5 - Deciphering delay

Summary

This chapter is devoted to codes with finite deciphering delay. Intuitively, codes with finite deciphering delay can be decoded, from left to right, with a finite lookahead. There is an obvious practical interest in this condition. Codes with finite deciphering delay form a family intermediate between prefix codes and general codes. There are two ways to define the deciphering delay, counting either codewords or letters. The first one is called verbal delay, or simply delay for short, and the second one literal delay.

The first section is devoted to codes with finite verbal deciphering delay. We present first some preliminary material. In particular we prove a characterization of the deciphering delay in terms of simplifying words.

In the second section, we prove Schützenberger's theorem (Theorem 5.2.4) saying that a finite maximal code with finite deciphering delay is prefix. We prove that any rational code with finite deciphering delay is contained in a maximal rational code with the same delay (Theorem 5.2.9).

The next section considers the literal deciphering delay, that is the deciphering delay counted in terms of letters instead of words of the code. A code with finite literal deciphering delay is called weakly prefix. We introduce the notion of automata with finite delay, also called weakly deterministic. We prove the equivalence between weakly prefix codes and weakly deterministic automata (Proposition 5.3.4). We use this characterization to give yet another proof of Schützenberger's theorem. Next, we show that a rational completion with the same literal deciphering delay exists (Theorem 5.3.7).