Finite Maximal Codes Research Articles

Finite maximal codes and triangle conjecture

In this paper, we give a property of the structure of finite maximal codes. By using this property, we obtain some results which are closely related to the results on the triangle conjecture proposed by Perrin and Schützenberger in 1981.

Maximal Bifix Codes of Degree 3

The construction of thin maximal bifix codes of degree 1 or 2 is clear and simple. In this paper, a class of thin maximal bifix codes of degree 3 which contains all finite maximal bifix codes of degree 3 is investigated. The construction of such codes is determined. It is well known that for any positive integers d and k, there are finitely many finite maximal bifix codes of degree d over a k-letter alphabet. But the enumeration problem is unsolved in general. In this paper, we show that for any positive integer k, there is a bijection from the set of all finite maximal bifix codes of degree 3 over a k-letter alphabet onto the set of all directed acyclic graphs with k vertices, and then, the enumeration problem of finite maximal bifix codes of degree 3 is solved.

A note on the factorization conjecture

We give partial results on the factorization conjecture on codes proposed by Schutzenberger. We consider finite maximal codes C over the alphabet A = {a, b} with C \cap a^* = a^p, for a prime number p. Let P, S in Z <A>, with S = S_0 + S_1, supp(S_0) \subset a^* and supp(S_1) \subset a^*b supp(S_0). We prove that if (P,S) is a factorization for C then (P,S) is positive, that is P,S have coefficients 0,1, and we characterize the structure of these codes. As a consequence, we prove that if C is a finite maximal code such that each word in C has at most 4 occurrences of b's and a^p is in C, then each factorization for C is a positive factorization. We also discuss the structure of these codes. The obtained results show once again relations between (positive) factorizations and factorizations of cyclic groups.

Permutation Cellular Automata

Permutation cellular automata are cellular automata defined by using finite maximal prefix codes. The overall dynamics of onesided and twosided permutation cellular automata is studied. For some classes of permutation cellular automata including the class of those defined by using finite maximal bifix codes, the overall dynamics is completely or partially described in terms of the codes used to define them.

Bifix codes and Sturmian words

We prove new results concerning the relation between bifix codes, episturmian words and subgroups of free groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set F of words (F-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let F be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an F-maximal bifix code of degree d behaves just as the set of words of F of length d. An F-maximal bifix code of degree d in a Sturmian set of words on an alphabet with k letters has (k−1)d+1 elements. This generalizes the fact that a Sturmian set contains (k−1)d+1 words of length d. Moreover, given an infinite word x, if there is a finite maximal bifix code X of degree d such that x has at most d factors of length d in X, then x is ultimately periodic. Our main result states that any F-maximal bifix code of degree d on the alphabet A is the basis of a subgroup of index d of the free group on A.

FINITE COMPLETIONS VIA FACTORIZING CODES

Several results relate finite maximal codes to factorizations of cyclic groups. In the case of factorizing codes C, i.e. finite maximal codes which satisfy the still open Schützenberger's factorization conjecture, special factorizations, discovered by Hajós, intervene. In particular, given a two-letter alphabet {a,b}, it is known that the set C1 = C ∩ a*ba* satisfies a structural property defined by means of the Hajós factorizations. Conversely, it is not true that a set satisfying this structural property can be embedded in a factorizing code and some partial results are known on the problem of finding additional hypotheses that guarantee the existence of such embedding. Let C be a factorizing code. Inspired by the recursive construction of the Hajós factorizations and starting with a special equation associated with C1 = C ∩ a*ba*, we define a family [Formula: see text] of subsets of a*ba*, each of them still satisfying the above-mentioned structural property. We prove that for each set [Formula: see text], there exists a factorizing code C with C1 = C ∩ a*ba* and as a consequence C1 is a code. C is obtained starting with prefix/suffix codes and by using two types of operations on codes — composition and substitution. We extend all these results to alphabets of size greater than two. We conjecture that for each factorizing code C, we have [Formula: see text]. We also give a method of finding solutions to the above-mentioned equation associated with C1 and we conjecture that this method constructs all these solutions.

