1,132,060 research outputs found
Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms
We relate two measures of complexity of regular languages. The first is
syntactic complexity, that is, the cardinality of the syntactic semigroup of
the language. That semigroup is isomorphic to the semigroup of transformations
of states induced by non-empty words in the minimal deterministic finite
automaton accepting the language. If the language has n left quotients (its
minimal automaton has n states), then its syntactic complexity is at most n^n
and this bound is tight. The second measure consists of the quotient (state)
complexities of the atoms of the language, where atoms are non-empty
intersections of complemented and uncomplemented quotients. A regular language
has at most 2^n atoms and this bound is tight. The maximal quotient complexity
of any atom with r complemented quotients is 2^n-1, if r=0 or r=n, and
1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} \binom{h}{n} \binom{k}{h}, otherwise. We
prove that if a language has maximal syntactic complexity, then it has 2^n
atoms and each atom has maximal quotient complexity, but the converse is false.Comment: 12 pages, 2 figures, 4 table
Quotient Complexity of Regular Languages
The past research on the state complexity of operations on regular languages
is examined, and a new approach based on an old method (derivatives of regular
expressions) is presented. Since state complexity is a property of a language,
it is appropriate to define it in formal-language terms as the number of
distinct quotients of the language, and to call it "quotient complexity". The
problem of finding the quotient complexity of a language f(K,L) is considered,
where K and L are regular languages and f is a regular operation, for example,
union or concatenation. Since quotients can be represented by derivatives, one
can find a formula for the typical quotient of f(K,L) in terms of the quotients
of K and L. To obtain an upper bound on the number of quotients of f(K,L) all
one has to do is count how many such quotients are possible, and this makes
automaton constructions unnecessary. The advantages of this point of view are
illustrated by many examples. Moreover, new general observations are presented
to help in the estimation of the upper bounds on quotient complexity of regular
operations
- …