Contribution
The book gives a mechanistic account of how a determinate machine can change from unadapted to adapted behaviour. It defines system, state, field, stability, parameter, step-function and independence by observable operations (Chs. 2–6, 12). It deduces that adaptation by trial and error needs a second, slower feedback through the essential variables to the parameters of the first (Ch. 7), builds the Homeostat to show it (Ch. 8), and then spends half the book on the difficulty that such a system, if large and richly joined, would take impossibly long (Chs. 11–17). The answer is that adaptation succeeds in practicable time only where partial adaptations can be retained, which needs temporary independence between parts.
Key insight
Stability belongs to the whole, and a whole made of many richly joined parts is almost never stable by chance. A system finds stability by changing its own parameters blindly until a configuration holds, and it can do this in useful time only when what it has already got right is kept apart from what it is still trying, so that success can accumulate part by part, as a crystal grows molecule by molecule rather than forming all at once.
Assumptions
- Determinacy of the significant units. Each part, “if in a particular state internally and affected by particular conditions externally, will behave in one way only” (S. 1/14).
- State-determined systems. The organism and its environment together form a state-determined system (S. 3/9); variables are dial readings, and the observer controls all states (S. 2/7).
- Survival as the criterion. Essential variables, with limits fixed by the species, define adaptation (Ss. 3/14, 5/3). Reproduction and the “racial” behaviour of Holmes’s table are set aside (S. 5/9).
- Essential variables affected by the environment only. The case taken is the hardest, in which the reacting part cannot act on the essential variables except through the environment (S. 7/3), and does not know how the environment is coupled to them (S. 7/4).
- No consciousness, no teleology. Neither is used (Ss. 1/13, 1/16).
- Idealized cases. The book follows “the strategy for the complex system” (S. 2/17): exact treatment of type-cases, with real systems assumed to behave like the nearest one.
Key results
- Adaptation is stability (Ss. 4/8, 5/8). “A region is stable if the lines of behaviour from all points in the region stay within the region.” “‘Adaptive’ behaviour is equivalent to the behaviour of a stable system, the region of the stability being the region of the phase-space in which all the essential variables lie within their normal limits.” Stability “is a property of the whole system and can be assigned to no part of it” (S. 4/18): two stable systems can join to an unstable whole, two unstable ones to a stable whole (S. 21/12 gives matrices).
- Parameters change fields (S. 6/3). “A change in the value of an effective parameter changes the field”, so “a system can, in general, show as many fields as its parameters can show combinations of values”.
- The veto (Ss. 6/8–6/9). “That a whole dynamic system should be in equilibrium at a particular state it is necessary and sufficient that each part should be in equilibrium at that state, in the conditions given to it by the other parts.” Every part “has a power of veto over the states of equilibrium of the whole”.
- The second feedback is necessary (Ss. 7/4–7/8). Against an environment that is a Black Box, only trial and error can elicit the needed information (S. 7/4). The basic rule is “If the trial is unsuccessful, change the way of behaving; when and only when it is successful, retain the way of behaving” (S. 7/7). So: when the essential variables are outside their limits, no state of the parameters S is equilibrial; when inside, every state is. Any adaptive system “must have a second feedback formally identical (isomorphic) with that described here” (S. 7/8); otherwise it would be “a candidate who gives the correct answers without being given the questions”.
- And sufficient (S. 7/11). By the veto, “all the possible equilibria of the whole have the essential variables within the given limits”. “The mechanism has been shown to be necessary by S. 7/8, and sufficient by the present section.”
- Step-functions (Ss. 7/13–7/23). A variable with “finite intervals of constancy separated by instantaneous jumps”, changing at “critical states”. With n two-valued step-mechanisms the main variables show 2ⁿ fields, and the whole stays determinate while its main variables appear to change “spontaneously” (S. 7/21).
