Recent trends and applications in definability theory
by
Ildiko Sain

Algebraic logic can be discussed in the broader perspective of the following two theories: definablity theory and duality theories.

The (logical) theory of definability was initiated by Alfred Tarski in 1934. Willem Blok and Don Pigozzi argue in [BP89, Appendic A, pp.60-62] that abstract algebraic logic can be based on definability theory. The case of "concrete" algebraic logic is analogous: to illustrate this we mention that the elements of a "generic" cylindric algebra are relations definable over some fixed model of first order logic. For more general considerations we refer to [HMT II, section 4.3].

By duality theories we mean theories like Stone Duality or adjoint functor pairs or Galois theoreies, Galois correspondences - cf. [AHS, p.80].

These two theories play an essential role in a new approach to relativity theory, recently persued by István Németi and co-workers. We will discuss and indicate how these applications induce applications in theoretical computer science.

The novelty in definablity theory we would like to discuss is definability of new sorts as opposed to (the classical approach of) definability of new relations. Definability of new sorts means definability of "new universes" populated with "new entities". This direction is related to Saharon Shelah's classification theory, or more precisely, Shelah's relative (to a predicate) classification theory. It is also strongly related to the computer science oriented approach of Dale Myers in [My97].

[AHS] J. Adámek, H. Herrlich and G. Strecker: Abstract and concrete categoriess, or the Joy of Cats. Wiley, 1990.

[BP89] W. J. Blok and D. L. Pigozzi: Algebraizable Logics. Memoirs Amer. Math. Soc., vol.77, 396, 1989, vi+78pp.

[HMT II] L. Henkin, J. D. Monk and A. Tarski: Cylindric Algebras Part II. North-Holland, Amsterdam, 1985.

[My97] D. Myers: An interpretive isomorphism between binary and ternary relations. In: Structures in logic and computer science, ed. Jan Mycielski, Grzegorz Rozenberg and Arto Salomaa, LNCS 1261, Springer, Berlin 1997, pp. 84-105.

Date received: January 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafo-57.