A general correction grammar for a language L is a program g that, for each (x, t) ∈ N2, 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 Σu−1 sets in Ershov’s hierarchy of sets for Δ02. Herein we study the relative succinctness between these classes of correction grammars. Example: Given u and v, transfinite elements of O with u 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.
CITATION STYLE
Case, J., & Royer, J. S. (2016). Program size complexity of correction grammars in the ershov hierarchy. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9709, pp. 240–250). Springer Verlag. https://doi.org/10.1007/978-3-319-40189-8_25
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