[Show abstract][Hide abstract] ABSTRACT: The state complexity of a regular language is the number of states in a
minimal deterministic finite automaton accepting the language. The syntactic
complexity of a regular language is the cardinality of its syntactic semigroup.
The syntactic complexity of a subclass of regular languages is the worst-case
syntactic complexity taken as a function of the state complexity $n$ of
languages in that class. We prove that $n^{n-1}$, $n^{n-1}+n-1$, and
$n^{n-2}+(n-2)2^{n-2}+1$ are tight upper bounds on the syntactic complexities
of right ideals and prefix-closed languages, left ideals and suffix-closed
languages, and two-sided ideals and factor-closed languages, respectively.
Moreover, we show that the transition semigroups meeting the upper bounds for
all three types of ideals are unique, and the numbers of generators (4, 5, and
6, respectively) cannot be reduced.
[Show abstract][Hide abstract] ABSTRACT: A regular language is most complex in its class if it meets the quotient
complexity (state complexity) bounds for boolean operations, product
(concatenation), star, and reversal, has the largest syntactic semigroup, and
has the maximal number of atoms, each of which has maximal quotient complexity.
It is known that there exist such most complex regular languages, and also
right ideals, left ideals, and two-sided ideals. In contrast to this, we prove
that there does not exist a most complex suffix-free regular language. However,
we do exhibit one ternary suffix-free witness that meets the bound for product
and whose restrictions to binary alphabets meet the bounds for star and boolean
operations. We also exhibit a quinary witness that meets the bounds for boolean
operations, reversal, size of syntactic semigroup, and atom complexities.
Moreover, we show that the bound for the product of two languages of quotient
complexities $m$ and $n$ can be met in the binary case for infinitely many $m$
and $n$.
Two transition semigroups play an important role for suffix-free languages:
semigroup $\mathbf{T}^{\le 5}$ is the largest suffix-free semigroup for $n\le
5$, while semigroup $\mathbf{T}^{\ge 6}$ is largest for $n=2,3$ and $n\ge 6$.
We prove that all witnesses meeting the bounds for the star and the second
witness in a product must have transition semigroups in $\mathbf{T}^{\le 5}$.
On the other hand, witnesses meeting the bounds for reversal, size of syntactic
semigroup and the complexity of atoms must have semigroups in $\mathbf{T}^{\ge
6}$.
[Show abstract][Hide abstract] ABSTRACT: A (left) quotient of a language $L$ by a word $w$ is the language
$w^{-1}L=\{x\mid wx\in L\}$. The quotient complexity of a regular language $L$
is the number of quotients of $L$; it is equal to the state complexity of $L$,
which is the number of states in a minimal deterministic finite automaton
accepting $L$. An atom of $L$ is an equivalence class of the relation in which
two words are equivalent if for each quotient, they either are both in the
quotient or both not in it; hence it is a non-empty intersection of
complemented and uncomplemented quotients of $L$. A right (respectively, left
and two-sided) ideal is a language $L$ over an alphabet $\Sigma$ that satisfies
$L=L\Sigma^*$ (respectively, $L=\Sigma^*L$ and $L=\Sigma^*L\Sigma^*$). We
compute the maximal number of atoms and the maximal quotient complexities of
atoms of right, left and two-sided regular ideals.
[Show abstract][Hide abstract] ABSTRACT: We solve an open problem concerning syntactic complexity: We prove that the
cardinality of the syntactic semigroup of a suffix-free language with $n$ left
quotients (that is, with state complexity $n$) is at most $(n-1)^{n-2}+n-2$ for
$n\ge 7$. Since this bound is known to be reachable, this settles the problem.
We also reduce the alphabet of the witness languages reaching this bound to
five letters instead of $n+2$, and show that it cannot be any smaller. Finally,
we prove that the transition semigroup of a minimal deterministic automaton
accepting such a witness language is unique for each $n\ge 7$.
[Show abstract][Hide abstract] ABSTRACT: We solve two open problems concerning syntactic complexity: We prove that the
cardinality of the syntactic semigroup of a left ideal or a suffix-closed
language with $n$ left quotients (that is, with state complexity $n$) is at
most $n^{n-1}+n-1$, and that of a two-sided ideal or a factor-closed language
is at most $n^{n-2}+(n-2)2^{n-2}+1$. Since these bounds are known to be
reachable, this settles the problems.
[Show abstract][Hide abstract] ABSTRACT: The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n; for 1 ≤ r ≤ n − 1 the bound is For each n ≥ 2, we exhibit a language with 2n atoms which meet these bounds.
