Knowledge Discovery Efficiency (KEDE) and Ashby's Law of Requisite Variety

In section 2, we provide a detailed overview of Ashby's Law of Requisite Variety, including its mathematical formulation and implications for system regulation. We also discuss the present day understanding of residual variety and its significance in the context of Ashby's Law. In section 3, we discuss the challenges of applying Ashby's Law to real-world systems, including combinatorial explosion, dual control dilemma, and resource constraints. We propose a solution to these challenges by treating the system as a black box, observing probability of successful outcomes to disturbances, and estimate the gap in its internal variety based on that. In section 4, we start establishing the solution by introducing the Knowledge Discovery Process, which is narrowing down and selecting the appropriate response from its set of alternative responses. We then label H(X|Y) as the knowledge to be discovered, which is the gap in internal variety that had to be compensated by selection. In section 5, we show how to quantify the knowledge to be discovered H(X|Y) based on the observable outcomes E. In section 6, we generalize the Knowledge-Discovery Efficiency (KEDE) - scalar metric that quantifies how efficiently a system closes the gap between the variety demanded by its environment and the variety embodied in its prior knowledge. Finally, in section 7, we explore applications of KEDE in various domains, demonstrating its utility as a diagnostic tool for evaluating system performance and adaptability.

Mathematical Formulation

In Ashby's terms, we can think of the system as a transducer:

Where:

• D be the set of disturbances, representing the possible states of disturbances that a system may experience.
• R be the set of responses, representing the possible regulatory actions that counteract disturbances.
• O be the set of realized outcomes, representing the possible outcomes that can result from the disturbance-response pairs (D, R), without any implication of desirability. D x R → O is the fixed transition rule of the environment.
• E be the set of values / essential-variable states that the system aims to achieve, induced by a valuation mapping v : O → E. The scale of values in E may be as simple as the 2-element set {good, bad}, and is commonly an ordered set.

Some subset of acceptable outcomes of E can be defined as the goal G ⊂ E[8]. Given the goal G then the inverse mapping of this subset will define, over O, the subset S of “acceptable outcomes”. Thus is defined a relation equivalent to “r_i, as response to d_i, gives an acceptable outcome s ∈ S”[8].

This can be represented as a Table of Outcomes (T) where:

• T be the pay-off matrix, i.e. a fixed mapping T : D x R → O or the fixed transition rule of the environment.
• Rows represent disturbances D
• Columns represent regulatory responses R
• Entries represent realized outcomes O = {o₁₁, o₁₂, o₁₃, ..., o₂₁, o₂₂, o₂₃, ..., o₃₁, o₃₂, o₃₃, ..., ...}
in the cells of the matrix, O = {o} = T(d,r) ∈ O, where d is a disturbance and r is a regulatory response.

The Table of Outcomes (T) is a fixed, structured space of possibilities, or the fixed transition rule of the environment, which directly determines the essential variables E. In simplified analyses, O and E are sometimes conflated, but they are conceptually distinct. In the deterministic pay-off-matrix case, the environment is the fixed function T : D x R → O, and evaluation is v : O → E. If D and R are treated as random variables, then the induced conditional distribution is P(E | D, R) determined by E = v(T(D,R)).

The regulator's accumulated structure M is its learned law of action. In the deterministic case, M : D → R. In general (learning, uncertainty), treat it as a policy P(R | D), i.e. how probability mass is allocated across responses for each disturbance.

Across time, the learning object M is not fixed. If the table O is the space of possibilities, then M is the mechanism that induces a measure over possible paths through that space.

Learning and rework do not alter the structure of the Table of Outcomes T. Instead, they modify the regulator's accumulated structure M. Changes in M alter which disturbance-action pairs become more or less probable and therefore how the system's realized trajectories are distributed over the table O.

Rework corresponds to a revision accumulated structure M: probability mass is shifted away from previously selected or ineffective responses and reassigned to alternative responses for the same disturbance. Thus, the Table of Outcomes (T) remains a fixed space of possibilities, while learning and rework change only the regulator's induced distribution of trajectories through that space, not the space itself.

Each input from D is transformed via selection into a response $r\in R$, which is then processed to produce an outcome $o\in O$. which is then evaluated as $e\in E$, which is the essential variable that the system aims to achieve, induced by a valuation mapping v : O → E.

