Parsing Unary Boolean Grammars Using Online Convolution

In contrast to context-free grammars, the extension of these grammars by explicit conjunction, the so-called conjunctive grammars can generate (quite complicated) non-regular languages over a single-letter alphabet (DLT 2007). Given these expressibility results, we study the parsability of Boolean grammars, an extension of context-free grammars by conjunction and negation, over a unary alphabet and show that they can be parsed in time O(|G| log^2(n) M(n)) where M(n) is the time to multiply two n-bit integers. This multiplication algorithm is transformed into a convolution algorithm which in turn is converted to an online convolution algorithm which is used for the parsing

Relaxing order basis computation

International audienceThe computation of an order basis (also called sigma basis) is a fundamental tool for linear algebra with polynomial coefficients. Such a computation is one of the key ingredients to provide algorithms which reduce to polynomial matrices multiplication. This has been the case for column reduction or minimal nullspace basis of polynomial matrix over a field. In this poster, we are interested in the application of order basis to compute minimal matrix generators of a linear matrix sequence. In particular, we focus on the linear matrix sequence used in the Block Wiedemann algorithm

Space complexity in on-line computation

AbstractA technique is developed for determining space complexity in on-line computation. It is shown that each of the following functions requires linear space: (i) the conversion of binary numbers into ternary numbers, (ii) the multiplication of integers and (iii) the translation of arithmetic expressions in infix notation into Polish notation

Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases

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Application of Computer Algebra in List Decoding

The amount of data that we use in everyday life (social media, stock analysis, satellite communication etc.) are increasing day by day. As a result, the amount of data needs to be traverse through electronic media as well as to store are rapidly growing and there exist several environmental effects that can damage these important data during travelling or while in storage devices. To recover correct information from noisy data, we do use error correcting codes. The most challenging work in this area is to have a decoding algorithm that can decode the code quite fast, in addition with the existence of the code that can tolerate highest amount of noise, so that we can have it in practice. List decoding is an active research area for last two decades. This research popularise in coding theory after the breakthrough work by Madhu Sudan where he used list decoding technique to correct errors that exceeds half the minimum distance of Reed Solomon codes. Towards the direction of code development that can reach theoretical limit of error correction, Guruswami-Rudra introduced folded Reed Solomon codes that reached at $1 - R - \epsilon.$ To decode this codes, one has to first interpolate a multivariate polynomial first and then have to factor out all possible roots. The difficulties that lies here are efficient interpolation, dealing with multiplicities smartly and efficient factoring. This thesis deals with all these cases in order to have folded Reed Solomon codes in practice

Fast On-Line Integer Multiplication

A Turing machine multiplies binary integers on-Zine if it receives its inputs low-order digits first and produces the jth digit of the product before reading in the (j+l)st digits of the two inputs. We present a general method for converting any off-line multiplication algorithm which forms the product of two n-digit binary numbers in time F(n) into an on-line method which uses time only O(F() log ), assuming that F is monotone and satisfies n F() F(2)/2 ! kF() for some constant k. Applying this technique to the fast multiplication algorithm of Schönhage and Strassen gives an upper bound of O(n (log n)² loglog n) for on-line multiplication of integers. A refinement of the technique yields an optimal method for on-line multiplication by certain sparse integers. Other applications are to the on-line computation of products of polynomials, recognition of palindromes, and multiplication by a constant