Electronic Notes in Theoretical Computer Science, 2009
This paper presents work in the context of the certification of a safety component for autonomous... more This paper presents work in the context of the certification of a safety component for autonomous service robots, and investigates the potential advantages offered by formally modelling the domain knowledge, specification and implementation in a theorem prover in higher-order logic. This allows safety properties to be stated in an abstract manner close to textbook mathematics. The automatic proof checking alleviates correctness concerns, and provides a seamless development process from high-level safety requirements down to concrete implementation. Moreover, the formalisation can be checked for correctness automatically, and the certification review process can focus on the correctness of the specification and safety cases.
This paper offers some new results on randomness with respect to classes of measures, along with ... more This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli
In this paper we analyze the notion of "stopping time complexity", the amount of information need... more In this paper we analyze the notion of "stopping time complexity", the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic . It turns out that plain stopping time complexity of a binary string x could be equivalently defined as (a) the minimal plain complexity of a Turing machine that stops after reading x on a one-directional input tape; (b) the minimal plain complexity of an algorithm that enumerates a prefix-free set containing x; (c) the conditional complexity C(x | x * ) where x in the condition is understood as a prefix of an infinite binary sequence while the first x is understood as a terminated binary string; (d) as a minimal upper semicomputable function K such that each binary sequence has at most 2 n prefixes z such that K(z) < n; (e) as max C X (x) where C X (z) is plain Kolmogorov complexity of z relative to oracle X and the minimum is taken over all extensions X of x. We also show that some of these equivalent definitions become non-equivalent in the more general setting where the condition y and the object x may differ, and answer some open question from Chernov, Hutter and Schmidhuber .
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2008
The main goal of this paper is to put some known results in a common perspective and to simplify ... more The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from saying that lim sup n C(x|n) (here C(x|n) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0 ′ (x), the plain Kolmogorov complexity with 0 ′ -oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of [4] about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of 0 ′ Martin-Löf randomness (called also 2-randomness) proved in [3]: a sequence ω is 2-random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y) |y|c. (In the 1960ies this property was suggested in [1] as one of possible randomness definitions; its equivalence to 2-randomness was shown in [3] while proving another 2-randomness criterion (see also ): ω is 2-random if and only if C(x) |x|c for some c and infinitely many prefixes x of ω. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.
The main goal of this article is to put some known results in a common perspective and to simplif... more The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result of Vereshchagin [13] saying that lim sup n C(x|n) (here C(x|n) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0 ′ (x), the plain Kolmogorov complexity with 0 ′-oracle. Then we use the same argument to prove similar results for prefix complexity, a priori probability on binary tree, to prove Conidis' theorem [3] about limits of effectively open sets, and also to improve the results of Muchnik [8] about limit frequencies. As a byproduct, we get a criterion of 0 ′ Martin-Löf randomness (called also 2-randomness) proved in Miller [7]: a sequence ω is 2-random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y) |y| − c. (In the 1960ies this property was suggested in Kolmogorov [5] as one of possible randomness definitions; its equivalence to 2-randomness was shown in Miller [7]). Miller [7] and Nies et al. [9] proved another 2randomness criterion: ω is 2-random if and only if C(x) |x| − c for some c and infinitely many prefixes x of ω. This criterion is also a consequence of the results mentioned above. [The original version of this work [2] contained a weaker (and cumbersome) version of Conidis' result, and the proof used low basis theorem (in quite a strange way). The full version was formulated as a conjecture. This conjecture was later proved by Conidis. Bruno Bauwens (personal communication) noted that the proof can be obtained also by a simple modification of our original argument, and we reproduce Bauwens' argument with his permission.]
Péter Gács showed (Gács 1974) that for every n there exists a bit string x of length n whose plai... more Péter Gács showed (Gács 1974) that for every n there exists a bit string x of length n whose plain complexity C (x) has almost maximal conditional complexity relative to x, i.e., C (C (x)|x) ≥ log n − log (2) n − O(1). (Here log (2) i = log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound log n − O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| − C (x) ≤ c and |x| + K (|x|) − K (x) ≥ log (2) |x| − c log (3) |x|. In fact, the results of Solovay and Gács are closely related. Using the result above, we provide a short proof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n − C (x) ≤ c and n + K (n) − K (x) ≥ K (K (n)|n) − 3 K (K (K (n)|n)|n) − c. We also prove a close upper bound K (K (n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) − K(x) ≥ log (2) |x| − O(log (3) |x|).
