In “Evo-Info 4: Non-Conservation of Algorithmic Specified Complexity,” I neglected to explain that algorithmic mutual information is essentially a special case of algorithmic specified complexity. This leads immediately to two important points:
- Marks et al. claim that algorithmic specified complexity is a measure of meaning. If this is so, then algorithmic mutual information is also a measure of meaning. Yet no one working in the field of information theory has ever regarded it as such. Thus Marks et al. bear the burden of explaining how they have gotten the interpretation of algorithmic mutual information right, and how everyone else has gotten it wrong.
- It should not come as a shock that the “law of information conservation (nongrowth)” for algorithmic mutual information, a special case of algorithmic specified complexity, does not hold for algorithmic specified complexity in general.
My formal demonstration of unbounded growth of algorithmic specified complexity (ASC) in data processing also serves to counter the notion that ASC is a measure of meaning. I did not explain this in Evo-Info 4, and will do so here, suppressing as much mathematical detail as I can. You need to know that a binary string is a finite sequence of 0s and 1s, and that the empty (length-zero) string is denoted The particular data processing that I considered was erasure: on input of any binary string the output is the empty string. I chose erasure because it rather obviously does not make data more meaningful. However, part of the definition of ASC is an assignment of probabilities to all binary strings. The ASC of a binary string is infinite if and only if its probability is zero. If the empty string is assigned probability zero, and all other binary strings are assigned probabilities greater than zero, then the erasure of a nonempty binary string results in an infinite increase in ASC. In simplified notation, the growth in ASC is
for all nonempty binary strings Thus Marks et al. are telling us that erasure of data can produce an infinite increase in meaning.
In Evo-Info 4, I observed that Marks et al. had as “their one and only theorem for algorithmic specified complexity, ‘The probability of obtaining an object exhibiting bits of ASC is less then [sic] or equal to ’” and showed that it was a special case of a more-general result published in 1995. George Montañez has since dubbed this inequality “conservation of information.” I implore you to understand that it has nothing to do with the “law of information conservation (nongrowth)” that I have addressed.
Now I turn to a technical point. In Evo-Info 4, I derived some theorems, but did not lay them out in “Theorem … Proof …” format. I was trying to make the presentation of formal results somewhat less forbidding. That evidently was a mistake on my part. Some readers have not understood that I gave a rigorous proof that ASC is not conserved in the sense that algorithmic mutual information is conserved. What I had to show was the negation of the following proposition:
for all binary strings and for all probability distributions over the binary strings, and for all computable [ETA 12/12/2019: total] functions on the binary strings. The variables in the proposition are universally quantified, so it takes only a counterexample to prove the negated proposition. I derived a result stronger than required:
Theorem 1. There exist computable function and probability distribution over the binary strings such that
for all binary strings and for all nonempty binary strings
Proof. The proof idea is given above. See Evo-Info 4 for a rigorous argument.
In the appendix, I formalize and justify the claim that algorithmic mutual information is essentially a special case of algorithmic specified complexity. In this special case, the gain in algorithmic specified complexity resulting from a computable transformation of the data is at most i.e., the length of the shortest program implementing the transformation, plus an additive constant.
Some of the following is copied from Evo-Info 4. The definitions of algorithmic specified complexity (ASC) and algorithmic mutual information (AMI) are:
where is a distribution of probability over the set of binary strings, and are binary strings, and binary string is a shortest program, for the universal prefix machine in terms of which the algorithmic complexity and the conditional algorithmic complexity are defined, outputting The base of logarithms is 2. The “law of conservation (nongrowth)” of algorithmic mutual information is:
for all binary strings and and for all computable [ETA 12/12/2019: total] functions on the binary strings. As shown in Evo-Info 4,
where the universal semimeasure on is defined for all binary strings Here we make use of the normalized universal semimeasure
where Now the algorithmic mutual information of binary strings and differs by an additive constant, from the algorithmic specified complexity of in the context of with probability distribution
Theorem 2. Let and be binary strings. It holds that
Proof. By the foregoing definitions,
Theorem 3. Let and be binary strings, and let be a computable [and total] function on Then
Proof. By Theorem 2,
The inequality in the last step follows from (1).