If R does nothing, i.e. keeps to one value, then the variety in D threatens to go through O to E, contrary to what is wanted. It may happen that O, without change by R, will block some of the variety and occasionally this blocking may give sufficient constancy at E. More commonly, a further suppression at E is necessary; it can be achieved only by further variety at R.

Under idealized assumptions, Ashby showed that a lower bound on the achievable variety in the outcomes is given by the smallest variety that can be achieved in the set of actual outcomes cannot be less than the quotient of the number of rows divided by the number of columns. $$\mathrm{V}\left(\mathrm{E}\right)\ge \frac{\mathrm{V}\left(\mathrm{D}\right)}{\mathrm{V}\left(\mathrm{R}\right)}$$

Where:

• V(D) = variety in the disturbances
• V(R) = variety in the regulatory responses
• V(E) = variety in the essential variables

This inequality shows that the variety in the essential variables cannot be reduced below the division between the variety in the disturbances and the variety in the regulatory responses.

Perfect regulation here means deterministic essential variable, not merely acceptable performance. For perfect regulation, we need V(E) to be stable or close to 1 (only one possible value) i.e. $\mathrm{V}\left(\mathrm{E}\right)\approx 1$

The law reflects a fundamental insight: control is about the ratio of disturbances to regulatory responses, not just their arithmetic difference.

Information-Theoretic Formulation

All acts of regulation can be related to the concepts of communication theory by noticing that the “disturbances” correspond to noise, and the “goal” is a message of zero entropy, because the target value E is constant. Thus, the law of Requisite Variety says that R 's capacity as a regulator cannot exceed R 's capacity as a channel of communication.

The variety is measured by the logarithm of its value. If the logarithm is taken to base 2, the unit is the bit of information. Applying this to Ashby's Law we get:

In practice, we use the Shannon information entropy, denoted by H. For a quantifiable variable, entropy is just another measure of variance. If we assume equiprobable disturbances/responses and treat variety as cardinality, then H reduces to ${\log }_{2}V$, yielding:

Ashby's Law can be interpreted as a cybernetic analogue of Shannon's "Noisy channel coding theorem" which states that communication through a channel that is corrupted by noise may be restored by adding a correction channel with a capacity equal to or larger than the noise corrupting that channel. The disturbance D, which threatens to get through to the outcome E, clearly corresponds to the noise; and the correction channel is the system R, which is supposed to restore the outcome E[8].

Ashby's law can thus be reformulated clearly:

The information-processing capacity (entropy) of a control system must be at least as large as the information (entropy) in the system it regulates.

It has been shown shown that the law of requisite variety can be extended to include knowledge or ignorance by simply adding this conditional uncertainty term[31] When buffering is present, part of the environmental variety is absorbed passively before reaching the regulator. This reduces the effective disturbance entropy by an amount K, which is the buffering capacity:

Where:

• H(E) is the residual variety i.e. the realized essential-variable distribution, not the size of the set E
• H(R) is the entropy of the regulator, representing its information-processing capacity.
• H(D) is the entropy of the disturbances, representing the complexity of the environment.
• H(R|D) is the conditional entropy of the regulator given disturbances, representing the lack of requisite knowledge i.e. the ignorance of the regulator about how to react correctly to each appearance of a disturbance D. Only a regulator that knows how to use the available regulatory variety H(R) to react correctly to each disturbance D will reach the optimal result of regulation,
• K is the buffering capacity measured in bits of disturbance variety absorbed before reaching the regulator. Buffering is the passive absorption or damping of disturbances i.e. the amount of noise that a system can absorb without requiring an active regulatory response.

A necessary but not sufficient condition for effective control (to make H(E) small) is: $H\left(R\right)\ge H\left(D\right)+ H\left(R\right|D) - K$ . Sufficiency additionally requires $H\left(R|D\right)\approx 0$

Successful (essential) outcomes E do not depend solely on the variety of responses H(R) available to a regulator R; the system must also know which response to select for a given disturbance. Effective compensation of disturbances requires that the system possess the ability to map each disturbance to an appropriate response from its repertoire. The absence or incompleteness of such knowledge can be quantified using the conditional entropy H(R|D)[31]. In other words, H(R|D) measures how much the regulator R lacks the requisite knowledge to match responses to disturbances. In the absence of such requisite knowledge, the system would have to select responses, until eliminating all disturbances. Thus, merely increasing the response variety H(R) is not sufficient; it must be complemented by a corresponding increase in selectivity, that is, reduction in H(R|D) i.e. increasing knowledge. H(R|D) = 0 represents the case of no uncertainty or complete knowledge, where the action is completely determined by the disturbance. This requirement may be called the law of requisite knowledge[29].

