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Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
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iii For my Mama, whose *minimal index is computable (because it’s finite).
An incomplete set of shortest descriptions
 The Journal of Symbolic logic
"... The truthtable degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domainrandom strings, and show that the truthtable degrees of these sets depend on the underlying acceptable numb ..."
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The truthtable degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domainrandom strings, and show that the truthtable degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truthtable incomplete versions of these sets, namely retraceability and approximability. We give priorityfree constructions of bounded truthtable chains and bounded truthtable antichains inside the truthtable complete degree by identifying an acceptable set of domainrandom strings within each degree. 1 Meyer’s Problem No algorithm can determine, even in the limit, whether two distinct programs represent the same function. But one can, relative to the set of shortest programs MINϕ = {e: (∀j < e) [ϕj 6 = ϕe]},
On the Turing Degrees of Minimal Index Sets
, 2007
"... We study generalizations of shortest programs as they pertain to Schaefer’s MIN ∗ problem. We identify sets of mminimal and Tminimal indices and characterize their truthtable and Turing degrees. In particular, we show MIN m ⊕ ∅ ′ ′ ≡T ∅ ′′ ′ , MIN T(n) ∅ (n+2) ≡T ∅ (n+4) , and that there exists ..."
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We study generalizations of shortest programs as they pertain to Schaefer’s MIN ∗ problem. We identify sets of mminimal and Tminimal indices and characterize their truthtable and Turing degrees. In particular, we show MIN m ⊕ ∅ ′ ′ ≡T ∅ ′′ ′ , MIN T(n) ∅ (n+2) ≡T ∅ (n+4) , and that there exists a Kolmogorov numbering ψ satisfying both MIN m ψ ≡tt ∅ ′′ ′ and MIN T(n) ψ ≡T ∅ (n+4). This Kolmogorov numbering also achieves maximal truthtable degree for other sets of minimal indices. Finally, we show that the set of shortest descriptions, SD, is 2c.e. but not co2c.e. Some open problems are left for the reader. 1 The MIN ∗ problem The set of shortest programs is fMIN: = {e: (∀j < e) [ϕj � = ϕe]}. In 1972, Meyer demonstrated that fMIN admits a neat Turing characterization, namely fMIN ≡T ∅ ′ ′ [10]. In Spring 1990 (according to the best recollection of the author), John Case issued a homework assignment with the following definition [1]: fMIN ∗: = {e: (∀j < e) [ϕj � = ∗ ϕe]}, 1 where = ∗ means equal except for a finite set. Case notes that fMIN ∗ is Σ2immune, although his assignment exclusively refers to the Σ2sets as “limr.e. ” sets. On October 1, 1996, six years after the initial homework assignment, Case introduced the set fMIN ∗ to Marcus Schaefer in an email. The following year, Schaefer published a master’s thesis on minimal indices [14], which became the first public account of fMIN ∗. In his survey thesis, Schaefer proved that fMIN ∗ ⊕ ∅ ′ ≡T ∅ ′′ ′ , leaving open the tantalizing question of whether or not fMIN ≡T ∅ ′′ ′. All that would be required to answer this question affirmatively is to show that fMIN ∗ ≥T ∅ ′ , care of Schaefer’s result. This is the “MIN ∗ problem. ” The reader is encouraged to attempt this reduction before proceeding. This concludes our historical remarks. Our approach in this paper is to study c.e. sets in place of p.c. functions. This allows us to consider equivalence relations other than = and = ∗ which are especially natural for sets, namely: Definition 1.1. For n ≥ 0: MIN: = {e: (∀j < e) [Wj � = We]},
WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 1.
Simplicity and Strong Reductions
, 2000
"... A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP # coNP # SUBEXP. However, we can exhibit ..."
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A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP # coNP # SUBEXP. However, we can exhibit a relativized world in which there is an NPsimple set that is complete under Turing (Cook) reductions, even conjunctive reductions. This raises the questions whether the result by Hartmanis, Li and Yesha generalizes to reductions of intermediate strength. We show that NPsimple sets are not complete for NP under positive bounded truthtable reductions unless UP # SUBEXP. In fact, NPsimple sets cannot be complete for NP under bounded truthtable reductions under the stronger assumption that UP # coUP ## SUBEXP (while there is an oracle relative to which there is an NPsimple set conjuntively complete for NP). We present several other results for di#erent types of reductions, a...
Electronic Colloquium on Computational Complexity, Report No. 71 (2004) Simplicity and Strong Reductions
, 2004
"... A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP ∩ coNP ⊆ SUBEXP. However, we can exhibit a rela ..."
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A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP ∩ coNP ⊆ SUBEXP. However, we can exhibit a relativized world in which there is an NPsimple set that is complete under Turing (Cook) reductions, even conjunctive reductions. This raises the questions whether the result by Hartmanis, Li and Yesha generalizes to reductions of intermediate strength. We show that NPsimple sets are not complete for NP under positive bounded truthtable reductions unless UP ⊆ SUBEXP. In fact, NPsimple sets cannot be complete for NP under bounded truthtable reductions under the stronger assumption that UP ∩ coUP ⊆ SUBEXP (while there is an oracle relative to which there is an NPsimple set conjunctively complete for NP). We present several other results for different types of reductions, and show how to prove a similar result for NEXP which does not require any assumptions. We also prove that all NEXPcomplete sets are Plevelable, extending work by Tran [Tra95]. Most of the results are derived by the use of inseparable sets. This technique turns out to be very powerful in the study of truthtable and even (honest) Turing reductions.
Notre Dame Journal of Formal Logic Cuppability of Simple and Hypersimple Sets
"... Abstract An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this p ..."
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Abstract An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this paper we settle all the remaining cases for the standard notions of simplicity and all the main strong reducibilities. There are two sides to every question.Protagoras, quoted in Diogenes Laertius, Lives of Eminent Philosophers. 1 Introduction In his approach to constructing an incomplete c.e. degree, Emil Post attempted to define structural properties of c.e. sets that would force their incompleteness. In his groundbreaking 1944 paper Recursively enumerablesets of positive integers and their decision problems ([24], reprinted in Davis's The Undecidable [1]) this goal led him to isolate many of the classical concepts of computability, including creativity, manyone reducibility, bounded and unbounded truthtable reducibility, simplicity, hypersimplicity, and hyperhypersimplicity.