International Journal of Foundations of Computer Science 02/2014; 24(07). DOI:10.1142/S0129054113400285 · 0.30 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that every regular language defines a unique nondeterministic finite automaton (NFA), which we call “átomaton”, whose states are the “atoms” of the language, that is, non-empty intersections of complemented or uncomplemented left quotients of the language. We describe methods of constructing the átomaton, and prove that it is isomorphic to the reverse automaton of the minimal deterministic finite automaton (DFA) of the reverse language. We study “atomic” NFAs in which the right language of every state is a union of atoms. We generalize Brzozowski's double-reversal method for minimizing a deterministic finite automaton (DFA), showing that the result of applying the subset construction to an NFA is a minimal DFA if and only if the reverse of the NFA is atomic. We prove that Sengoku's claim that his method always finds a minimal NFA is false.
[Show abstract][Hide abstract] ABSTRACT: The syntactic complexity of a regular language is the size of its syntactic
semigroup. This semigroup is isomorphic to the transition semigroup of the
minimal deterministic finite automaton accepting the language, that is, to the
semigroup generated by transformations induced by non-empty words on the set of
states of the automaton. In this paper we search for the largest syntactic
semigroup of a star-free language having $n$ left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
$n$ states.
We introduce two new aperiodic transition semigroups. The first is generated
by transformations that change only one state; we call such transformations and
resulting semigroups unitary. In particular, we study complete unitary
semigroups which have a special structure, and we show that each maximal
unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary
semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by
transformations that map a non-empty subset of states to a single state; we
call such transformations and semigroups semiconstant. In particular, we
examine semiconstant tree semigroups which have a structure based on full
binary trees. The semiconstant tree semigroups are at present the best
candidates for largest aperiodic semigroups.
We also prove that $2^n-1$ is an upper bound on the state complexity of
reversal of star-free languages, and resolve an open problem about a special
case of state complexity of concatenation of star-free languages.
[Show abstract][Hide abstract] ABSTRACT: A right ideal is a language L over an alphabet A that satisfies L = LA*. We
show that there exists a stream (sequence) (R_n : n \ge 3) of regular right
ideal languages, where R_n has n left quotients and is most complex under the
following measures of complexity: the state complexities of the left quotients,
the number of atoms (intersections of complemented and uncomplemented left
quotients), the state complexities of the atoms, the size of the syntactic
semigroup, the state complexities of the operations of reversal, star, and
product, and the state complexities of all binary boolean operations. In that
sense, this stream of right ideals is a universal witness.
[Show abstract][Hide abstract] ABSTRACT: The quotient complexity of a regular language L is the number of left
quotients of L, which is the same as the state complexity of L. Suppose that L
and L' are binary regular languages with quotient complexities m and n, and
that the transition semigroups of the minimal deterministic automata accepting
L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively.
Denote by o any binary boolean operation that is not a constant and not a
function of one argument only. For m,n >= 2 with (m,n) not in
{(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn
if and only either (a) m is not equal to n or (b) m=n and the bases (ordered
pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in
{(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In
proving these results we generalize the notion of uniform minimality to direct
products of automata. We also establish a non-trivial connection between
complexity of boolean operations and group theory.
[Show abstract][Hide abstract] ABSTRACT: We explore the relationship between the transition semigroup of the minimal
deterministic finite automaton (DFA) of a regular language, and the
state/quotient complexities of the language's atoms. The atoms of a language
are non-empty intersections of complemented and uncomplemented quotients of the
language. Tight upper bounds on the number of atoms of a language and on the
quotient complexities of atoms are known. We introduce a new class of regular
languages called the \emph{maximally atomic languages} -- languages with the
maximal number of atoms, for which every atom has the maximal possible quotient
complexity. We establish several relationships between transition semigroups
and atoms; in particular, we give necessary and sufficient conditions a
transition semigroup must meet for the associated language to be maximally
atomic. Our methods of proof give new insights into the structures of minimal
DFAs of atoms.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We examine the NFA minimization problem in terms of atomic NFA's, that is,
NFA's in which the right language of every state is a union of atoms, where the
atoms of a regular language are non-empty intersections of complemented and
uncomplemented left quotients of the language. We characterize all reduced
atomic NFA's of a given language, that is, those NFA's that have no equivalent
states. Using atomic NFA's, we formalize Sengoku's approach to NFA minimization
and prove that his method fails to find all minimal NFA's. We also formulate
the Kameda-Weiner NFA minimization in terms of quotients and atoms.
