Contribution
It takes Szostak’s (2003) “functional information”, proposed for RNA aptamers, and offers it as a general complexity measure for any system of many agents with a quantifiable function. It applies it to three cases and reports, for Avida, reproducible steps in the curve of information against degree of function.
Key insight
Complexity is measured not in the configuration but in the configuration space relative to a task: how rare, among all configurations, is doing this well? A highly structured object can carry zero functional information for a function it does not perform, and an undefined one for a function nobody can perform.
Assumptions
- A finite (or countable) space of N configurations, each with a measurable degree of function E_x for one stated function x.
- Uniform weighting: “we assign equal probability to all possible sequences”; unequal weights are said to be “a straightforward extension” of Shannon’s treatment, not given.
- The distribution of function must be “derived from the statistical properties of the system as a whole [as opposed … to … populations evolving in a fitness landscape]”.
Key results
- Definition. I(E_x) = −log2[F(E_x)] = −log2[M(E_x)/N], M the number of configurations with degree ≥ E_x. I(E_max) = log2 N for a single best configuration; I(E_min) = 0.
- Letters. For about 1,000 ten-letter strings that summon a fire engine, I ≈ −log2(1000/26¹⁰), given as ”≈ 36 bits”; the maximum for n letters is log2 26ⁿ ≈ 4.7n bits.
- Avida. Random 100–500-instruction genomes from 26 instructions are functional with P ≈ 10⁻³ at ~10² lines; none of four or fewer instructions is. For 10⁷ random 300-line genomes the NAND curve is continuous for E between 2 and 48 with reproducible gaps at 48/49 and 58/59; first addition gaps at 19, 39, 59 and 69 executions for genome lengths 100, 200, 300 and 500.
- Control. Counting 1s in random 100-digit strings gives a smooth, unstepped curve (Figure 4).
- Aptamers (from Carothers et al.). 11 classes of GTP aptamers; a 10-fold gain in binding costs about 10 bits, a ~1,000-fold drop in abundance; a “suggestion” of steps, from data “too few to draw definitive conclusions”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Functional information is defined only relative to a specified function and degree | strong | definition |
| C2 | In Avida the I–E curve has reproducible steps | moderate | repeated random samples (Figs. 1–2); repetitions not counted |
| C3 | The steps come from distinct classes of solutions with different maxima | weak | a schematic model (Fig. 3) and one example of two unrelated subtraction genes |
| C4 | The measure can be applied to physical systems such as sand ripples | weak | proposed, not done |
Concepts
- Degree of function E_x: any quantitative measure of how well a configuration performs function x; units arbitrary.
- Functional information I(E_x): −log2 of the fraction of all configurations with degree at least E_x.
- Islands of function: distinct classes of configurations solving the same task, each with its own maximum.
Connections
The definition is Szostak (2003, Nature 423:689) and Carothers et al. (2004); neither is registered. It is a surprisal over configuration space, so it shares the form of a self-information measure but fixes the measure (uniform) and the event (function ≥ E_x) by hand.
Bearing on the record
- CASE-102, the college program that keeps producing good teams. The LIT records A10’s reading: “function-relative, and silent on whether successive teams are one individual.” The reading confirms both. It adds a third point the case should note: the measure is taken over all configurations under a uniform weighting, explicitly not over what a selective process produces. So it can say how rare a good roster is among all rosters, but not why one program keeps drawing from the rare region. That is the mechanism the case asks about, and functional information leaves it untouched. A slow, member-read variable such as a program’s standing (CLAIM-327) is the kind of thing that would.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- An arithmetic slip. −log2(1000/26¹⁰) = 10·log2 26 − log2 1000 ≈ 47.0 − 10.0 = 37.0 bits, not ”≈ 36 bits”.
- A wording slip. The abstract says functional information “represents the probability” that a configuration achieves the function; it is minus the log of that probability.
- Eq. (9) writes the denominator as a sum ”∑_{1−n_opt}(4ⁿ)”, a garbled index, for the count of all strings up to the optimal length.
- Unequal weights are declared a straightforward extension and not given; for any real system (an evolved population, a sand bed) the uniform measure is the strong assumption.
- No count of repetitions for the “robust” Avida gaps.
- The higher-dimensional and origin-of-life sections are proposals.
Open questions
- Under which weighting over configurations does the measure track what selection or assembly actually produces, and does the stepped structure survive it?