String Matching: Communication, Circuits, and Learning

String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings. - Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. - Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k. - Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Improved texture image classification through the use of a corrosion-inspired cellular automaton

In this paper, the problem of classifying synthetic and natural texture images is addressed. To tackle this problem, an innovative method is proposed that combines concepts from corrosion modeling and cellular automata to generate a texture descriptor. The core processes of metal (pitting) corrosion are identified and applied to texture images by incorporating the basic mechanisms of corrosion in the transition function of the cellular automaton. The surface morphology of the image is analyzed before and during the application of the transition function of the cellular automaton. In each iteration the cumulative mass of corroded product is obtained to construct each of the attributes of the texture descriptor. In a final step, this texture descriptor is used for image classification by applying Linear Discriminant Analysis. The method was tested on the well-known Brodatz and Vistex databases. In addition, in order to verify the robustness of the method, its invariance to noise and rotation were tested. To that end, different variants of the original two databases were obtained through addition of noise to and rotation of the images. The results showed that the method is effective for texture classification according to the high success rates obtained in all cases. This indicates the potential of employing methods inspired on natural phenomena in other fields.Comment: 13 pages, 14 figure

An automaton-theoretic approach to the representation theory of quantum algebras

We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatorics. We show that for a fixed number $m$, the torus-invariant primitive ideals in $m\times n$ quantum matrices can be seen as a regular language in a natural way. Using this description and a semigroup approach to the set of Cauchon diagrams, a combinatorial object that paramaterizes the primes that are torus-invariant, we show that for $m$ fixed, the number of torus-invariant primitive ideals in $m\times n$ quantum matrices satisfies a linear recurrence in $n$ over the rational numbers. In the $3\times n$ case we give a concrete description of the torus-invariant primitive ideals and use this description to give an explicit formula for the number P(3,n).Comment: 31 page