Institution (computer Science) - Definition

Definition

The theory of institutions does not assume anything about the nature of the logical system. That is, models and sentences may be arbitrary objects; the only assumption being that there is a satisfaction relation between models and sentences, telling whether a sentence holds in a model or not. Satisfaction is inspired by Tarski's truth definition, but can in fact be any binary relation. A crucial feature of institutions now is that models, sentences and their satisfaction are always being considered to live in some vocabulary or context (called signature) that defines the (non-logical) symbols that may be used in sentences and that need to be interpreted in models. Moreover, signature morphisms allow to extend signatures, change notation etc. Nothing is assumed about signatures and signature morphisms except that signature morphisms can be composed; this amounts to having a category of signatures and morphisms. Finally, it is assumed that signature morphisms lead to translations of sentences and models in a way that satisfaction is preserved. While sentences are translated along with signature morphisms (think of symbols being replaced along the morphism), models are translated (or better: reduced) against signature morphisms: for example, in case of a signature extension, a model of the (larger) target signature may be reduced to a model of the (smaller) source signature by just forgetting some components of the model.

Formally, an institution consists of

• a category of signatures,
• a functor Set giving, for each signature, the set of sentences, and for each signature morphism, the sentence translation map, where often is written as ,
• a functor Cat giving, for each signature, the category of models, and for each signature morphism, the reduct functor, where often is written as ,
• a satisfaction relation for each ,

such that for each in the following satisfaction condition holds:

if and only if

for each and .

The satisfaction condition expresses that truth is invariant under change of notation (and also under enlargement or quotienting of context).

Strictly speaking, the model functor ends in the quasi-category of all large categories.