Multiset and Set Decipherable Codes

We extend some results of Lempel and Restivo on multiset decipherable codes to set decipherable codes

Monoids and Maximal Codes

In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal code. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code.Comment: In Proceedings WORDS 2011, arXiv:1108.341

The Degree of a Finite Set of Words

We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y?Z is a composition of finite sets of words with Y complete, then d(X) = d(Y) ? d(Z), where d(T) is the degree of T. We also show that a finite set is synchronizing if and only if its degree equals one. This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X^* with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations

Coding Partitions of Regular Sets *

Abstract A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the factorizations of finite sequences. We prove that the canonical coding partition of a regular set contains a finite number of regular classes and we give an algorithm for computing this partition. From this we derive a canonical decomposition of a regular monoid into a free product of finitely many regular monoids

Unique Decipherability in Formal Languages

We consider several language-theoretic aspects of various notions of unique decipherability (or unique factorization) in formal languages. Given a language L at some position within the Chomsky hierarchy, we investigate the language of words UD(L) in L^* that have unique factorization over L. We also consider similar notions for weaker forms of unique decipherability, such as numerically decipherable words ND(L), multiset decipherable words MSD(L) and set decipherable words SD(L). Although these notions of unique factorization have been considered before, it appears that the languages of words having these properties have not been positioned in the Chomsky hierarchy up until now. We show that UF(L), ND(L), MSD(L) and SD(L) need not be context-free if L is context-free. In fact ND(L) and MSD(L) need not be context-free even if L is finite, although UD(L) and SD(L) are regular in this case. We show that if L is context-sensitive, then so are UD(L), ND(L), MSD(L) and SD(L). We also prove that the membership problem (resp., emptiness problem) for these classes is PSPACE-complete (resp., undecidable). We finally determine upper and lower bounds on the length of the shortest word of L^* not having the various forms of unique decipherability into elements of L

Benefiting from disorder: source coding for unordered data

The order of letters is not always relevant in a communication task. This paper discusses the implications of order irrelevance on source coding, presenting results in several major branches of source coding theory: lossless coding, universal lossless coding, rate-distortion, high-rate quantization, and universal lossy coding. The main conclusions demonstrate that there is a significant rate savings when order is irrelevant. In particular, lossless coding of n letters from a finite alphabet requires Theta(log n) bits and universal lossless coding requires n + o(n) bits for many countable alphabet sources. However, there are no universal schemes that can drive a strong redundancy measure to zero. Results for lossy coding include distribution-free expressions for the rate savings from order irrelevance in various high-rate quantization schemes. Rate-distortion bounds are given, and it is shown that the analogue of the Shannon lower bound is loose at all finite rates.First author draf