Showing papers by "Juris Hartmanis published in 1963"

Regularity preserving modifications of regular expressions

Abstract: This paper is concerned with the problem of determining whether a set of sequences R \t', obtained by some given rule from a regular set of sequences R , is again a regular set. A number of such problems are solved in this paper and a basic technique is used which is easy to apply to problems of this type. This technique yields an explicit construction of the sequential machine which recognizes the sequences in R \t', if R \t' is a regular set. Among other things it is shown that: (i) the derivative of a regular set R with respect to any set of sequences W is a regular set, although a regular expression designating this set cannot in general be computed; (ii) the set of sequences obtained from a regular set R by removing “arbitrary halves” of the sequences in R and the set of these removed “halves” are both regular sets; (iii) the set of sequences, obtained from a regular set R by making no more than k changes in any m consecutive digits in sequences from R , is regular; (iv) the set of sequences which can be concatenated in one and only one way from sequences in R is regular.

A Study of Feedback and Errors in Sequential Machines

Abstract: The object of this paper is to study feedback in sequential machines, to classify (according to their seriousness) and analyze errors which arise in the state transitions of machines, and to establish some relations between feedback and errors. It is shown that the previously developed algebraic methods1,2 supply the necessary tools and a rigorous basis for this theory, and relate these new results to previously obtained results about the structure of sequential machines. For example, this work yields the necessary methods to detect the existence of a decomposition of machines into component machines so that the most ``serious'' errors of the computation can occur only in an isolated component machine. This leads to the possibility of imposing selectively different reliability conditions on the component machines to achieve high over-all reliability of the realizations.

Further Results on the Structure of Sequential Machines

Abstract: Abstr(~ct. Recently it has been showa how we can obJ~ain all possible realizations of a given sequential machine :]i (ks a submachine of a machine reMized) from interconnected sets of smaller machines. In this paper we shall derive further results about the realization of a sequential machine M from two smaller machines M~ and M= which are eotmeeted in series or parallel. We shall relate the structural properties of the machine M to the structural properties of the component machines M, and M~ and derive results about the uniqtte-hess and economy of such realizat.ions, The possibility of realizing a given sequential machine M from interconnected sets of smaller machines has been discussed in [1, 2, 3], where necessary ~md sufficient conditions were given for the decomposition of M into a serial or parMlel connection of two machines M~ and M2. We are considering only the case when M is a sub-machine of the machine obtahmd from M~ and M~. In this paper we continue the investigation of the structure of sequential machines by deriving further results on the serial and parallel decompositions. We shall assume that the reader is familiar with the notation, definitions and the basic results of [3]. Furthermore, we shall discuss only Moore-type sequential machines in this paper since the results can be easily extended to Mealy-type machines [4, 5]. First., we discuss the serial realization of M from 21:I1 and 2br2 and then deal with the properties of the parallel realizations. Let M be a reduced, completely specified sequential machine which we want to realize as u sub-machine of a machine realized from two serially connected machines M1 and M2, each having fewer states than M (i.e. to every state of M there wilt correspond a pair of states from M~ and M2 and this correspondence is preserved by the operation of the machines). We know from [3] that such a realization of M exists if and only if there exists a nomtrivial partition ~r with the substitution property on the set of states of M. If such a partition 7r exists, then the first machine M~ will have as its states the blocks of ~r and its state behavior is determined by the machine M. The second machine M2 is determined by a second partition r on the set of states of the machine M such that rr.r = O. In the terminology of [3] …