- The ultrastable system defined (S. 7/26). “Two systems of continuous variables (that we called ‘environment’ and ‘reacting part’) interact, so that a primary feedback … exists between them. Another feedback, working intermittently and at a much slower order of speed, goes from the environment to certain continuous variables which in their turn affect some step-mechanisms, the effect being that the step-mechanisms change value when and only when these variables pass outside given limits. The step-mechanisms affect the reacting part; by acting as parameters to it they determine how it shall react.” Ashby adds that this form “is only one of a large class of related forms”, kept as a type.
- Random construction is no obstacle (S. 7/25). The second feedback “is in no way put out of order if R is initially constructed at random or if the successive values at S occur at random”.
- The Homeostat (Ch. 8). Four units, each a pivoted magnet driving a current to the others through commutators and potentiometers set by a 25-position uniselector wired from Fisher and Yates’ random numbers (390,625 combinations); a relay F closes when a needle passes about 45° and advances the uniselector, tested every 1–10 seconds (S. 8/2). Reversing a connection by hand produces a runaway, then uniselector changes until a compensating reversal is found (Figures 8/4/1, 8/5/1), as in Marina’s and Sperry’s crossed-muscle and crossed-nerve monkeys. It adapts to a glass fibre joining two magnets and becomes unstable when the fibre is removed (S. 8/11); with an alternating commutator it keeps the first setting stable for both (S. 8/10). “Faults” are shared with organisms: defenceless against a tasteless poison or congenital insensitivity to pain (S. 8/13), against sudden discontinuities such as traps (S. 8/14), and dependent on a trial duration neither too short nor too long (S. 8/15).
- Bimodal disturbance (Ss. 9/14, 9/19). The ultrastable form is “the optimal mechanism” when disturbances are of two classes: “frequent … small impulsive disturbances to the main variables” and “occasional changes, of step-function form, to its parameters”.
- Goals can be moved (S. 9/16). Whatever moves the critical states, or transforms the input, moves the goal.
- No accumulation in the Homeostat (Ss. 10/5–10/9). Given A, then B, then A again, “it treats A as if it had never encountered A before”; retroactive inhibition is “complete”. For adaptations to accumulate, “the step-mechanisms must be divisible into non-overlapping sets”, with a “gating mechanism” Γ, controlled by the situation P, connecting one set at a time to the second feedback.
- Adaptation time (Ss. 11/2–11/6). With 100 essential variables each brought within limits with probability ½ per trial, the expected number of trials is 2¹⁰⁰, about 10²² years at one per second. For N wheels each showing A with probability p: all at once takes (1/p)ᴺ spins; one at a time, about N/p; spinning only the failures, a little more than the single-wheel time. With p = ½ and N = 1,000 at one spin a second: about 10²⁹³ years, about 8 minutes, a few seconds. “What had made the processes … so excessively time-consuming is that partial successes go for nothing.”
- Crystallisation (S. 11/8). “If a visible crystal could form only when all its molecules happened simultaneously to be properly related … then crystallisation could never occur … But in fact crystallisation can occur by succession.” “A compound event that is impossible if the components have to occur simultaneously may be readily achievable if they can occur in sequence or independently.”
- Accumulation needs disconnection (S. 11/10). If adapting essential variable 3 leaves 1 and 2 adapted, there can be no channel from the step-mechanisms changed for 3 to the parts that serve 1 and 2. “For the accumulation of adaptations to be possible the system must not be fully joined.” The Homeostat “is too well integrated, cannot accumulate adaptations” (S. 11/12).
- Walls of constancy (Ss. 12/10, 12/17, 22/14). “That A and C should be independent and state-determined it is necessary and sufficient that the variables B should be null-functions”: “a wall of constancies”. “If a sufficient proportion of its variables go constant”, a connected system “may … be temporarily equivalent to a set of unconnected subsystems. Constancies, in other words, can cut a system to pieces.” Thresholds readily give such constancies (S. 12/15).