H(R|D) reminds us that response alone is not sufficient: if the regulator does not know which response is appropriate for the given disturbance, it can only try out regulatory actions at random, in the hope that one of them will be effective and that none of them would make the situation worse. The larger the H(R|D), the larger the probability that the regulator would choose a wrong regulatory response, and thus fail to reduce the variety in the outcomes H(E). Therefore, this term H(R|D) has a “+” sign in the inequality: more uncertainty (less knowledge) produces more variation in the essential variables E[54].

To achieve control, the regulator R must possess sufficient information-processing capacity (entropy) such that the following is achieved: $$H\left(R\right)\ge H\left(D\right)$$ $$H\left(R|D\right)=0$$

In other words, the complexity of the environment D can not exceed that of the system R, which means the system R fully matches the environment D[7].

Since $H\left(R\right)-H\left(R|D\right)=I(R:D)$ the law simplifies to: $$H\left(E\right)\ge H\left(D\right)-I\left(R:D\right)-K$$

The mutual information I(R:D) represents the requisite knowledge of the regulator R about how to react correctly to each disturbance D, i.e. the amount of regulatory variety that is effectively correlated with and therefore absorbs the variety in the disturbances. Such knowledge may be realized structurally as the regulator's learned law of action M represented by a mapping M: D → R , by which disturbances are mapped to regulatory responses[54]. The mutual information I(R:D) quantifies how much of the regulator's learned law of action M: D → R effectively couples disturbances to responses, while the remaining uncertainty H(R|D) quantifies the lack of requisite knowledge[29].

Solution

These challenges significantly hinder the practical application of Ashby's Law. Whether considering a human, an AI model, or an organization, we are typically limited to observing external behavior rather than internal mechanisms—unless we are able to "open the box."

Ashby himself emphasized that all real systems can be considered black boxes. He argued that while black boxes mimic the behavior of real objects, in practice, real objects are black boxes: we have always interacted with systems whose internal workings are, to some extent, unknown.

This leads to what Ashby termed the black box identification approach [2], which involves:

1. Perturbing the system by applying external disturbances,
2. Measuring the system's responses to these perturbations, and
3. Inferring the internal variety or capacity from the observed input-outcome relationships.

In most practical scenarios, we are only able to observe the outcomes of a system. These observable outcomes can be used to infer bounds on the system's internal variety—specifically, the extent of variety it must possess or lack in order to exhibit the observed behavior.

We propose such an approach: to treat the system as a black box, observe the probability of successful outcomes to disturbances, and estimate the gap in its internal variety based on that. Let E denote the event that the system gives an response to disturbance D, and let R be the regulator's action. In information-theoretic terms, perfect regulation requires H(R|D) = 0[31]. Using our novel information-theoretic estimator, empirical estimates of P(E=1) are used to quantify H(R|D) in bits of information. This quantifies how much disorder the regulator still permits and, conversely, how close the system comes to meeting Ashby's requisite-variety condition.

Multi-stage selection

We denote the amount of selection achieved at each stage i as: ${\sigma }_1,{\sigma }_2,\dots {\sigma }_k$. When mapping one stage of selection with one selection signal Z_i we have:

As selections accumulate, the remaining uncertainty $H\left(X|Y,Z_1,\dots ,Z_i\right)$ shrinks. The total amount of selection Ashby describes is mathematically identical to the total mutual information accumulated across stages.

Shannon's chain rule for mutual information is:

If stages share information or impose overlapping constraints, summing their marginal “reductions” overcounts. The correct decomposition credits each stage only for the reduction of residual uncertainty left by previous stages. Therefore the per-stage contributions must be conditional (incremental) to avoid double counting, because stages may share information or constrain overlapping parts of X. The correct staged accounting is the mutual-information chain rule over the selection signals Z:

After k selections we have:

Given the success criterion of determining the response X for a given disturbance Y, Ashby's Law of Requisite Variety for a staged selection process is:

Or equivalently, the sum of the bits removed by selection by every stage must at least equal the bits of uncertainty injected by the original range of possibilities or by disturbances.

