5 research outputs found
Improved bounds for testing Dyck languages
In this paper we consider the problem of deciding membership in Dyck
languages, a fundamental family of context-free languages, comprised of
well-balanced strings of parentheses. In this problem we are given a string of
length in the alphabet of parentheses of types and must decide if it is
well-balanced. We consider this problem in the property testing setting, where
one would like to make the decision while querying as few characters of the
input as possible.
Property testing of strings for Dyck language membership for , with a
number of queries independent of the input size , was provided in [Alon,
Krivelevich, Newman and Szegedy, SICOMP 2001]. Property testing of strings for
Dyck language membership for was first investigated in [Parnas, Ron
and Rubinfeld, RSA 2003]. They showed an upper bound and a lower bound for
distinguishing strings belonging to the language from strings that are far (in
terms of the Hamming distance) from the language, which are respectively (up to
polylogarithmic factors) the power and the power of the input size
.
Here we improve the power of in both bounds. For the upper bound, we
introduce a recursion technique, that together with a refinement of the methods
in the original work provides a test for any power of larger than .
For the lower bound, we introduce a new problem called Truestring Equivalence,
which is easily reducible to the -type Dyck language property testing
problem. For this new problem, we show a lower bound of to the power of
A generalized mixed type of quartic, cubic, quadratic and additive functional equation
We determine the general solution of the functional equation f(x+ky)+f(x−ky) = g(x+y)+g(x−y)+
+h(x)+h˜(y) for fixed integers k with k 6= 0, ±1 without assuming any regularity condition on the unknown
functions f, g, h, h˜. The method used for solving these functional equations is elementary but exploits an
important result due to Hosszu. The solution of this functional equation can also be determined in certain type ´
of groups using two important results due to SzekelyhidiВизначено загальний розв’язок функцiонального рiвняння f(x + ky) + f(x − ky) = g(x + y) +
+ g(x − y) + h(x) + h˜(y) для фiксованих цiлих k при k 6= 0, ±1 без припущення наявностi будь-якої
умови регулярностi для невiдомих функцiй f, g, h, h˜. Метод, що використано для розв’язку цих
функцiональних рiвнянь, елементарний, але базується на важливому результатi Хозу. Розв’язок цього
функцiонального рiвняння може бути визначений у певному типi груп з використанням двох важливих
результатiв Чекелiхiдi