Showing papers by "Juris Hartmanis published in 1959"

Lattice Theory of Generalized Partitions

Abstract: In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.

Linear Multivalued Sequential Coding Networks

Abstract: Linear multivalued sequential coding networks are circuits whose input and output are synchronized sequences of nonnegative integers less than some fixed number m . The output depends linearly on the present input and a finite number of previous inputs and outputs. The transfer characteristics of such a network are described by a ratio of polynomials in the delay operator, where the multiplication and addition are performed with respect to the fixed modulus m . An algebraic theory of the delay polynomials is obtained. It is shown that a polynomial has a complete set of null sequences if, and only if, its first and last coefficients are prime to the modulus m. The polynomials with no null sequences are characterized. It is shown when common null sequences imply that the polynomials have common factors and that a complete set of null sequences defines the polynomial. It is also shown that a transfer function can be realized if the denominator contains a constant term prime to m and explicit constructions are given. A network is stable if the polynomial in the denominator of the transfer junction has no null sequence. Thus any nontrivial polynomial or its inverse is unstable if we are working modulo a prime. If the modulus is not prime, stable networks with stable inverses are constructed. Finally it is indicated how polynomials with no null sequences can be used to simplify the construction of coding networks.

The application of some basic inequalities for entropy

Abstract: This paper is part of a general study of efficient information selection, storage and processing. It is assumed that the information is contained in binary time series generated by a stochastic source. The main problem is to determine how to approximate the statistical properties of this information source by lower order probability distributions. First, it is determined what restrictions are imposed by known lower order probability distributions on the higher order distributions which are to be determined or estimated. This study suggests an optimal way to estimate the higher order distributions. In the second part the entropy changes which occur in going from lower to higher order probability distributions are studied. The upper and lower bounds for the entropy of the higher order distributions are computed in terms of the entropies of the lower order distributions. These results allow the computation of the “strength” of the conditions which are imposed on the higher order distribution and are not induced by the lower order ones. From this one can compute the importance of knowing the higher order probability distributions for information processing. In conclusion, these results are applied to a coding problem to determine the delay required before encoding a message in order to achieve a prescribed fraction of the optimal comparison given by Shannon's coding theorem.