Abstract
This paper constitutes a first attempt at constructing semantic theories over institutions and examining the logical relations holding between different such theories. Our results show that this approach can be very useful for theoretical computer science (and may also contribute to the current philosophical debate regarding the semantic and the syntactic presentation of scientific theories). First we provide a definition of semantic theories in the institution theory framework - in terms of a set of models satisfying a given set of sentences - using the language-independent satisfaction relation characterizing institutions (Definition 17.3). Then we give a proof of the logical equivalence holding between the syntactic and the semantic presentation of a theory, based on the Galois connection holding between sentences and models (Theorem 17.1). We also show how to integrate and combine semantic theories using colimits (Theorem 17.2). Finally we establish when the output of a model-based software verification method applied to a semantic theory over an institution also holds for a semantic theory defined over a different institution (Theorem 17.3).
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Notes
- 1.
In the terminology of axiomatic set theory, \(E^*\) is a class of \(\varSigma \)-models, rather than a set [12]. For the sake of simplicity, this paper will nonetheless consider \(E^*\) as a set.
- 2.
\(E^\bullet \) is a model-theoretic closure of the set of \(\varSigma \)-sentences. For some institutions, including equational logic, a corresponding proof-theoretic notion can be given, insofar as there is a complete set of inference rules [13].
- 3.
In the equational institution, closed sets of models are usually called varieties. A set of models is called closed iff its objects are all the models of some set of sentences.
- 4.
Closure Lemma: For any morphism \(\phi : \varSigma \rightarrow \varSigma ^{'}\) and sets \(F, F^{'}\) of \(\varSigma -\) sentences: \(\phi (F^\circ ) \subseteq \phi (F)^\circ \).
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Angius, N., Dimarogkona, M., Stefaneas, P. (2017). Building and Integrating Semantic Theories over Institutions. In: Lambropoulou, S., Theodorou, D., Stefaneas, P., Kauffman, L. (eds) Algebraic Modeling of Topological and Computational Structures and Applications. AlModTopCom 2015. Springer Proceedings in Mathematics & Statistics, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-68103-0_17
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