Context Directed Reversals and the Ciliate Decryptome

TL;DR

This paper investigates the properties of context directed reversal (cdr) and context directed swap (cds) operations on permutations, proving new theorems about their behavior and applications to ciliate decryption models and combinatorial games.

Contribution

It introduces the CDR Fixed Point and CDR Parity Theorems, advancing understanding of permutation sorting and its biological and mathematical implications.

Findings

Any cdr fixed point of a cdr sortable permutation is cds sortable.

The parity of the number of cdr operations leading to different fixed points is invariant.

The results solve decision problems in combinatorial game theory.

Abstract

Prior studies of the efficiency of the block interchange (swap) and the reversal sorting operations on (signed) permutations identified specialized versions of the these operations. These specialized operations are here called context directed reversal, abbreviated cdr, and context directed swap, abbreviated cds. Prior works have also characterized which (signed) permutations are sortable by cdr or by cds. It is now known that when a permutation is cds sortable in n steps, then any application of n consecutive applicable cds operations will sort it. Examples show that this is not the case for cdr. This phenomenon is the focus of this paper. It is proven that if a signed permutation is cdr sortable, then any cdr fixed point of it is cds sortable (the cds Rescue Theorem). The cds Rescue Theorem is discussed in the context of a mathematical model for ciliate micronuclear decryption. It is also proven that though for a given permutation the number of cdr operations leading to different cdr fixed points may be different from each other, the parity of these two numbers is the same (the cdr Parity Theorem). This result provides a solution to two previously formulated decision problems regarding certain combinatorial games.