J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, pages 439--445, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= w and the computation takes # g( p ) steps . Natural choices for g would be: polynomials, functions of order flog 2 f, where f is a polynomial, or functions of order 2 cn etc. For information on the use of these complexity measures in computer science, the reader may consult the references [36], 59] and [90] 2a . 5.2 Kolmogorov s program In [50,34] Kolmogorov writes The idea that randomness consists in a lack of regularity is thoroughly traditional. But apparently only now has it become possible to found directly on this simple idea precise formulations of conditions for the ....

* J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proc. 24 FOCS (1983), 439-445.

....that E=NE i# there is no sparse set in NP P [15] This paper shows some inherent limitations of the technique. The main result of this paper is the construction of an oracle relative to which there are extremely sparse sets in NP P, but NEE = EE; this is in contradiction to a result claimed in [14, 16]. Thus, although the upward separation technique is useful in relating the existence of sets of polynomial (and greater) density in NP P to the NTIME(T (n) DTIME(T (n) problem, the existence of sets of very low density in NP P can not be shown to have any bearing on on this problem using proof ....

....The technique seems to be powerful and applicable to other situations. A number of generalizations of this technique are worked out in [5] It is also worth mentioning that the oracle construction makes use of Kolmogorov complexity as a tool for simplifying the combinatorial arguments. Hartmanis [14] was the first to use Kolmogorov complexity as a tool in oracle constructions; a number of applications of this technique are presented in [13] Although the main result of this paper is a demonstration that certain generalizations of the upward separation results do not hold, it is useful to ....

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J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proc. 24th IEEE Symposium on Foundations of Computer Science, 1983, pages 439--445.

....a Turing machine U such that for any Turing machine M and for any word w # A # the following inequality holds: KU (w) # KM (w) CM , where CM is a constant which does not depend on w. Similar results hold for space bounded and time bounded Kolmogorov complexities. Invariance Theorem II [9]. There exists a Turing machine U such that for any Turing machine M , for any bound # : N # N , and for any word w # A # the following inequalities hold: KSU (w, CM #) # KM (w, #) CM , KTU (w, CM # log(#) # KM (w, #) CM , where CM is a constant which does not depend on w. The theorems ....

J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proc. 24th IEEE Simposium on Foundations of Computer Science, 1983, pp. 439-- 445.

....the sets containing all of the random strings. For our purposes, it will su#ce to consider the following two sets: RKT is defined to be x KT(x) # x 2 . R Kt is defined to be x Kt(x) # x 2 . Similar (but not identical) notions of time bounded randomness have been considered before [BT01,Ko91,Har83,KC00]. RKT is in coNP, and R Kt is in E. It seems natural to conjecture that these upper bounds are essentially optimal. However, there are significant obstacles to showing that there are no smaller complexity classes containing these languages. Most significantly, there is reason to believe that ....

Juris Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations (preliminary report). In IEEE Symposium on Foundations of Computer Science (FOCS), pages 439--445, 1983.

.... This concept has been studied extensively and has found many applications in computer science; the reader will find more topics on Kolmogorov complexity in the survey paper by Li and Vitanyi [LV88] A modification of the general idea of Kolmogorov complexity has been developed by Hartmanis [Ha83]: consider not only the length of a program but also, and simultaneously, the running time of the program. One can define a generalized, two parameter Kolmogorov complexity measure for finite strings which measures how far and how fast a string can be compressed: given functions d and t, a string ....

J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, in "Proc. 24th IEEE Sympos. Foundations of Computer Sciences", IEEE (1983), 439-445.

....we let functions de ned on strings sometimes act on numbers where the length of the number is the logarithm of its value. We de ne the Kolmogorov complexity function C(xjy) see [LV97] by C(xjy) minfjpj : U(p; y) xg. We de ne unconditional Kolmogorov complexity by C(x) C(xj ) Hartmanis [Har83] de ned a time bounded version of Kolmogorov complexity, but resource bounded versions of Kolmogorov complexity date back as far as [Bar68] See also [LV97] So C s and C t maybe de ned as space and time bounded versions of C in the usual way. We de ne an enumerator for a function (e.g. C) ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439-445, 1983.

....role in complexity theory. For example, Kadin [25] has proven that if some sparse set is p T complete for NP then PH = P NP jj . Hemachandra and Wechsung [20] have shown that the theory of randomness (in the form of the resource bounded Kolmogorov complexity theory of Adleman [1] Hartmanis [18], and Sipser [39] is deeply tied to the question of whether P NP jj = P NP , i.e. whether parallel and sequential access to NP coincide. Buss and Hay [10] have shown that P NP jj exactly captures the class of sets acceptable 4 via multiple rounds of parallel queries to NP and also exactly ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 439-445. IEEE Computer Society Press, 1983.