[Show abstract][Hide abstract] ABSTRACT: The syntactic complexity of a regular language is the cardinality of its
syntactic semigroup. The syntactic complexity of a subclass of the class of
regular languages is the maximal syntactic complexity of languages in that
class, taken as a function of the state complexity n of these languages. We
study the syntactic complexity of R- and J-trivial regular languages, and prove
that n! and floor of [e(n-1)!] are tight upper bounds for these languages,
respectively. We also prove that 2^{n-1} is the tight upper bound on the state
complexity of reversal of J-trivial regular languages.
International Journal of Foundations of Computer Science 08/2012; 25(07). DOI:10.1007/978-3-642-39310-5_16 · 0.30 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Regular languages that are most complex under common complexity measures are studied. In particular, certain ternary languages Un(a,b,c), n ≥ 3, over the alphabet {a,b,c} are examined. It is proved that the state complexity bounds that hold for arbitrary regular languages are also met by the languages Un(a,b,c) for union, intersection, difference, symmetric difference, product (concatenation) and star. Maximal bounds are also met by Un(a,b,c) for the number of atoms, the quotient complexity of atoms, the size of the syntactic semigroup, reversal, and 22 combined operations, 5 of which require slightly modified versions. The language Un(a,b,c,d) is an extension of Un(a,b,c), obtained by adding an identity input to the minimal DFA of Un(a,b,c). The witness Un(a,b,c,d) and its modified versions work for 14 more combined operations. Thus Un(a,b,c) and Un(a,b,c,d) appear to be universal witnesses for alphabets of size 3 and 4, respectively.
Proceedings of the 17th international conference on Implementation and Application of Automata; 07/2012
[Show abstract][Hide abstract] ABSTRACT: We study the state complexity of boolean operations and product
(concatenation, catenation) combined with star. We derive tight upper bounds
for the symmetric differences and differences of two languages, one or both of
which are starred, and for the product of two starred languages. We prove that
the previously discovered bounds for the union and the intersection of
languages with one or two starred arguments, for the product of two languages
one of which is starred, and for the star of the product of two languages can
all be met by the recently introduced universal witnesses and their variants.
[Show abstract][Hide abstract] ABSTRACT: We study the state complexity of boolean operations, concatenation and star
with one or two of the argument languages reversed. We derive tight upper
bounds for the symmetric differences and differences of such languages. We
prove that the previously discovered bounds for union, intersection,
concatenation and star of such languages can all be met by the recently
introduced universal witnesses and their variants.
[Show abstract][Hide abstract] ABSTRACT: We study the syntactic complexity of finite/cofinite, definite and reverse
definite languages. The syntactic complexity of a class of languages is defined
as the maximal size of syntactic semigroups of languages from the class, taken
as a function of the state complexity n of the languages. We prove that (n-1)!
is a tight upper bound for finite/cofinite languages and that it can be reached
only if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We prove
that the bound is also (n-1)! for reverse definite languages, but the minimal
alphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)!\rfloor is a
lower bound on the syntactic complexity of definite languages, and conjecture
that this is also an upper bound, and that the alphabet size required to meet
this bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove the
conjecture for n\le 4.
[Show abstract][Hide abstract] ABSTRACT: An atom of a regular language L with n (left) quotients is a non-empty
intersection of uncomplemented or complemented quotients of L, where each of
the n quotients appears in a term of the intersection. The quotient complexity
of L, which is the same as the state complexity of L, is the number of
quotients of L. We prove that, for any language L with quotient complexity n,
the quotient complexity of any atom of L with r complemented quotients has an
upper bound of 2^n-1 if r=0 or r=n, and 1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r}
C_{h}^{n} \cdot C_{k}^{h} otherwise, where C_j^i is the binomial coefficient.
For each n\ge 1, we exhibit a language whose atoms meet these bounds.
[Show abstract][Hide abstract] ABSTRACT: A ternary algebra is a De Morgan algebra (that is, a distributive lattice with 0 and 1 and a complement operation that satisfies De Morgan's laws) with an additional constant Φ satisfying , , and . We provide a characterization of finite ternary algebras in terms of "subset-pair algebras," whose elements are pairs (X, Y) of subsets of a given base set ℰ, which have the property X ∪ Y = ℰ, and whose operations are based on common set operations.
International Journal of Algebra and Computation 11/2011; 07(06). DOI:10.1142/S0218196797000319 · 0.51 Impact Factor