Nobuki Tokura

On the capabilities of while, repeat, and exit statements

A well-formed program is defined as a program in which loops and if statements are properly nested and can be entered only at their beginning. A corresponding definition is given for a well-formed flowchart. It is shown that a program is well formed if and only if it can be written with if, repeat, and multi-level exit statements for sequence control. It is also shown that if, while, and repeat statements with single-level exit do not suffice. It is also shown that any flowchart can be converted to a well-formed flowchart by node splitting. Practical implications are discussed.

On the weight structure of Reed-Muller codes

The following theorem is proved. Let f(x_1,\cdots, x_m) be a binary nonzero polynomial of m variables of degree \nu . H the number of binary m -tuples (a_1,\cdots, a_m) with f(a_1, \cdots, a_m) = 1 is less than 2^{m-\nu+1} , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. \begin{equation} f = y_1 \cdots y_{\nu - \mu} (y_{\nu-\mu+1} \cdots y_{\nu} + y_{\nu+1} \cdots y_{\nu+\mu}), \end{equation} where m \geq \nu+\mu and \nu \geq \mu \geq 3 . \begin{equation} f = y_1 \cdots y_{\nu-2}(y_{\nu-1} y_{\nu} + y_{\nu+1} y_{\nu+2} + \cdots + y_{\nu+2\mu -3} y_{\nu+2\mu-2}), \end{equation} This theorem completely characterizes the codewords of the \nu th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloanes results.

On the weight enumeration of weights less than 2.5d of Reed—Muller codes

Let Pr be the set of all polynomials of degree r in m variables over GF(2). Polynomial ƒ in Pr is said to be affine equivalent to polynomial g in Pr, if ƒ is transformable to g by an invertible affine transformation of the variables. Any polynomial of weight less than 2m−r+1 + 2m−r−1 in Pr is shown to have a simple structure. By using this fact, we find out a set of representative polynomials such that any polynomial of weight less than 2m−r+1 + 2m−r−1 in Pr is affine equivalent to one and only one polynomial of the set. By counting the number of polynomials which are affine equivalent to each representative polynomial in the set, we derive explicit formulas for the enumerators of all weights less than 2.5d of Reed—Muller codes, where d is the minimum weight.

Some Remarks on BCH Bounds and Minimum Weights of Binary Primitive BCH Codes

It is shown that if m \neq 8, 12 and m > 6 , there are some binary primitive BCH codes (BCH codes in a narrow sense) of length 2^{m} - 1 whose minimum weight is greater than the BCH bound. This gives a negative answer to the question posed by Peterson [1] of whether or not the BCH bound is always the actual minimum weight of a binary primitive BCH code. It is also shown that for any even m \geq 6 , there are some binary cyclic codes of length 2^{m} - 1 that have more information digits than the primitive BCH codes of length 2^{m} - 1 with the same minimum weight.

Minimal Negative Gate Networks

A negative gate is a gate that can realize an arbitrary negative function (or monotone decreasing function) and a positive gate is one that can realize an arbitrary positive function (or monotone increasing function). This paper discusses methods of realizing a given logical function or a given set of logical functions using a minimum number of negative gates alone or using a minimum number of negative and positive gates.

Optimal Fault-Tolerant Distributed Algorithms for Election in Complete Networks with a Global Sense of Direction

This paper considers the leader election problem (LEP) in asynchronous complete networks with undetectable fail-stop failures. Especially, it is discussed whether presence of a global sense of direction affects the message complexity of LEP in faulty networks. For a complete network of n processors where k processors start the algorithm spontaneously and at most f p (<n/2) processors are faulty, this paper shows

Flow languages equal recursively enumerable languages

SummaryRecently, A.C. Shaw introduced a new class of expressions called flow expressions, and conjectured that the formal descriptive power of flow expressions lies somewhat below context-sensitive grammers. In this paper, we give a negative answer for his conjecture, that is, we show that all recursively enumerable languages may be denoted by flow expressions.

Failsafe Logic Nets

The basic properties and realization of failsafe combinational logic nets and failsafe sequential machines are discussed. The present note concerns totally asymmetric faiiure nets which are composed only of (+) elements and/ or (-) elements, where a (+) element is an asymmetric failure element which never fails to logical zero and a (-) element is defined similarly.

Efficient distributed algorithms solving problems about the connectivity of network

This paper presents efficient distributed algorithms on an asynchronous network for the following problems: finding bi-connected components, finding cutpoints, finding bridges, testing for bi-connectedness and finding strongly connected components of a directed graph defined on a network. All these distributed algorithms use a depth-first search tree having an arbitrary processor in the network as its root. The communication complexity of these algorithms is O(nlogn+e) and their ideal-time complexity is O(nG(n)), where n and e represent the numbers of processors and links, respectively, and G(n) is almost a constant. It is shown also that a lower bound for the communication complexity of the five problems is O(e) and a lower bound for their ideal-time complexity is O(n).

An optimal time algorithm for the k -vertex-connectivity unweighted augmentation problem for rooted directed trees

Abstract For a given digraph G=(V,A) and a positive integer k, the k-vertex-connectivity unweighted augmentation problem (k-VCUAP) for G is to find a minimum set of arcs A′ (A′⊆(V×V−A)) such that the digraph (V,A∪A′) is k-vertex-connected. It is known that the time-complexity of 1-VCUAP for every digraph is θ(|V|+|A|). However, it remains still open whether or not there exist polynomial time algorithms for k-VCUAPs (k≥2) for digraphs. This paper shows that the time-complexity of k-VCUAP (k≥2) is θ(k|V|) for every rooted directed tree.