Abstract
A general correction grammar for a language L is a program g that, for each \((x,t)\in \mathbb {N} ^2\), issues a yes or no (where when \(t=0\), the answer is always no) which is g’s t-th approximation in answering “\(x{\in } L?\)”; 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 \(u\in O\), 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 \(\varSigma _u^{-1}\) sets in Ershov’s hierarchy of sets for \(\varDelta _2^0\). Herein we study the relative succinctness between these classes of correction grammars. Example: Given u and v, transfinite elements of O with \(u<_ov\) (Kleene’s ordering on O), for each \(\emptyset ^{(2)}\)-computable \(H:\mathbb {N} \rightarrow \mathbb {N} \), there is a v-correction grammar \(i_v\) for a finite (alternatively, a co-finite) set A such that the smallest u-correction grammar for A is \({>}H({i_{v}})\). 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.
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Notes
- 1.
For more discussion, see [6].
- 2.
A Dollop of Standard Terminology: \(W_i\) is the i-th c.e. set, where i codes a program for generating or for accepting \(W_i\) [30]. \(\langle \cdot ,\cdot \rangle \) is a pairing function, i.e., a computable isomorphism from \(\mathbb {N} \times \mathbb {N} \) to \(\mathbb {N} \) [30], where \(\mathbb {N} \) = the natural numbers. For \(A\subseteq \mathbb {N} \), the A -computable (respectively, partial A-computable) functions are the total (respectively, partial) functions over \(\mathbb {N} \) that are computable relative to oracle A. \(W_i^A\) is the i-th A-c.e. set, where i codes a relativized program that, with oracle A, generates or accepts \(W_i^A\) [30]. For each \(k\in \mathbb {N} \), let \(A^{(k)}\) be the k-th jump [30] of A, i.e.: \(A^{(0)}=A\) and
. Thus, \(\emptyset ^{(1)}=\) the halting problem, \(\emptyset ^{(2)}=\) the jump of \(\emptyset ^{(1)}\), etc. For \(A\subseteq \mathbb {N} \), \(\overline{A}=\mathbb {N}-A\). Any unexplained terminology below is from [30]. - 3.
For each \(k\in \mathbb {N} \), the system of \(\underline{k}\)-correction grammars turns out to be equivalent to the system of k-fold correction grammars introduced above.
- 4.
- 5.
\(\langle W_p^u \rangle _{p\in \mathbb {N}}\)’s acceptability provides s-m-n and recursion theorems that are needed in some of our proofs.
Let w be an O-notation for \(\omega \). While our w-c.e. sets are precisely \(\varSigma _{w}^{-1}\), they are distinct from the well known omega-c.e. sets [37] (each of which has a computable approximation analogous to an w-c.e. set except the number of mind changes is bounded by some computable function). The omega-c.e. sets are in \(\varDelta _{w}^{-1}\subsetneq \varSigma _w^{-1}\).
- 6.
Meyer and Bagchi’s conjecture had \(s_A\) as a computable size measure and
. - 7.
Another dollop of terminology.For partial function \(\psi , ~\psi (x){\mathclose {\downarrow }}\) means that \(\psi \) is defined on x and \(\psi (x) {\mathclose {\uparrow }}\) means that \(\psi \) is not defined on x. Let \(\langle \varphi ^A_p \rangle _{p\in \mathbb {N}}\) be an acceptable programming system for the A-computable partial functions over \(\mathbb {N} \). For this \(\varphi ^A\) we have (i) the s-1-1 theorem for \(\varphi ^A\) : There is a computable \(s:\mathbb {N} ^2\rightarrow \mathbb {N} \) such that, for each p, x, y: \(\varphi _{s(p,x)}^A(y) = \varphi _p^A(\langle x,y \rangle )\); and (ii) the Kleene parametric recursion theorem for \(\varphi ^A\) : There is a computable \(r:\mathbb {N} ^2\rightarrow \mathbb {N} \) such that, for each p, x, y: \(\varphi _{r(p,x)}^A(y) = \varphi _p^A(\langle r(p,x),y \rangle )\). (A, B) double m-reduces [35] to (C, D) (written: \((A,B)\le _m(C,D)\)) if and only if there is an computable f such that, for each x, \([x\in A\Leftrightarrow f(x)\in C\) and \(x\in B\Leftrightarrow f(x)\in D]\); such an f witnesses the reduction.
- 8.
One may wonder why \(\langle \!\langle u \text { and } v \in O \rangle \!\rangle \) (or the same thing but with ‘settled’ in place of ‘\(\in O\)’) is in the conclusion of Theorem 8. The simple answer is that \(\emptyset ^{(2)}\) is not a strong enough oracle to remove either one from inside the \(\langle \!\langle \cdots \rangle \!\rangle \) in the conclusion.
. Thus, 
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Acknowledgments
Thanks to Frank Stephan for alerting us to our earlier blunder of not noticing the need for even/odd cases in Theorem 6. Grant support was received by J. Case from NSF grant CCR-0208616, and by J. Royer from NSF grants CCR-0098198 and CCF-1319769.
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Case, J., Royer, J.S. (2016). Program Size Complexity of Correction Grammars in the Ershov Hierarchy. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_25
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