Realizability of Inductive Logic

Abstract

The basic model is a two-way communication system in which observer O transmits axioms A, interprets received message S* by rules R of a Post normal logic. O's strategy is to generate (applying R to A) derivations S that minimze d(S, S*), subject, among other things, to R being Turing universal. This implies1 that (A, R: S*) are analogs of complementary observables and interaction potential in quantum mechanics. Here they represent words of binary information symbols (±1): R is a dictionary of pairs (gi : ki), which still can be universal with the restriction, length m(gi) = m0. If m- is the maximum of m(ki), then all k words in R are made up to this length by additions of a neutral symbol (O), so that R is an m0-to-m- function fR on the three values (O, ±1), realizable n fold redundantly by a nm0-to-nm probabilistic net with connexion matrices Mαij and thresholds θj, where θ(m) is random with Poisson distribution. If d(S,S*) is a scalar product, suitable learning algorithm reinforces all connections contributing positively, etc., where input is a current segment of nm0 bits of S*. The quantum condition is realized, essentially, by making Mij periodic in m(S) with period m0.