- Polystable systems (Ch. 13). In a richly joined system of parts rich in equilibria, a near-total equilibrium (999 of 1,000 parts) falls back to the average: it “is thus typically unable to retain partial or local successes” (S. 13/9). In a poorly joined one the count of parts at equilibrium “cannot fall with time” and the whole progresses by “accumulation of local equilibria” (Ss. 13/10, 13/13). A state-determined subsystem at equilibrium among constant neighbours is “trapped in equilibrium”: “change can come only from change” (S. 13/4).
- Habituation (Ch. 14). Under a repeated stimulus a polystable system sticks “at a state from whose confluent the stimulus cannot shift it”, so responses shrink; under small random disturbances it goes to equilibria “immune to the action of small irregular disturbances” (S. 14/8), and “roughing it” removes fields with equilibria near the edge of the region (S. 14/10).
- Serial and iterated adaptation (Ch. 15). The terrestrial environment is mostly part-functions, constant much of the time, each affecting few others (S. 15/2). Iterated subsystems adapt in about the time of one; a chain adapts in its natural order, “A first, then B, then C”, and only if the channels down the chain have low capacity (S. 15/11).
- The multistable system (Ch. 16). Subsystems, each ultrastable, weakly joined: “subsystem adapts to subsystem in exactly the same way as animal adapts to environment” (S. 16/7). Its step-mechanisms stop when, and only when, every combination that occurs is stable (S. 16/8). With few active variables per stimulus, two stimuli activate overlapping sets in proportion to the product of the fractions, so different stimuli use different step-mechanisms, a “functional equivalent of the gating mechanism” (Ss. 16/12–16/13). As stimuli grow alike the overlap grows, and “the more the new learning resembled the old the more marked was the interference” (S. 16/14; Robinson, Skaggs). Retroactive inhibition and accumulation are “opposed” demands; an intermediate multistable system shows both partially (S. 16/15).
- Communication is not always good (Ss. 17/3–17/7). Too fast a change of step-mechanisms, any channel from the essential variables to the stores of other adaptations, and every added connection between subsystems all harm adaptation. Coordination between parts can pass through the environment, as the server’s thrown ball guides the striking arm (S. 17/4). Richness of connection has costs and benefits and must lie between limits.
- Ancillary regulations (Ss. 17/8–17/13). Trial duration, the hunt-at-bad, stick-at-good relation, the gating mechanism, the richness of equilibria, the degree of connection, and the distribution of the second-order feedbacks to the parts that caused the trouble must all be set. A second essential variable F, the time E takes to adapt, when exceeded should change “those conditions that have continued unchanged throughout E’s hundred trials”, so that trials are drawn “from a new statistical population”. All such selection obeys the law of requisite variety and comes ultimately from natural selection.
- Selection is the rule (Ss. 18/1–18/7). Every isolated state-determined system is selective, for “two lines of behaviour may become one, but one line cannot become two”. “Every single-valued operation tends to select forms that are peculiarly able to resist its change-inducing action.” Learning is indirect regulation, which can “amplify” regulation beyond what the gene-pattern transmits, as teaching a child to use a dictionary beats ten minutes of definitions.
- Probability of stability (S. 20/10). For random linear systems with entries uniform on −9 to +9, tested by Hurwitz’s rule: 2×2, 77 of 320 stable (24%); 3×3, 12 of 100; 4×4, 1 of 100, “compatible … with the hypothesis that the probability for a matrix of order n × n is 1/2ⁿ”. With intrinsically stable diagonals, 2×2 72% and 3×3 55%. “Large linear systems should be assumed to be unstable unless evidence to the contrary is produced.”
