Incompleteness,Complexity, Randomness and Beyond

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.     

Kolmogorov-Loveland Stochasticity and Kolmogorov Complexity

Merkle et al. (Ann. Pure Appl. Logic 138(1–3):183–210, 2006) showed that all Kolmogorov-Loveland stochastic infinite binary sequences have constructive Hausdorff dimension 1. In this paper, we go even further, showing that from an infinite sequence of dimension less than H(\frac 12+d)\mathcal {H}(\frac {1}{2}+\delta) (ℋ being the Shannon entropy function) one can extract by an effective selection rule a biased subsequence with bias at least δ. We also prove an analogous result for finite strings.     

Schnorr Randomness

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. Since its initial development in the 1970s, the notion has received considerably less attention than Martin-Löf randomness. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.     

Randomness in cryptography

Although more formal definitions of randomness exist, a colloquial one will suffice here: a random process is one whose consequences are unknown. Intuitively, this is why randomness is crucial in cryptographic applications - because it provides a way to create information that an adversary can't learn or predict. It's then the task of a good protocol designer to leverage this power in the best possible way to protect data and communication. In this paper, we'll look at some basic uses of randomness in cryptography and briefly review the process of securely generating randomness.     

随机性及其应用研究

本文首先讨论"随机性"与"高效计算"之间的关系,并强调引入"随机性"于问题求解的意义与重要性.随后给出产生"随机性"的现实途径及为计算引入"随机性"的两种不同方式,即"在线"方式与"离线"方式;通过对概率图灵机求解判定问题的讨论,来说明两种引入"随机性"方式之间的等价关系.最后,本文指出现实的随机算法设计与实现并没有为计算引入真正的"随机性".     

Bounded Queries, Approximations, and the Boolean Hierarchy

This paper investigates nondeterministic bounded query classes in relation to the complexity of NP-hard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NP-hard approximation problems. The results in this paper take advantage of this machine-based model to prove that in many cases, NP-approximation problems have the upward collapse property. That is, a reduction between NP-approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level. For example, if M C reduces to (log n)-approximating M C using many–one reductions, then the Traveling Salesman Problem (TSP) is equivalent to M C under many–one reductions. Several upward collapse theorems are presented in this paper. The proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes. In fact, these results depend on a surprising connection between the Boolean hierarchy and nondeterministic bounded query classes.     

Randomness in Distribution Protocols

Randomness is a useful computation resource due to its ability to enhance the capabilities of other resources. Its interaction with resources such as time, space, interaction with provers and its role in several areas of computer science has been extensively studied. In this paper we give a systematic analysis of the amount of randomness needed by secret sharing schemes and secure key distribution schemes. We give both upper and lower bounds on the number of random bits needed by secret sharing schemes. The bounds are tight for several classes of secret sharing schemes. For secure key distribution schemes we provide a lower bound on the amount of randomness needed, thus showing the optimality of a recently proposed key distribution protocol.     

Randomness in interactive proofs

This paper initiates a study of the quantitative aspects of randomness in interactive proofs. Our main result, which applies to the equivalent form of IP known as Arthur-Merlin (AM) games, is a randomness-efficient technique for decreasing the error probability. Given an AM proof system forL which achieves error probability 1/3 at the cost of Arthur sendingl(n) random bits per round, and given a polynomialk=k(n), we show how to construct an AM proof system forL which, in the same number of rounds as the original proof system, achieves error 2 ^–k(n) at the cost of Arthur sending onlyO(l+k) random bits per round.Underlying the transformation is a novel sampling method for approximating the average value of an arbitrary functionf:{0,1} ^l [0,1]. The method evaluates the function onO(^–2 log ^–1) sample points generated using onlyO(l + log ^–1) coin tosses to get an estimate which with probability at least 1- is within of the average value of the function.     

Depth as Randomness Deficiency

Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program in absence of auxiliary information. Variants known as logical depth and computational depth are expressed in Kolmogorov complexity theory. We derive quantitative relation between logical depth and computational depth and unify the different depth notions by relating them to A. Kolmogorov and L. Levin’s fruitful notion of randomness deficiency. Subsequently, we revisit the computational depth of infinite strings, study the notion of super deep sequences and relate it with other approaches.     

随机性测试的研究与实现

介绍随机性测试方法的数理统计原理,给出16种常见的随机性测试,研究密码算法随机性测试的流程,讨论测试ID的编排方法,并用这些方法对欧洲加密标准——Camellia算法进行随机性测试,实验结果表明,该算法3轮以上的缩减轮版本所产生的密文具有较高的随机性。     

Martin-L?f Randomness, Invariant Measures and Countable Homogeneous Structures

We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fraïssé limits) to study closed amenable subgroups G of the symmetric group S _?? of a countable set, where S _?? has the topology of pointwise convergence. We construct G-invariant measures on the universal minimal flows associated with these groups G in, moreover, an algorithmic manner. This leads to an identification of the generic elements, in the sense of being Martin-Löf random, of these flows with respect to the constructed invariant measures. Along these lines we study the random elements of S _??, which are permutations that transform recursively presented universal structures into such structures which are Martin-Löf random.     

Deletions That Preserve Randomness

This paper discusses dynamic properties of data structures under insertions and deletions It is shown that, in certain circumstances, the result of n random insertions and m random deletions will be equivalent to n-m random insertions, under various interpretations of the world "random" and under various constraints on the order of insertions and deletions.     

Effective Randomness of Unions and Intersections

We investigate the ??-randomness of unions and intersections of random sets under various notions of randomness corresponding to different probability measures. For example, the union of two relatively Martin-Löf random sets is not Martin-Löf random but is random with respect to the Bernoulli measure $\lambda_{\frac{3}{4}}$ under which any number belongs to the set with probability $\frac{3}{4}$ . Conversely, any $\lambda_{\frac{3}{4}}$ random set is the union of two Martin-Löf random sets. Unions and intersections of random closed sets are also studied.     

Walrasian Equilibrium: Hardness,Approximations and Tractable Instances

We study the complexity issues for Walrasian equilibrium in a special case of combinatorial auction, called single-minded auction, in which every participant is interested in only one subset of commodities. Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004) showed that it is NP-hard to decide the existence of a Walrasian equilibrium for a single-minded auction and proposed a notion of approximate Walrasian equilibrium called relaxed Walrasian equilibrium. We show that every single-minded auction has a relaxed Walrasian equilibrium that satisfies at least two-thirds of the participants, proving a conjecture posed in Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004). Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show NP-completeness and hardness of approximation results for weak Walrasian equilibria. In search of positive results, we restrict our attention to the tollbooth problem (Guruswami et al. in Proceedings of the Symposium on Discrete Algorithms (SODA), pp. 1164–1173, 2005), where every participant is interested in a single path in some underlying graph. We give a polynomial time algorithm to determine the existence of a Walrasian equilibrium and compute one (if it exists), when the graph is a tree. However, the problem is still NP-hard for general graphs.     

Verifying Relative Safety,Accuracy, and Termination for Program Approximations

Approximate computing is an emerging area for trading off the accuracy of an application for improved performance, lower energy costs, and tolerance to unreliable hardware. However, developers must ensure that the leveraged approximations do not introduce significant, intolerable divergence from the reference implementation, as specified by several established robustness criteria. In this work, we show the application of automated differential verification towards verifying relative safety, accuracy, and termination criteria for a class of program approximations. We use mutual summaries to express relative specifications for approximations, and SMT-based invariant inference to automate the verification of such specifications. We perform a detailed feasibility study showing promise of applying automated verification to the domain of approximate computing in a cost-effective manner.