.... Chaitin s and G acs independent papers on prefix complexity and m [24, 17] Solomonoff s work on inductive inference helped to inspire less general yet practically more feasible principles of minimum description length [68, 46] as well as time bounded restrictions of Kolmogorov complexity, e.g. [28, 69, 39], as well as the concept of logical depth of x, the runtime of the shortest program of x [6] Equation (14) makes predictions of the entire future, given the past. This seems to be the most general approach. Solomonoff [60] focuses just on the next bit in a sequence. Although this provokes ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

.... and G acs independent papers on prefix complexity and m [35, 27] Solomonoff s work on inductive inference helped to inspire less general yet practically more feasible principles of minimum description length [95, 66, 44] as well as time bounded restrictions of Kolmogorov complexity, e.g. [42, 2, 96, 56], as well as the concept of logical depth of x, the runtime of the shortest program of x [8] Equation (15) makes predictions of the entire future, given the past. This seems to be the most general approach. Solomonoff [83] focuses just on the next bit in a sequence. Although this provokes ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....it. The pre x Kolmogorov complexity of a string x, K(x) is the length of the shortest self delimiting program that produces x: The Kolmogorov complexity of nite strings classi es them as more or less random without consider the time necessary to produce them from the minimal program. Hartmanis [Har83] introduced a generalization of Kolmogorov complexity for nite objects. Intuitively, x 2 f0; 1g n is in the Kolmogorov class K[f(n) g(n) if there is a string y such that jyj f(n) from which x can be computed in time g(n) The advantage of this approach is that it measures the amount of ....

....computed in time g(n) The advantage of this approach is that it measures the amount of randomness detectable in a given time, i.e. measures how far (f(n) and how fast (g(n) a string can be compressed. Note that this is closely related to the time bounded Kolmogorov complexity. De nition 2. 2 ([Har83]) Let M be a Turing machine and f; g : N N then KM [f(n) g(n) fx : 9y(jyj f(jxj) and M(y) x in time g(jxj)g This measure has the advantage of consider, not only, the program size, but also the time necessary to produce the string from it, however time and program length are disjoint, ....

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J. Hartmanis. Generalized kolmogorov complexity and the structure of feasible computations. In Proceedings, 24 IEEE Symposium on Foundations of Computing, pages 439-445, 1983.

....this can be strengthened to show that this set is also truth table complete. The resource bounded version of the random strings was rst studied by Ko [Ko91] The polynomial time bounded Kolmogorov complexity C p (x) for p a polynomial is the smallest program that prints x in p(jxj) steps ( Har83] Ko showed that there exists an oracle such that the set of random strings with respect to this time bounded Kolmogorov complexity is complete for coNP under strong nondeterministic polynomial time reductions. He also constructed an oracle where this set is not complete for coNP under ....

....isomorphism problem, GI, is in NP R CND t and that if for some polynomial t, R CND t 2 NP coNP then GI 2 NP coNP. The s(n) space bounded Kolmogorov complexity, CS s (xjy) is de ned as the size of the smallest program that prints x, given y and uses at most s(jxj jyj) tape cells [Har83] Likewise we de ne cR CS s = f x; y : CS s (xjy) jxjg for s(n) a polynomial. We show that PSPACE NP cR CS s . For the rst two results we use the oblivious sampler construction of Zuckerman [Zuc96] a lemma [BF97] that measures the size of sets in terms of CD complexity, and we ....

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439-445, 1983.

....computer science. A simple counting argument showing that for each length there exist random strings, i.e. strings with no regularity, has had many applications (see [LV97] Early in the history of computational complexity resource bounded notions of Kolmogorov complexity were studied [Har83, Lon90, Lon86]. In particular Sipser [Sip83] introduced a new version of resource bounded Kolmogorov complexity, CD complexity, where one considers the size of the smallest program that accepts the given string and no others. Sipser showed that one can approximate the size of sets using CD complexity with ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....space bounded Kolmogorov complexity. 1 A set is said to have small generalized Kolmogorov complexity if all of its strings are highly compressible and easily restorable. Generalized time bounded Kolmogorov complexity and generalized space bounded Kolmogorov complexity are introduced in [Ha83] and [S83] Several researchers [R86, BB86, HH88] show that P printable sets are exactly the sets in P with small generalized time bounded Kolmogorov complexity. AR88] show that a set has small generalized time bounded Kolmogorov complexity if and only if it is P isomorphic to a tally set. Using ....

