Amount of Nonconstructivity in Finite Automata

When D. Hilbert used nonconstructive methods in his famous paper on invariants (1888), P.Gordan tried to prevent the publication of this paper considering these methods as non-mathematical. L. E. J. Brouwer in the early twentieth century initiated intuitionist movement in mathematics. His slogan was "nonconstructive arguments have no value for mathematics". However, P. Erdos got many exciting results in discrete mathematics by nonconstructive methods. It is widely believed that these results either cannot be proved by constructive methods or the proofs would have been prohibitively complicated. R.Freivalds [7] showed that nonconstructive methods in coding theory are related to the notion of Kolmogorov complexity. We study the problem of the quantitative characterization of the amount of nonconstructiveness in nonconstructive arguments. We limit ourselves to computation by deterministic finite automata. The notion of non constructive computation by finite automata is introduced. Upper and lower bounds of nonconstructivity are proved.