Results 1  10
of
14
Randomness is Hard
 SIAM Journal on Computing
, 2000
"... We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomi ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity, CS introduced by Hartmanis. For all of these measures we dene the set of random strings R CD t , R CND t , and R CS s as the set of strings x such that CD t (x), CND t (x), and CS s (x) is greater than or equal to the length of x, for s and t polynomials. We show the following: MA NP R CD t , where MA is the class of MerlinArthur games dened by Babai. AM NP R CND t , where AM is the class of ArthurMerlin games. PSPACE NP cR CS s . In the last item cR CS s is the set of pairs <x; y> so that x is random given y. These results show that the set of random strings for various resource bounds is hard for...
Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
iii For my Mama, whose *minimal index is computable (because it’s finite).
On strings with trivial Kolmogorov complexity
 Int J Software Informatics
"... Abstract The Kolmogorov complexity of a string is the length of the shortest program that generates it. A binary string is said to have trivial Kolmogorov complexity if its complexity is at most the complexity of its length. Intuitively, such strings carry no more information than the information th ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
(Show Context)
Abstract The Kolmogorov complexity of a string is the length of the shortest program that generates it. A binary string is said to have trivial Kolmogorov complexity if its complexity is at most the complexity of its length. Intuitively, such strings carry no more information than the information that is inevitably coded into their length (which is the same as the information coded into a sequence of 0s of the same length). We study the set of these trivial sequences from a computational perspective, and with respect to plain and prefixfree Kolmogorov complexity. This work parallels the well known study of the set of nonrandom strings (which was initiated by Kolmogorov and developed by Kummer, Muchnik, Stephan, Allender and others) and points to several directions for further research.
On the computational power of random strings
 Annals of Pure and Applied Logic
"... Abstract. There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of nonrandom strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and All ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of nonrandom strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and Allender and coauthors. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not ttcomplete. This paper answers this question in the negative by proving that the overgraph of any optimal monotone machine, or any optimal process machine, is ttcomplete. The monotone results are shown for both descriptional complexity Km and KM, the complexity measure derived from algorithmic probability. A distinction is drawn between two definitions of process machines that exist in the literature. For one class of process machines, designated strict process machines, it is shown that there is a universal machine whose set of nonrandom strings is not ttcomplete. 1.
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 ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
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]},
Contrasting plain and prefixfree Kolmogorov complexity
"... Abstract. Let SCRc = {σ ∈ 2n: K(σ) ≥ n + K(n) − c}, where K denotes prefixfree Kolmogorov complexity. These are the strings with essentially maximal prefixfree complexity. We prove that SCRc is not a Π01 set for sufficiently large c. This implies Solovay’s result that strings with maximal plai ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let SCRc = {σ ∈ 2n: K(σ) ≥ n + K(n) − c}, where K denotes prefixfree Kolmogorov complexity. These are the strings with essentially maximal prefixfree complexity. We prove that SCRc is not a Π01 set for sufficiently large c. This implies Solovay’s result that strings with maximal plain Kolmogorov complexity need not have maximal prefixfree Kolmogorov complexity, even up to a constant. We show that if Q ⊆ SCRc is an infinite Π01 set, then Q is hyperimmune. Furthermore, assuming that Q ∈ Π01 contains strings of every length, we derive a bound on the least element of Qr SCRc, matching the bound Solovay gave for Q = KRk = {σ ∈ 2n: C(σ) ≥ n − k}. We also give short derivations of Solovay’s formulae relating plain and prefixfree complexity and An. A. Muchnik’s result that these two complexity measures can disagree on the relative complexity of strings. 1.
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 ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
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]},
Immunity and hyperimmunity for sets of minimal indices
 Notre Dame Journal of Formal Logic
"... We extend Meyer’s 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarch ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
We extend Meyer’s 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal index sets located in Π3−Σ3 with distinct levels of immunity and that certain immunity properties depend on the choice of underlying acceptable numbering. We show that minimal index sets are never hyperimmune, however they can be immune against the arithmetic sets. Lastly, we investigate Turing degrees for sets of random strings defined with respect to Bagchi’s sizefunction s. 1 A short introduction to shortest programs The set of shortest programs is {e: (∀j < e) [ϕj 6 = ϕe]}. (1.1) In 1967, Blum [4] showed that one can enumerate at most finitely many shortest programs. Five years later, Meyer [13] formally initiated the investigation of minimal index sets with questions on the Turing and truthtable degrees of (1.1). Meyer’s research parallels inquiry from Kolmogorov complexity where one searches for shortest programs generating single numbers or strings. The clearest confluence
MSc in Logic
, 2011
"... verfocht. Das war konfus. ≫Das Objekt ≪ sagte der eine, und ..."
(Show Context)
DOI IOS Press 1
"... Abstract. A real number which equals the probability that a universal prefixfree machine halts when its input bits are determined by tosses of a fair coin is known as an Omega number. We present a new characterization of Omega numbers: a real in the unit interval is an Omega number if and only if i ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A real number which equals the probability that a universal prefixfree machine halts when its input bits are determined by tosses of a fair coin is known as an Omega number. We present a new characterization of Omega numbers: a real in the unit interval is an Omega number if and only if it is the weight of the strings that some universal prefixfree machine compresses by at least a certain constant number of bits. For any universal prefixfree machine U, any a, and any sufficiently large b, the weight of the strings that U compresses by at least a bits but not by b bits is again an Omega number. In fact, we can characterize the Omega numbers by finite intervals of compressibility as well.