....be considered a form of compression. Another approach to compression is found in the study of Kolmogorov complexity; a string is said to have low information content if it it has low Kolmogorov complexity. We are interested in the space bounded Kolmogorov complexity class defined by Hartmanis [Ha83]. Definition 8 Let M v be a Turing machine, and let f and s be functions on the natural numbers. Then we define KS v [f(n) s(n) fw : jwj = n and 9y(jyj f(n) and M v (y) w and M v uses s(n) space)g: Following the notation of [AR88] we refer to y as the compressed string, f(n) as the ....

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J. Hartmanis. Generalized Kolmogorov Complexity and the Structure of Feasible Computations. IEEE Proceedings of 24th Symposium Foundations of Computer Science, 439--445, 1983.

....in formulating a satisfactory notion of hard instances. However the diculty can be overcome by taking also the sizes of programs into account. This led to the notion of instance complexity [15] which uses ideas stemming from the study of time bounded Kolmogorov complexity (introduced by Hartmanis [5]) 2 Kolmogorov Complexity Kolmogorov complexity measures the information content of strings. Originally, it was introduced independently by Kolmogorov [8] and Chaitin [3] to give precise computational meaning to the concept of randomness and information contained in a nite string. An excellent ....

.... log n. Another program that prints s is print s which has length 6 n. This shows that the Kolmogorov complexity of a string has about the length of the string as upper bound. If one chooses any universal programming language, it does up to an constant summand not depend on that choice (see [5, 15, 11] for details) We leave out details like the choice of a universal Turing machine with respect to which the size and the computation time of the programs are measured, and suppose that an optimal interpreter is used to run the programs. The existence, robustness, and invariance of such an ....

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 439-445, 1983.

....computer science. A simple counting argument showing that for each length there exist random strings, i.e. strings with no regularity, has had many applications (see [LV97] Early in the history of computational complexity resource bounded notions of Kolmogorov complexity were studied [Har83, Lon90, Lon86]. In particular Sipser [Sip83] introduced a new version of resource bounded Kolmogorov complexity, CD complexity, where one considers the size of the smallest program that accepts the given string and no others. Sipser showed that one can approximate the size of sets using CD complexity with ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....in NP that is Turing complete for NP SPARSE, in fact the given set is tally) and ask whether the result can be improved to the many one case. Hartmanis has also shown that the set of satisfiable formulas with small Kolmogorov complexity SAT K[log; n 2 ] is Turing complete for NP SPARSE [8], but the completeness of this set under many one reductions would imply unexpected consequences in the exponential time hierarchy. More recently Schoning has proven that there are sets that are complete for this class under many one randomized reductions [15] 2 Basic Notions We assume some ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science (FOCS'83), pp. 439--445, 1983

....Machine is used, since the Kolmogorov complexity would change only by an additive constant. Proposition 11. For each length n, and each natural number m, there are at most 2 n Gammam 1 Gamma1 strings w , with w n, having Kolmogorov complexity K(w) n Gamma m. Hartmanis introduces in [3] a tool we will use in this paper: resource bounded Kolmogorov Complexity. We follow the notation in [3] Using U and functions f and g : N N define the classes of bounded Kolmogorov complexity sets K[f(n) g(n) and KS[f(n) g(n) as follows: Definition 12. i L 2 K[f(n) g(n) iff 8x 2 L ....

....11. For each length n, and each natural number m, there are at most 2 n Gammam 1 Gamma1 strings w , with w n, having Kolmogorov complexity K(w) n Gamma m. Hartmanis introduces in [3] a tool we will use in this paper: resource bounded Kolmogorov Complexity. We follow the notation in [3]. Using U and functions f and g : N N define the classes of bounded Kolmogorov complexity sets K[f(n) g(n) and KS[f(n) g(n) as follows: Definition 12. i L 2 K[f(n) g(n) iff 8x 2 L 9w; jwj f(jxj) such that U(w) x in time g(jxj) ii L 2 KS[f(n) g(n) iff 8x 2 L 9w; jwj f(jxj) ....

J. Hartmanis, Generalized Kolmogorov Complexity and the Structure of Feasible Computations, Proc. 24th IEEE Symposium on Foundations of Computer Science, 1983, pp.439--445.

