Algorithmic information theory and computational complexity

Abstract

We present examples where theorems on complexity of computation are proved using methods in algorithmic information theory. The first example is a non-effective construction of a language for which the size of any deterministic finite automaton exceeds the size of a probabilistic finite automaton with a bounded error exponentially. The second example refers to frequency computation. Frequency computation was introduced by Rose and McNaughton in early sixties and developed by Trakhtenbrot, Kinber, Degtev, Wechsung, Hinrichs and others. A transducer is a finite-state automaton with an input and an output. We consider the possibilities of probabilistic and frequency transducers and prove several theorems establishing an infinite hierarchy of relations. We consider only relations where for each input value there is exactly one allowed output value. Relations computable by weak finite-state transducers with frequency km/kn but not with frequency m/n are presented in a non-constructive way using methods of algorithmic information theory. © 2013 Springer-Verlag Berlin Heidelberg.