Probabilistic automata with time-variant final state sets monitored, in an intuitive manner, by autonomous probabilistic automata are considered. It is shown that they include a scheme suggested by Turakainen as a particular case, and that they accept all the stochastic languages. If the monitors are started in stationary distributions, it is proved that such schemes will accept all and only the stochastic languages; other cases in which the languages can be shown to be stochastic follow, including the case in which the state transition matrix of the monitor is deterministic.

Scientific writing has long been adversely affected by the common misconception that "objective" and "dull" must be synonyms. That this is not the case Professor Arbib has once more proven by producing a sound and scientifically precise book that is also a pleasure to read. It will elicit more than just an occasional chuckle from its readers, and may even send some of them scurrying for their dictionaries with Greek polysyllables as "prolegomenon," "idiosyncratic," "palindrome," "polycephalic," and others; their search is doomed to failure with "possibilistic," for the author has apparently coined this one himself. Sometimes it is difficult to separate his wit from misprints and misspellings as in "... partial recursive with respect to one radix, they are partial recursive with respect to all radishes," granted, even, that "radish" comes from the same Latin word for "root," "radix." The flavor of the book is certainly unique. Can the reader think of any other book on automata theory closing with a phrase even remotely resembling "That is why both nature and nurture are important"?

Stochastic sequential machines produced by the insertion of noise in a deterministic machine are considered; the question of the relationship between the state reduction of the machine so formed and that of the underlying deterministic machine is raised. The theorem states that, for the case of observational noise only, if the stochastic machine has stochastically distinguishable outputs, then it is reduced iff the underlying machine is reduced. Machines with additive noise are used as an example of application.

In an effort to relate various existent schools of thought, this paper describes four models of probabilistic finite-state systems, giving necessary and sufficient conditions for reciprocal describability. An algorithm for the realization (synthesis) of a stochastic sequential machine by a deterministic machine and additive noise is presented. Different definitions of equivalence are listed and related by theorems, the most important of which states that "behavioral equivalence" with the same cutpoints for sequence recognizers is identical to "stochastic equivalence" (indistinguishability) for sequence transducers. Corollaries show the import of these results to the state minimization problem. The paper composes a strong case in favor of the adoption of J. W. Carlyle's model as a unifying one.

Abstract—The problem of embedding nonlinear machines into larger, linear ones is considered. The notions involved are carefully defined, and a set of necessary and sufficient conditions for a submachine of a linear machine to be nonlinear is given, for the case in which the submachine utilizes the full input alphabet. An example illustrates the most important theorems.

It would be very simple to yield to the temptation for cheap wit and call this work "an introduction to an introduction." However, it is certainly much more than this, even though it may be advisable for the novice to read it first, before attempting to unravel the intricacies of the author's Introduction to Metamathematics. Rather, it represents an honest effort to present parts of the former work to a more general audience, and in this the author succeeds admirably, producing a very readable and lucid work, in many ways superior to other books written with the same intention.1

A study of linear estimators that minimize the variance of the error squared is reported. A projection principle and a sufficiency test establish and verify minimality. When the signal process and the data process are jointly Gaussian, the problem is completely solved for unique absolutely minimizing operators, which are found to be a cascade of the classical minimum-mean-squared error estimator and an amplifier.