CC BY
20
On the task of extracting the root from the language
B. F. Melnikov, S. Yu. Korabelshchikova, V. N. Dolgov
Abstract—In this paper, we consider a special case of constructing the inverse morphism for the given finite language: we try to extract the root from this language. For such extracting, we consider some special simple cases.
We shall firstly look at some special class of languages, for which the problem is solved easily. Namely, we describe this class, consider some examples, and give some obtained results of computer experiments. Then we define antiderivative roots and give some propositions that show, how to use antiderivative roots for obtaining any root of the given degree.
Then we give an interesting example, which refutes a possible simple algorithm for this task. Therefore, if there exists a polynomial-time algorithm that solves this problem, then it should be formulated more complicated.
We hope that the further development of the theory described in this paper will provide an opportunity to describe a polynomial-time algorithms for solving the general case of this problem.
Keywords—formal languages, exponential algorithm, polynomial-time algorithm, root extraction.
I. INTRODUCTION
We use terminology, notation, and some results from [1]. However, this paper is not a continuation of that one: here, we only consider one of the problems set there. It would be more correct to say, that this paper is a continuation of the topics of [2], [3], [4], [5].
In this paper, we look at an important case for the hypothesis considered in [1] (we called it "the second hypothesis"): we consider a special case of constructing the inverse morphism for the given finite language, i.e., we try to extract the root from the given finite language. Let us note, that our task is much more complicated than, for example, the simple algorithms of extracting the root from a given word; the last one can be formulated, e.g., by [6, Ch. 1, Ex.4,5].
Thus, we extract the root from the given finite language. We can formulate our problem in the following way. For the given language A, we build such a language B over the given alphabet E, for which
A = Bn. (1)
(All notation associated with the theory of formal languages is standard. We can, for example, assume, that we are applying notation of [6] using the "new" symbol for the empty word, i.e. e.)
We already used the term "a root of the language"; by (1), B can be defied as "a root of the language of n-th
Received December 31, 2018.
Boris F. Melnikov, Russian State Social University (email: bf-melnikov@ yandex.ru).
Svetlana Yu. Korabelshchikova, Northern (Arctic) Federal University named after M. V. Lomonosov (email: [email protected]).
Vasily N. Dolgov, independent researcher (email: [email protected]).
degree". However, this term requires further definitions, we shall consider only some of them in this paper. The main note is that the root, generally speaking, is not the only one (see details below). Therefore, the simple notation VA is used:
• sometimes for the set B such that Bn = A;
• and sometimes for the set of such sets. For this reason, we shall not use this notation.
For such extracting, we consider some special simple cases, and then give an interesting example, which refutes a possible simple algorithm for this task. (This incorrect algorithm can be considered as an analogous to the mentioned algorithm of extracting the root from the given word.) Therefore, if there exists a polynomial-time algorithm that solves this problem, then it should be formulated more complicated. We hope that the further development of the theory described in this paper will provide an opportunity to describe a polynomial-time algorithms for solving this problem and the other problems set in [1].
This paper has the following structure. In Section II, we give some reasons that explain the need to consider our task. In Section III, we consider the formal problem setting and give an example.
In Sections IV and V, we shall look at some special class of languages, for which the problem is solved easily. Namely, in Section IV, we describe this class, consider some examples, and give some obtained results of computer experiments. And in Section V, we define antiderivative roots and give some propositions that show, how to use antiderivative roots for obtaining any root of the given degree.
In Section VI, we consider the general statement of the problem again. We formulate an method for extracting roots; however, as we said before, this method leads to an incorrect algorithm. But we believe that it is to be formulated, because it reflects possible directions for further work. Some of these directions are shortly described in Conclusion (Section VII).
II. Motivation: why do we consider this task?
Some reasons that explain the need to consider this task were already given in our previous papers, for example, in [1], [4], [5], etc. Let us cite some of them, as well as the motivation not met in these papers.
• In some subclasses of the context-free language class the equivalence problem is decidable, unlike the whole class; see [7], [8], [9].
