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Context in Computing pp 381–399Cite as

Multi-context Logics—A General Introduction

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Abstract

Multi-context logics (MCLs) constitute a family of formalisms that allow one to integrate multiple logical theories (contexts) into an articulated structure, where different theories can affect one another via so-called bridge-rules. In the past 20 years multi-context logics has been developed, for contexts in propositional logics, first order logics, description logics and temporal logic. Each of these logics has been developed, in an independent manner, for representing and reasoning about contextual knowledge in a specific application domain instead of originating from a single general formal framework. The absence of such a general formal framework for Multi-Context Systems (MCS), from which to extract tailored versions for the different application domain, has led to the development of a rather heterogeneous family of formal systems, whose comparison is sometimes very difficult. Being able to represent all these systems as specifications of a general class would be very useful as, for instance, one could reuse results proven in one MCS in another one. In this chapter, the authors provide an a-posteriori, systematic, and homogeneous description of the various MCSs introduced in the past. The authors do this firstly by providing a general definition of the MCS framework with its main components, which is general enough to capture the various versions of MCSs. Then, an account of the main logical specialisations of the MCS framework is provided, with an explanation of the domain of application they have been developed for.

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Notes

  1. 1.

    The term “chain” is slightly misleading, as it suggests that the set of contexts are structured in a total order (i.e., a chain) which might not be the case. Historically total ordered context structure was the first form of multi-context logic that has been studied. This made it natural to use the term “chain” for\(c\in C\). For the sake of notation this terminology was maintained also in more complex MCLs with different context structures as the one depicted in Fig. 25.2.

  2. 2.

    The domain relation is used to represent the overlapping between the domains of two contexts. Usually, in databases, or in ontology integration scenarios, the overlapping between two domains Δ i and Δ j is represented by imposing that\(\Delta_i\cap\Delta_j\) contains a set of elements which are supposed to exists both in the domain of theith context and of thejth context. The usage of a domain relation turns out to be more flexible than assuming domain intersection since it allows to integrate knowledge defined over overlapping but heterogeneous domains of interpretation. The typical case is the one of two databases that adopt a different level of abstraction to represent a specific domain. For instance, time at the level of day, and time at the level of hours.

  3. 3.

    Wherel,c andr stand forleft,center andright, respectively.

  4. 4.

    In the definition of multi-context logical consequence there is an implicit existential quantification of the free variables in φ which are not free in Γ. This is similar to what happens for the semantics of rules in logic programming, where variables that appear in the head of a rule (the consequence) which are not contained in the body are usually interpreted existentially.

  5. 5.

    Notice that, if\(|c_i|> 1\), i.e., there is more than one local model, it is possible that\(C\not\models i:\phi\) and\(C\not\models i:\neg\phi\).

  6. 6.

    The most general setting semantic matches are associated with weights (confidence value) but when mappings are crisp (i.e., confidence value is equal to 1) then they can be fruitfully formalized in two valued logics.

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  1. Fondazione Bruno Kessler, via Sommarive 18, 38123, Trento, Italy

    Chiara Ghidini & Luciano Serafini

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  1. Chiara Ghidini
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  2. Luciano Serafini
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Correspondence to Chiara Ghidini .

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Editors and Affiliations

  1. University Pierre and Marie Curie (UPMC), Paris, France

    Patrick Brézillon

  2. University of Central Florida, Orlando, Florida, USA

    Avelino J. Gonzalez

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Ghidini, C., Serafini, L. (2014). Multi-context Logics—A General Introduction. In: Brézillon, P., Gonzalez, A. (eds) Context in Computing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1887-4_24

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  • DOI: https://doi.org/10.1007/978-1-4939-1887-4_24

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