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The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 29 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
What can be efficiently reduced to the Kolmogorovrandom strings
 Annals of Pure and Applied Logic
, 2004
"... We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorovrandom strings RC. This question arises because PSPACE ⊆ P RC and NEXP ⊆ NP RC, and no larger complexity classes are known to be reducibl ..."
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Cited by 15 (5 self)
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We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorovrandom strings RC. This question arises because PSPACE ⊆ P RC and NEXP ⊆ NP RC, and no larger complexity classes are known to be reducible to RC in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results. • Although Kummer showed that, for every universal machine U there is a time bound t such that the halting problem is disjunctive truthtable reducible to RCU in time t, there is no such time bound t that suffices for every universal machine U. We also show that, for some machines U, the disjunctive reduction can be computed in as little as doublyexponential time. • Although for every universal machine U, there are very complex sets that are ≤P dttreducible to RCU, it is nonetheless true that P = REC ∩ ⋂ {A: U A ≤P dtt RCU}. (A similar statement holds for paritytruthtable reductions.) This is an extended version of a paper that appeared in Proceedings of the 21 st Symposium on
On approximating realworld halting problems
 Reischuk (Eds.), Proc. FCT 2005, in: Lectures Notes Comput. Sci
, 2005
"... Abstract. No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most ’ instances and thus solve it at least approximately. Whether and how ..."
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Cited by 6 (0 self)
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Abstract. No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most ’ instances and thus solve it at least approximately. Whether and how well such approximations are feasible highly depends on the underlying encodings and in particular the Gödelization (programming system) which in practice usually arises from some programming language. We consider BrainF*ck (BF), a simple yet Turingcomplete realworld programming language over an eight letter alphabet, and prove that the natural enumeration of its syntactically correct sources codes induces a both efficient and dense Gödelization in the sense of [Jakoby&Schindelhauer’99]. It follows that any algorithm M approximating the Halting Problem for BF errs on at least a constant fraction εM> 0 of all instances of size n for infinitely many n. Next we improve this result by showing that, in every dense Gödelization, this constant lower bound ε to be independent of M; while, the other hand, the Halting Problem does admit approximation up to arbitrary fraction δ> 0byan appropriate algorithm M δ handling instances of size n for infinitely many n. The last two results complement work by [Lynch’74]. 1
Program Size, Oracles, And The Jump Operation
, 1977
"... There are a number of questions regarding the size of programs for calculating natural numbers, sequences, sets, and functions, which are best answered by considering computations in which one is allowed to consult an oracle for the halting problem. Questions of this kind suggested by work of T. Kam ..."
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Cited by 5 (4 self)
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There are a number of questions regarding the size of programs for calculating natural numbers, sequences, sets, and functions, which are best answered by considering computations in which one is allowed to consult an oracle for the halting problem. Questions of this kind suggested by work of T. Kamae and D. W. Loveland are treated. 2 G. J. Chaitin 1. Computer Programs, Oracles, Information Measures, and Codings In this paper we use as much as possible Rogers' terminology and notation [1, pp. xvxix]. Thus N = f0; 1; 2; : : :g is the set of (natural) numbers; i, j, k, n, v, w, x, y, z are elements of N ; A, B, X are subsets of N ; f , g, h are functions from N into N ; ', / are partial functions from N into N ; hx 1 ; : : : ; x k i denotes the ordered ktuple consisting of the numbers x 1 ; : : : ; x k ; the lambda notation x[: : : x : : :] is used to denote the partial function of x whose value is : : : x : : :; and the mu notation ¯x[: : : x : : :] is used to denote the least x...
Short lists with short programs in short time
"... Abstract—Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed t ..."
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Cited by 1 (1 self)
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Abstract—Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a O(logx)short program for x. We also show that there exist computable functions that map every x to a list of size O(x  2) containing a O(1)short program for x and this is essentially optimal because we prove that such a list must have size Ω(x  2). Finally we show that for some machines, computable lists containing a shortest program must have length Ω(2 x ).
Programming Language Constructs and Program Size
"... Intuitively, the richer the set of control structures provided by a programming language (language constructs), the better its expressibility. Equating the ease of expression of an algorithm with the succinctness (size) of the resultant program, we show that all commonlyused programming languages p ..."
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Intuitively, the richer the set of control structures provided by a programming language (language constructs), the better its expressibility. Equating the ease of expression of an algorithm with the succinctness (size) of the resultant program, we show that all commonlyused programming languages provide all implementable control structures. We also study the class of size functions witnessing programming language constructs. 312 1 Introduction The design philosophy employed for engineering a software project and the choice of the programming language to be used to implement the project are mutually influential. The features provided by a programming language are typically used to subjectively classify that language as being either a highlevel or a lowlevel programming language. This classification, albeit informal, is widely accepted. The level of a programming language seems to coincide with the relative ease of expressing algorithms in the language. Herein, we attempt to for...
Time Cutoff and the Halting Problem
, 2010
"... Abstract. This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts suppor ..."
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Abstract. This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut–off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.
KOLMOGOROV COMPLEXITY AND THE ASYMPTOTIC BOUND FOR ERROR–CORRECTING CODES
, 1203
"... ABSTRACT. The set of all error–correcting block codes over a fixed alphabet with q letters determines a recursively enumerable set of rational points in the unit square with coordinates (R,δ): = (relative transmission rate, relative minimal distance). Limitpointsof thisset form a closedsubset, defin ..."
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ABSTRACT. The set of all error–correcting block codes over a fixed alphabet with q letters determines a recursively enumerable set of rational points in the unit square with coordinates (R,δ): = (relative transmission rate, relative minimal distance). Limitpointsof thisset form a closedsubset, defined by R ≤ αq(δ), where αq(δ) is a continuous decreasing function called asymptotic bound. Its existence was proved by the first–named author in 1981 ([Man1]), but no approaches to the computation of this function are known, and in [Man5] it was even suggested that this function might be uncomputable in the sense of constructive analysis. In this note we show that the asymptotic bound becomes computable with the assistance of an oracle producing codes in the order of their growing Kolmogorov complexity. Moreover, a natural partition function involving complexity allows us to interpret the asymptotic bound as a curve dividing two different thermodynamic phases of codes.