On factorizing codes: Structural properties and related decision problems

The algebraic theory of variable-length codes was initiated by Schützenberger in the 1950s. Almost all the subsequently stated results in this theory are constructive and therefore lead to algorithms. However, there are some basic problems that are still open. For instance, we still do not know how to decide whether a finite code can be embedded in a finite maximal one. We answer this question under additional hypotheses. Precisely, let A be a finite alphabet with a ∈ A , let X ⊆ a ∗ ( A ∖ a ) a ∗ . Let n ∈ N , let Ω ( n ) be the number of factors in the prime factorization of n. We give an algorithm to decide whether there exists n, with Ω ( n ) ⩽ 2 , and a finite maximal code C which is also factorizing such that X ∪ a n ⊆ C . We recall that a factorizing code is a finite maximal code which satisfies the factorization conjecture, proposed by Schützenberger. The above-mentioned statement is a consequence of another result proved in this paper. Namely, given a factorizing code C, it is known that the words in C 1 = C ∩ a ∗ ( A ∖ a ) a ∗ satisfy a property P defined by using factorizations of cyclic groups. In this paper we give an algorithm to decide whether a set X ⊆ a ∗ ( A ∖ a ) a ∗ can be embedded in a set C 1 satisfying P . Furthermore, we prove that, conversely, for each set C 1 satisfying P , under additional hypotheses, there exists a factorizing code C such that C 1 = C ∩ a ∗ ( A ∖ a ) a ∗ and, as a consequence, C 1 is a code. In this case, C can be constructed starting with prefix/suffix codes and by using two types of operations on codes ( composition and substitution). The additional required hypotheses concern the structure of the factorizations involved and are always satisfied when, for each b ∈ A ∖ a , we have | C 1 ∩ a ∗ b a ∗ | = n , with Ω ( n ) ⩽ 2 . In addition, we prove that there exist sets C 1 which satisfy P and which are not codes.

Some decompositions of Bernoulli sets and codes

A decomposition of a set X of words over a d-letter alphabet A = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd of subsets of A* such that the sets Xi, i = 1,...,d, are pairwise disjoint, their union is X, and for all i, 1 ≤ i ≤ d, Xi ~ aiYi, where ~ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.

Finite Completion of comma-free codes Part 2

This paper is a sequel to an earlier paper of the present author, in which it was proved that every finite comma-free code is embedded into a so-called (finite) canonical comma-free code. In this paper, it is proved that every (finite) canonical comma-free code is embedded into a finite maximal comma-free code, which thus achieves the conclusion that every finite comma-free code has finite completions.

Finite Completion of comma-free codes Part 1

This paper is the first step in the solution of the problem of finite completion of comma-free codes. We show that every finite comma-free code is included in a finite comma-free code of particular kind, which we called, for lack of a better term, canonical comma-free code. Certainly, finite maximal comma-free codes are always canonical. The final step of the solution which consists in proving further that every canonical comma-free code is completed to a finite maximal comma-free code, is intended to be published in a forthcoming paper.

Product of Finite Maximal P-Codes. This paper was supported in part by HK UGC grants 9040596, 9040511 and City U Strategic Grants 7001189, 7001060, and by the Natural Science Foundation of China (project No. 60073056) and the Guangdong Provincial Natural Science Foundation (project No. 001174)

Several properties of the products of finite maximal prefix, maximal biprefix, semaphore, synchronous, maximal infix and maximal outfix codes are discussed respectively. We show that, for two nonempty subsets X and Y of A * such that the product XY being thin, if XY is a maximal biprefix code, then X and Y are maximal biprefix codes. Also, it is shown that, for two finite nonempty subsets X and Y of A * such that the product XY being unambiguous, if XY is a semaphore code then X and Y are semaphore codes. Finally, two open problems to the product of finite semaphore and maximal infix codes are presented.

