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Parsimony Hierarchies for Inductive Inference
 JOURNAL OF SYMBOLIC LOGIC
, 2004
"... Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsi ..."
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Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A limcomputable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "notsonearly" minimal size, e.g., to be within a limcomputable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are limcomputable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of limcomputability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the limcomputable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.
There Exists A Maximal 3C. E. Enumeration Degree
"... We construct an incomplete 3c.e. enumeration degree which is maximal among the nc.e. enumeration degrees for every n with 3 # n # #. Consequently the nc.e. enumeration degrees are not dense for any such n. We show also that no low nc.e. edegree can be maximal among the nc.e. edegrees, for 2 # ..."
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We construct an incomplete 3c.e. enumeration degree which is maximal among the nc.e. enumeration degrees for every n with 3 # n # #. Consequently the nc.e. enumeration degrees are not dense for any such n. We show also that no low nc.e. edegree can be maximal among the nc.e. edegrees, for 2 # n # #.
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"... Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at most a bits? Let 0 ≤ a ≤ n − 1. We show that any deterministic protocol, as well as any errorfree quantum protocol (C ∗ version), for this problem requires at least n − 2 bits of communication. We ..."
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Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at most a bits? Let 0 ≤ a ≤ n − 1. We show that any deterministic protocol, as well as any errorfree quantum protocol (C ∗ version), for this problem requires at least n − 2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly a bits. We also prove a lower bound of n/2 − 1 for errorfree Q ∗ quantum protocols. Our results are obtained by lowerbounding the ranks of the appropriate matrices. 1
ON Σ1STRUCTURAL DIFFERENCES AMONG ERSHOV HIERARCHIES
"... Abstract. We show that the structure R of recursively enumerable degrees is not a Σ1elementary substructure of Dn, where Dn (n> 1) is the structure of nr.e. degrees in Ershov hierarchy. 1. ..."
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Abstract. We show that the structure R of recursively enumerable degrees is not a Σ1elementary substructure of Dn, where Dn (n> 1) is the structure of nr.e. degrees in Ershov hierarchy. 1.
On the Finiteness of the Recursive Chromatic Number
"... A recursive graph is a graph whose vertex and edges sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: ..."
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A recursive graph is a graph whose vertex and edges sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: Let A be a set. An Arecursive graph is a recursive graph that also has the following property: one can recursivelyinA determine the neighbors of a vertex. We show that, if A is r.e. and not recursive, then there exists Arecursive graphs that are 2colorable but not recursively kcolorable for any k. This is false for highlyrecursive graphs but true for recursive graphs. Hence Arecursive graphs are closer in spirit to recursive graphs then to highly recursive graphs.