Reziprog

Reziprog, often written reziprog, is a theoretical construct in mathematics and computer science describing a family of reversible, polynomial-based transforms used for representing and processing finite sequences. The name combines ideas of reversal, compression-like data representation, and programmable transforms. In practice, reziprog transforms aim to map an input sequence to an alternative representation in a way that is invertible, enabling lossless recovery.

For a fixed length n, a reziprog transform is defined by a set of polynomials P_0, ..., P_{n-1}, and the transform of a sequence x = (x_0, ..., x_{n-1}) yields y_k = sum_{i=0}^{n-1} P_k(i) x_i. The collection A = [P_k(i)] forms an n-by-n transform matrix. Reziprog is invertible if A is nonsingular; specific choices of P_k yield structured matrices (for example, P_k(i) = i^k gives a Vandermonde-type matrix). Efficient evaluation and inversion can exploit polynomial recurrences or convolution-like structure, achieving near-linear time in favorable cases.

Variants include localized reziprog, which uses only short neighborhoods to improve locality, and probabilistic reziprog, which samples bases to improve numerical stability. Applications discussed in literature range from reversible data encoding and error-detection to signal processing and research in reversible computing. The framework emphasizes reversibility, numerical stability, and the ability to tailor the basis to the data.

Because reziprog is not standardized, different sources adopt different polynomial bases and notations. The concept has been explored mainly in theoretical contexts and occasionally in speculative or educational materials as a tool for illustrating reversible transforms based on polynomial bases.