• An example of such a class is the "notorious" class of deterministic context-free languages, the equivalence issue for which was formulated back in the late of 1960s (see, for example, [10, Sect. 4.2]), and then was subsequently resolved. It is not known to the authors of
this paper, that anyone found errors in the proof given in a series of parers of L. Staneviciene (see [11], [12] and some papers cited there). Besides, it is possible, apparently, to bring to the end a not completed proof of V.Meytus, see [13]: that paper does not contain obvious errors. But, as usual in such situations, the author of the proof for the problem of equality of deterministic CF-languages (i.e., equivalence of deterministic MP-automata) "is generally recognized" not a Russian scientist, but G. Senizergues, see [14].1 Thus, we are referring to subclasses that do not coincide with the class of deterministic CF-languages. However, some real programming languages can be described using these subclasses, see some examples in [9]. To describe these subclasses, we often consider pairs of languages satisfying the conditions A = B. (As we said before, the corresponding definition was given in [1], [2] etc.)
Based on the conditions of the relationship A = B, we can formulate some necessary and sufficient conditions for the commutation in the global monoid of the free monoid2 and some of its submonoids, see [3], [5], [15]. In other words, we can formulate some criteria for the fulfillment of equality AB = BA, where A, B c £*. The possible association between the conditions of the relationship A = B and some other algebraic problems can be derived from the results of the papers [16], [17], [18], where w- and 2w-languages are considered. 3 In our problems, such languages are usually generated by infinite sequences of reflections of a point from the sides of a certain polygon (in other words, they are generated by special billiards). These results are associated with some algorithmic problems for monomial algebras (i.e. associative algebras defined by so-called obstruction languages). This relationship follows from the results of the classic works [19], [20], etc. Note that in [1], we could consider the same w-automata instead of usual nondeterministic finite automata, and formulate equivalence conditions for them. The tasks discussed in this paper arise in the graphical description of the CF-languages class and some its subclasses, see [11], [21], [22], [23]. In particular, we obtain them when solving the equivalence problems in these subclasses. 4
And oddly enough, so-called "infinite" case (i.e., when we allow infinite languages A or B in the condition A = B) is less complicated and less interesting than the "finite" one; some results for the infinite case were given in [24]. However, in some tasks, the need to consider infinite languages A and B arises: for example, when we consider the problem of description of conditions of equivalence of morphic images of so-called bracketed
1 Apparently, an even more obvious example is the following. Yu. Med-vedev published a paper in 1956, where, in fact, nondeterministic finite automata were defined and considered. However, the authors of NFAs "are generally recognized" M. Rabin and D. Scott, who described them later, i.e. in 1959, but received the Turing Prize for describing such automata.
2 Sometimes it is called "supermonoid", but that is not quite true.
3 For example, we can define 2w-word a over the given alphabet £ as any mapping of the type a : Z ^ £.
4 Moreover, the development of this graphic approach gives an opportu-
nity to describe (also graphically) languages, having types 1 and 0 in the hierarchy of Chomsky.
languages. (See papers [8], [9] cited before; we should note once again, that in these works we have considered some other variants of using morphisms of infinite languages for description of some subclasses of the CF-languages class.) However, as we noted in [1], the most interesting question is to establish a connection between the issues we are considering and the possible equality P=NP. In the mentioned paper, the possible plan of research of this problem is given, namely, the plan of proof of inequality of these classes.
III. The formal problem setting and an example
Thus, we shall extract here the root of the language; i.e., for the given language A, we build such a language B over the given alphabet E, for which equation (1) holds. About (1), we note the following:
• either n > 2 is set a priori; then, in case of absence of the required language B, we return the negative answer, also in the polynomial time;
• or n is not given; then we have to construct the language B satisfied (1) for maximum possible n > 1.
It is obviously, that two these tasks almost coincide (and do not change the fact of the possibility or impossibility of constructing a polynomial-time algorithm): if there is a solution to the first one, then by considering all the values n from
max lul
ueA
"downto" 1, we get the solution for the second of these tasks.
