Finite Automata and the Set of Squares
Journal of the ACM (JACM)
Unrecognizable Sets of Numbers
Journal of the ACM (JACM)
A Remark on Acceptable Sets of Numbers
Journal of the ACM (JACM)
Some applications of a technique of Sakoda and Sipser
ACM SIGACT News
Absolutely parallel grammars and two-way deterministic finite-state transducers
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Functions Definable by Arithmetic Circuits
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Tape-reversal bounded turing machine computations
Journal of Computer and System Sciences
Absolutely parallel grammars and two-way finite-state transducers
Journal of Computer and System Sciences
On the complexity of a family of k-context-free sequences
Theoretical Computer Science
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A study of the problem of recognizing the set of primes by automata is presented. A simple algebraic condition is derived which shows that neither the set of primes nor any infinite subset of the set of primes can be accepted by a pushdown or finite automaton.In view of this result an interesting open problem is to determine the “weakest” automaton which can accept the set of primes. It is shown that the linearly bounded automaton can accept the set of primes, and it is conjectured that no automaton whose memory grows less rapidly can recognize the set of primes. One of the results shows that if this conjecture is true, it cannot be proved by the use of arguments about the distribution of primes, as described by the Prime Number Theorem. Some relations are established between two classical conjectures in number theory and the minimal rate of memory growth of automata which can recognize the set of primes.