Catálogo de publicaciones - libros

Gödel’s modal logic approach to analyzing provability attracted a great deal of attention and eventually led to two distinct mathematical models. The first is the modal logic GL, also known as the Provability Logic, which was shown in 1979 by Solovay to be the logic of the formal provability predicate. The second is Gödel’s original modal logic of provability S4, together with its explicit counterpart, the Logic of Proofs LP, which was shown in 1995 by Artemov to provide an exact provability semantics for S4. These two models complement each other and cover a wide range of applications, from traditional proof theory to λ -calculi and formal epistemology.

This paper begins by briefly indicating the principal, non-standard motivations of the author for his decades of work in Computability Theory (CT), a.k.a. Recursive Function Theory.

The intuitive notion of computability was formalized in the XXth century, which strongly affected the development of mathematics and applications, new computational technologies, various aspects of the theory of knowledge, etc. A rigorous mathematical definition of computability and algorithm generated new approaches to understanding a solution to a problem and new mathematical disciplines such as computer science, algorithmical complexity, linear programming, computational modeling and simulation databases and search algorithms, automatical cognition, program languages and semantics, net security, coding theory, cryptography in open systems, hybrid control systems, information systems, etc.

Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity—including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.

What does the future hold for mathematical logic? In the early 1950’s I learned all the logic then existing. Until the mid eighties I read everything published in logic and related computer science. I am a “quick study,” but the quantity of papers become enormous, and I now limit my reading. I have watched all the well-known logicians and their subjects evolve for fifty-six years. Can I say anything beyond truisms about future trends?

In this paper, we present recent results in the region-based theory of space that concern algebras of regions, the corresponding topological and discrete models, and representation theory. We also discuss applications to Qualitative Spatial Reasoning (QSR), an actively developing branch of AI and Knowledge Representation (KR). In particular, we show how new results in some practically motivated areas of QSR and KR can be obtained by combining methods from such established classical disciplines as Boolean algebras, topology and logic.