Time-bounded Kolmogorov Complexity Research Articles

Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations

We call any consistent and sufficiently powerful formal theory that enables to algorithmically verify whether a text is a proof algorithmically verifiable mathematics (av-mathematics). We study the fundamental question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of av-mathematics.Our goal is to show strong indications that nondeterminism is more powerful than determinism for polynomial time computations. To do that, we do not consider decision problems only, but also compression algorithms. We show that at least one of the following three claims must be true:(i)▪(ii)non-determinism is more powerful than determinism for polynomial-time compression(iii)for each polynomial-time compression algorithm there exists another one of the same asymptotic time complexity that compresses infinitely many strings logarithmically strongerAnother surprising consequence of P = NP would be that time-bounded Kolmogorov complexity for any polynomial bound can be computed by deterministic algorithms in polynomial time.

ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS

AbstractCook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:•There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the $\bigvee $ -hardness, and show that one specific gadget generator is the $\bigvee $ -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite $\mathcal {N}\mathcal {P}$ sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite $\mathcal {N}\mathcal {P}$ sets.

Approximating Kolmogorov complexity

It is well known that the Kolmogorov complexity function (the minimal length of a program producing a given string, when an optimal programming language is used) is not computable and, moreover, does not have computable lower bounds. In this paper we investigate a more general question: can this function be approximated? By approximation we mean two things: firstly, some (small) difference between the values of the complexity function and its approximation is allowed; secondly, at some (rare) points the values of the approximating function may be arbitrary. For some values of the parameters such approximation is trivial (e.g., the length function is an approximation with error d except for a O ( 2 − d ) fraction of inputs). However, if we require a significantly better approximation, the approximation problem becomes hard, and we prove it in several settings. Firstly, we show that a finite table that provides good approximations for Kolmogorov complexities of n-bit strings, necessarily has high complexity. Secondly, we show that there is no good computable approximation for Kolmogorov complexity of all strings. In particular, Kolmogorov complexity function is neither generically nor coarsely computable, as well as its approximations, and the time-bounded Kolmogorov complexity (for any computable time bound) deviates significantly from the unbounded complexity function. We also prove hardness of Kolmogorov complexity approximation in another setting: the mass problem whose solutions are good approximations for Kolmogorov complexity function is above the halting problem in the Medvedev lattice. Finally, we mention some proof-theoretic counterparts of these results. A preliminary version of this paper was presented at CiE 2019 conference (In Computing with Foresight and Industry – 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings (2019) 230–239 Springer).

Cryptographic hardness under projections for time-bounded Kolmogorov complexity

A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete.Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤mNC0 reductions. In this paper, we improve this, to show that MKTP‾ is hard for the (apparently larger) class NISZKL under not only ≤mNC0 reductions but even under projections. Also MKTP‾ is hard for NISZK under ≤mP/poly reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZKL is the non-interactive version of the class SZKL that was studied by Dvir et al.As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP).

Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

We survey recent developments related to the Minimum Circuit Size Problem and time-bounded Kolmogorov Complexity.

Polylog depth, highness and lowness for E

We study the relations between the notions of highness, lowness and logical depth in the setting of complexity theory. We introduce a new notion of polylog depth based on time bounded Kolmogorov complexity. We show polylog depth satisfies all basic logical depth properties, namely sets in P are not polylog deep, sets with (time bounded)-Kolmogorov complexity greater than polylog are not polylog deep, and only polylog deep sets can polynomially Turing compute a polylog deep set. We prove that if NP does not have p-measure zero, then NP contains polylog deep sets. We show that every high set for E contains a polylog deep set in its polynomial Turing degree, and that there exist Low(E,EXP) polylog deep sets. Keywords: algorithmic information theory; Kolmogorov complexity; Bennett logical depth.

New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems

The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deal with time-bounded Kolmogorov complexity are prominent candidates for NP-intermediate status. We show that, under very modest cryptographic assumptions (such as the existence of one-way functions), the problem of approximating the minimum circuit size (or time-bounded Kolmogorov complexity) within a factor of n 1 − o (1) is indeed NP-intermediate. To the best of our knowledge, these problems are the first natural NP-intermediate problems under the existence of an arbitrary one-way function. Our technique is quite general; we use it also to show that approximating the size of the largest clique in a graph within a factor of n 1 − o (1) is also NP-intermediate unless NP⊆ P/poly. We also prove that MKTP is hard for the complexity class DET under non-uniform NC 0 reductions. This is surprising, since prior work on MCSP and MKTP had highlighted weaknesses of “local” reductions such as ≤ NC 0 m . We exploit this local reduction to obtain several new consequences: — MKTP is not in AC 0 [ p ]. — Circuit size lower bounds are equivalent to hardness of a relativized version MKTP A of MKTP under a class of uniform AC 0 reductions, for a significant class of sets A . — Hardness of MCSP A implies hardness of MCSP A for a significant class of sets A . This is the first result directly relating the complexity of MCSP A and MCSP A , for any A .

