Referenced automata and metaregular families

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It is well-known that the set of strings (over some alphabet) definable by a read-only Turing machine in time intervals equal to the lengths of the strings is just a regular set over the given alphabet. The situation changes however if the machine is permitted to interrogate an oracle in the manner prescribed by Turing. We recall that in Turing machine theory [2] a computed function f is said to be recursive in a predicate ~ , if the change of state of the machine is permitted to depend upon whether or not the current number n (on the tape) is in ~ . As formalized in [2], there is no loss of computing time involved in consulting and responding to the oracle, but this desirable feature of the oracle may be lost if a particular model of realization (of the oracular operation) is adopted. In what follows, we shall be concerned with the notion of definability (of classes of sets) which results when a one-way read-only Turing machine is permitted to interrogate--in a style different from that of Turing--an oracle in the form of an additional tape R on which the oracular information is stored. R may be viewed as a (one-way infinite) sequence 7a7~73 over an alphabet F. ( I f / ' i s considered as a subset of{0, 1} n then/ may be taken to represent a finite set of predicates 9~i, i = 1,..., n.) We permit the read-only Turing machine to ask, at the ith step of its operation, not whether the number on the input tape is in the oracle's predicate, but whether i is in the oracle's predicate(s). Thus we seek to model our construction on the type of computation which relies upon a table look-up, but which, at each next step of computation, may know only the next table entry. Thus, in contradistinction to Turing's oracle, the device here (which we call a referenced automaton) may be regarded as a real-time oracle. With respect to some initial positioning of the machine (with respect to the reference tape) a set of input strings is defined (accepted); if the initial position is changed, the corresponding machine-reference tape combination defines another set of input strings. Our interest is in the family of sets thus defined. (Such families we call metaregular families.) The referenced automata have recently been considered by Agasandyan [1] and Salomaa [5]. They studied the generalization of regular sets obtained from a time-