Multimodal logic

Multimodal logic

A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.

A modal logic with n primitive unary modal operators \Box_i, i\in \{1,\ldots, n\} is called an n-modal logic. Given these operators and negation, one can always add \Diamond_i modal operators defined as \Diamond_i P if and only if \lnot \Box_i \lnot P.

The first substantive example of a 2-modal logic is perhaps Arthur Prior's tense logic, with two modalities, F and P, corresponding to “sometime in the future” and “sometime in the past”. A logic[1] with infinitely many modalities is (propositional) dynamic logic, introduced in 1976 and having a separate modal operator for every regular expression. A version of temporal logic introduced in 1977 and intended for program verification has two modalities, corresponding to dynamic logic's [A] and [A*] modalities for a single program A, understood as the whole universe taking one step forwards in time. The term "multimodal logic" itself was not introduced until 1980. Another example of a multimodal logic is the Hennessy-Milner logic, itself a fragment of the more expressive modal μ-calculus, which additionally is also a fixed-point logic.

Multimodal logic can be used also to formalize a kind of knowledge representation: the motivation of epistemic logic is allowing several agents (they are regarded as subjects capable of forming beliefs, knowledge); and managing the belief or knowledge of each agent, so that epistemic assertions can be formed about them. The modal operator \Box must be capable of bookkeeping the cognition of each agent, thus \Box_i must be indexed on the set of the agents. The motivation is that \Box_i \alpha should assert "The subject i has knowledge about α being true". But it can be used also for formalizing "the subject i believes α". For formalization of meaning based on the possible world semantics approach, a multimodal generalization of Kripke semantics can be used: instead of a single "common" accessibility relation, there is a series of them indexed on the set of agents.[2]

Notes

  1. ^ Sergio Tessaris; Enrico Franconi; Thomas Eiter (2009). Reasoning Web. Semantic Technologies for Information Systems: 5th International Summer School 2009, Brixen-Bressanone, Italy, August 30 - September 4, 2009, Tutorial Lectures. Springer. pp. 112. ISBN 9783642037535. http://books.google.com/books?id=JdyeU7zs4-AC&pg=PA112. 
  2. ^ Ferenczi 2002: 257

References

  • Ferenczi, Miklós (2002) (in Hungarian). Matematikai logika. Budapest: Műszaki könyvkiadó. ISBN 963 16 2870 1. 
  • Dov M. Gabbay, A. Kurucz, F. Wolter, M. Zakharyaschev (2003). Many-dimensional modal logics: theory and applications. Elsevier. ISBN 9780444508263. 
  • Walter Carnielli; Claudio Pizzi (2008). Modalities and Multimodalities. Springer. ISBN 9781402085895. 

External links

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