On the lattice of prefix codes

The natural correspondence between prefix codes and trees is explored, generalizing the results obtained in Giammarresi et al. (Theoret. Comput. Sci. 205 (1998) 1459) for the lattice of finite trees under division and the lattice of finite maximal prefix codes. Joins and meets of prefix codes are studied in this light in connection with such concepts as finiteness, maximality and varieties of rational languages. Decidability results are obtained for several problems involving rational prefix codes, including the solution to the primeness problem.

Locally complete sets and finite decomposable codes

We are interested in the concept of locally complete set: A subset X of the free monoid is locally complete if a code Y⊂A ∗ exists, with Y≠A, X ∗⊂Y ∗ , and such that both the sets X ∗ and Y ∗ have the same sets of factors. Our contribution is based on the three following results: • A characterization of local completeness for very thin sets in terms of morphic images. • A polynomial time algorithm for deciding whether a finite code is locally complete. • A polynomial time algorithm for deciding whether a finite maximal code is decomposable.

Finite maximal solid codes

Solid codes, a special class of bifix codes, were introduced recently in the connection with formal languages. However, they have a much earlier history and more important motivation in information transmission dating back to the 1960s. In this paper, they are studied as an independent subject in the theory of variable-length codes. It is shown that every finite solid code is contained in a finite maximal one; based on further analysis of the structure of finite maximal solid codes, an algorithm is proposed to construct all of them starting from the most simple and evident ones.

Palindrome words and reverse closed languages

A word which is equal to its mirror image is called a palindrome word. Any language consisting of palindrome words is called a palindrome language. In this paper we investigate properties of palindrome words and languages. We show that there is no dense regular language consisting of palindrome words. A language contains all the mirror images of its elements is called a reverse closed language. Clearly, every palindrome language is reverse closed. We show that whether a given regular or context-free language is reverse closed is decidable. We study certain properties concerning reverse closed finite maximal prefix codes in this paper. Properties of languages that commute with reverse closed languages are investigated too.

SOME PROPERTIES OF MAXIMAL COMMA-FREE CODES

A code $A$ over a finite alphabet $X$ is a comma-free code if $A^2\cap X^+AX^+=\emptyset$, where $X$ is a finite alphabet containing more than one letter. This paper is a study of some algebraic properties of finite maximal comma-free codes. We give several characterizations on two-element comma-free codes and maximal comma-free codes. Let $X_n^m= X^n\cup X^{n+1}\cup \cdots \cup X^m$. We prove that for $n \ge 3$ , a maximal comma-free code in $X^n$ is a maximal comma-free code in the region $X_1^m\cup X^n$, $m &lt; n/2$. We also obtain that for $X = \{a,b\}$, a maximal comma-free code in $X^3$ is a maximal comma-free code; a maximal comma-free code in $X^4$ is a maximal comma-free code in $X_1^4$; for every $n\ge 4$, there is a maximal comma-free code in $X^n$ which is not a maximal comma-free code.

A note on codes having no finite completions

We search for codes having no finite completions containing a n , for n≤4. We show that such codes cannot be found among n-bayonet codes, namely, every n-bayonet code, n≤4, is shown to be completed to a finite maximal code. We give examples of finitely incompletable codes containing a n , for n≤4, by a more detailed analysis of the factorization technique introduced in [6].

Synchronization and decomposability for a family of codes: Part 2

This paper is the second part of a work dealing with two Schützenberger's conjectures on variable-length codes. It contains the detailed proofs of the characterization that we obtained for the degree and the decomposability of finite maximal codes over the alphabet { a, b} with at most 3 occurrences of the letter b in each word.

A partial result about the factorization conjecture for finite variable-length codes

We construct a family of finite maximal codes over the alphabet { a,b} which verify the factorization conjecture on codes proposed by Schützenberger. This family contains any finite maximal code with at most three occurences of the letter b by word.

SYNCHRONIZATION AND DECOMPOSABILITY FOR A FAMILY OF CODES

In the framework of two Schützenberger’s conjectures on codes, we characterize the degree of finite maximal codes over the alphabet {a, b} with at most 3 occurrences of the letter b in each word; it is strongly related to the number of factorizations of the code. We also describe all the indecomposable codes inside this family.