Randomness is Hard

This paper proposes new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions i.e., both trivial and random sequences are not monotone poly deep, monotone poly depth satisfies the slow growth law i.e., no simple process can transform a non deep sequence into a deep one, and monotone poly deep sequences exist (unconditionally).We give two natural examples of deep sets, by showing that both the set of Levin-random strings and the set of Kolmogorov random strings are monotone poly deep.KeywordsKolmogorov ComplexityRandom StringPolynomial VersionUniversal Turing MachineLogical DepthThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Markov chain embedding technique is commonly used to study the distribution of statistics associated with regular patterns (i.e. set of strings described by a regular expression) in random strings. In this extended abstract, we formalize the concept Markov chain embedding for random strings produced by a possibly non-stationary Markov source. A notion of memory conveyed by the states of a deterministic finite automaton is introduced. This notion is used to characterize the smallest state-space size Markov chain required to specify the distribution of the count statistic of a given regular pattern. The research finds applications in problems associated with regular patterns in random strings that demand exponentially large state spaces.

Approximating shortest superstrings with constraints

We present a methodology for quantifying biodiversity at the sequence level by developing the probability theory on a set of strings. Further, we apply our methodology to the problem of quantifying the population diversity of microorganisms in several extreme environments and digestive organs and reveal the relation between microbial diversity and various environmental parameters.

The Closest Substring problem (CSP), where a short string is sought that minimizes the number of mismatches between it and each of a given set of strings, is a minimization problem with polynomial time approximation schemes. In this paper, a lower bound on approximation algorithms for the CSP problem is developed. We prove that unless the Exponential Time Hypothesis (ETH Hypothesis, i.e., not all search problems in SNP are solvable in subexponential time) fails, the CSP problem has no polynomial time approximation schemes of running time f(1/Ɛ)|x|O(1/Ɛ) for any function f, where |x| is the size of input instance. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound Ɛ. As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the CSP problem for small error bound Ɛ.

Motivated by Dvir et al. and Wigderson and Yehudayoff [3,10], we examine the question of discovering the set of heavy hitters of a distribution on strings (i.e., the set of strings with a certain minimum probability) from lossy or noisy samples. While the previous work concentrated on finding both the set of most probable elements and their probabilities, we consider enumeration, the problem of just finding a list that includes all the most probable elements without associated probabilities. Unlike Wigderson and Yehudayoff [10], we do not assume the underlying distribution has small support size, and our time bounds are independent of the support size. For the enumeration problem, we give a polynomial time algorithm for the lossy sample model for any constant erasure probability μ < 1 , and a quasi-polynomial algorithm for the noisy sample model for any noise probability ν < 1/2 of flipping bits. We extend the lower bound for the number of samples required for the reconstruction problem from [3] to the enumeration problem to show that when μ = 1 − o(1), no polynomial time algorithm exists.

With the emerging of highly automated and flexible manufacturing systems in semiconductor fabrication, reliability and optimal productivity of such systems require very intelligent and complex control systems. Deadlock issue arises easily in these systems due to shared equipment usage and high production flexibility. This paper extends the event-based deadlock avoidance supervisor for semiconductor manufacturing systems from scenario without choices in process flows to cases with choices which are common in modern manufacturing systems. To allow choices in process flows improves operational flexibility of these systems, it also requires more sophisticated supervisory control techniques. Built upon a directed graph model of process flows, the extended supervisor is able to efficiently avoid the deadlock state space explosion problem. Concepts such as compound events, operation strings and deadlock strings are restated. New concepts such as broken circuits, choice circuits, split events, merge events and so on are introduced. These concepts help to dramatically decrease the number of circuits that the supervisor has to check thus increasing efficiency of the method. Major features in the proposed method include: (i) it enables the optimal deadlock free operation of regular systems; and (ii) it runs in polynomial time (fast online computation) provided that the set of deadlock strings is calculated offline. Examples are provided to show the effectiveness of the method.

Hardness of computing the Kolmogorov complexity of a given string is closely tied to a security proof of hitting set generators, and thus understanding hardness of Kolmogorov complexity is one of the central questions in complexity theory. In this paper, we develop new proof techniques for showing hardness of computing Kolmogorov complexity under surprisingly efficient reductions, which were previously conjectured to be impossible. It is known that the set R K of Kolmogorov-random strings is PSPACE-hard under polynomial-time Turing reductions, i.e., PSPACE ⊂ P R K , and that NEXP ⊂ NP R K , which was conjectured to be tight by Allender (CiE 2012). We prove that EXP NP ⊂ P R K , which simultaneously improves these hardness results and refutes the conjecture of Allender under the plausible assumption that EXP NP ≠ NEXP. At the core of our results is a new security proof of a pseudorandom generator via a black-box uniform reduction, which overcomes an impossibility result of Gutfreund and Vadhan (RANDOM/APPROX 2008). Our proof techniques have further consequences, including: 1. Applying our proof techniques to the case of resource-bounded Kolmogorov complexity, we obtain NP-hardness of the problem MINcKT SAT of computing conditional polynomial-time-bounded SAT-oracle Kolmogorov complexity under polynomial-time deterministic reductions. In contrast, the Minimum SAT-Oracle Circuit Size Problem, which is a version of sublinear-time-bounded Kolmogorov complexity, cannot be NP-hard under polynomial-time deterministic reductions without resolving EXP ≠ ZPP. Our hardness result is the first result that overcomes the non-NP-hardness results of MCSP. We also prove DistNP-hardness of MINKT SAT , which is a partial converse of the approach of Hirahara (FOCS 2018) for proving the equivalence between worst-case and average-case complexity of NP. 2. We prove S 2 p -hardness of Kolmogorov complexity under quasi-polynomial-time nonadaptive reductions. This is the first result that overcomes a P/poly barrier result of Allender, Buhrman, Friedman, and Loff (MFCS 2012). We also establish a firm link between non-trivial satisfiability algorithms and immunity of random strings, and obtain the following unconditional lower bounds. 1. It has been a long-standing open question whether the set of subexponential-time-bounded Kolmogorov-random strings is decidable in P. We resolve this open question, by showing that the set of super-polynomial-time-bounded Kolmogorov-random strings is P-immune, which is a much stronger lower bound than an average-case lower bound. 2. The set of Levin’s Kolmogorov-random strings is (P-uniform ACC)-immune.

