Institution (computer science)

The notion of institution was created by Joseph Goguen and Rod Burstall in the late 1970s, in order to deal with the "population explosion among the logical systems used in computer science". The notion attempts to "formalize the informal" concept of logical system.

The use of institutions makes it possible to develop concepts of specification languages (like structuring of specifications, parameterization, implementation, refinement, and development), proof calculi, and even tools in a way completely independent of the underlying logical system. There are also morphisms that allow to relate and translate logical systems. Important applications of this are re-use of logical structure (also called borrowing), and heterogeneous specification and combination of logics.

The spread of institutional model theory has generalized various notions and results of model theory, and institutions themselves have impacted the progress of universal logic.

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 is 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 is that models, sentences, and their satisfaction, are always considered to live in some vocabulary or context (called signature) that defines the (non-logic) symbols that may be used in sentences and that need to be interpreted in models. Moreover, signature morphisms allow to extend signatures, change notation, and so on. 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 the 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.

Let ${𝐂𝐚𝐭}^{\mathrm{o}\mathrm{p}}$ denote the opposite of the category of small categories. An institution formally consists of

• a category $𝐒𝐢𝐠𝐧$ of signatures,
• a functor $𝑆𝑒𝑛:𝐒𝐢𝐠𝐧\to$ $𝐒𝐞𝐭$ giving, for each signature $\mathrm{\Sigma }$, the set of sentences $𝑆𝑒𝑛(\mathrm{\Sigma })$, and for each signature morphism $\sigma :\mathrm{\Sigma }\to {\mathrm{\Sigma }}^{\prime }$, the sentence translation map $𝑆𝑒𝑛(\sigma ):𝑆𝑒𝑛(\mathrm{\Sigma })\to 𝑆𝑒𝑛({\mathrm{\Sigma }}^{\prime })$, where often $𝑆𝑒𝑛(\sigma )(\phi )$ is written as $\sigma (\phi )$,
• a functor $𝐌𝐨𝐝:𝐒𝐢𝐠𝐧\to {𝐂𝐚𝐭}^{\mathrm{o}\mathrm{p}}$ giving, for each signature $\mathrm{\Sigma }$, the category of models $𝐌𝐨𝐝(\mathrm{\Sigma })$, and for each signature morphism $\sigma :\mathrm{\Sigma }\to {\mathrm{\Sigma }}^{\prime }$, the reduct functor $𝐌𝐨𝐝(\sigma ):𝐌𝐨𝐝({\mathrm{\Sigma }}^{\prime })\to 𝐌𝐨𝐝(\mathrm{\Sigma })$, where often $𝐌𝐨𝐝(\sigma )(M^{\prime })$ is written as $M^{\prime }{|}_{\sigma }$,
• a satisfaction relation ${\models }_{\mathrm{\Sigma }}\subseteq |𝐌𝐨𝐝(\mathrm{\Sigma })|\times 𝑆𝑒𝑛(\mathrm{\Sigma })$ for each $\mathrm{\Sigma }\in 𝐒𝐢𝐠𝐧$,

such that for each $\sigma :\mathrm{\Sigma }\to {\mathrm{\Sigma }}^{\prime }$ in $𝐒𝐢𝐠𝐧$, the following satisfaction condition holds:

$$M^{\prime }{\models }_{{\mathrm{\Sigma }}^{\prime }}\sigma (\phi )\text{if and only if}{M}^{\prime }{|}_{\sigma }{\models }_{\mathrm{\Sigma }}\phi$$

for each $M^{\prime }\in 𝐌𝐨𝐝({\mathrm{\Sigma }}^{\prime })$ and $\phi \in 𝑆𝑒𝑛(\mathrm{\Sigma })$.

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 "category" of all large categories.

Examples of institutions

• Common logic
• Common Algebraic Specification Language (CASL)
• First-order logic
• Higher-order logic
• Intuitionistic logic
• Modal logic
• Propositional logic
• Temporal logic
• Web Ontology Language (OWL)

See also

• Abstract model theory
• Institutional model theory
• Universal logic

Further reading

• Template:Citation. This was the first publication on institution theory and the preliminary version of Goguen and Burstall (1992).
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External links

• Template:Citation
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• Formalism, Logic, Institution - Relating, Translating and Structuring. Includes large bibliography.
• Template:Citation. Contains recent work on institutional model theory.