There have been several studies related to a reduction of the amount of computational resources used by Turing machines. As consequences, Linear speed-up theorem", tape compression theorem" and reversal reduction theorem" have been obtained. In this paper, we discuss a leaf reduction theorem on alternating Turing machines. Recently, the result that one can reduce the number of leaves by a constant factor without increasing the space complexity was shown for space- and leaf-bounded alternating Turing machines. We show that for time- and leaf-bounded alternating Turing machines, the number of leaves can be reduced by a constant factor without increasing time used by the machine. Therefore, our result says that a constant factor on the leaf complexity does not affect the power of time- and leaf-bounded alternating Turing machines.

A chordal ring network is a processor network on which n processors are arranged to a ring with additional chords. We study a distributed leader election algorithm on chordal ring networks and present trade-offs between the message complexity and the number of chords at each processor and between the message complexity and the length of chords as follows:For every d(1dlog* n1) there exists a chordal ring network with d chords at each processor on which the message complexity for leader election is O(n(log(d1)nlog* n)).For every d(1dlog* n1) there exists a chordal ring network with log(d1)nd1 chords at each processor on which the message complexity for leader election is O(dn).For every m(2mn/2) there exists a chordal ring network whose chords have at most length m such that the message complexity for leader election is O((n/m)log n).

In this paper, some classes of arithmetic circuits are introduced that capture the computational complexity of computing the determinant of matrices with entries either indeterminates or constants from a field. An arithmetic circuit is just like a Boolean circuit, except that all AND and OR gates (with fan-in two) are replaced by gates realizing a multiplication and an addition, respectively, of two polynomials over some indeterminates with coefficients from the field, and the circuit computes a (formal multivariate) polynomial in the obvious sense. An arithmetic circuit is said to be skew if at least one of the inputs of each multiplication gate is either an indeterminate or a constant. Then it is shown that for all square matrices M of dimension q, the determinant of M can be computed by a skew arithmetic circuit of (q20) gates, and is shown that for all skew arithmetic circuits C of size q, the polynomial computed by C can be defined as the determinant of a square matrix M of dimension (q). Thus the size of skew arithmetic circuit is polynomially related to the dimension of square matrices when it is considered to represent multivariate polynomials in both arithmetic circuits and the determinant. The results are extended to some other classes of arithmetic circuits less restricted than skew ones, and by using such an extended result, a difference and a similarity are demonstrated between polynomials represented as the determinant of matrix of relatively small dimension and those polynomials computed by arithmetic formulas and arithmetic circuits of relatively small size and degree.