Finite Prefix Codes Research Articles

Algebraic synchronization criterion and computing reset words

We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain new upper bounds for automata with a short word of small rank. The results are applied to make several improvements in the area.In particular, we improve the upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(nlog3n). In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary n-state decoder is at most O(nlog n). We prove the Černý conjecture for n-state automata with a letter of rank ≤6n−63. In another corollary, we show that the probability that the Černý conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and that the expected value of the reset threshold is at most n3/2+o(1).Moreover, all of our bounds are constructible. We present suitable polynomial algorithms for the task of finding a reset word of length within our bounds.

Conjugacy relations of prefix codes

It is shown that, if X and Y are prefix codes and Z is a non-empty language satisfying the condition XZ=ZY, then Z is the union of a non-empty family {Pn}i∈I of pairwise disjoint prefix sets such that XPi=PiY for all i∈I. Consequently, the conjugacy relations of prefix codes are explored and, under the restriction that both of X and Y are prefix codes, the solutions of the conjugacy equation XZ=ZY for languages are determined. Also, the decidability of the conjugacy problem for finite prefix codes is confirmed.

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.

A Generalization of Girod’s Bidirectional Decoding Method to Codes with a Finite Deciphering Delay

Girod’s encoding method has been introduced in order to efficiently decode from both directions messages encoded by using finite prefix codes. In the present paper, we generalize this method to finite codes with a finite deciphering delay. In particular, we show that our decoding algorithm can be realized by a deterministic finite transducer. We also investigate some properties of the underlying unlabeled graph.

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.

PREFIX PICTURE CODES: A DECIDABLE CLASS OF TWO-DIMENSIONAL CODES

A two-dimensional code of pictures is defined as a set X ⊆ Σ** such that any picture over Σ is tilable in at most one way with pictures in X. It is proved that in general it is undecidable whether a finite set of picture is a code. The subclass of prefix codes is introduced and it is proved that it is decidable whether a finite set of pictures is a prefix code. Further a polynomial time decoding algorithm for finite prefix codes is given. Maximality and completeness of finite prefix codes are studied.

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.

On the dynamics of cellular automata induced from a prefix code

In this paper we study some general dynamical properties of the class of one-dimensional permutation cellular automata induced by maximal finite prefix codes defined on the one-sided full shift A N . Then we define families of codes every member of which induces an onto permutation cellular automaton F, and we investigate some properties of the (topological) dynamical system ( A N , F ) such as positive expansiveness, entropy and periodic points. We also define and study a special case of permutation cellular automata, namely, elector automata, which are canonically constructed from codes, and provide classes of cellular automata which attain their limit sets in finite time, and other families without that property.

On syntactic groups

We prove that for any finite prefix code X with n elements, the non special subgroups in the syntactic monoid of X have degree at most n − 1. This implies in particular that the groups in the syntactic monoid of X are all cyclic when X is a prefix code with three elements.

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.

Free submonoids in the monoid of languages

The monoid of language M is the family of languages M = {A ¦0 ≠ ⊆ X + or A = {1}} , where X + is the free semigroup generated by the finite alphabet X containing more than one letter and l is the empty word. This paper is a study of free submonoids in M. We found a free submonoid of M which contains the family of all finite prefix codes.

A three-word code which is not prefix-suffix composed

In this paper, we are concerned with finite p-s-composed codes, that is with codes obtained by composition of finite prefix and suffix codes. We give a method to decompose a finite prefix-suffix composed code in a minimal number of prefix and suffix codes. Using this method, we establish that every prefix-suffix composed n-word code ( n ⩾ 3) can be expressed as the composition of at most 2 n - 3 prefix and suffix codes. We show that for all n, this limit is reached, that is, there exists a p-s-composed n-word code that cannot be expressed as the composition of less than 2 n - 3 prefix and suffix codes. Then we give an example of a three-word code which is not prefix-suffix composed, refuting a conjecture proposed by Restivo et al. in 1989.