We also note the following fact. Of course, the task of extracting the root from a given word is quite simple; but still now, some of related questions are considered in the famous monograph [6] as not quite trivial exercises. However, it is unlikely that summarizing the relevant results for the words to the case of the languages5 can give interesting results: it is, in fact, the reformulating the tasks for words in the terms of languages.
In this regard, we also note the following fact from [1]. The algorithm of constructing all the variants of representation of the given word u in the form of concatenation of words of some given finite language A is not polynomial-time. In this regard, let us consider a trivial example
A = {■
on this issue, we have the following facts:
• the size of the problem (depending on the n) is polynomial;
• the number of variants for a representation required in our problem is exponential (also depending on n);
• and, therefore, the number of variants for the consideration (depending on the size of the task, we use the brute force method here) is also exponential.
However, this fact does not prove the non-existing polynomial-time algorithm for the whole problem we are considering, i.e. for the construction of inverse morphism required in our problem.
5 From the algebraic point of view, it is a transition from monoids to supermonoids. (The last term is considered in some sources not quite successful, but we will not pay attention to such algebraic subtleties.)
2
n
u = a
a
Tab. 1. The count of roots
n / k 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 1 2 3 5 9 15 28 50 95 174 337 637 1231 2373 4618 8974 17567
3 1 2 4 7 13 25 49 95 185 365 721 1430 2844 5655 11259 22462 44829
4 1 2 4 8 15 29 57 113 225 447 889 1773 3537 7062 14109 28195 56361
5 1 2 4 8 16 31 61 121 241 481 961 1919 3833 7661 15317 30625 61233
6 1 2 4 8 16 32 63 125 249 497 993 1985 3969 7935 15865 31725 63445
7 1 2 4 8 16 32 64 127 253 505 1009 2017 4033 8065 16129 32255 64505
8 1 2 4 8 16 32 64 128 255 509 1017 2033 4065 8129 16257 32513 65025
9 1 2 4 8 16 32 64 128 256 511 1021 2041 4081 8161 16321 32641 65281
10 1 2 4 8 16 32 64 128 256 512 1023 2045 4089 8177 16353 32705 65409
IV. ABOUT ROOTS OF SPECIAL-TYPE LANGUAGES: SOME SIMPLE EXAMPLES AND COMPUTER EXPERIMENTS
In this section and in the next one, we shall look at some special class of languages, for which the problem is solved easily. Namely, we investigate the question of extracting the root of a given degree from the languages of the type
y where ti,t2 e N, ti < t2.
t1
(2)
Obviously, the root extraction operation is, generally speaking, ambiguous: let us consider an example which is an evolution of one considered in Section III.
Example 1. Let
A = {
a8, a9,
,23 „24
.
Square roots can be considered, among some others, languages
„4 a5 „11 „12
fi
Bi = { a*, a"
, a", a" } ,
B2
we have besides
Bi \ { a6 } , and B3 = Bi \ { ai0 } 5
Bi2 = B22 = B32 = A,
Bi = { a2, a3, a4, a5, a6 }2 = { a1, a2, a3 }4 , and, at the same time, we have no language C, for which
C2 = B2 or C2 = B3.
Let us consider another interesting example. Example 2. Let
{<■0 {<•0
A B
a", ai, a2, a6, a8, a9
a", ai, a3, a6, a8, a9
} , } .
Then
a2 however
b2
a0, ai,
,ai8} \ { a5,ai3} ,
(A u B)2 = { a0,a\...,ai8} = A2.
Let us also note another obvious fact: in order for the root of n-th degree from the language of the type (2) to be extracted, it is necessary and enough to both ti and t2 are divided into n. Let us formulate a simple generalization of the last fact in the form of the following statement.
Proposition 1: Language
U si
ieM
is the root of the n-th degree for language
A = U E (3)
nn 1 ^i^nn 2
if and only if
M c {n1,n1 + 1,... ,n2}, for which the following condition holds:
(vi € N) (n • n1 ^ i ^ n • n2 ^ i = a1 + a2 + • • • + an).
Here, a1, a2, ..., an are some elements of the set M, not necessarily different ones.