Regulation

An episode represents a specific interaction or event in time, characterized by a structure designed for learning and pattern recognition:

• A disturbance (Y): An external stimulus, perturbation, or environmental input that acts upon the system.
• A response (X): The system's internal action or selection made or outcome generated in response to the disturbance Y.
• A within-episode sequence of evidence (Z₁, Z₂, …, Z_k): A sequence of constraints, tests, feedback signals, or rules that are applied to the system in response to the disturbance.
• The system's internal structural coupling (M): A mapping of disturbances to responses.

• A Start Time (t_start): The time at which the window starts (e.g. the start of the first episode).
• A End Time (t_end): The time at which the window ends (e.g. the end of the last episode).

A disturbance Y is fixed and fully observed at episode start and is treated as given. All remaining uncertainty is only about which response X will be selected; subsequent Z is evidence about X, not new disturbance info. This initial uncertainty is the lack of requisite knowledge, measured as the initial conditional entropy H(X|Y). The system then applies a sequence Z₁, Z₂, …, Z_k of constraints, tests, feedback signals, or rules, each of which removes some possibilities and therefore reduces the lack of requisite knowledge. In Ashby's terms this is “selection”; in Shannon's terms each stage i contributes conditional mutual information I(X ; Z_i | Y, Z_<i). “Non-outcome time” is entirely spent on discriminating information about X for the current episode, and outcome emissions are atomic and don't hide extra selection. “episode closes with an acceptable outcome” event occurs only after the regulator has already fixed the exact response X (so the closure event certifies that the episode's X has been identified/committed). Therefore, the episode closes with an acceptable outcome event certifies that the episode's X has been identified/committed.

Let M_t be the system's internal stored structural coupling (mapping) at the start of episode t. In Ashby's terms, M is not a separate object but the regulator's law of action i.e. the learned functional relation by which disturbances are mapped to responses to produce acceptable outcomes[54]. In contrast, the Table of Outcomes (T) is a fixed strucured space of possible outcomes and it specifies what outcome would result from each disturbance'-response pair $T:Y\times X\to O$ .

For each episode t in a window, the stored structural coupling (mapping) M_t is treated as a parameter (not a random variable): Therefore we do not write entropies or mutual informations “conditioned on M_t”. Instead, we write them as quantities induced by the mapping:

When we need a quantity that also depends on within-episode evidence Z, we use the same convention, e.g. I_t(X;Z|Y) means I_Mt(X;Z|Y).

Here "induced by M_t" means that the distribution we use to compute entropy is the one produced by the policy/model state M_t. This aligns with the “ignorance term” H(D|R) changing under adaptation (variable structure) without treating the structure itself as a random variable inside the episode. $$P_t\left(x\right|y):=P(x|y;M_t)$$ $$H_t\left(X\right|Y):=H_{P_t}(X\left|Y\right)$$ $$I_t(X;Y):=I_{P_t}(X;Y)$$

Regulation uses staged evidence $Z_{1:k}$ to eliminate uncertainty within an episode t, where k is the number of selection signals within the episode t. During an episode staged selection supplies $Z_{1:k}$ until the response X is effectively determined:

Multiple acceptable responses (compatibility note).
In this theory, X denotes the specific response actually selected/emitted by the regulator (a random variable), not the equivalence class of acceptable responses. Therefore, even if multiple responses x∈X are acceptable for a given disturbance y∈Y, H(X|Y) do not measure “uncertainty about success.” H(X|Y) measures the remaining uncertainty about which response will be emitted under the regulator's current policy P_t(X|Y). Learning can still drive H(X|Y) toward 0 by structuralizing tie-breaking.

Learning (across episodes)

Regulation can use within-episode evidence Z to determine the response X for a given disturbance Y during an episode t. Learning makes that success persistent by updating the stored structural coupling (mapping) M so future episodes start with less esidual uncertainty about which response will be selected for the same class of disturbances.

Rework, retraction, and net learning

A mapping update may include both the the addition of improved structure and withdrawal of previously stored structure. These internal components must be distinguished from observable rework and from the net epistemic effect of the update.

Let M_t be the regulator's stored structural coupling at the start of episode t, and let M_t+1 = Update(M_t, Z_t) be the mapping after the episode's evidence Z_t has been incorporated. We do not assume any particular memory mechanism (overwrite, patching, versioning, or full replacement). We track only the effect of the update on the regulator's induced coupling measures.

