Efsane Bilgisayar Bilimci Prof. Rusins Freivalds'ın Boğaziçi'ndeki Semineri

Nonconstructive methods in finite automata

Rusins Freivalds
(University of Latvia)

A.Ambainis and R.Freivalds proved in 1998 that for recognition of some languages quantum finite automata can have smaller number of states than deterministic ones, and this difference can even be exponential. The proof contained a slight non-constructiveness, and the exponent was not shown explicitly. For probabilistic finite automata, the exponentiality of such a distinction was not yet proved. The best (smaller) gap was proved by Ambainis in 1996. The languages considered by Ambainis/Freivalds were presented explicitly, but the exponent was not. In a very recent paper by R.Freivalds the non-constructiveness is modified, and an explicit (and seemingly much better) exponent is obtained at the expense of having only non-constructive description of the languages used.

Moreover, the best estimate proved in this paper is proved under assumption of the well-known Artin's Conjecture (1927) in Number Theory. The paper contains also a theorem that does not depend on any open conjectures but the estimate is worse, and the description of the languages used is even less constructive. This seems to be the first result in finite automata depending on open conjectures in Number Theory.