.... Kolmogorov [18] and Chaitin [6] The extraordinary power and scope of this notion have recently been surveyed by Kolmogorov and Uspenskii [19] and Li and Vitanyi [21] In this paper we are primarily concerned with resource bounded Kolmogorov complexities, which have been investigated by Hartmanis [10], Sipser [39] Ko [17] Longpr e [22] Balc azar and Book [3] Huynh [13] Lutz [24] Allender and Watanabe [2] and many others. Martin Lof [29] showed that K(xjn) the conditional Kolmogorov complexity of infinite binary sequences x, exhibits a strong Shannon effect. Specifically, Martin Lof ....

J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, 1983, pp. 439--445.

....in NP that is Turing complete for NP SPARSE, in fact the given set is tally) and ask whether the result can be improved to the many one case. Hartmanis has also shown that the set of satisfiable formulas with small Kolmogorov complexity SAT K[log; n 2 ] is Turing complete for NP SPARSE [8], but the completeness of this set under many one reductions would imply unexpected consequences in the exponential time hierarchy. More recently Schoning has proven that there are sets that are complete for this class under many one randomized reductions [16] 2 Basic Notions We assume some ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science (FOCS'83), pp. 439--445, 1983

....generalized space bounded Kolmogorov complexity. A set is said to have small generalized Kolmogorov complexity if all of its strings are highly compressible and easily restorable. Generalized time bounded Kolmogorov complexity and generalized space bounded Kolmogorov complexity are introduced in [Har83] and [Sip83] Several researchers [Rub86, BB86, HH88] show that P printable sets are exactly the sets in P with small generalized time bounded Kolmogorov complexity. AR88] show that a set has small generalized time bounded Kolmogorov complexity if and only if it is P isomorphic to a tally set. ....

....be considered a form of compression. Another approach to compression is found in the study of Kolmogorov complexity; a string is said to have low information content if it it has low Kolmogorov complexity. We are interested in the space bounded Kolmogorov complexity class defined by Hartmanis [Har83]. Definition 2.8. Let M v be a Turing machine, and let f and s be functions on the natural numbers. Then we define KS v [f(n) s(n) fw : jwj = n and 9y(jyj f(n) and M v (y) w and M v uses s(n) space)g: Following the notation of [AR88] we refer to y as the compressed string, f(n) as the ....

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J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, in Proc. 24th IEEE Symposium on Foundations of Computer Science, 1983, pp. 439--445.

....; is defined by: n 1 = 2, n i is triply exponential in n i Gamma1 , for i 1; if x 2 K and jxj = n, then x has Kolmogorov complexity n. The inclusion of QBF gives machines with access to B the full power of PSPACE. It is straightforward to prove that P B 6= NP B using the techniques in [14]. To show that no invulnerable generators exist relative to B, let G i;j be a generation scheme that has access to the oracle, and assume that it is ff invulnerable, for some constant ff in (0; 1) We derive a contradiction by producing an adversary f in PF B that can crack a higher fraction ....

J. Hartmanis. "Generalized Kolmogorov Complexity and the Structure of Feasible Computations, " Proceedings of the 24 th FOCS, IEEE, 1983, 439--445.

....[Cha66] Roughly speaking, the Kolmogorov complexity of a finite binary string x is the length of a shortest program that generates x. Intuitively, if a string x can be generated by a program shorter than x itself, then x can be compressed. The notion of generalized Kolmogorov complexity [Adl79, Har83, Sip83] is a version of Kolmogorov complexity that provides information about not only whether and how far a string can be compressed, but also how fast it can be restored. We now give the definition of (unconditional and conditional) generalized Kolmogorov complexity. Definition 2.3 ( Har83] see ....

....Har83, Sip83] is a version of Kolmogorov complexity that provides information about not only whether and how far a string can be compressed, but also how fast it can be restored. We now give the definition of (unconditional and conditional) generalized Kolmogorov complexity. Definition 2. 3 ([Har83], see also [Adl79, Sip83] For any Turing machine T and functions s and t mapping IN to IN, define K T [s(n) t(n) df = fx j (9y) jxj = n and jyj s(n) and T (y) outputs x in at most t(n) steps ]g: It was shown in [Har83] that there exists a universal 1 Turing machine U such that for any ....

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 439--445. IEEE Computer Society Press, 1983.

