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A non-ambiguous language factorization problem.
Abstract
Given languages Z,L⊆Σ * , we say that Z is L-decomposable (resp., finitely L-decomposable) if there exists a non-trivial couple (A,B) of languages (resp., finite languages), such that Z=B∪AL and the operations are non-ambiguous. We prove that it is decidable whether Z is L-decomposable and whether Z is finitely L-decomposable, in the case Z and L are regular languages, and we provide a set of minimal decompositions (A,B) in the affirmative case. In the particular case Z=L, Z is L-decomposable if there exist languages A,B⊆Σ * , A≠∅, {1}, such that L=A * B with non-ambiguous operations. The result in this case has an important application, in which it allows to decide whether given a finite language S⊆Σ * , there exist finite languages C, P such that SC * P=Σ * and the operations are non-ambiguous, also providing a solution (C,P) in the affirmative case. This problem is strictly related to the factorization conjecture on codes stated by Schützenberger in 60ths and still open. We also show how to construct an infinite family of factorizing codes starting from one of them.
... Algorithms exist for some sub-classes: prefix codes, suffix codes, biprefix codes and n-codes, with n ≤ 3 (see [17] and references inside). Further, starting from a factorizing code C one can construct an infinite family of factorizing codes C , applying composition ( [6]) or substitution, introduced in [4,5]. Another way of constructing factorizing codes starting from a related class is given in [19]. ...
... On the other hand, by means of the substitution operation, if S appears in some equation S C * P = A * then an infinite family of couples (C , P ) can be constructed s.t. S C * P = A * ( [4,5]). ...
... Firstly in [4,5] it is proved that it is decidable whether a finite language S is strong factorizing and a construction for related couples (C, P ) is given, if any. We will say that a language S is right-context-closed if s = s t with s, s ∈ S implies t ∈ S. For example, any suffix-closed language is right-context-closed. ...
Schützenberger Conjecture claims that any finite maximal code C is factorizing, i.e. SC*P = A* in a non-ambiguous way, for some S, P. Let us suppose that Schützenberger Conjecture holds. Two problems arise: the construction of all (S, P) and the construction of C starting from (S, P). Regarding the first problem we consider two families of possible languages S: S prefix-closed and S s.t. S {1} is a code. For the second problem we present a method of constructing C from (S, P), that is relied on the construction of right- and left-factors of a language. Results are based on a combinatorial characterization
of right- and left-factorizing languages.
... A preliminary and partial version of this paper is contained in [1]. ...
... The decidability of this property follows from the decidability of inclusion between regular languages. Proposition 6 solves main problem (1). Note that the decomposition (u; Z \ uL) provided in the proof of Proposition 6, is often not a ÿnite decomposition. ...
... This is the problem more extensively considered in the next sections. Also note that in Section 7 we will present another way of solving main problem (1), that yet provides special decompositions. ...
Given languages Z,L⊆Σ∗,Z is L-decomposable (finitely L-decomposable, resp.) if there exists a non-trivial pair of languages (finite languages, resp.) (A,B), such that Z=AL+B and the operations are non-ambiguous. We show that it is decidable whether Z is L-decomposable and whether Z is finitely L-decomposable, in the case Z and L are regular languages. The result in the case Z=L allows one to decide whether, given a finite language S⊆Σ∗, there exist finite languages C,P such that SC∗P=Σ∗ with non-ambiguous operations. This problem is related to Schützenberger's Factorization Conjecture on codes. We also construct an infinite family of factorizing codes.
... Nghiên cứu các tính chất không nhập nhằng liên quan đến sự phân tích mã là bài toán cơ bản của lý thuyết mã. Các tính chất đại số của mã dựa trên tích không nhập nhằng được nghiên cứu sâu sắc bởi Schützenberger (1955), Gilbert and Moore (1959) và các tác giả khác (xem [1][2][3][4][5][6]). Gần đây, nghiên cứu lý thuyết mã có xu hướng đưa vào các yếu tố điều khiển, đa trị, nhập nhằng để mở rộng khái niệm tích, từ đó xây dựng những lớp mã mới. ...
The classification of languages based on unambiguous product of words and codes contains a gap. Our work aims at investigations to fill up thegap. The classes of k-unambiguous languages is considered as extensions of codes, in which a code is kunambiguous for all k ≥ 0 , the unambiguous product can be used to define k-unambiguous languages with k ≤ 2. Given a regular language X, its unambiguous value k can be determined by an O(n2) time complexity algorithm, where n is the finite index of the syntactic congruence of X. The k-unambiguous languages with kis large enough, can be used in information encryption, and can provide us an encryption schemawith high enough security since their ambiguous characteristics.
... Let us mention that under the hypotheses of Proposition 5.5 when X and Y are factorizing codes, then Z is also a factorizing code [1]. This, trivially, implies that Z is commutatively preÿx. ...
A Bernoulli set is a set X of words over a finite alphabet A such that for any positive Bernoulli distribution π in A∗ one has that π(X)=1. In the case of a two-letter alphabet A={a,b} a characterization of finite Bernoulli sets is given in terms of the function xi,j counting the number of words of X having i occurrences of the letter a and j occurrences of the letter b. Moreover, we also derive a necessary and sufficient condition on the distribution xi,j which characterizes Bernoulli sets which are commutatively equivalent to prefix codes.
In this paper we mainly deal with factorizing codes C over A, i.e., codes verifying the famous still open factorization conjecture formulated by Schützenberger. Suppose A = a,b and denote a
n the power of a in C. We show how we can construct C starting with factorizing codes C∼ with a
n ∈ C∼ and n n, under the hypothesis that all words a
i
waj in C, with w ∈ bA*b ∪b, satisfy i; j. The operation involved, already introduced in [1], is also used to show that all maximal codes C = P(A - 1)S + 1 with P; S ∈ Z〈A〉 and P or S in Z〈a〉 can be constructed by means of this operation starting from prefix and sufix codes. Inspired by another early Schützenberger
conjecture, we propose here an open problem related to the results obtained and to the operation introduced in [1] and considered in this paper.
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