We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that the... more We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete and Computational Geometry 20 (1998) 265-279]. By the same techniques we show that Ammann A2 tilings are not robust in the sense of [B. Durand, A. Romashchenko, A. Shen. Fixed-point tile sets and their applications, Journal of Computer and System Sciences, 78:3 (2012) 731-764].
Fix an optimal Turing machine U and for each n consider the ratio ρ U n of the number of halting ... more Fix an optimal Turing machine U and for each n consider the ratio ρ U n of the number of halting programs of length at most n by the total number of such programs. Does this quantity have a limit value? In this paper, we show that it is not the case, and further characterise the reals which can be the limsup of such a sequence ρ U n. We also study, for a given optimal machine U , how hard it is to approximate the domain of U from the point of view of coarse and generic computability.
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of ... more Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter p. By the law of large numbers, the frequency of zeros in the sequence tends to p, and thus we can get better and better approximations of p as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that p is a computable real, and one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter p (in the form of its Turing code). Can one do such a thing uniformly for all sequences that are random for computable Bernoulli measures, or even for a 'large enough' fraction of them? In this paper, we give a negative answer to this question. In fact, we prove a very general negative result which extends far beyond the class of Bernoulli measures. We do however provide a weak positive result, by showing that looking at a sequence X generated according to some computable probability measure, we can eventually guess a sequence of measures with respect to which X is random in Martin-Löf's sense.
Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophi... more Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there is no good model? If yes, how often these bad (non-stochastic) data appear "in real life"? Another, more technical motivation comes from algorithmic information theory. In this theory a notion of complexity of a finite object (=amount of information in this object) is introduced; it assigns to every object some number, called its algorithmic complexity (or Kolmogorov complexity). Algorithmic statistic provides a more fine-grained classification: for each finite object some curve is defined that characterizes its behavior. It turns out that several different definitions give (approximately) the same curve. 1 In this survey we try to provide an exposition of the main results in the field (including full proofs for the most important ones), as well as some historical comments. We assume that the reader is familiar with the main notions of algorithmic information (Kolmogorov complexity) theory. An exposition can be found in [44, chapters 1, 3, 4] or [22, chapters 2, 3], see also the survey [37].
We study the minimal complexity of tilings of a plane with a given tile set. We note that every t... more We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity at least n. This adds a quantitative angle to classical results on non-recursivity of tilings-that we also develop in terms of Turing degrees of unsolvability.
It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable... more It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to an intuitive idea of a geometric construction as a special kind of an "algorithm" using restricted means (straightedge and/or compass). However, the formalization is not obvious, and different descriptions existing in the literature are far from being complete and clear. We discuss the history of this notion and a possible definition in terms of a simple game.
For additional information and updates on this book, visit w w w .am s.org/bookpages/surv-220 ibr... more For additional information and updates on this book, visit w w w .am s.org/bookpages/surv-220 ibrary o f C on gress C a ta lo g in g -in -P u b lic a tio n D a ta Names: Shen, A. (Alexander), 1958-| Uspenskiï, V. A.
Leonid Levin [21] published a new (and very nice) proof of the Kučera-Gács theorem that occupies ... more Leonid Levin [21] published a new (and very nice) proof of the Kučera-Gács theorem that occupies only a few lines when presented in his style. We try to explain more details and discuss the connection of this proof with image randomness theorems, making explicit some result (see Proposition 4) that is implicit in [21]. Then we review the previous work about the oracle use when reducing a given sequence to another one, and its connection with algorithmic dimension theory.
We consider the notion of information distance between two objects x and y introduced by Bennett,... more We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [1] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. In the above paper, it was shown that the prefix version of information distance equals max(K(x|y), K(y|x) up to additive logarithmic terms. It was claimed by Mahmud [12] that this equality holds up to additive O(1)-precision. We show that this claim is false, but does hold if the distance is at least logarithmic. This implies that the original definition provides a metric on strings that are at superlogarithmically separated.
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