There are several sorts of Kolmogorov complexity, better to say several Kolmogorov complexities: decision complexity, simple complexity, prefix complexity, monotonie complexity,a priori complexity. The last three can and the first two cannot be used for defining randomness of an infinite binary sequence. All those five versions of Kolmogorov complexity were considered, from a unified point of view, in a paper by the first author which appeared in Watanabe's book [23]. Upper and lower bounds for those complexities and also for their differences were announced in that paper without proofs. (Some of those bounds are mentioned in Section 4.4.5 of [16].) The purpose of this paper (which can be read independently of [23]) is to give proofs for the bounds from [23]. The terminology used in this paper is somehow nonstandard: we call “Kolmogorov entropy” what is usually called “Kolmogorov complexity.” This is a Moscow tradition suggested by Kolmogorov himself. By this tradition the term “complexity” relates toany mode of description and “entropy” is the complexity related to anoptimal mode (i.e., to a mode that, roughly speaking, gives theshortest descriptions).

In 1975 Chaitin introduced his Ω number as a concrete example of random real. The real Ω is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω solve the halting problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation Θ of Chaitin Ω number. The real Θ is defined based on the set of all compressible strings. We investigate the properties of Θ and show that Θ is random. In addition, we generalize Θ to two directions Θ(T) and \(\overline{\Theta}(T)\) with a real T > 0. We then study their properties. In particular, we show that the computability of the real Θ(T) gives a sufficient condition for a real T ∈ (0,1) to be a fixed point on partial randomness, i.e., to satisfy the condition that the compression rate of T equals to T.

We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.

Solovay functions and their applications in algorithmic randomness

We consider an extension of first-order logic by modular quantifiers of a fixed modulus $q$. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1's is divisible by a prime $p$ that does not divide $q$. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class $ACC$ from $NC^1$. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.

This special issue of Theory of Computing Systems contains articles which relate to talks given at the 8th International Conference on Computability, Complexity and Randomness (CCR) in Moscow from the 23rd to the 27th of September 2015. The CCR conference series is devoted to algorithmic randomness, Kolmogorov complexity, and their relationship with computability theory, complexity theory, logic and reverse mathematics. The contributions to this special issue reflect both the interests of reserachers in Russia, in particular Kolmogorov and computational complexity, and new directions in the field of algorithmic randomness such as the focus on normal numbers and connections to ergodic theory. No conference proceedings have been published; this special issue is the only publication relating to the confernece. All contributions are a significant extension of the conference presentations, and underwent the usual rigorous journal reviewing process. The article Polynomial-time algorithms for checking some properties of Boolean functions given by polynomials by Svetlana Selezneva and Anton Bukhman presents a polynomial time algorithm, which tests whether a Boolean function represented by a multivariate polynomial over GF(2) is even, that is, has zero derivative in the direction of (1, . . . , 1). The approach is based on the notion of the characteristic of monomials in the polynomial representation of Boolean functions. In Normality and finite-state dimension of Liouville numbers Satyadev Nandakumar gives a new construction of a Liouville number which is normal for a given base. Vladimir V’yugin in On Stability of Probability Laws with Respect to Small Violations of Algorithmic Randomness examines the robustness of ergodic theorems when the assumption of randomness is relaxed only a little. For example he shows that

Under a standard hardness assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all polynomial-time samplable distributions. More precisely we show that if exponential time is not infinitely often in subexponential space, then the following are equivalent for any algorithm $A$: \begin{itemize} \item For all $\p$-samplable distributions $\mu$, $A$ runs in time polynomial on $\mu$-average. \item For all polynomial $p$, the running time for A is bounded by $2^{O(K^p(x)-K(x)+\log(|x|))}$ for \emph{all} inputs $x$. \end{itemize} where $K(x)$ is the Kolmogorov complexity (size of smallest program generating $x$) and $K^p(x)$ is the size of the smallest program generating $x$ within time $p(|x|)$. To prove this result we show that, under the hardness assumption, the polynomial-time Kolmogorov distribution, $m^p(x)=2^{-K^p(x)}$, is universal among the P-samplable distributions.