1,107 research outputs found
Quotient Complexities of Atoms in Regular Ideal Languages
A (left) quotient of a language by a word is the language
. The quotient complexity of a regular language
is the number of quotients of ; it is equal to the state complexity of ,
which is the number of states in a minimal deterministic finite automaton
accepting . An atom of 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 . A right (respectively, left
and two-sided) ideal is a language over an alphabet that satisfies
(respectively, and ). We
compute the maximal number of atoms and the maximal quotient complexities of
atoms of right, left and two-sided regular ideals.Comment: 17 pages, 4 figures, two table
Maximally Atomic Languages
The atoms of a regular 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 maximally atomic
languages, consisting of all languages meeting these bounds. We prove the
following result: If L is a regular language of quotient complexity n and G is
the subgroup of permutations in the transition semigroup T of the minimal DFA
of L, then L is maximally atomic if and only if G is transitive on k-subsets of
1,...,n for 0 <= k <= n and T contains a transformation of rank n-1.Comment: In Proceedings AFL 2014, arXiv:1405.527
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
Most Complex Regular Ideal Languages
A right ideal (left ideal, two-sided ideal) is a non-empty language over an alphabet such that (, ). Let for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences () of right, left, and two-sided regular ideals, where has quotient complexity (state complexity) , such that is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of , the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations. In that sense, these ideals are "most complex" languages in their classes, or "universal witnesses" to the complexity of the various operations.Natural Sciences and Engineering Research Council of Canada [OGP0000871
Asymptotic approximation for the quotient complexities of atoms
In a series of papers, Brzozowski together with Tamm, Davies, and Szykuła studied the quotient complexities of atoms of regular languages [6, 7, 3, 4]. The authors obtained precise bounds in terms of binomial sums for the most complex situations in the following five cases: (G): general, (R): right ideals, (L): left ideals, (T): two-sided ideals and (S): suffix-free languages. In each case let κc(n) be the maximal complexity of an atom of a regular language L, where L has complexity n ≥ 2 and belongs to the class C ϵ {G, R, L, T , S}. It is known that κT(n) ≤ κL(n) = κR(n) ≤ κG(n) 3 if and only if κC(n+1)/κC(n) < 3
Most Complex Regular Right-Ideal Languages
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.Comment: 19 pages, 4 figures, 1 tabl
Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages
A language over an alphabet is suffix-convex if, for any words
, whenever and are in , then so is .
Suffix-convex languages include three special cases: left-ideal, suffix-closed,
and suffix-free languages. We examine complexity properties of these three
special classes of suffix-convex regular languages. In particular, we study the
quotient/state complexity of boolean operations, product (concatenation), star,
and reversal on these languages, as well as the size of their syntactic
semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with
arXiv:1605.0669
Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages
We study the state complexity of binary operations on regular languages over
different alphabets. It is known that if and are languages of
state complexities and , respectively, and restricted to the same
alphabet, the state complexity of any binary boolean operation on and
is , and that of product (concatenation) is . In
contrast to this, we show that if and are over different
alphabets, the state complexity of union and symmetric difference is
, that of difference is , that of intersection is , and
that of product is . We also study unrestricted complexity of
binary operations in the classes of regular right, left, and two-sided ideals,
and derive tight upper bounds. The bounds for product of the unrestricted cases
(with the bounds for the restricted cases in parentheses) are as follows: right
ideals (); left ideals ();
two-sided ideals (). The state complexities of boolean operations
on all three types of ideals are the same as those of arbitrary regular
languages, whereas that is not the case if the alphabets of the arguments are
the same. Finally, we update the known results about most complex regular,
right-ideal, left-ideal, and two-sided-ideal languages to include the
unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of
the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3.
The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59,
2017, the issue of selected papers from DCFS 2016. This version corrects the
proof of distinguishability of states in the difference operation on p. 12 in
arXiv:1609.04439v
- …