Under the mapping M_t, define:

Three distinct notions

1. Observable rework. Rework is any externally visible correction, undo, replacement, deletion, or repair of prior commitments. Observable rework is defined at the level of artifacts and execution behavior.

2. Internal retraction and accretion. A mapping M update may contain:

• a retraction component: previously stored structure is withdrawn, weakened, or invalidated, and
• an accretion component: new or improved structure is added or strengthened.

These are internal aspects of the update of M. They should not individually be identified with “negative learning” or “positive learning”. In particular, removing false structure is often part of successful learning.

3. Net learning. Learning is evaluated by the net change in the induced coupling measures after the whole update has been completed, not by whether some internal fragment was deleted.

Net learning criterion

Define the net change in stored requisite coupling:

Equivalently, under the uncertainty view:

Then:

• Net positive learning iff Δ_tI > 0 (equivalently, a net decrease in lack of requisite knowledge under the chosen uncertainty view).
• Zero net learning iff Δ_tI = 0.
• Net negative learning iff Δ_tI < 0, i.e. the final mapping is worse than the initial one with respect to stored requisite coupling.

Thus, negative learning is reserved for net worsening. It should not be used merely because part of the update involved deletion or withdrawal of prior structure.

Exact knowledge ledger of net change

For knowledge accounting, define the positive and negative parts of the net coupling change:

By definition, the net ledger is exact:

This ledger records only the net epistemic effect of the update. It does not assert that the update process itself consisted of a pure gain or a pure loss. A single episode may contain both correction of prior commitments and improved learning, yet still end with G_t > 0.

Why rework is not the same as negative learning

An episode may contain observable rework and still produce net positive learning. For example, a software developer may correct a previously committed line of code by deleting a wrong symbol and replacing it with the correct one. This is clearly rework at the artifact level. But if the correction improves the developer's future mapping from disturbance to response, then the episode yields net positive learning, not negative learning.

Accordingly:

• Rework + net positive learning is possible.
• Rework + zero net learning is possible.
• Rework + net negative learning is also possible.

Therefore, observable rework does not by itself imply Δ_tI < 0.

Observable rework as external evidence

Operationally, we usually cannot observe the internal decomposition of the mapping update. What we can observe is whether the window contains explicit undo/repair commitments. Let W_t denote the number of such observable rework outcomes in window t.

A nonzero W_t > 0 is evidence that correction activity occurred. It may indicate that some prior structure was inadequate and that the update contains a retraction component. But it is not sufficient to conclude that the episode produced net negative learning. That conclusion requires the stronger condition Δ_tI < 0.

In summary:

• Rework is an observable artifact-level phenomenon.
• Retraction / accretion are internal components of a mapping update.
• Learning is evaluated only by the net change in stored requisite coupling or, equivalently, by the net change in lack of requisite knowledge.

Defining Knowledge to be Discovered

Ashby's Law of Requisite Variety concerns capacity: the regulator must have a sufficiently large repertoire of possible responses[1]. Heylighen's Law of Requisite Knowledge adds the missing condition for effective regulation: it is not enough to have many possible actions; the regulator must know which action to select for the given disturbance. Otherwise, increased action variety increases the chance of choosing the wrong action, forcing trial-and-error selection[29].

In information-theoretic terms, this “knowing which action to select” is exactly the reduction of uncertainty about the response X once the disturbance Y is observed. That uncertainty is the conditional entropy H(X|Y).

Knowledge To Be Discovered. We define Knowledge To Be Discovered as the conditional entropy H(X|Y): the expected uncertainty about which response(s) will achieve success, given that the disturbance is known. Operationally, it is the amount of selection (in bits) that must still be supplied in order to determine a successful response.

Knowledge to be discovered H(X|Y) is Ashby's ignorance term H(R|D) (same mathematical object), and it appears additively in the lower bound on residual outcome entropy.

Thus, Knowledge To Be Discovered is the precise difference between the action variety available and the coupling/selectivity actually achieved:

• H(X) is the regulator's variety of responses available to be selected, or the prior knowledge.
• I(X;Y) is the regulator's requisite knowledge: how much the disturbance actually informs the choice of action, or the required knowledge.
• H(X|Y) is the remaining uncertainty after the disturbance is known, i.e. the lack of requisite knowledge, or the knowledge to be discovered.