Kolmogorov One-Way Functions Revisited

We study characterizations of one-way functions in terms of time-bounded Kolmogorov complexity. As the main contribution, we propose definitions for strong and weak Kolmogorov one-way functions and show that these are equivalent to classical strong and weak one-way functions, respectively. The new definitions were motivated by the fact that the expected value approach is not able to characterize strong one-way functions as we prove in the paper.

Reductions to the set of random strings: The resource-bounded case

This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.

One-Way Functions Using Algorithmic and Classical Information Theories

We prove several results relating injective one-way functions, time-bounded conditional Kolmogorov complexity, and time-bounded conditional entropy. First we establish a connection between injective, strong and weak one-way functions and the expected value of the polynomial time-bounded Kolmogorov complexity, denoted here by E(K t (x|f(x))). These results are in both directions. More precisely, conditions on E(K t (x|f(x))) that imply that f is a weak one-way function, and properties of E(K t (x|f(x))) that are implied by the fact that f is a strong one-way function. In particular, we prove a separation result: based on the concept of time-bounded Kolmogorov complexity, we find an interval in which every function f is a necessarily weak but not a strong one-way function. Then we propose an individual approach to injective one-way functions based on Kolmogorov complexity, defining Kolmogorov one-way functions and prove some relationships between the new proposal and the classical definition of one-way functions, showing that a Kolmogorov one-way function is also a deterministic one-way function. A relationship between Kolmogorov one-way functions and the conjecture of polynomial time symmetry of information is also proved. Finally, we relate E(K t (x|f(x))) and two forms of time-bounded entropy, the unpredictable entropy H unp, in which one-wayness of a function can be easily expressed, and the Yao+ entropy, a measure based on compression/decompression schema in which only the decompressor is restricted to be time-bounded.

Time-Bounded Kolmogorov Complexity and Solovay Functions

A Solovay function is an upper bound g for prefix-free Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n. We obtain natural examples of computable Solovay functions by showing that for some constant c 0 and all computable functions t such that c 0 n?t(n), the time-bounded version K t of K is a Solovay function. By unifying results of Bienvenu and Downey and of Miller, we show that a right-computable upper bound g of K is a Solovay function if and only if ? g =?2?g(n) is Martin-Lof random. We obtain as a corollary that the Martin-Lof randomness of the various variants of Chaitin's ? extends to the time-bounded case in so far as $\Omega _{ \textnormal{K}^{t}}$ is Martin-Lof random for any t as above. As a step in the direction of a characterization of K-triviality in terms of jump-traceability, we demonstrate that a set A is K-trivial if and only if A is O(g(n)?K(n))-jump traceable for all computable Solovay functions g. Furthermore, this equivalence remains true when the universal quantification over all computable Solovay functions in the second statement is restricted either to all functions of the form K t for some function t as above or to a single function K t of this form. Finally, we investigate into the plain Kolmogorov complexity C and its time-bounded variant C t of initial segments of computably enumerable sets. Our main theorem here asserts that every high c.e. Turing degree contains a c.e. set B such that for any computable function t there is a constant c t >0 such that for all m it holds that C t (B?m)?c t ?m, whereas for any nonhigh c.e. set A there is a computable time bound t and a constant c such that for infinitely many m it holds that C t (A?m)?logm+c. By similar methods it can be shown that any high degree contains a set B such that C t (B?m)?+ m/4. The constructed sets B have low unbounded but high time-bounded Kolmogorov complexity, and accordingly we obtain an alternative proof of the result due to Juedes et al. (Theor. Comput. Sci. 132(1---2):37---70, 1994) that every high degree contains a strongly deep set.