....of the argument string is given. More precisely, let V be a Turing transducer. For s; t : N N, KC V [s(n) t(n) is the set fx 2 Sigma j (9 y) jyj s(jxj) and V (y; jxj) outputs x in time t(jxj)g. Similar to the result of Hartmanis for the unconditional time bounded Kolmogorov complexity [Har83], there is a universal Turing transducer U such that for all transducers V there is a constant c such that for all functions s and t, KC V [s(n) t(n) KCU [s(n) c; ct(n) log t(n) c] We fix such a machine U and abbreviate KCU [s(n) t(n) by KC[s(n) t(n) Our notion of large set is ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 439--445. IEEE Computer Society Press, 1983.

....choice of a universal Turing machine with respect to which the size and the computation time of the programs are measured, and suppose that an optimal interpreter is used to run the programs. The existence, robustness, and invariance of such an interpreter and more details can be found e.g. in [7,14,10]. For simplicity and w.l.o.g. we assume that the program = denoted by the empty string halts on every input without any computation and output. The Kolmogorov complexity of a string x 2 Sigma relative to oracle A is C A (x) min program fj j j A ( xg : The notion of ....

.... = denoted by the empty string halts on every input without any computation and output. The Kolmogorov complexity of a string x 2 Sigma relative to oracle A is C A (x) min program fj j j A ( xg : The notion of time bounded Kolmogorov complexity was introduced by Hartmanis [7]. The t time bounded Kolmogorov complexity is C A;t (x) min program fj j j A ( x and A ( makes at most t(jxj) stepsg : We use C t (x) to denote C ; t (x) Note that C A;t (seen as function Sigma N) has neither a limit superior nor a limit inferior for every ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 439--445, 1983.

....generalized space bounded Kolmogorov complexity. A set is said to have small generalized Kolmogorov complexity if all of its strings are highly compressible and easily restorable. Generalized time bounded Kolmogorov complexity and generalized space bounded Kolmogorov complexity are introduced in [Har83] and [Sip83] Several researchers [Rub86, BB86, HH88] show that P printable sets are exactly the sets in P with small generalized time bounded Kolmogorov complexity. AR88] show that a set has small generalized time bounded Kolmogorov complexity if and only if it is P isomorphic to a tally set. ....

....can be considered a form of compression. Another approach to compression is found in the study of Kolmogorov complexity; a string is said to have low information content if it has low Kolmogorov complexity. We are interested in the space bounded Kolmogorov complexity class defined by Hartmanis [Har83]. Definition 2.8. Let M v be a Turing machine, and let f and s be functions on the natural numbers. Then we define KS v [f(n) s(n) w : w = n and #y( y # f(n) and M v (y) w and M v uses s(n) space) Following the notation of [AR88] we refer to y as the compressed string, f(n) ....

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<F3.746e+05> J.<F3.786e+05> Hartmanis,<F3.373e+05> Generalized Kolmogorov complexity and the structure of feasible computations,<F3.786e+05> in Proc. 24th IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1983, pp. 439--445.

....clarify the content of the above statement. Our result involves the variant of time bounded Kolmogorov complexity. More precisely, let V be a Turing transducer. For s; t : N N, K V [s(n) t(n) is the set fx 2 Sigma j (9 y) jyj s(jxj) and V (y) outputs x in time t(jxj)g. It is known (see [Har83]) that there is a universal Turing transducer U such that for all transducers V there is a constant c such that for all functions s and t, K V [s(n) t(n) KU [s(n) c; ct(n) log t(n) c] We fix such a machine U and abbreviate KU [s(n) t(n) by K[s(n) t(n) For a comprehensive discussion ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 439--445. IEEE Computer Society Press, 1983.

....A; B such that A p t in B, and A 6 p tt B, and A 62 P B [O(log n) In the proof of the Proposition we will make use of the information encoded into strings, which is measured by means of Kolmogorov complexity. Time bounded generalized Kolmogorov complexity were introduced by Hartmanis [Har83] We give a slightly modified definition for notational convenience. For a Turing machine M and integers l and s let KTM [l; t] fx j 9y : jyj l and M (y) outputs x using at most t steps g be the time bounded Kolmogorov complexity (relative to M ) For fixed integers this is a finite set, but ....

....M and integers l and s let KTM [l; t] fx j 9y : jyj l and M (y) outputs x using at most t steps g be the time bounded Kolmogorov complexity (relative to M ) For fixed integers this is a finite set, but we can still let l and t be unbounded functions. The following fact can be easily verified [Har83] There exists a Universal Turing machine U such that, for every Turing machine M , there is a constant c such that, for every l and t, KTM [l; t] KTU [l c; c Delta t Delta log t c] In the following we fix such a Universal machine U and omit the subscript U when no confusion arises. ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 439--445, 1983.