Requisite knowledge is not an independent thing but the total selectivity actually available at decision time makes H(X|Y) small which can be stored structure M or online evidence Z:

• Offline (stored) coupling: learning updates the mapping M so that future stages start with higher I_t(X;Y) and therefore lower H_t(X|Y).
• Online (within-stage) selection: evidence Z (tests, observations, constraints) supplies additional conditional information I(X;Z|Y), driving H(X|Y,Z) toward 0 during regulation.

In the ideal case of complete adaptation for the task class, the regulator's selectivity is perfect: H(X|Y) = 0 (the response is effectively determined by the disturbance). If multiple responses are acceptable, we will replace X with the equivalence class of acceptable actions or measure success directly via the goal variable.

This yields the intended conceptual pairing:

• Residual variety H(E): what remains uncontrolled in outcomes.
• Knowledge To Be Discovered H(X|Y)≡H(R|D): what remains undetermined about which action succeeds, given the disturbance.

This framing suggests that reducing "knowledge to be discovered" is a pathway to reducing "residual variety" - which is exactly the cybernetic insight that better information about disturbances enables better control. It emphasizes the constructive, learning-oriented aspect of cybernetic systems rather than just their current limitations.

Information-theoretic bound

For each $Y=y$ if the regulator’s within-episode binary selection strategy is optimal (minimizes $E\left(k|Y=y\right)$ for the given posterior $P\left(X|Y=y\right)$), then the expected number of binary selections required to identify X satisfies the Shannon bound:

Now introduce the latent (information-theoretic) quantity of the Knowledge Discovery Process: for each disturbance realization $Y=y_i$ , the regulator faces a finite candidate set ${\Omega }_{y_i}$ of admissible responses $X$. We assume optimal binary selection: the multi-stage selection process Z₁...Z_k is implemented by the regulator through an optimal sequence of binary selections for identifying the response $X$ given $Y=y_i$.

The expected number of selections per episode is: $E\left(k|Y=y_i\right)$ . Although k is inferred as a concrete count per episode, it varies with which response X is correct; hence the latent expected count for that disturbance class $E\left(k|Y=y_i\right)$ is taken over the randomness of the $X|Y=y_i$ under the fixed questioning strategy.

Crucially, $E\left[k\right]$ is a model quantity, while $Q$ is an externally measurable execution count.

Compatibility note (multiple acceptable responses). Even if multiple responses could lead to acceptable outcomes for a given y, this theory keeps X as the specific response selected/emitted by the regulator. Thus H(X∣Y) measures uncertainty about which response will be emitted under P(X∣Y), not uncertainty about success. Learning may structuralize tie-breaking so H(X∣Y)→0 for a stable task class.

Operational window accounting (black-box observable proxy)

Let $\left\{b_t\right\}=\{b_1,\dots ,b_n\}$ denote the complete time-ordered sequence of events over a fixed window of n discrete time units, where each unit is fully utilized by exactly one atomic action on a single shared execution channel. Each interval is classified by the binary event-type indicator B_t which is 1 if the interval contains an episode-closing commitment and 0 otherwise.

The total number of counted episode closures in the window is

In this base section, $B_t$ marks a closure that the ledger presently counts as an acceptable episode closure. Later, when rework is introduced, the accounting can be refined into gross closures, wasted closures, and net closures, without changing the basic meaning of $B_t$ as an event-type marker.

Partition the event stream $\left\{b_t\right\}$ into the S consecutive non-overlapping episodes $\left\{w_i\right\}$, where episode w_i corresponds to responding to one disturbance realization $Y=y_i$, consists of exactly one outcome $o_i$ that closes the episode and contains exactly k_i selections preceding that outcome, By construction, the inferred total number of selections for the window satisfy $Q=\sum _{i=1}^S{k}_i$. The window begins immediately after an episode-closing outcome and ends on an episode-closing outcome.

If the minimum outcome duration is one unit of time, equal to the time it takes to make one selection, then each action occupies one unit of time t. Hence the sum of S and Q equals the length $n=\frac{T}{t}$ of the time series. The total available time T is therefore: $Qt+St=T$. We define $r=t^{-1}$ as the execution rate of the channel, and since $n=\frac{T}{t}=T{t}^{-1}=Tr$ then $Q+S\mathit{=}Tr$. We now drop usage of n and instead use N as maximum action capacity (selections + outcomes) per window. $N=Tr$

Consequently, over any observation window with maximum action capacity $N$ and observed number of outcomes $S$, the inferred number of selections is:

Observable rework

Under fixed execution capacity, observable rework reduces net throughput because some channel capacity is spent on corrective or undo/repair commitments rather than on net new episode closures. This section is purely operational: it concerns externally visible execution counts, not the internal decomposition of the mapping update.