The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory

We continue an investigation into resource-bounded Kolmogorov complexity (Allender et al., 2006 [4]), which highlights the close connections between circuit complexity and Levin's time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resource-bounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) (Allender et al., 2006 [4]). Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resource-bounded Kolmogorov complexity are two-fold: • to demonstrate that other complexity measures such as branching-program size and formula size can also be discussed in terms of Kolmogorov complexity, and • to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity (Buhrman et al., 2002 [15]) also fit well into this framework. The main theorems that we provide using this new approach to resource-bounded Kolmogorov complexity are: • A complete set ( R KNt ) for NEXP / poly defined in terms of strings of high Kolmogorov complexity. • A lower bound, showing that R KNt is not in NP ∩ coNP . • New conditions equivalent to the conditions “ NEXP ⊆ nonuniform NC 1 ” and “ NEXP ⊆ L / poly ”. • Theorems showing that “distinguishing complexity” is closely connected to both FewEXP and to EXP. • Hardness results for the problems of approximating formula size and branching program size.

Time-bounded incompressibility of compressible strings and sequences

For every total recursive time bound t, a constant fraction of all compressible (low Kolmogorov complexity) strings is t-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length n is compressible to log n yet t-bounded incompressible below 1 4 n − log n ; and there is a countably infinite number of recursive infinite sequences of which every initial segment is similarly t-bounded incompressible. These results and their proofs are related to, but different from, Barzdins's lemma.

Power from Random Strings

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual many-one reductions. Let ${{R_{\rm C}}}, {{R_{\rm Kt}}}, {{R_{\rm KS}}}, {{R_{\rm KT}}}$ be the sets of strings x having complexity at least $|x|/2$, according to the usual Kolmogorov complexity measure ${\mbox{\rm C}}$, Levin's time-bounded Kolmogorov complexity ${\mbox{\rm Kt}}$ [L. Levin, Inform. and Control, 61 (1984), pp. 15-37], a space-bounded Kolmogorov measure ${\mbox{\rm KS}}$, and a new time-bounded Kolmogorov complexity measure ${\mbox{\rm KT}}$, respectively. Our main results are as follows: \begin{remunerate} \item ${{R_{\rm KS}}}$ and ${{R_{\rm Kt}}}$ are complete for ${{\rm{PSPACE}}}$ and {\mbox{\rm EXP}}, respectively, under ${\mbox{\rm P/poly}}$-truth-table reductions. Similar results hold for other classes with ${{\rm{PSPACE}}}$-robust Turing complete sets. \item ${\mbox{\rm EXP}} = {\mbox{\rm NP}}^{{{R_{\rm Kt}}}}.$ \item ${{\rm{PSPACE}}} = {\mbox{\rm ZPP}}^{{{R_{\rm KS}}}} \subseteq {\mbox{\rm P}}^{{{R_{\rm C}}}}$. \item The Discrete Log, Factoring, and several lattice problems are solvable in ${\mbox{\rm BPP}}^{{{R_{\rm KT}}}}$. \end{remunerate} Our hardness result for ${{\rm{PSPACE}}}$ gives rise to fairly natural problems that are complete for ${{\rm{PSPACE}}}$ under ${\mbox{$\leq^{\rm p}_{\rm T}$}}$ reductions, but not under ${\mbox{$\leq^{\rm log}_{\rm m}$}}$ reductions. Our techniques also allow us to show that all computably enumerable sets are reducible to ${{R_{\rm C}}}$ via ${\mbox{\rm P/poly}}$-truth-table reductions. This provides the first "efficient" reduction of the halting problem to ${{R_{\rm C}}}$.

Resource bounded symmetry of information revisited

The information contained in a string x about a string y is the difference between the Kolmogorov complexity of y and the conditional Kolmogorov complexity of y given x, i.e., I ( x : y ) = C ( y ) - C ( y | x ) . The Kolmogorov–Levin Theorem says that I ( x : y ) is symmetric up to a small additive term. We investigate if this property also holds for several versions of polynomial time-bounded Kolmogorov complexity. We study symmetry of information for some variants of distinguishing complexity CD where CD ( x ) is the length of a shortest program which accepts x and only x. We show relativized worlds where symmetry of information does not hold in a strong way for deterministic and nondeterministic polynomial time distinguishing complexities CD poly and CND poly . On the other hand, for nondeterministic polynomial time distinguishing complexity with randomness, CAMD poly , we show that symmetry of information holds for most pairs of strings in any set in NP. Our techniques extend work of Buhrman et al. (Language compression and pseudorandom generators, in: Proc. 19th IEEE Conf. on Computational Complexity, IEEE, New York, 2004, pp. 15–28) on language compression by AM algorithms, and have the following application to the compression of samplable sources, introduced in Trevisan et al. (Compression of sample sources, in: Proc. 19th IEEE Conf. on Computational Complexity, IEEE, New York, 2004, pp. 1–15): any element x in the support of a polynomial time samplable source X can be given a description of size - log Pr [ X = x ] + O ( log 3 n ) , from which x can be recovered by an AM algorithm.

