Pursuit of the Universal

Abstract

Abstract.A general correction grammar for a language L is a program g that, for each (x, t) ∈ N 2 , issues a yes or no (where when t = 0, the answer is always no) which is g's t-th approximation in answering "x∈L?"; moreover, g's sequence of approximations for a given x is required to converge after finitely many mind-changes. The set of correction grammars can be transfinitely stratified based on O, Kleene's system of notation for constructive ordinals. For u ∈ O, a u-correction grammar's mind changes have to fit a count-down process from ordinal notation u; these u-correction grammars capture precisely the Σ −1 u sets in Ershov's hierarchy of sets for Δ 0 2 . Herein we study the relative succinctness between these classes of correction grammars. Example: Given u and v, transfinite elements of O with u <o v (Kleene's ordering on O), for each ∅ (2) -computable H : N → N, there is a v-correction grammar iv for a finite (alternatively, a co-finite) set A such that the smallest u-correction grammar for A is >H(iv). We also exhibit relative succinctness progressions in these systems of grammars and study the "information-theoretic" underpinnings of relative succinctness. Along the way, we verify and improve slightly a 1972 conjecture of Meyer and Bagchi.