The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and order-sortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...

Multi-algebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multi-algebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multi-algebras. Contents 1 Introduction 2 2 A short recap on multi-algebras 4 3...

For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of re-using proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various well-known institutions of total, order-sorted and partial algebras and first-order structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, order-sorted and partial algebras and first-order structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of so-called parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd Krieg-Br\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.

Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions.

State charts are finite state machines with hierarchical structuring and several models of communication. They are widely used in hardware, software, and communication industries to specify reactive and concurrent systems. In absence of a common semantics for state charts, we use Evolving Algebras, that showed to be powerful enough to model all types of sequential, parallel, distributed and real-time applications of dynamic systems. The generality of evolving algebras allows to mimic all state chart dialects. We use a many-sorted higher-order logic algebraic specification language with loose semantics to describe the states of evolving algebras. We have found that the hierarchical structuring operations of state charts coincide with the basic composition operations of the used specification language SLANG. We do not have to provide the complicated communication mechanisms existing in many state chart dialects, since SLANG is powerful enough to specify them upon need. SLANG supports con...

. The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of `classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has influenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review t...

. A single pushout approach to the transformation of attributed partial graphs based on categories of partial algebras and partial morphisms is introduced. A sufficient condition for pushouts in these categories is presented. As the synchronization mechanism we use amalgamation of rules and show how synchronization can be minimized. We point out how the results obtained can be employed in order to define an operational semantics for object specification languages. 1 Introduction Graphs and graph grammars usually yield intuitive descriptions of complex phenomena in computer science. Therefore, numerous approaches to graph grammars have been put forward, among them the logical approach [6], the set theoretic approach [29], and the algebraic approach [9]. Graph-based techniques have for instance been successfully applied in the realm of software engineering development environments [13, 14], for object-oriented languages based on asynchronous communication [22, 24, 20, 21] and in logic p...

ions are defined both for objects and layers. There are several compatibility requirements for the definition of these functions. The set of objects contains a specific ?-element which allows the source and target functions to be total on the set of objects. Up to now there exists no implementation of general colimits in the AGG-system. This problem is currently fixed by the integration of the colimit library. Again we can use the colimit computation for Alpha algebras. For this purpose we have to find an Alpha representation of AGG-graphs. Here we will outline the idea. r0 r1 r2 object layer label v0 r4 r0 Item Data The picture above presents a possible Alpha type algebra for AGG-graphs. r 0 ; r 1 and r 2 correspond to the abstraction, source and target functions, r 4 represents the assignment of layers to objects and v 0 is the labelling function. Note that although not shown in the picture, since all references are total, r 1 ; r 2 and r 3 are defined also for layer. This shows ...

Graph transformation systems have been introduced for the formal specication of software systems. States are thereby modeled as graphs, and computations as graph derivations according to the rules of the specication. Operations on graph derivations provide means to reason about the distribution and composition of computations. In this paper we discuss the development of an algebra of graph derivations as a descriptive model of graph transformation systems. For that purpose we use a categorical three level approach for the construction of models of computations based on structured transition systems. Categorically the algebra of graph derivations can then be characterized as a free double category with nite horizontal colimits.

. We propose a new approach to algebraic semantics of functional languages based on higher-order partial algebras and conditional existence equations. After dicussing more generally the relation ships between certain features of functional languages and special algebraic concepts and techniques the paper presents the theoretical basis of our approach. The main result concerns the existence of higher-order partial algebras freely generated by a set of variables and a set of existence equations. This result ensures the existence of free functor semantics in our approach. 1 Introduction We propose in this paper a new pragmatical approach to algebraic semantics of functional languages. The development of this approach was guided by the following heuristic principles: -- The basis should be a well-developed first-order algebraic specification formalism providing initial/free semantics and compositional operations on specifications. -- Higher-order features should be incorporated...