. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

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...

. The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures -- viewed as multialgebras -- reveals the one with the most desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to sets (thus being relations). Finally, we indicate some consequences of our results for partial algebras which are special case of multialgebras. 1 Introduction In the study of universal algebra, the central place occupies the pair of "dual" notions of congruence and homomorphism: every congruence on an algebra induces a homomorphism into a quotient and every homomorphism induces a congruence on the source algebra. Categorical approach attempts to express all (internal) properties of algebras in (external) terms of homomorphisms. When passing to relational structures, however, the close correspondence of these internal and external aspects seems to get lost. The most common, and natural, gene...

. Multi-algebras allow to model 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 \Sigma as cartesian functors from the algebraic theory of \Sigma , Th(\Sigma), to Set. The functors we introduce are based on variations of the notion of theory, having a structure weaker than cartesian, and their target is Rel, the category of sets and relations. We argue that this functorial presentation provides an original abstract syntax for partial and multi-algebras. 1 Introduction Nondeterminism is a fundamental concept in Computer Science. It arises not only from the study of intrinsically nondeterministic computational models, like Turing machines and various kinds of automata, but also in the study of the behaviour of deterministic sys...

Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multi-algebras based on gs-monoidal theories, we argue that speci cations for multi-algebras should be based on the notion of term graphs instead of on standard terms. We consider the simplest case of (term graph) equational specification, showing that it enjoys an unrestricted form of substitutivity. We discuss the expressive power of equational specification for multialgebras, and we sketch possible extensions of the calculus.

This paper is an attempt to bring some order into this chaos. Instead of listing and defending new definitions we hope that approaching the problem from a more algebraic perspective may bring at least some clarification. Section 2 addresses the question of composition of homomorphisms. In subsection 2.1 we give a characterization of relational homomorphisms which are closed under composition -- in fact, most of the suggested defintions, like most of those in table 1, do not enjoy this property which we believe is crucial. We also characterize equivalences associated with various compositional homomorphisms. Then, in section 3 we introduce multialgebras which are relational structures with composition of relations (1.2) reflecting the traditional way of composing functions. Subsection 3.1 sketches the relation between multialgebras and their quotients by (congruence)

This paper introduces a framework for rapid prototyping of object oriented programming languages and corresponding analysis tools. It is based on formal definitions of language features in rewrite logic, a simple and intuitive logic for concurrency with powerful tool support. A domain-specific front-end consisting of a notation and a technique, called K, allows for compact, modular, expressive and easy to understand and change definitions of language features. The framework is illustrated by first defining KOOL, an experimental concurrent object-oriented language with exceptions, and then by discussing the definition of JAVA. Generic rewrite logic tools, such as efficient rewrite engines and model checkers, can be used on language definitions and yield interpreters and corresponding formal program analyzers at no additional cost.

Exactly solving first-order constraints (i.e., first-order formulas over a certain prede ned structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate solutions instead of exact ones. However, the quantifiers of the first-order predicate language are an obstacle to allowing approximations to arbitrary small error bounds. In this paper we solve the problem by modifying the first-order language and replacing the classical quantifiers with approximate quantifiers. These also have two additional advantages: First, they are tunable, in the sense that they allow the user to decide on the trade-off between precision and efficiency. Second, they introduce additional expressivity into the first-order language by allowing reasoning over the size of solution sets.

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Summary. We introduce a geometrical 3 setting which seems promising for the study of computation in multiset rewriting systems, but could also be applied to register machines and other models of computation. This approach will be applied here to membrane systems (also known as P systems) without dynamical membrane creation. We discuss the rôle of maximum parallelism and further simplify our model by considering only one membrane and sequential application of rules, thereby arriving at asynchronous multiset rewriting systems (AMR systems). Considering only one membrane is no restriction, as each static membrane system has an equivalent AMR system. It is further shown that AMR systems without a priority relation on the rules are equivalent to Petri Nets. For these systems we introduce the notion of asymptotically exact computation, which allows for stochastic appearance checking in a priori bounded (for some complexity measure) computations. The geometrical analogy in the lattice N d 0, d ∈ N, is developed, in which a computation corresponds to a trajectory of a random walk on the directed graph induced by the possible rule applications. Eventually this leads to symbolic dynamics on the partition generated by shifted positive cones C + p, p ∈ N d 0, which are associated with the rewriting rules, and their intersections. Complexity measures are introduced and we consider non–halting, loop–free computations and the conditions imposed on the rewriting rules. Eventually, two models of information processing, control by demand and control by availability are discussed and we end with a discussion of possible future developments. 1