Resource-Bounded Kolmogorov Complexity Revisited

We take a fresh look at CD complexity, where CDt(x) is the size of the smallest program that distinguishes x from all other strings in time t(|x|). We also look at CND complexity, a new nondeterministic variant of CD complexity, and time-bounded Kolmogorov complexity, denoted by C complexity. We show several results relating time-bounded C, CD, and CND complexity and their applications to a variety of questions in computational complexity theory, including the following: Showing how to approximate the size of a set using CD complexity without using the random string as needed in Sipser's earlier proof of a similar result. Also, we give a new simpler proof of this result of Sipser's. Improving these bounds for almost all strings, using extractors. A proof of the Valiant--Vazirani lemma directly from Sipser's earlier CD lemma. A relativized lower bound for CND complexity. Exact characterizations of equivalences between C, CD, and CND complexity. Showing that satisfying assignments of a satisfiable Boolean formula can be enumerated in time polynomial in the size of the output if and only if a unique assignment can be found quickly. This answers an open question of Papadimitriou. A new Kolmogorov complexity-based proof that BPP\subseteq\Sigma_2^p$. New Kolmogorov complexity based constructions of the following relativized worlds: There exists an infinite set in P with no sparse infinite NP subsets. EXP=NEXP but there exists a NEXP machine whose accepting paths cannot be found in exponential time. Satisfying assignments cannot be found with nonadaptive queries to SAT.

On resource-bounded instance complexity

The instance complexity of a string x with respect to a set A and time bound t, ic t ( x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where “do not know” answers are permitted). The instance complexity conjecture of Ko, Orponen, Schöning, and Watanabe (1986) states that for every recursive set A not in P and every polynomial t there is a polynomial t′ and a constant c such that for infinitely many x, ic t ( x : A) ⩾ C t′ ( x) − c, where C t′ ( x) is the t′-time bounded Kolmogorov complexity of x. In this paper the conjecture is proved for all recursive tally sets and for all recursive sets which are NP-hard under honest reductions, in particular it holds for all natural NP-hard problems. The method of proof also yields the polynomialspace bounded and the exponential-time bounded versions of the conjecture in full generality. On the other hand, the conjecture itself turns out to be oracle dependent: In any relativized world where P = NP the conjecture holds, but there are also relativized worlds where it fails, even if C-complexity is replaced by Sipser's CD-complexity. Additionally it is proved that the instance complexity measure is noncomputable and it is investigated whether for every polynomial t there is a polynomial t′ such that C t′ -complexity is bounded above by CD t -complexity.

On the complexity of ranking

This paper structurally characterizes the complexity of ranking. A set A is (strongly) P-rankable if there is a polynomial time computable function f so that for all x, f(x) computes the number of elements of A that are lexicographically ⩽ x, i.e., the rank of x with respect to A. This is the strongest of three notions of P-ranking we consider in this paper. We say a class C is P-rankable if all sets in C are P-rankable. Our main results show that with the same certainty with which we believe counting to be complex, and thus with at least the certainty with which we believe P ≠ NP, P has no uniform, strong, weak, or enumerative ranking functions. We show that: 1. • P and NP are equally likely to be P-rankable, i.e., P is P-rankable if and only if NP is P-rankable. 2. • P is P-rankable if and only if P = P #P. This extends work of Blum, Goldberg, and Sipser. 3. • Even the two weaker notions of P-ranking that we study are hard if P ≠ P #P. 4. •If P has small ranking circuits, then it has small ranking circuits of relatively low complexity. 5. • If P has small ranking circuits then counting is in the polynomial hierarchy, i.e., P #P ⊆ Σ 2 p = PH. 6. • P/poly has small ranking circuits if and only if P #P/poly = P #P/poly = P/poly. 7. • If P is P-rankable, then P/poly has small ranking circuits. This links the ranking complexity of uniform and nonuniform classes. 8. • The ranks of some strings in easy sets are of high relative time-bounded Kolmogorov complexity unless P = P #P. It follows that even a type of approximate ranking, enumerative ranking, is hard unless P = P #P. 9. • The complexity of generating “the next largest” element in a set has clear structural characterizations. In particular, (1) we can efficiently find some element of polynomial hierarchy sets at an input length if and only if P = PH ∩ P/poly, and (2) we can efficiently find some element of a polynomial hierarchy set greater than an input if and only if all sets in NP have infinite P-printable subsets.