Maximal Prefix Codes Research Articles

State complexity of binary coded regular languages

For the given non-unary input alphabet Σ, a maximal prefix code h mapping strings over Σ to binary strings, and an optimal deterministic finite automaton (DFA) A with n states recognizing a language L over Σ, we consider the problem of how many states we need for an automaton A′ that decides membership in h(L), the binary coded version of L. Namely, A′ accepts binary inputs belonging to h(L) and rejects binary inputs belonging to h(LC), where LC is the complement of L. The outcome on inputs that are not valid binary codes for any string in Σ⁎ can be arbitrary: A′ may accept, reject, or halt in a “don't care” state. We show that every optimal deterministic don't care finite automaton (dcDFA) A′ solving this promise problem uses at most (‖Σ‖−1)⋅n states but at least n states. In addition, we have established a kind of pseudo-isomorphism between such DFAs and dcDFAs. We also show that, for each non-unary input alphabet Σ, there exists a maximal binary prefix code h such that, for each n≥2 and for each N in range from n to (‖Σ‖−1)⋅n, there exists a language L over Σ such that the optimal DFA recognizing L uses exactly n states and every optimal dcDFA for solving the above promise problem uses exactly N states. Thus, we have the complete state hierarchy for deciding membership in the binary coded version of L, with no magic numbers in between the lower and upper bounds.

On the decomposition of prefix codes

In this paper we focus on the decomposition of rational and maximal prefix codes. We present an effective procedure that allows us to decide whether such a code is decomposable. In this case, the procedure also produces the factors of some of its decompositions. We also give partial results on the problem of deciding whether a rational maximal prefix code decomposes over a finite prefix code.

Structure and properties of strong prefix codes of pictures

A setX⊆ Σ** of pictures is a code if every picture over Σ is tilable in at most one way with pictures inX. The definition ofstrong prefix codeis introduced. The family of finite strong prefix codes is decidable and it has a polynomial time decoding algorithm. Maximality for finite strong prefix codes is also studied and related to the notion of completeness. We prove that any finite strong prefix code can be embedded in a unique maximal strong prefix code that has minimal size and cardinality. A complete characterization of the structure of maximal finite strong prefix codes completes the paper.

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.

Some property-preserving homomorphisms

This paper aims to Investigate properties of homomorphisms which preserve some languages. We study the characteristic of maximal codes, maximal hypercodes, maximal prefix codes, primitive words, noncounting languages and left-noncounting languages.

On the size of transducers for bidirectional decoding of prefix codes

In a previous paper [L. Giambruno and S. Mantaci, Theoret. Comput. Sci. 411 (2010) 1785–1792] a bideterministic transducer is defined for the bidirectional deciphering of words by the method introduced by Girod [ IEEE Commun. Lett. 3 (1999) 245–247]. Such a method is defined using prefix codes. Moreover a coding method, inspired by the Girod’s one, is introduced, and a transducer that allows both right-to-left and left-to-right decoding by this method is defined. It is proved also that this transducer is minimal. Here we consider the number of states of such a transducer, related to some features of the considered prefix code X . We find some bounds of such a number of states in relation with different notions of “size” of X . In particular, we give an exact formula for the number of states of transducers associated to maximal prefix codes. We moreover consider two special cases of codes: maximal uniform codes and a class of codes, that we name string-codes. We show that they represent, for maximal codes, the extreme cases with regard to the number of states in terms of different sizes. Moreover we prove that prefix codes corresponding to isomorphic trees have transducers that are isomorphic as unlabeled graphs.

Languages and P0L schemes

Many systems involve substitutions between some sets of elements. The 0L system is a known technique which can help us to investigate properties of substitutions systematically. The aim of this paper is to establish some properties of the P0L schemes which preserve some types of properties of languages. Characterizations of pure-language-preserving, dense-preserving and palindrome-preserving P0L schemes are proposed. s-Injective, primitivity preserving, d-primitivity preserving, prefix code preserving and maximal prefix code preserving substitutions are also studied. Properties of dense-generating 0L schemes are also investigated.

