A Canonical Form of Vector Machines

We introduce RS-vector machines (RS-VMs) as a canonical form of vector machines. They are based on vector operations called repeat and stretch. Repeat enlarges a vector (a1 , a2 , ..., am) to (a1 , a2 , ..., am , a1 , a2, ..., am) and stretch enlarges (a1, a2, ..., am) to (a1, a1, a2, a2, ..., am, am), when the expansion factor d(m)=2. It is shown that we can change the power of RS-VMs depending on this single parameter d(m): (i) Polynomial-time RS-VMs of d(m)=2 have the same power as polynomial-space TMs, and (ii) polynomial-time RS-VMs of d(m)=m have the same power as exponential-time, polynomialalternation, alternating TMs. The more general results are: (iii) RS-VMs of d(m)=k (k 2 is constant) have the same power as RS-VMs of d(m)=2; (iv) a wide variety of d(m)s such as d(m)=m, m, ..., cm, 2 m, ..., A(2, m) m, have at least the same power as d(m)=m, where A(i, j) is Ackermann's function; and (v) any polynomial d(m) in m cannot surpass the power of d(m)=m. ] 1998 Academic Press