Approximation Theorems on Some Classes of Automata
A. de Korvin · SIAM Journal on Control · 1968
This paper considers a machine as a pair $(G,M)$, where G is a group or a semigroup and where M is a state-space. The first part of the paper considers the case where G is a locally compact group and M is any locally compact space. The essential requirement is that $(x,p) \to x(p)$ be continuous where $x \in G$, $p \in M$ and $x(p) \in M$: i.e., we require that the next state function be continuous. The notion of projective limit is discussed and a criterion is given as to when G is the projective limit of some of its quotient groups. Next an infinitesimal element is defined. An identification is then made near the respective identities of G and the set of infinitesimal operations. The second part of the paper treats the case when G is a so-called amenable semigroup, having a representation of bounded operators on a Hilbert space. In the case in which the representation is an isometry, weakly continuous, a decomposition theorem is given. On a particular subspace the representation turns out to be a direct sum of finite-dimensional operations. Diverse characterizations of that space are given. Next the notion of coordinates of a representation is defined and two orthogonality theorems are stated. The whole paper might be considered as an attempt at giving approximation theorems on essentially infinite automata.