Accordingly, we distinguish:

• Observable rework: externally visible corrective, undo, rollback, replacement, or repair outcomes.
• Net negative learning: a worsening of the stored mapping in the coupling ledger, i.e. Δ_tI < 0.

These are related but not identical. Observable rework may accompany net positive learning, zero net learning, or net negative learning. Therefore the counting model below should be interpreted as an execution-capacity accounting model, not as a direct observation of the internal knowledge ledger.

Per window t, let:

• N_t be anchored action capacity in the window (selections + outcome-emissions).
• S_gross,t be the number of gross outcome-emissions (all episode-closing commitments observed in the window).
• W_t be the number of those outcome-emissions that are observable rework outcomes, i.e. commitments whose purpose is to undo, repair, replace, or correct prior commitments, counted in the same window they occur and never retroactively, with 0 ≤ W_t ≤ S_gross,t.
• S_net,t be the number of net realized outcomes: $S_{\text{net,t}}=S_{\text{gross,t}}-W_t$ .
• Q_t be the number of binary selections (non-outcome action units): $Q_t=N_t-S_{\text{gross,t}}$ .

Here W_t is an observable artifact-level count. It is not equal to the negative part of the internal coupling change L_t = max(-Δ_tI, 0). A nonzero W_t is evidence that corrective activity occurred, but it does not by itself imply Δ_tI < 0.

Including W yields an operational equivocation rate per net outcome under non-retroactive windowing; it is not claiming that undo/repair commits are literally binary questions about X, only that they consume the same constrained channel capacity.

Selection-equivalent debit under fixed capacity

Because observable rework outcomes consume the same constrained channel capacity while not contributing to net new closures, we treat them operationally as a selection-equivalent debit. This does not mean that rework outcomes are literally binary questions about X; it means only that they occupy atomic action units on the same bounded execution channel.

Since wasted outcomes consume the same constrained channel capacity while not increasing $S\mathrm{net}$ , define the effective non-net-progress selection-equivalent effective debit:

Using $Q_t=N_t-S_{\text{gross,t}}$ and $S_{\text{net,t}}=S_{\text{gross,t}}-W_t$ , we obtain:

Thus observable rework reduces net closures and correspondingly increases the effective debit counted against net progress.

Weak result: operational upper bound. Under the assumptions that (i) each non-outcome atomic unit contributes at most 1 bit of discriminating information relevant to fixing the selected response for the current disturbance, (ii) outcome emission contains no hidden selection, and (iii) episode closure occurs only after the response is fixed, the latent conditional uncertainty satisfies ${\widehat{k}}_{\text{eff}}\le E\left(k\right)$ . Therefore any large-window empirical estimator of $E\left(k\right)$ , such as Ĥ(X|Y) is an operational conservative proxy for the latent knowledge-to-be-discovered, subject to sampling and nonstationarity error.

Strong result: one-bit tightness under optimal binary selection. If, in addition, the within-episode discrimination strategy is optimal among binary decision procedures for identifying the selected response, then the standard question-depth bound in (1) applies. Hence, when the observation window is sufficiently large and stable so that the empirical mean depth approximates $E\left(k\right)$, the operational proxy ${\widehat{k}}_{\text{eff}}$ will typically lie within about one bit of the latent conditional entropy H(X|Y).

${\widehat{k}}_{\text{eff}}$ is an operationally observable proxy for the (latent) expected within-episode decision depth $E\left(k\right)$ . Under large windows / stability assumptions (so empirical averages represent expectations), ${\widehat{k}}_{\text{eff}}\approx E\left(k\right)$ . Empirically, ${\widehat{k}}_{\text{eff}}$ approximates E[k] with sampling error that shrinks with window size N increasing. Combined with the information-theoretic bound above, this links the abstract “knowledge to be discovered” H X | Y to externally observable execution counts through an expected-depth proxy.