Besides, we have the following bijection between the roots of the n-th degree for language A defined before and the roots of the n-th degree for language
B = U Ei :
the set of indices M of the root of language B corresponds the set of indices
L = M u {n1 - 1}
of the root of language A. □
Note that the number of roots of the n-th degree depends on the n and the difference n2 - n1 ; the last value is the difference of boundary values of the set of root indices. The notation A (which, certainly, depends on n, n1, and n2, that we shall not explicitly mark) will be used in the remainder of the paper.
For further, we denote also k = n2 - n1 + 1; this value is the cardinality of the set {n1, n1 + 1,..., n2}. We implemented an algorithm of finding all the possible roots for the given n, t1, and t2. The complexity of the calculation of the obtained implementation is equal to
O(2(fc-4) • n2 • k2);
it allows to find all the roots for k < 30. Table 1 (see before) shows the count of roots for some n and k that our computer program obtained. (For k € { 1,2, 3 }, the values are equal to 1.)
V. DEFINITION AND BASIC PROPERTIES OF ANTIDERIVATIVE ROOTS FROM LANGUAGES
For the future, let S be the set of set of indices for the roots of the n-th degree for the language A defined by (3). Definition 1: The root of the type
U e
ieM
will be called an antiderivative one, if M is minimum on inclusion of the set S.
The cardinality of the set of indices will be called the weight of the root. □
Example 3. For the given language
u
we have the following 5 different square roots; they have sets of indices
{1, 2, 3,4, 5, 6, 7}, {1, 2, 3,4,6, 7}, {1, 2, 4, 5, 6, 7}, {1, 2,4,6, 7}, and {1,2, 3, 5,6,7}.
The two last sets are minimum ones, they corresponds to antiderivative roots with the weights 5 and 6 respectively.
Let us formulate the obvious properties of the antiderivative in the form of following statements.
Proposition 2: Let M g S, besides let M be a set of indices of minimum cardinality. Then the root corresponding to M is an antiderivative one. □
Note that, generally speaking, the assertion reversed to the last one is incorrect: the antiderivative root is not required to have a minimum weight, see Example 3. Proposition 3: Let
u
ieM
be an antiderivative root of n-th degree for the language A (3) with the set of indices M. Let
(«1, «2, .. ., «k)
be the binary characteristic vector (see [25] etc.) for subset M of the set
{ni, ni + 1, . . . , n2}. Then the symmetrical vector
(ak, ak-u..., «1)
also sets the set of indices of (another) antiderivative root.
□
Proposition 4: If the antiderivative root of the given language is the only one, then the characteristic vector of its set of indices is symmetrical. □
Proposition 5: Let
u
iem
be an antiderivative root of n-th degree for the language A (3) with the set of indices M. Then for each set M', such that
M C M' C {n1,n1 + 1,.. .,n2},
the language
u
ieM'
is the root of n-th degree for the same language. □
Proposition 6: If the weight of the antiderivative root is equal to w < k and it is the only, then the common number of the roots for the considered language is equal to 2k-w. Proof. This fact follows from the Proposition 4. □
Thus, as follows from the properties of antiderivative roots, to get all the roots of n-th degree for the language
u
n ni ^i^n n2
there is enough to know all the original roots. This coincides with obtaining their set of indices.
Example 4. Let us construct all the root of 3-rd degree of the language
U E*.
The set of indices should be the subset of the set
{9,10,11,12,13,14},
i.e. k = 6. For n = 3 and k = 6, we have the only antiderivative root, its characteristic vector is (110011). Then we obtain 4 roots, having the following sets of indices:
{9,10,13,14}, {9,10,11,13,14}, {9,10,12,13,14}, and {9,10,11,12,13,14}.
Thus, the task of extracting the root from this type of languages can be reduced to the task of representing some natural numbers from the specified interval [ti, t2] as the sum of n natural terms from the interval [t1/n, t2/n]. Therefore, such a task can be considered as a special case of the knapsack problem. Some possible algorithms for solving this problem for situations we are considering in this paper, were considered by one of the authors in [26]. The building set of antiderivative of n-th degree allows to find all the roots from the languages of the type (2). On the other hand, if the set of all roots of a given language is large enough, then the building its antiderivative roots can be considered as the best way for description of this set.
