The Entropy Rate of Some Pólya String Models

On the Reverse-Complement String-Duplication System

“…Fix some string-duplication system S = S(Σ, s, T ). Depending on the application, one can view S in two ways: either as a generative model which describes what possible strings may be derived from s (e.g., see [2], [6], [9], [12], [15], [25]), or as a channel which describes what corrupted versions of the transmitted s may be received (e.g., see [16]- [18], [20], [32], [34], [38]- [40]). While the latter view calls for the construction of suitably tailored error-correcting codes, the former inspires the following properties of S to be studied.…”

“…The last property of S is its entropy rate (see [9]). Unlike the deterministic nature of the previous two properties, here we describe a stochastic process.…”

“…In this section we study the capacity and entropy rate of the reverse-complement string-duplication system. Loosely speaking, while the capacity focuses on what is possible [12], [15], the entropy rate looks at what is probable [9]. Thus, the capacity is an upper bound on the entropy rate (see [9]).…”

“…In contrast, the system with a fixed length tandem duplication, where the copy is inserted next to its original position, has zero capacity and is never fully expressive, but when the length of the tandem duplication is only lower-bounded, full expressiveness and positive capacity are obtained. In [9], the exact entropy rate of both end-and tandem-duplication systems were found, in the case where the duplication size is k = 1 and the alphabet is binary. Furthermore, a noisy scenario was investigated where the duplicated bit has a probability of being complemented.…”

“…It was shown there that the system with a fixed-length duplication has full expressiveness and positive capacity. In [9], reverse-complement duplication was studied, but only for a duplication of length k = 1. Upper and lower bounds on the entropy rate of the system were found there, but a gap between them remains.…”