T - Complexity and T - Information Theory – an Executive Summary

This paper describes the derivation of the T-Complexity and T-Information Theory from the decomposition of finite strings, based on the duality of strings and variable-length T-Codes. It further outlines its similarity to the string parsing algorithm by Lempel and Ziv. It is intended as a summary of work published mainly by Titchener and Nicolescu. 1 A brief Introduction to T-Codes This paper first gives an introduction to T-Codes and their construction technique, as this is fundamental for the understanding of the T-decomposition algorithm that underpins the T-complexity measure. T-Codes [3, 4, 11] are similar to Huffman codes in that they are codes with variable-length codewords. Depending on the source symbol probabilities, a Huffman code may also be a T-Code. In fact, any T-Code set could theoretically have been constructed as a result of a Huffman code construction algorithm. However, rather than letting the source symbol probabilities determine the length of the codewords and the structure of the decoding tree, T-Codes are constructed with no regard to symbol probabilities. Their construction focuses instead on a recursive tree structure. A T-Code set is constructed as follows: 1. Start with a finite alphabet S (e.g., the binary alphabet, where S = {0, 1}). Every finite alphabet is a (trivial) T-Code set by default, with the letters being primitive codewords. 2. Given a T-Code set, another T-Code set may be derived from it by a process called “T-augmentation”. This involves picking a codeword from the existing T-Code set, which is called the “T-prefix”, and a positive integer called the “T-expansion parameter”. Any T-Code set may thus be derived from an alphabet in a series of n T-augmentations using a