....etc. 31] Shannon information, however, is not the right concept to exploit the potential benefits of algorithmic redundancy. Instead we need to look at Kolmogorov complexity or algorithmic information [8] 34] 2] 11] 3] and especially at its computationally tractable generalizations (e.g. [4] [10] 12] 35] to properly treat general (as opposed to conventional statistical) sources of redundant information. This is a recent focus of my research [25, 26, 32, 31] 6 ACKNOWLEDGMENTS Thanks to Peter Dayan, Richard Zemel and Alex Pouget for sharing their insight regarding the equivalence ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....M is such a machine we denote the output of M on input (p 1 ; p 2 ; x) by M(p 1 ; p 2 ; x) and the number of steps in the computation by timeM (p 1 ; p 2 ; x) We assume that t(n) n for all time bounds t. The following notion of time bounded Kolmogorov complexity was introduced by Hartmanis [3] and Sipser [17] Definition 1 For any time bound t = t(n) and x; y 2 f0; 1g the t bounded Kolmogorov complexity of x conditional to y using M is defined as C t M (xjy) minfjpj : M(p; y) x timeM (p; y) t(jxj)g: The t bounded Kolmogorov complexity of x using M is defined as C t M ....

.... t M (xj) Thus, unconditional CD complexity can be considered as a special case of instance complexity; for all natural interpreters M we have CD t M (x) ic t M (x : fxg) There is a universal Turing machine U (an optimal interpreter ) such that the following invariance property holds (see [3], 15, Theorem 2.1] Fact 4 For every Turing machine M there is a constant c such that for all sets A, all time bounds t, and all strings x; y: ic t 0 U (x : A) ic t M (x : A) c, C t 0 U (xjy) C t M (xjy) c; CD t 0 U (xjy) CD t M (xjy) c; where t 0 (n) ct(n) log t(n) ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computation. In: Proc. 24th IEEE Symp. Foundations of Computer Science, pp. 439--445, 1983.

....2.3 Kolmogorov Complexity Fix a Universal Turing machine U . The Kolmogorov complexity of a string w (resp. the Kolmogorov complexity relative to y) is the length of the shortest program (resp. pair hprogram; yi) which, when given as input to U , will lead U to write down w as output. Hartmanis [14] and Sipser [25] modified the original idea of Kolmogorov complexity to include the running time or space used by the Universal Turing machine, in order to produce an output. Ko, in [18] 1 , followed the same approach but applying it to the notion of infinite sequences with respect to ....

....are in P log (see [16] for more details) Since P=log 6= P=poly (see for instance [11] we have: Theorem 9 P T (P=log) is not included in P=log. Moreover, one can see that even Pm(P=log) is not included in P=log. see again [16] 1 Preliminary versions of [18] circulated simultaneously to [14] and [25] 3 The classes Full P=log and Pref P=log Since the logarithmic analog to P poly is not closed under most usual reducibilities, an alternative approach was introduced by Ko [19] Ko s class, although with a different name, is introduced in the following definition. Definition 10 A set ....

J. Hartmanis. Generalized Kolmogorov Complexity and the Structure of Feasible Computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....definitions, allowing one to define, for each function f , a f(n) time bounded Kolmogorov complexity measure K f , where K f (x) is the length of the shortest description of x from which x can be produced in f(jxj) steps. A related (and much more influential) definition due to Hartmanis [Har83] yields sets of the form K[g(n) G(n) consisting of all strings x that can be produced from a description of length g(jxj) in time G(jxj) Pointers to other approaches to time bounded Kolmogorov complexity may be found in [All89a, LV90] Supported in part by the National Science Foundation ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. Foundations of Computer Science, pages 439--445, 1983.

....1 shorter programs, there are Kolmogorov random strings of every length. Generalized Kolmogorov complexity, a two parameter version of Kolmogorov complexity that includes information about not only how far a string can be compressed, but how fast it can be restored, was introduced by Hartmanis ([Har83]) whose definition is presented here. Definition 2.2 For a (deterministic) Turing machine M and functions g and G mapping natural numbers to natural numbers, let KM [g(n) G(n) fx j (9y) jyj g(jxj) and M(y) x in G(jxj) or fewer steps]g: It was shown in [Har83] that there exists a universal ....

