Affiliation:
1. Massachusetts Institute of Technology, Cambridge, Massachusetts and Both of Department of Electrical Engineering and Project MAC
Abstract
When is a set
A
of positive integers, represented as binary numbers, “regular” in the sense that it is a set of sequences that can be recognized by a finite-state machine? Let π
A
(
n
) be the number of members of
A
less than the integer
n
. It is shown that the asymptotic behavior of π
A
(
n
) is subject to severe restraints if
A
is regular. These constraints are violated by many important natural numerical sets whose distribution functions can be calculated, at least asymptotically. These include the set
P
of prime numbers for which π
P
(
n
) @@@@
n
/log
n
for large
n
, the set of integers
A
(
k
) of the form
n
k
for which π
A
(
k
)
n
) @@@@
n
P/k
, and many others. The technique cannot, however, yield a decision procedure for regularity since for every infinite regular set
A
there is a nonregular set
A′
for which | π
A
(
n
) — π
A′
(
n
) | ≤ 1, so that the asymptotic behaviors of the two distribution functions are essentially identical.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Reference3 articles.
1. Finite automata and their decision problems;RABIN M.;IBM J. Res. Develop.,1959
2. Representation of events in nerve nets and finite automata. In Automata Studies, Shannon, C. E., and McCarthy, J. (Eds.), Princeton U. Press, Princeton;KIEENE S.C;N. J.,1956
3. Finite Automata and the Set of Squares
Cited by
31 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献