Bounds on the variety generated by completely regular syntactic monoids from finite prefix codes

Recent work by Lallement has shown that if the syntactic monoid S of C ∗, where C is a finite prefix code, is completely regular then the components or D -classes of S must form a chain. The purpose of this article is to consider how general this class is within the class of all completely regular semigroups. It is shown that the variety (pseudo-variety) that it generates is a proper subvariety (subpseudo-variety) of the variety (or pseudo-variety) of all completely regular semigroups but does contain the variety of bands and even the variety of local orthogroups.

Equations over finite sets of words and equivalence problems in automata theory

Several observations concerning equations over monoids of finite sets of words are made. Monoids considered are the monoid of finite prefix codes, the monoid of all finite sets and the monoid of all finite multisets, as well as some others. As an application, some known decidability results in automata theory are extended. Finally, several open questions are raised.

Biprefix codes, inverse semigroups and syntactic monoids of injective automata

We prove for injective automata an analogue of Pin's result (1978) that any finite monoid divides the syntactic monoid of P ∗ , for some finite prefix code P. Pin's result for arbitrary automata is strengthened in that we show divides' can be replaced by is a subsemigroup of. We give some conditions sufficient to guarantee that if a syntactic monoid is inverse, then the automaton is injective. A direct proof is given of Reutenaur's result (1979, 1981) that if the language recognized by an injective automaton is a subsemigroup, then it is of the form B ∗ for some biprefix code B.

Unions of groups of small height

A semigroup S is called a union of groups if each of its elements lies in a (maximal) subgroup of S. It is well known that in a union of groups Green’s relation 3 (defined by a/‘b if and only if SaS=%S) is a congruence whose classes are completely simple subsemigroups of S and such that S/g is a semilattice (i.e. satisfies x1=x, xu=yx) [I]. In this paper we present a construction of all unions of groups S having the following properties: (1) S is the syntactic semigroup of a language of the form C+ = Un,O Cn, where C is a finite prefix code; (2) S/f is the two-element semilattice. Using the terminology of Clifford and Preston the semigroups studied here are ideal extensions of a completely simple semigroup by another, or unions of groups of height two. All the completely simple semigroups satisfying condition (1) above have been obtained in [6], using the concept of team tournament, All unions of groups S satisfying (1) and having a non-trivial group of units are constructible using certain factorizations of B,, the group of integers modulo n [4]. The two construction techniques are combined here to give all the unions of groups satisfying (1) and (2) as transition semigroups of what we call a ‘standard amalgamation’ of the automata of two team tournaments (Theorem 3.2). It is likely that the process given here generalizes to a construction of all unions of groups satisfying (1) provided some convenient representation of the two 2-classes case is found. We conjecture that all these semigroups are chains of length n of completely simple semigroups, and that they are in general of group complexity n, in the sense of J. Rhodes [3] (cf. also Tilson’s Chapter 12 in [2]). The complexity conjecture, at least in the case n = 2, can be verified on examples using the results of [ll]. Directly related to these considerations is the problem of describing the variety I of all languages whose syntactic semigroups are unions of groups (see [2], [lo]). In view of the recent results of J.E. Pin [9], the following question arises naturally: Is

On varieties of rational languages and variable-length codes

This article is a continuation of the work of the second author on the connections between the theory of varieties of languages and the theory of codes. We show that every variety of languages closed under concatenation product is described by its finite prefix codes. We also consider the operation which associates to any variety of monoids V the variety V. W generated by all semidirect products of a monoid of V by a monoid of W, for various varieties W, and we describe the corresponding operation on varieties of languages.

A combinatorial property of finite prefix codes

A necessary and sufficient condition is given under which a finite prefix code A (A ⊆ X*) is maximal. This condition, which does not hold in general for infinite rational prefix codes, is derived from a main theorem obtained by means of a combinatorial method consisting of the construction for any prefix code and word f ε X* of a suitable sequence of conjugate words of f. Further, some auxiliary results and consequences of the theorem are shown.