....was introduced by Hartmanis ( Har83] whose definition is presented here. Definition 2. 2 For a (deterministic) Turing machine M and functions g and G mapping natural numbers to natural numbers, let KM [g(n) G(n) fx j (9y) jyj g(jxj) and M(y) x in G(jxj) or fewer steps]g: It was shown in [Har83] that there exists a universal Turing machine M u such that for any other Turing machine M there exists a constant c such that KM [g(n) G(n) KMu [g(n) c; cG(n) log G(n) c] Dropping the subscript, K[g(n) G(n) will actually denote KMu [g(n) G(n) This relativizes in a straightforward ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, Tucson, Arizona, pages 439--445, 1983.

....R n a GammaT (SPARSE) for a 1. This proof is based on resource bounded Kolmogorov complexity. It has as its precursor the methods of Gavalda and Watanabe in [GW91] The following is a standard definition for Kolmogorov time bounded complexity [BDG90] The original notion is due to Hartmanis [Hart83]. Fix any reasonable universal transducer, U and let U(x) denote the output of U on input x. Definition 2 The Kolmogorov time bounded complexity set is K[f; g] fu j 9w(jwj f(juj) U(w) u and this result is obtained in at most g(juj) steps of U g: Let K i be the time bounded Kolmogorov ....

....in at most g(juj) steps of U g: Let K i be the time bounded Kolmogorov complexity measure based on the i th Turing machine in some standard enumeration of Turing transducers. The following central fact justifies our use of a fixed universal transducer in the above definition. Lemma 2. 1 [Hart83] Let i be any index and g(n) t(n) be time constructible functions. Then there exists a c 0 such that K i [g(n) t(n) K[g(n) c; c t(n)logt(n) c] Intuitively K[f; g] is the set of strings u each of which can be retrieved from some string w that represents a compression of u by a factor ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc.24th Annual FOCS Conference, pages 439--445, 1983.

....really be framed in terms of programs to some fixed, sufficiently efficient universal machine. In the body of the paper we will use the correct definition, but the above suffices for purposes of discussion. Technically, our definition is obviously inspired by the notion of Kolmogorov complexity [10, 18, 19], which provides a measure for the complexity of an individual string. Recall that the t bounded Kolmogorov complexity of a string x is defined (roughly) as K t (x) minfjM j : time M ( t(jxj) and M( xg; where denotes the empty string. There is also an interesting variant of this, ....

....(Invariance) There exists an interpreter U such that corresponding to any other interpreter M there is a constant c, such that for all sets A, time bounds t and strings x, ic t 0 U (x : A) ic t M (x : A) c; K t 0 U (x) K t M (x) c; where t 0 (n) ct(n) log t(n) c. Proof. See [10, 18, 19]; this is the standard result on the invariance of time bounded Kolmogorov complexity, using the efficient Hennie Stearns simulation (see [12, Sec. 12] of multitape machines by two tape machines. 2 Because the complexities obtained using U essentially minorize the complexities obtained using ....

[Article contains additional citation context not shown here]

Hartmanis, J. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th Annual Symposium on the Foundations of Computer Science (Tucson, Az., November). IEEE, New York, N.Y., 1983, pp. 439--445.

....of SAT. It is well known that there exist Universal Turing machines able to simulate any other Turing machine given to it encoded as part of the input, with reasonable time and space overheads. Fix such a universal machine U . Using U , we introduce the resource bounded Kolmogorov Complexity [26, 14]. The original idea of Kolmogorov complexity is modified in order to include the running time spent by the universal Turing machine to produce an output. Kolmogorov complexity strings K[f(n) g(n) are defined as follows: Definition 3 K[f(n) g(n) fx j 9y; jyj f(jxj) U(y) x in at most ....

J. Hartmanis. Generalized Kolmogorov Complexity and the Structure of Feasible Computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....definitions, allowing one to define, for each function f , a f(n) time bounded Kolmogorov complexity measure K f , where K f (x) is the length of the shortest description of x from which x can be produced in f(jxj) steps. A related (and much more influential) definition due to Hartmanis [Har83] yields sets of the form K[g(n) G(n) consisting of all strings x that can be produced from a description of length g(jxj) in time G(jxj) Pointers to other approaches to time bounded Kolmogorov complexity may be found in [All89a, LV90] Supported in part by the National Science Foundation ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. Foundations of Computer Science, pages 439--445, 1983.

.... a machine we denote the output of M on input (p 1 ; p 2 ; x) by M(p 1 ; p 2 ; x) and the number of steps in the computation by timeM (p 1 ; p 2 ; x) We assume that t(n) n for all time bounds t = t(n) The following notion of time bounded Kolmogorov complexity was introduced by Hartmanis [6], Ko [9] and Sipser [21] Intuitively, the t bounded Kolmogorov complexity of x is the length of the shortest program which computes x in t(jxj) steps from the empty input. Definition 1 For any time bound t and x; y 2 Sigma the t bounded Kolmogorov complexity of x conditional to y using M is ....

