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1 Basic Concepts

First, before considering the particular concepts underlying CASL, here is a brief reminder of how specification frameworks in general may be formalized in terms of so-called institutions (some category-theoretic details are omitted) and proof systems.

A basic specification framework may be characterized by:

a class Sig of signatures Sigma, each determining the set of symbols |Sigma| whose intended interpretation is to be specified, with morphisms between signatures;
a class Mod(Sigma) of models, with homomorphisms between them, for each signature Sigma;
a set Sen(Sigma) of sentences (or axioms), for each signature Sigma;
a relation |= of satisfaction, between models and sentences over the same signature; and
a proof system, for inferring sentences from sets of sentences.

A basic specification consists of a signature Sigma together with a set of sentences from Sen(Sigma). The signature provided for a particular sentence in a specification is called its local environment. It may be a restriction of the entire signature of the specification, e.g., determined by an order of presentation for the signature declarations and the sentences with linear visibility, where symbols may not be used before they have been declared; or it may be the entire signature, reflecting non-linear visibility.

The (loose) semantics of a basic specification is the class of those models in Mod(Sigma) which satisfy all the specified sentences. A specification is said to be consistent when there are some models that satisfy all the sentences, and inconsistent when there are no such models. A sentence is a consequence of a basic specification if it is satisfied in all the models of the specification.

A signature morphism sigma:Sigma -> Sigma' determines a translation function Sen(sigma) on sentences, mapping Sen(Sigma) to Sen(Sigma'), and a reduct function Mod(sigma) on models, mapping Mod(Sigma') to Mod(Sigma).¹ Satisfaction is required to be preserved by translation: for all S in Sen(Sigma), M' in Mod(Sigma'),

Mod(sigma)(M') |= S <=> M' |= Sen(sigma)(S).

The proof system is required to be sound, i.e., sentences inferred from a specification are always consequences; moreover, inference is to be preserved by translation.

Sentences of basic specifications may include constraints that restrict the class of models, e.g., to reachable ones.

The rest of this section considers many-sorted basic specifications of the CASL specification framework, and indicates the underlying signatures, models, and sentences.² Consideration of the extra features concerned with subsorts is deferred to Chapter 3.

The syntax of the language constructs used for expressing many-sorted basic specifications is described in Chapter 2; subsorting constructs are deferred to Chapter 4. The abstract syntax of any well-formed basic specification determines a signature and a set of sentences, the models of which provide the semantics of the basic specification.

CoFI Document: CASL/Summary --Version 0.99-- 21 April 1998.
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