VI. Why do not simple algorithms work?
Let us return to the general statement of the problem.
When considering our task (as well as some tasks related to it), we can suggest the following simple "algorithm" for its solving; let us note in advance, that this algorithm is incorrect, this can be seen from the title of this section. However, we believe that it is to be formulated, because it reflects possible directions for further work, in particular, to obtaining the correct polynomial-time algorithm for solving the problem we are looking at.
Thus, we will consider the problem of extracting the root of the n-th degree, when the language from which the root is extracted (let it be A c E*) and the value n > 1 are given.
Definition 2: The word u g E* is a potential root if un g A. _
The set of all potential roots will be denoted VA.6 (We shall never omit n.) □
Example 5. Let
A = { abab, abaab, aabab, aabaab } . Then _
= { ab, aab } .
Let us remark, that
2 = A.
Example 6. Let
A = { aa, abab, baba, babbab } .
6 As we said in Introduction, we do not use the simple notation VA.
Then let us denote
B = ^A = { a, ab, ba, bab } ; let us remark, that
(B)2 = A,
because, for example,
B 3 aab, and aab g A. However, there exists B, such that B2 = A, that is
B = { a, bab } ;
let us remark, that B c B. The following simple statement is given without proof. Proposition 7: Let B" = A. Then
B c vA. □
Based on the last proposition, we obtain a trivial algorithm for building any B, such that B" = A: we should consider all the subsets of the set VA. Evidently, this algorithm is not a polynomial-time one.
Besides, we could formulate a simple "algorithm" based on the following hypothesis:
1) we consider a word of VA having the maximum possible length; if there are 2 or more such words, then we choose an arbitrary one; let this word be u;
2) then we consider a word v = u"-1;
3) if vw g A, then we include w in the formed language (let it be C);
4) C has no more words;
5) then C" = A.
However, this hypothesis is false, and we can show this fact in the following example.
Example 7. Let
B = { ab, abb, bba } ,
and let A = B2 be the given language. Then A contains the word abbba. Besides:
1) B = #A;
2) abb is a word of B2-1 = B, such that it has the maximum possible length, i.e. 3;
3) abbba = abb • ba, but ba g B.
We believe, that this example shows the need to build more complex algorithms.
VII. Conclusion. Some directions
FOR FURTHER WORK
Thus, we need to build more complex algorithms. However, of course, all algorithms for this problem should be polynomial-time ones: the trivial exponential algorithm was given in previous section.
(In describing the remaining problems for further solution, we again use the terminology of [1] and other cited works. Namely, we use the terms = and mp+ defined there.)
We hope that in the future, we shall be able to describe polynomial-time algorithms for solving both the problem formulated in this paper and also one of the two following more complex problems:
• for two given languages A and B, such that A = B, but A = B, we have to build language D, such that A, B € mp+ (D);
• for the given finite language A, we have to build language D, such that A € mp+ (D), besides, language D is minimal according to a metrics; the possibilities for its choices for the language D are "natural", some of possible metrics are also discussed in [1].
It is important to note that according to [2], [3], [5], the solution of one of them automatically involves the solution of the other. Moreover, it is easy to show that the description of the polynomial-time algorithm for solving one of them will give a description of a comparable algorithm for solving the other.
Acknowledgment
We express our gratitude for some consultations to A. Ve-revkin (Ulyanovsk State University, Russia).
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Boris Feliksovich MELNIKOV, Professor of Russian State Social University,
http://www.rgsu.net/, email: [email protected], mathnet.ru: personid=27967, elibrary.ru: authorid=15715, scopus.com: authorId=55 954 04 030 0.
Svetlana Yuryevna KORABELSHCHIKOVA, Associated Professor (Docent) of Northern (Arctic) Federal University named after M. V. Lomonosov, Russia,
https://narfu.ru/en/university/, email: [email protected], elibrary.ru: authorid=711810.
Vasily Nikolaevich DOLGOV, Independent Researcher, Russia, email: [email protected], elibrary.ru: authorid=7 00 0 0 0.
