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Data Integration
Published in Praveen Kumar, Jay Alameda, Peter Bajcsy, Mike Folk, Momcilo Markus, Hydroinformatics: Data Integrative Approaches in Computation, Analysis, and Modeling, 2005
The first framework comes from an area of mathematics known as model theory [29] and consists of (1) formal languages to characterize the data acquisition (the world and the sensing process), (2) models to represent input data, operations on data, and relations among the data, and (3) theories to capture symbolic knowledge about sensing [26]. Model theory concerns itself with expressing logical statements in a formal language, and then analyzing mathematical structures (known as models) that interpret these languages.
Logical dual concepts based on mathematical morphology in stratified institutions: applications to spatial reasoning
Published in Journal of Applied Non-Classical Logics, 2019
There exists a profusion of logics but all of them satisfy the same structure defined by a syntax, a semantics and a calculus. Syntax gives both the language (signatures) and the formal rules that define well-formed formulas and theories. Semantics, so-called model theory, gives the mathematical meaning of all these syntactic notions, among others the rules that associate truth values to formulas. Finally, calculus, so-called proof theory, gives the inference rules that govern the reasoning and thus translate semantics into syntax as correctly as possible. To cope with the explosion of logics, a categorical abstract model-theory, the theory of institutions (Diaconescu, 2008; Goguen & Burstall, 1992), has been proposed, that generalises Barwise's ‘Translation Axiom’ (Barwise, 1974). Institutions then define both syntax and semantics of logics at an abstract level, independently of commitment to any particular logic. Later, institutions have been extended to propose a syntactic approach to truth (Diaconescu, 2006, 2008; Fiadeiro & Sernadas, 1988; Meseguer, 1989). For the sake of generalisation, in institutions signatures are simply defined as objects of a category and formulas built over signatures are simply required to form a set. All other contingencies such as inductive definition of formulas are not considered. However, in concrete logics (anyway all the particular logics considered in this paper as examples), the reasoning (both syntactic and semantic) is defined by induction on the structure of formulas. Indeed, usually, formulas are built from ‘atomic’ formulas by applying iteratively operators such as connectives, quantifiers or modalities. What we can then observe is that most of these logical operators come through dual pairs (conjunction and disjunction ∧ and ∨, quantifiers ∀ and ∃, modalities and ⋄).