Number Of Input Symbols Research Articles

A Polynomial Formula for Finite-Length BATS Code Performance

Batched sparse (BATS) codes have been proposed for communication through networks with packet loss. BATS codes include a matrix generalization of fountain codes as the outer code and random linear network coding at the intermediate network nodes as the inner code. BATS codes, however, do not possess a universal degree distribution that achieves the optimal rate for any distribution of the transfer matrix ranks, so that fast performance evaluation of finite-length BATS codes is important for optimizing the degree distribution. The state-of-the-art finite-length performance evaluation method has a computational complexity of $\mathcal {O}(K^{2}n^{2}M)$ , where $K$ , $n$ , and $M$ are the number of input symbols, the number of batches, and the batch size, respectively. We propose a polynomial-form formula for finite-length BATS codes performance evaluation with the computational complexity of $\mathcal {O}(K^{2}n\ln n)$ . Numerical results demonstrate that the polynomial-form formula can be significantly faster than the previous methods.

Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design

In this paper we analyze LT and Raptor codes under inactivation decoding. A first order analysis is introduced, which provides the expected number of inactivations for an LT code, as a function of the output distribution, the number of input symbols and the decoding overhead. The analysis is then extended to the calculation of the distribution of the number of inactivations. In both cases, random inactivation is assumed. The developed analytical tools are then exploited to design LT and Raptor codes, enabling a tight control on the decoding complexity vs. failure probability trade-off. The accuracy of the approach is confirmed by numerical simulations.

Belief propagation decoding assisted on-the-fly Gaussian elimination for short LT codes

Belief propagation (BP) decoding has been widely used for decoding Luby transform (LT) codes which perform very well for a large number of input symbols. However, in reality, small numbers of input...

Deterministic One-Way Turing Machines with Sublinear Space

Deterministic one-way Turing machines with sublinear space bounds are systematically studied. We distinguish among the notions of strong, weak, and restricted space bounds. The latter is motivated by the study of P automata. The space available on the work tape depends on the number of input symbols read so far, instead of the entire input. The class of functions space constructible by such machines is investigated, and it is shown that every function f that is space constructible by a deterministic two-way Turing machine, is space constructible by a strongly f space-bounded deterministic one-way Turing machine as well. We prove that the restricted mode coincides with the strong mode for space constructible functions. The known infinite, dense, and strict hierarchy of strong space complexity classes is derived also for the weak mode by Kolmogorov complexity arguments. Finally, closure properties under AFL operations, Boolean operations and reversal are shown.

Transmission Control for Fast Recovery of Rateless Codes

Luby Transform (LT) codes are more important in communication applications due to the characteristics of fast encoding/decoding process, and low complexity. However, LT codes are optimal only when the number of input symbols is close to infinity. Some researches modified the degree distribution of LT codes to make LT codes suitable for short symbol length. In this article, we propose an optimal coding algorithm to recover all of the encoded symbols for LT codes quickly. The proposed algorithm observes the coding status of each client and increases the coding performance by changing the transmission sequence of low-degree and high-degree encoding packets. Simulation results show that the resulting decoding overhead of our algorithm are lower than the traditional LT codes, and our algorithm is very appropriate to server various clients in the broadcasting channel environment.

Diversity-based inference of finite automata

We present new procedures for inferring the structure of a finite-state automaton (FSA) from its input/output behavior, using access to the automaton to perform experiments.Our procedures use a new representation for finite automata, based on the notion of equivalence betweentests. We call the number of such equivalence classes thediversityof the automaton; the diversity may be as small as the logarithm of the number of states of the automaton. For the special class ofpermutation automata, we describe an inference procedure that runs in time polynomial in the diversity and log(1/δ), where δ is a given upper bound on the probability that our procedure returns an incorrect result. (Since our procedure uses randomization to perform experiments, there is a certain controllable chance that it will return an erroneous result.) We also discuss techniques for handling more general automata.We present evidence for the practical efficiency of our approach. For example, our procedure is able to infer the structure of an automaton based on Rubik's Cube (which has approximately 1019states) in about 2 minutes on a DEC MicroVax. This automaton is many orders of magnitude larger than possible with previous techniques, which would require time proportional at least to the number of global states. (Note that in this example, only a small fraction (10-14) of the global states were even visited.)Finally, we present a new procedure for inferring automata of a special type in which the global state is composed of a vector of binary local state variables, all of which are observable (orvisible) to the experimenter. Our inference procedure runs provably in time polynomial in the size of this vector (which happens to be the diversity of the automaton), even though the global state space may be exponentially larger. The procedure plans and executes experiments on the unknown automaton; we show that the number of input symbols given to the automaton during this process is (to within a constant factor) the best possible.