- Trials to a terminal field (S. 22/7). If each new field is stable with independent probability p, the expected number of fields is 1/p.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Adaptive behaviour is stability of the organism–environment system within the region where essential variables stay within limits | moderate | a definition, defended by examples, Ch. 5 |
| C2 | Any system that adapts by trial and error to an unknown environment must have a second feedback from essential variables to the parameters of its reacting part | strong | deduction under the stated assumptions, Ss. 7/4–7/8 |
| C3 | Such a system’s reachable equilibria are all adapted | strong | the veto theorem, Ss. 6/8, 7/11 |
| C4 | A machine with this organization adapts to reversals and unforeseen constraints | strong | the Homeostat records, Ch. 8 |
| C5 | Blind search over a richly joined system takes time exponential in its size; retention of partial successes reduces it to roughly linear | strong | probability argument, Ss. 11/2–11/6 |
| C6 | Accumulation of adaptations requires that the system not be fully joined | moderate | deduction from the operational test of independence, S. 11/10 |
| C7 | Two subsystems are independent and state-determined iff the variables between them are constant | strong | proof, Ss. 12/10, 22/14 |
| C8 | New learning interferes with old in proportion to the overlap of the parts it uses, which grows with similarity | moderate | argued for the polystable system, matched to retroactive-inhibition data, Ss. 16/12–16/14 |
| C9 | Repetition selects states the repeated stimulus cannot displace, so responses diminish | moderate | argument and Homeostat demonstration, Ch. 14 |
| C10 | The probability that a random linear system is stable falls roughly as 1/2ⁿ | weak | 520 sampled matrices up to 4×4, S. 20/10 |
| C11 | Natural brains approximate multistable systems | weak | qualitative fit to ablation, habituation and interference data, Chs. 13–16 |
Method
The Homeostat (S. 8/2; equations S. 19/11): four units, magnet deviations xᵢ obeying, in the high-viscosity limit, dxᵢ/dt = Σⱼ aᵢⱼxⱼ, with the aᵢⱼ step-functions drawn from a rectangular distribution on −1 to +1 when the uniselector of unit j advances, which it does when |xⱼ| crosses its critical surface (S. 22/6). Relays F stand for essential variables; uniselectors for the step-mechanisms S; the four magnets for environment and reacting part together, divided arbitrarily.
Concepts
- State-determined system. One in which a given state, under given conditions, is always followed by the same transition (S. 2/15); equivalently dxᵢ/dt = fᵢ(x₁, …, xₙ) (S. 19/9).
- Field. All lines of behaviour from all initial states under one set of conditions (S. 2/14).
- Parameter. A variable not in the system that affects it; its value selects the field (Ss. 6/2–6/3).
- Essential variables. Variables closely linked to survival, with limits fixed by the species (S. 3/14).
- Step-function, step-mechanism, critical state. A variable constant between instantaneous jumps; the mechanism showing it; the states at which it jumps (Ss. 7/13, 7/18).
- Ultrastable system. S. 7/26, quoted above.
- Part-function, null-function. A variable with intervals of change and of constancy; one that never changes (S. 7/13).
- Independence, dominance. Y is independent of X if two transitions from states differing only in X give the same Y (S. 12/3); X dominates Y if Y depends on X and not conversely (S. 12/6).
- Polystable system. Parts rich in equilibria, taken and joined at random (S. 13/2).
- Multistable system. Weakly joined ultrastable subsystems (S. 16/6).
- Dispersion. Different inputs activating different sets of variables (S. 13/14).
- Confluent. The basin of one equilibrium or cycle (S. 14/2).
- Ancillary regulation. A regulation of the parameters on which the efficiency of the basic adaptation depends (Ch. 17).
Connections
The book builds on Cannon’s homeostasis, Sherrington’s reflexes, Pavlov and the learning literature, and the wartime servomechanism work; it acknowledges Sommerhoff’s Analytical Biology as parallel (fn. to S. 5/12), and refers its foundations to An Introduction to Cybernetics (1956), which states the law of requisite variety. Its stability results for random linear systems anticipate May (1972; LIT-930, held unread) and its gating of separate stores anticipates later accounts of catastrophic interference.
