Finite automata are considered in this paper a s instruments for classifying finite tapes. Each onetape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms. 7. J. C. Shepherdson, "The reduction of two-way automata to one-way automata,"

An Information Dispersal Algorithm (IDA) is developed that breaks a file F of length L = ↿ F ↾ into n pieces F i , l ≤ i ≤ n , each of length ↿ F i ↾ = L / m , so that every m pieces suffice for reconstructing F . Dispersal and reconstruction are computationally efficient. The sum of the lengths ↿ F i ↾ is ( n / m ) · L . Since n / m can be chosen to be close to l, the IDA is space efficient. IDA has numerous applications to secure and reliable storage of information in computer networks and even on single disks, to fault-tolerant and efficient transmission of information in networks, and to communications between processors in parallel computers. For the latter problem provably time-efficient and highly fault-tolerant routing on the n -cube is achieved, using just constant size buffers.

We present randomized algorithms to solve the following string-matching problem and some of its generalizations: Given a string X of length n (the pattern) and a string Y (the text), find the first occurrence of X as a consecutive block within Y. The algorithms represent strings of length n by much shorter strings called fingerprints, and achieve their efficiency by manipulating fingerprints instead of longer strings. The algorithms require a constant number of storage locations, and essentially run in real time. They are conceptually simple and easy to implement. The method readily generalizes to higher-dimensional patternmatching problems. IntroductionText-processing systems must allow their users to search for a given character string within a body of text. Database systems must be capable of searching for records with stated values in specified fields. Such problems are instances of the following string-matching problem: For a specified set {{X{i), Y(i)}\ of pairs of strings, determine, if possible, an r such that X(r) = Y(r). Usually the set is sjiecified not by explicit enumeration of the pairs, but rather by a rule for computing the pairs (X(i), Y{i)} from some given data. *<^opjTight 1987 by International Business Machines Corporation. Copying in printed form for private use is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) the Journal reference and IBM copyright notice are included on the first page. The title and abstract, but no other portions, of this paper may be copied or distributed royalty free without further permission by computer-based and other information-service systems. Permission to republish any other portion of this paper must be obtained from the Editor.We present a randomized algorithm to solve this problem. The algorithm associates with each string X a fingerprint {X) which is much shorter than the string itself The search for a match then compares short fingerprints instead of long strings.The algorithm selects the fingerprint function at random from a family of easy-to-compute functions. No matter which input \{X(i), F(/)>) is presented, the algorithm is unlikely to produce afiilse match, in which two fingerprints agree even though the original strings do not.The most widely studied pattern-matching problem is the following: Given a pattern X of length n and a text Y of length m'>n, find the first occurrence of X as a consecutive substring of Y. Several linear time algorithms have been given for this problem. The algorithms of Knuth, Morris, and Pratt [ 1 ] (KMP in the sequel) and of Boyer and Moore [2] require, for fast implementation, 0{n) registers to store a table of pyointers. The characters of the text Y can come in a stream and require no storage. But for fast implementation it is useful to have portions of Y in main memory. The algorithm of Galil and Seiferas [3] requires only C>(log n) registers. Recently Galil and Seiferas [4] have found a realtime algorithm using a constant number of registers.Our method,...

We efficiently combine unpredictability and verifiability by extending the Goldreich-Goldwasser-

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