Modal languages are natural process formalisms, but the question what makes
them tick at a suitably abstract level can be, and has been, answered in many
different ways, including tree model properties, guarded quantification, and
coalgebra. In this talk, we explore a new road via abstract model theory,
presenting some new Lindström theorems for basic modal logic and the
guarded fragment, while exploring connections with the modal invariance
and interpolation theorems. We also discuss new prospects for an abstract
model theory of weak languages below first-order logic. Two references:
http://www.illc.uva.nl/Publications/Researchreports/PP-2006-06.text.pdf
(to appear in Universal Logic 2006);
and Balder ten Cate, 2005, Model
Theory for Extended Modal Languages, dissertation, ILLC Amsterdam.
The standard models of computation like Turing machines or register machines are based on natural numbers concerning memory contents (Space) and available computing time (Time). I present a research program to extend the classical models from natural numbers to ordinal numbers. Using natural number space and ordinal time, J. Hamkins et. al. defined ``infinite time Turing machines''. I have shown that Turing machines with an ordinal length tape and ordinal time compute exactly the constructible sets of ordinals, i.e., the sets of ordinals in Gödel's constructible model of set theory. The same holds for register machines with ordinal register contents and ordinal time. On the other hand, register machines with natural number registers and ordinal time compute exactly the hyper-arithmetic reals. I shall present these results and discuss some details of their proofs.