Barwise’s Information Frames and Modal Logics

V.V. Rybakov

The aim of our research is to develop mathematical theory of Barwise’s information frames [1]. These frames, in mathematical sense, are special multi-based models, or algebraic systems, with the relations

s which express the context’s depending consequence relations on classification types, i.e. they are aimed to handle information about situations which occur as a base set in these frames. We answer the Jon Barwise question from [1]: to find axiomatization for modal logics generated by information frames:

Theorem 1. (i) Modal logic L(K_{c}) of all complete information frames is the logic K. (ii) Modal logic L(K_{c,s}) of all complete and sound information frames is the logic T. (iii) Modal logic L(K_{c,h}) of all complete and hereditary information frames is the logic K4. (iv) Modal logic L(K_{c,h,s}) of all complete hereditary and sound information frames is the logic S4. (v) Modal logic L(K_{c,con}) of all complete and consistent information frames is the logic D.

Theorem 2. (i) The weak modal logic L_{w}(K_{c}) of all information frames is the logic K. (ii) The weak modal logic L_{w}(K_{h}) of all hereditary information frames is the logic K4.

Theorem 3. Any Kripke complete normal modal logic is complete w.r.t. appropriate class of complete information frames, i.e. is information frames complete.

Theorem 4.For any rarefied information frame IF the context’s logical consequence relation _{s} can be presented by accessibility context’s consequence of usual Kripke frames.

Theorem 5. All modal logics generated by arbitrary classes of complete, rarefied and fully classified information frames are Kripke complete.

[1] Barwise J. Information and impossibilities.- Notre Dame Journal of Formal Logic, Vol.38, No. 4 (1997), 488 - 515.

[2] Barwise J., Seligman J. Information Flow in Distributed Systems. - Book, Cambridge University Press, Tracts in Theoretical Computer Science 44, 1997.

[3] Dretske F. Knowledge and the Flow of Information. - Bradford Books, MIT Press, 1981.

Address: Mathematics Department, Science Faculty of Hacettepe University, Beytepe-Ankara, Turkey, and Mathematical Department, Krasnoyarsk University, pr. Svobodnyi 79, 660 062, Krasnoyarsk, Russia

E-mail: rybakov@math.kgu.krasnoyarsk.su and rybakov@eti.cc.hun.edu.tr