Andrei E. Romashchenko

Fixed-point tile sets and their applications

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many fields, ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene@?s fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann@?s self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. This construction is rather flexible, so it can be used in many ways. We show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (in which any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, to characterize effectively closed one-dimensional subshifts in terms of two-dimensional subshifts of finite type (an improvement of a result by M. Hochman), to construct a tile set that has only complex tilings, and to construct a robust aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter, we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: There exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed. Some of these results were included in the DLT extended abstract (Durand et al., 2008 [9]) and in the ICALP extended abstract (Durand et al., 2009 [10]).

Effective closed subshifts in 1D can be implemented in 2D

In this paper we use fixed point tilings to answer a question posed by Michael Hochman and show that every one-dimensional effectively closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construction of an aperiodic tile set and its extensions.

Variations on Muchnik's Conditional Complexity Theorem

Muchniks theorem about simple conditional descriptions states that for all strings a and b there exists a short program p transforming a to b that has the least possible length and is simple conditional on b . In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one, based on extractors, can be generalized to prove a version of Muchniks theorem for space-bounded Kolmogorov complexity.

On essentially conditional information inequalities

In 1997, Z. Zhang and R.W. Yeung found the first example of a conditional information inequality in four variables that is not “Shannon-type”. This linear inequality for entropies is called conditional (or constraint) since it holds only under condition that some linear equations are satisfied for the involved entropies. Later, the same authors and other researchers discovered several unconditional information inequalities that do not follow from Shannons inequalities for entropy. In this paper we show that some non Shannon-type conditional inequalities are “essentially” conditional, i.e., they cannot be extended to any unconditional inequality. We prove one new essentially conditional information inequality for Shannons entropy and discuss conditional information inequalities for Kolmogorov complexity.

Variations on Muchnik’s Conditional Complexity Theorem

Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a program p transforming a to b that has the least possible length and is simple conditional onxa0b. In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one, based on extractors, can be generalized to prove a version of Muchnik’s theorem for space-bounded Kolmogorov complexity. Another version of Muchnik’s theorem is proven for a resource-bounded variant of Kolmogorov complexity based on Arthur–Merlin protocols.

Conditional Information Inequalities for Entropic and Almost Entropic Points

We study conditional linear information inequalities, i.e., linear inequalities for Shannon entropy that hold for distributions whose joint entropies meet some linear constraints. We prove that some conditional information inequalities cannot be extended to any unconditional linear inequalities. Some of these conditional inequalities hold for almost entropic points, while others do not. We also discuss some counterparts of conditional information inequalities for Kolmogorov complexity.

Pseudo-Random Graphs and Bit Probe Schemes with One-Sided Error

We study probabilistic bit-probe schemes for the membership problem. Given a set A of at most n elements from the universe of size m we organize such a structure that queries of type “x∈A?u2009” can be answered very quickly. H.xa0Buhrman, P.B.xa0Miltersen, J.xa0Radhakrishnan, and S.xa0Venkatesh proposed a randomized bit-probe scheme that needs space of O(nlogm) bits. That scheme has a randomized algorithm processing queries; it needs to read only one randomly chosen bit from the memory to answer a query. For every x the answer is correct with high probability (with two-sided errors).In this paper we slightly modify the bit-probe model of Buhrman et al. and consider schemes with a small auxiliary information in “cache” memory. In this model, we show that for the membership problem there exists a bit-probe scheme with one-sided error that needs space of O(nlog2m+poly(logm)) bits, which cannot be achieved in the model without cache. We also obtain a slightly weaker result (space of size n1+δpoly(logm) bits and two bit probes for every query) for a scheme that is effectively encodable.

On the non-robustness of essentially conditional information inequalities

We show that two essentially conditional linear inequalities for Shannons entropies (including the Zhang-Yeung97 conditional inequality) do not hold for asymptotically entropic points. This means that these inequalities are non-robust in a very strong sense. This result raises the question of the meaning of these inequalities and the validity of their use in practice-oriented applications.

Reliable computations based on locally decodable codes

We investigate the coded model of fault-tolerant computations introduced by D. Spielman. In this model the input and the output of a computational circuit is treated as words in some error-correcting code. A circuit is said to compute some function correctly if for an input which is a encoded argument of the function, the output, been decoded, is the value of the function on the given argument. n nWe consider two models of faults. In the first one we suppose that an elementary processor at each step can be corrupted with some small probability, and faults of different processors are independent. For this model, we prove that a parallel computation running on n elementary non-faulty processors in time t = poly(n) can be simulated on O(nlogn / log log n) faulty processors in time O(tlog log n). Note that we get a sub-logarithmic blow up of the memory, which cannot be achieved in the classic model of faulty boolean circuit, where the input is not encoded. n nIn the second model, we assume that at each step some fixed fraction of elementary processors can be corrupted by an adversary, who is free to chose these processors arbitrarily. We show that in this model any computation can be made reliable with an exponential blow up of the memory. n nOur method employs a sort of mixing mappings, which enjoy some properties of expanders. Based on mixing mappings, we implement an effective self-correcting procedure for an array of faulty processors.

Pseudo-random graphs and bit probe schemes with one-sided error

We study probabilistic bit-probe schemes for the membership problem. Given a set A of at most n elements from the universe of size m we organize such a structure that queries of type x ∈ A? can be answered very quickly. n nH. Buhrman, P.B. Miltersen, J. Radhakrishnan, and S. Venkatesh proposed a bit-probe scheme based on expanders. Their scheme needs space of O(n log m) bits, and requires to read only one randomly chosen bit from the memory to answer a query. The answer is correct with probability 2/3 with two-sided errors. n nIn this paper we show that for the same problem there exists a bitprobe scheme with one-sided error that needs space of O(n log2m + poly(logm)) bits. The difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh is that we consider a bit-probe scheme with an auxiliary word. This means that in our scheme the memory is split into two parts of different size: the main storage of O(n log2m) bits and a short word of logO(1) m bits that is pre-computed once for the stored set A and cached. To answer a query x e A? we allow to read the whole cached word and only one bit from the main storage. For some reasonable values of parameters (e.g., for poly(log m) ≪ n ≪ m) our space bound is better than what can be achieved by any scheme without cached data (the lower bound Ω(n2 log m/log n) was proven in [11]). n nWe obtain a slightly weaker result (space of size n1+δpoly(logm) bits and two bit probes for every query) for a scheme that is effectively encodable. n nOur construction is based on the idea of naive derandomization, which is of independent interest. First we prove that a random combinatorial object (a graph) has the required properties, and then show that such a graph can be obtained as an outcome of a pseudo-random generator. Thus, a suitable graph can be specified by a short seed of a PRG, and we can put an appropriate value of the seed into the cache memory of the scheme.