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...

. 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...

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this paper. Study of PSPs has long tended in the direction of PDTs [1, 2, 3, 5]. One of the problems is that, while the former continued the tradition of working with classes axiomatized by (possibly conditional) equations, the latter require a precise grasp on individual algebras (which, for modeling purposes, can be identified with programs): a program P taking as a parameter another program X cannot change X -- X functions in the context of P , that is in P [X ], in the same way as it would in isolation. This intuition of "preserving actual parameter" has been identified as one of the semantic requirements on PSP in form of the persistency requirement on the functors from # Email: yngvel@ii.uib.no + Email: michal@ii.uib.no (1), e.g., [3, 15, 2]. However, in the purely equational context, there was hardly any syntactic counterpart of this semantic requirement. Thus, no syntactic/logical means were available for reasoning about correctness of such implementations

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The paper presents a new way to model partial operations by use of nondeterminism: a partial operation is modeled as a nondeterministic operation returning, possibly, any value of the carrier. We introduce an institution of multialgebras MA (modelling nondeterministic operations by set-valued functions) and illustrate the flexibility of our approach by examples showing uniform treatment of strictness, non-strictness and various error handling. We present a methodology for specification development from an abstract specification to low level error handling. We also study the conditions for MA specifications to ensure the existence of initial models. Finally we relate MA to other institutions, in particular, we show embedding of partial algebras and membership algebras into MA. Applying institution transformation instead of embedding leads to the possibility of resuing partial algebra specifications in the proposed framework -- a partial algebra specification can be conservatively (prese...

All kinds of coherence and immanent theories of truth and meaning notwithstanding, the basic intuition remains the same: linguistic expressions refer, for the most, to some non-linguistic world and language’s meaning is constituted bya relation to this world. What this relation consists of is the painful question, even more so, as it is hard to saywhat the world is without saying it in... some language. A schematic illustration of this follows.

The paper attempts a systematic study of homomorphisms of relational structures. Such structures are modeled as multialgebras (i.e., relation is represented as a set-valued function). The first, main, result is that, under reasonable restrictions on the form of the definition of homomorphism, there are exactly nine compositional homomorphisms of multialgebras. Then the comparison of the obtained categories with respect to the existence of finite limits and co-limits reveals two of them to be finitely complete and co-complete. Without claiming that compositionality and categorical properties are the only possible criteria for selecting a definition of homomorphism, we nevertheless suggest that, for many purposes, these criteria actually might be acceptable. For such cases, the paper gives an overview of the available alternatives and a clear indication of their advantages and disadvantages.