134

11

Randomness and Complexity

conserved (and have a nonuniform probability distribution), whereas the noncoding

bits are fugitive (and have a uniform probability distribution). The information about

ee contained in the ensembleupper SS of copies is then the Shannon indexupper I left parenthesis upper S right parenthesis minus upper I left parenthesis upper S vertical bar e right parenthesisI(S) −I(S|e). In

finite ensembles, the quantity

I (S|e) = −

Σ

s

p(s|e) log p(s|e)

(11.35)

can be estimated by sampling the distribution p left parenthesis s vertical bar e right parenthesisp(s|e).

Computational complexity reflects how the number of elementary operations

required to compute a number increases with the size of that number. Hence, the

computational complexity of “011011011011011011 ellipsis011011011011011011 . . .” is of order unity, since

one merely has to specify the number of repetitions.

Algorithmic and computational complexity are combined in the concept of logical

depth, ¹⁴ defined as the number of elementary operations (machine cycles) required

to calculate a string from the shortest possible program. Hence, the numberpiπ, whose

specification requires only a short program, has considerable logical depth because

that program has to execute many operations to yield piπ.

Problem. A deep notion is generally held to be more meaningful than a shallow one.

Could one, then, identify complexity with meaning? Discuss the use of the ways of

quantifying complexity, especially effective complexity, as a measure of meaning

(cf. Sect. 6.3.2).

A very simple measure of complexity, subsuming many variables, is to compare

the specific price of a manufactured object with its scrap value. Thus, a Eurofighter

Typhoon aircraft, which costs about 124 MUSD and weighs 11 t, has a specific price

of 11,272 USD/kg; assuming that it could be sold for (aluminium) scrap at a price of

0.84 USD/kg, the complexity ratio is 13,420. In contrast, a gold bar costing about 58

kUSD/kg would be sold for “scrap” at the same price, hence yielding a complexity

ratio of 1. The latest TSMC 3 nm wafer costs 20 kUSD; with a diameter of 300

mm and a thickness of about 0.775 mm it weighs 127.6 g, if made solely of silicon.

The specific price is, therefore, 156,740 USD/kg. Its scrap value is negligible but

let us suppose it equals the price of sand, typically costing 0.05 USD/kg; hence the

complexity ratio is about 3,134,800.

In contrast, a similar calculation applied to living organisms yields far lower com-

plexity ratios. For example, a racehorse weighing about 500 kg might cost 20 kUSD,

yielding a specific price of 400 USD/kg. This relatively low value presumably reflects

the simplicity of generating replicas—in contrast to the intricate manufacturing pro-

cesses required for aircraft and semiconductors, in which precision complexity has

to be explicitly engineered; from an embryo, in itself a highly complex object, much

greater complexity, especially when viewed at the nanoscale, spontaneously devel-

ops without explicit human intervention. The “scrap” value of the horse could be

taken as that of the carcass sold for its meat, priced at about 4 USD/kg. Hence, the

14 Due to Bennett (1988).