Unification without occurs check on a cellular tree

Abstract This paper presents the design of a special‐purpose cellular tree architecture for the unification algorithm. The unification algorithm either finds the most general substitution which makes a set of terms identical, or else returns failure. The resulting substitution is permitted to contain loops. Many of the more recent pure logic programs make use of such recursive substitutions to store data structures which are accessed in a lazy fashion. So lack of such an occurs check is intentional. The algorithm involves only integer comparison and shifting operations between adjacent cells in a data‐driven computation mode. The physical design of a VLSI chip, therefore, should be quite easy. Each unification problem runs on the cellular tree in worst‐case time O (n · logn), where n is the number of input symbols. Many unification problems can be run simultaneously on the same chip with no slowdown.

Symmetry groups of automata

Symmetry transformations on the input and output code spaces of deterministic finite automata (DFA) are introduced. We show that the symmetry groups of transformations are produced by group DFA (gDFA) whose set of states and set of inputs are subgroups of the symmetric groups S q and S k , respectively ( q is the number of states and k the number of input symbols). The set of transitions of a gDFA is also a group. The symmetries of the n-moment delay DFA, relevant for cellular automata, are studied in detail. In particular, we show that the n-moment delay DFA on two symbols are self-symmetric. The symmetry gDFA of the 2-moment delay DFA on two symbols is displayed in detail. An algorithm to construct the symmetry gDFA of arbitrary DFA is given. An application of gDFA to cellular automata dynamics is mentioned.

A fast digital search algorithm using a double‐array structure

AbstractThis paper presents an efficient digital search algorithm by introducing a new internal array structure called a double‐array that combines the fast access of a matrix form with the compactness of a list form. Each arc of a digital search tree called a DS‐tree can be computed from the double‐array in O(1) time; that is, the worst‐case time complexity for retrieving a key becomes O(k) for the length k of that key. The double‐array is modified to make the size compact while maintaining fast access and algorithms for retrieval, insertion and deletion are presented. Suppose that the size of the double‐array is n + cm, where n is the number of nodes of the DS‐tree, m is the number of input symbols, and c is a constant depending on each double‐array. Then it is proved theoretically that the worst‐case times of deletion and insertion are proportional to cm and cm2, respectively, independent of n. From the experimental results of building the double‐array incrementally for various sets of keys, it is shown that the constant c has an extremely small value, ranging from 0.17 to 1.13.

An efficient digital search algorithm by using a double-array structure

An efficient digital search algorithm that is based on an internal array structure called a double array, which combines the fast access of a matrix form with the compactness of a list form, is presented. Each arc of a digital search tree, called a DS-tree, can be computed from the double array in 0(1) time; that is to say, the worst-case time complexity for retrieving a key becomes 0(k) for the length k of that key. The double array is modified to make the size compact while maintaining fast access, and algorithms for retrieval, insertion, and deletion are presented. If the size of the double array is n+cm, where n is the number of nodes of the DS-tree, m is the number of input symbols, and c is a constant particular to each double array, then it is theoretically proved that the worst-case times of deletion and insertion are proportional to cm and cm/sup 2/, respectively, and are independent of n. Experimental results of building the double array incrementally for various sets of keys show that c has an extremely small value, ranging from 0.17 to 1.13. >

On a Novel Approach of Fault Detection in an Easily Testable Sequential Machine with Extra Inputs and Extra Outputs

By augmenting a given sequential machine M with two extra input symbols Ie1 and Ie2 and with one extra output terminal Z', a method is developed in this paper for designing an efficient checking experiment for the machine M and for its fault diagnosis to cover the types of faults which may result in an increase in the number of states of the original machine M. The method is an extension of the works of Fujiwara et al.[6], and the scheme of modification never alters the modified machine M' from its easily testable nature as defined by Fujiwara et al [16]. The checking sequences exhibit remarkable savings of the number of input symbols for many sequential machines having a large number of states as compared to that of Fujiwara et al [16].