Bearing on the record
- LIT-787’s entry is right on ultrastability and misses the book’s
second half. The summary’s two feedback loops and step-functions are
as Ashby defines them (S. 7/26). Three additions matter:
- The second feedback is offered as necessary for any system that adapts by trial and error (S. 7/8), not as one mechanism among others; in the second edition the particular form is “only one of a large class of related forms” (S. 7/26).
- The parameter changes are blind. The step-mechanisms take new values “at random”, and the mechanism works only because the essential variables veto bad ones (Ss. 7/25, 8/3). The owner’s essay reads ultrastability as regulation against “alteration of the parameters of regulation”; in Ashby that alteration is undirected search, not a choice of new parameters.
- Chapters 10–17, more than half the main text, are about why a large ultrastable system would take impossibly long and what lets it not: retained partial adaptations, temporary independence, and separate stores for separate situations.
- CLAIM-185 has a precise ancestor here. Its thesis, that structure evolves only on persistent anchors, is Ashby’s result that adaptation finishes in practicable time only if “partial successes … can be conserved and accumulated” (S. 11/6), with crystallisation by succession as his own example (S. 11/8). Ashby adds the condition the claim does not state: retention requires that the retained part be cut off, by constancies, from what is still changing (Ss. 11/10, 12/17). A persistent structure that later changes can reach is not retained.
- QUESTION-038, what turns persistence into an anchor. Ashby’s answer is functional isolation plus access: the adapted step-mechanisms must be closed to the second feedback while other problems are worked on, and re-connected when their situation recurs (Ss. 10/8–10/9). In record terms, an Active entry that new filing cannot overwrite, and a way to reach it by its situation. Retirement by status, never deletion, is the first; codes and indexes are the second. This is analogy, and the record should treat it as such.
- CLAIM-176, recurrence. Ch. 10 is about the recurrent situation: the benefit of a recurrence is that the second response can be immediate, and that requires the store for that situation to be selected by the situation itself. Ch. 14 adds that a repeated disturbance selects states immune to it. Neither says that recurrence should trigger capture; both say what a system must have for recurrence to be exploited.
- CLAIM-180, drift and interference. Ashby’s interference grows with the similarity of the situations, because similar inputs use the same parts (S. 16/14). Read across, a close restatement of an idea is the one most likely to overwrite its earlier form, the converse of CLAIM-180’s concern that a distant one is taken for new. Both follow from content addressing; a code avoids both.
- CLAIM-179, the May note. Ashby’s sampling (S. 20/10) is an earlier, empirical form of the same conclusion: stability of a randomly joined system falls steeply with size, roughly as 1/2ⁿ in his samples, and “large linear systems should be assumed to be unstable”. It also gives the escape the note asks about: weak joining and many constancies (Chs. 13, 16).
- THEORY-217 and THEORY-153. Barandiaran’s essential variables and Beer’s requisite variety are Ashby’s (Ss. 3/14, 17/13); this reading is their common source.
- Filed from this reading: THEORY-324, the ultrastability account, and THEORY-307, adaptation time and the retention of partial adaptations. No existing THEORY held either.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The account of the brain is a theory of possible mechanism. Ashby says of the step-mechanisms “we have at present practically no idea of where to look, nor what to look for” (S. 9/4).
- The fit to real brains (Chs. 13–16) is qualitative. Adaptation times are calculated for type-cases with independent probabilities, not measured.
- The Homeostat is linear, with one equilibrium and four units.
- The stability statistics are 520 small matrices.
- Reward is set aside: training is treated as punishment (S. 8/8).
Open questions
- Whether the time of adaptation in real organisms scales with the degree of connection as Chapters 11–16 predict; Ashby says the study “should not be beyond the powers of the present-day experimenter” (S. 16/11).
- How the ancillary regulations are set in an individual, beyond “by the gene-pattern or … under an ultrastability of their own” (S. 17/13).
- The probability of stability for large random matrices, which Ashby calls unsolved (S. 20/10) and May later bounded.