : : : : : : : : : : : : : : 7 1.3.1 Def