.... M (xjffl) Thus, unconditional CD complexity can be considered as a special case of instance complexity; for all natural interpreters M we have CD t M (x) ic t M (x : fxg) There is a universal Turing machine U (an optimal interpreter ) such that the following invariance property holds (see [6], 18, Theorem 2.1] Fact 4 For every Turing machine M there is a constant c such that for all sets A, all time bounds t, and all strings x; y: ic t 0 U (x : A) ic t M (x : A) c, C t 0 U (xjy) C t M (xjy) c; CD t 0 U (xjy) CD t M (xjy) c; where t 0 (n) ct(n) log t(n) ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computation. In: Proc. 24th IEEE Symp. Foundations of Computer Science, pp. 439--445, 1983.

.... NP has sparse Turing complete sets if and only if NP has sparse truth table complete sets this can be seen either directly, via the parallel census technique discussed later in this subsection, or as a consequence of Hartmanis s sparse set that is truth table complete for the sparse sets in NP ( Har83] only Turing completeness is stated, but Hartmanis s set is clearly truth table complete) Kadin s result applies equally well to the (seemingly stronger) truth table case. The boolean hierarchy [CGH 88,CGH 89] is the closure of NP under boolean operations; equivalently, it is the class ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 439--445. IEEE Computer Society Press, 1983.

....by Levin [Lev74] and G acs [G ac74] see also [LV88] Using resource bounded Kolmogorov complexity, we address the issue of the validity of symmetry of information in a resource bounded environment. Resource bounded versions of Kolmogorov complexity have recently been studied (see for example [Har83, Sip83, Ko86, Lon86]) For a specific (universal) Turing machine M , time bound T (n) and integer m, define KT (x; T (n) minfl j l = jyj and M(y) x; using at most T (jxj) timeg KT (x j m; T (n) minfl j l = jyj and M(hy; mi) x; using at most T (jxj) timeg S(n) space bounded Kolmogorov complexity KS(x; ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983.

....of the argument to N. Pippenger. Levin introduced Kt complexity and the universal optimal search algorithm (see e.g. Levin, 1973b) and (Levin, 1984) where related ideas are attributed to Adleman see also (Adleman, 1979) Other generalizations of Kolmogorov complexity have been proposed, e.g. (Hartmanis, 1983), but see the contributions in (Watanabe, 1992) for more. Easily computable approximations of the MDL principle were formulated by Wallace and Boulton (1968) and Rissanen (1978, 1983, 1986) Such approximations build the basis of most if not all current machine learning applications, e.g. Quinlan ....

Hartmanis, J. (1983). Generalized Kolmogorov complexity and the structure of feasible computations.

..... p posS;negS Figure 1: The strength of monotonic reducibilities In the proof of the Proposition we will make use of the information encoded into strings, which is measured by means of Kolmogorov complexity. Time bounded generalized Kolmogorov complexity were introduced by Hartmanis [Har83] We give a slightly modified definition for notational convenience. For a Turing machine M and integers l and s let KTM [l; t] fx j 9y : jyj l and M(y) outputs x using at most t steps g be the time bounded Kolmogorov complexity (relative to M ) For fixed integers this is a finite set, but ....

....M and integers l and s let KTM [l; t] fx j 9y : jyj l and M(y) outputs x using at most t steps g be the time bounded Kolmogorov complexity (relative to M ) For fixed integers this is a finite set, but we can still let l and t be unbounded functions. The following fact can be easily verified [Har83] There exists a Universal Turing machine U such that, for every Turing machine M , there is a constant c such that, for every l and t, KTM [l; t] KTU [l c; c Delta t Delta log t c] In the following we fix such a Universal machine U and omit the subscript U when no confusion arises. ....

J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 439--445, 1983.

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, pages 439--445, 1983.

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J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, 1983, pp. 439--445.

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 439-445, 1983. 12

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science (FOCS'83), pp. 439--445, 1983

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Juris Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, pages 439--445, 1983. 33

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J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, In Proc. 24th IEEE Simposium on Foundations of Computer Science, pp. 439--445, 1983.

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J. Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, 1983, pp. 439--445. Institute of Electrical and Electronics Engineers.

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J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on the Foundations of Computer Science, pages 439--445, 1983.

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