Editing Moore machine (section)

== Gedanken-experiments ==

In Moore's 1956 paper "[[Thought experiment|Gedanken-experiments]] on Sequential Machines",<ref name="gedanken"/> the <math>(n;m;p)</math> automata (or machines) <math>S</math> are defined as having <math>n</math> states, <math>m</math> input symbols and <math>p</math> output symbols. Nine theorems are proved about the structure of <math>S</math>, and experiments with <math>S</math>. Later, "<math>S</math> machines" became known as "Moore machines".

At the end of the paper, in Section "Further problems", the following task is stated:
<blockquote>Another directly following problem is the improvement of the bounds given at the theorems 8 and 9.</blockquote>

Moore's Theorem 8 is formulated as:
<blockquote>Given an arbitrary <math>(n;m;p)</math> machine <math>S</math>, such that every two of its states are distinguishable from one another, then there exists an experiment of length <math>\tfrac{n(n-1)}{2}</math> which determines the state of <math>S</math> at the end of the experiment.</blockquote>

In 1957, [[Anatolii Alexeevitch Karatsuba|A. A. Karatsuba]] proved the following two theorems, which completely solved Moore's problem on the improvement of the bounds of the experiment length of his  "Theorem 8".

<blockquote>'''Theorem A.''' If <math>S</math> is an <math>(n;m;p)</math> machine, such that every two of its states are distinguishable from one another, then there exists a branched experiment of length at most <math>\tfrac{(n-1)(n-2)}{2} + 1</math> through which one may determine the state of <math>S</math> at the end of the experiment.</blockquote>

<blockquote>'''Theorem B.''' There exists an <math>(n;m;p)</math> machine, every two states of which are distinguishable from one another, such that the length of the shortest experiments establishing the state of the machine at the end of the experiment is equal to <math>\tfrac{(n-1)(n-2)}{2} + 1</math>.</blockquote>

Theorems A and B were used for the basis of the course work of a student of the fourth year, A. A. Karatsuba, "On a problem from the automata theory", which was distinguished by testimonial reference at the competition of student works of the faculty of mechanics and mathematics of [[Moscow State University]] in 1958. The paper by Karatsuba was given to the journal ''Uspekhi Mat. Nauk'' on 17 December 1958 and was published there in June 1960.<ref>{{cite journal| last=Karatsuba| first=A. A.| title=Solution of one problem from the theory of finite automata | pages=157–159| journal=Uspekhi Mat. Nauk| issue=15:3| year=1960}}</ref>

Until the present day (2011), Karatsuba's result on the length of experiments is the only exact nonlinear result, both in automata theory, and in similar problems of [[computational complexity theory]].