Farey codes and languages

A word w is central if it has a minimal period π w such that | w | − π w + 2 is a period of w coprime with π w . Central words are in a two-letter alphabet A and play an essential role in combinatorics of Sturmian words. We study some new structural properties of the set PER of central words which are based on the existence of two basic bijections ψ and φ of A ∗ in PER , related to two different methods of generation, and two natural bijections θ (the ratio of periods) and η (the rate) of PER in the set of all positive irreducible fractions. In this paper we are mainly interested in sets of central words which are codes. In particular, for any positive integer n we consider the set Δ n of all central words w such that the period | w | − π w + 2 is not larger than n + 1 and | w | ≥ n . In a previous paper we proved that for each n , Δ n is a maximal prefix central code called the Farey code of order n since it is naturally associated with the Farey series of order n + 1 . New structural properties of Farey codes are given as well as of their pre-codes P n . In particular one has PER = ∪ n ≥ 0 Δ n . Moreover, for each n two languages of central words L n and M n are introduced. The language L n (resp., M n ) is called the Farey (resp., dual Farey) language of order n . The name is motivated by the fact that L n and M n give faithful representations of the set of Farey’s fractions of order n . Finally, two total orderings of PER are naturally defined in terms of maps θ and η . The notion of order of a central word relative to a language of central words is given and some general results are proved. In the case of Farey’s languages one has that the Riemann hypothesis on the Zeta function can be restated in terms of a combinatorial property of these languages.

Codes of central Sturmian words

A central Sturmian word, or simply central word, is a word having two coprime periods p and q and length equal to p + q - 2 . We consider sets of central words which are codes. Some general properties of central codes are shown. In particular, we prove that a non-trivial maximal central code is infinite. Moreover, it is not maximal as a code. A central code is called prefix central code if it is a prefix code. We prove that a central code is a prefix (resp., maximal prefix) central code if and only if the set of its ‘generating words’ is a prefix (resp., maximal prefix) code. A suitable arithmetization of the theory is obtained by considering the bijection θ , called ratio of periods, from the set of all central words to the set of all positive irreducible fractions defined as: θ ( ε ) = 1 / 1 and θ ( w ) = p / q (resp., θ ( w ) = q / p ) if w begins with the letter a (resp., letter b), p is the minimal period of w, and q = | w | - p + 2 . We prove that a central code X is prefix (resp., maximal prefix) if and only if θ ( X ) is an independent (resp., independent and full) set of fractions. Finally, two interesting classes of prefix central codes are considered. One is the class of Farey codes which are naturally associated with the Farey series; we prove that Farey codes are maximal prefix central codes. The other is given by uniform central codes. A noteworthy property related to the number of occurrences of the letter a in the words of a maximal uniform central code is proved.

Improving Chinese Storing Text Retrieval Systems' Security via a Novel Maximal Prefix Coding

We have seen that Huffman coding has been widely used in data, image, and video compression. In this paper novel maximal prefix coding is introduced. Relationship between the Huffman coding and the optimal maximal prefix coding are discussed. We show that all Huffman coding schemes are optimal maximal prefix coding schemes and that conversely the optimal maximal prefix coding schemes need not be the Huffman coding schemes. Moreover, it is proven that, for any maximal prefix code C, there exists an information source I = (∑ P) such that C is exactly a Huffman code for I. Therefore, it is essential to show that the class of Huffman codes is coincident with one of the maximal prefix codes. A case study of data compression is also given. Comparing the Huffman coding, the maximal prefix coding is used not only for statistical modeling but also for dictionary methods. It is also good at applying to a large information retrieval system and improving its security.

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.

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.

Insertion and deletion closure of languages

To a given language L, we associate the sets ins(L) (resp. del(L)) consisting of words with the following property: their insertion into (deletion from) any word of L yields words which also belong to L. Properties of these sets and of languages which are insertion (deletion) closed are obtained. Of special interest is the case when the language is ins-closed (del-closed) and finitely generated. Then the minimal set of generators turns out to be a maximal prefix and suffix code, which is regular if L is regular. In addition, we study the insertion-base of a language and languages which have the property that both they and their complements are ins-closed.

The syntactic monoid of the semigroup generated by a maximal prefix code

The syntactic monoid of the semigroup generated by a maximal prefix code