In the idealized regime where the regulator’s within-episode questioning strategy is optimal, the window is large and stable so Ĥ(X|Y) ≈ E[k], the posteriors are close to uniform/power-of-two (balanced trees), then Ĥ(X|Y) will typically sit within about one bit of the true equivocation because E[k] does:

Knowledge To Be Discovered Conservative Estimator

This is the observed selection-equivalent decision depth per net realized outcome under non-retroactive windowing. It is not itself the true entropy H(X|Y), but a black-box observable quantity that increases whenever net stage closures are crowded out under fixed capacity. The atomic time unit for the rates is chosen fine-grained enough that each unit reveals ≤ 1 bit of discriminating information (i.e., any higher-bit test is decomposed into multiple counted units).

Ĥ(X|Y) measures the empirically observed expected decision depth per outcome. It is therefore an operational conservative proxy for the latent Knowledge To Be Discovered H(X|Y) under the counting model above. This upper-bound interpretation requires only that each counted non-outcome unit contribute at most one discriminating bit and that closure occur only after response fixation. A stronger near-tightness claim—namely that the proxy lies within about one bit of H(X|Y)—requires the additional assumption of optimal binary within-episode selection and representative large-window averaging. Whatever the true H(X|Y) is, it cannot exceed what is implied by the observed execution capacity.

Ĥ(X|Y) is an operational, black-box measure of the “knowledge to be discovered”: the observed selection-equivalent depth per net realized outcome. Rework increases Ĥ(X|Y) mechanically by reducing S_net under fixed N, capturing the crowd-out effect of wasted commitments on net episode closures.

The result is operational: it links an abstract information-theoretic quantity $H\left(X|Y\right)$ to directly observable execution counts $(N,S,W)$, without assuming access to the true distribution. It provides an operational upper bound on conditional entropy under idealized assumptions (optimal binary selection, single shared channel, no idle capacity). It does not claim to recover the true entropy exactly in arbitrary systems.

We can rearange the bounds to be:

If there is no knowledge to be discovered, i.e. there is no need to make selections at all, then S equals N and H(X|Y) is zero. This is not redefining entropy — it is measuring it operationally.

Extra operational burden attributable to observable rework

For decomposition purposes, define the corresponding no-observable-rework baseline while holding S_gross,t fixed:

This is a conditional comparison only: it asks how much the operational proxy increases when some of the gross observed closures are reclassified as observable rework outcomes, while total gross outcome-emissions are held fixed. It is not a universal causal counterfactual about what would have happened in a different world.

The extra operational burden attributable to observable rework under that fixed-gross comparison is:

Therefore the operational burden induced by observable rework is strictly increasing in W_t and grows convexly as W_t → S_gross,t.

The role of W_t in this model is purely operational. Observable rework consumes capacity that could otherwise have produced net new closures, thereby increasing the observed selection-equivalent depth per net outcome. This is a throughput and capacity-crowding effect.

It should not be read as an identity between:

• artifact-level observable rework W_t, and
• internal net coupling loss L_t.

A window with W_t > 0 may still exhibit net positive learning in the internal ledger, and a window with Δ_tI < 0 need not expose all of that loss through visible repair outcomes in the same window.

In summary, observable rework affects the operational proxy $\widehat{H}\left(X|Y\right)$ implicitly through bounded capacity and reduced net closures; it does not by itself identify the sign of the internal knowledge update.

Anchoring KEDE to Natural Constraints

In our model, N is always the theoretical maximum action rate (selections + outcomes) in an unconstrained environment, and S is the observed outcome rate under specific conditions over a given interval.

A key question is how to assign a natural constraint to N. That is, what constitutes an appropriate reference value for the maximum action rate (selections + outcomes)?

We may turn to physics for an instructive analogy. A quantum (plural: quanta) represents the smallest discrete unit of a physical phenomenon. For instance, a quantum of light is a photon, and a quantum of electricity is an electron. In this context, the speed of light in a vacuum serves as a fundamental upper bound for N. However, identifying an analogous natural constraint for human activity—particularly knowledge work—presents greater challenges.

Consider the example of typing. Here, the quantum can reasonably be defined as a symbol, since it is the smallest discrete unit of text. A symbol may be a letter, number, punctuation mark, or whitespace character. To determine the appropriate bin width Δt, we refer to empirical data on the minimum time required to produce a single symbol. Typing speed has been subject to considerable research. One of the metrics used for analyzing typing speed is inter-key interval (IKI), which is the difference in timestamps between two keypress events. We see that IKI is defined equal to the symbol duration time t. Hence we can use the research of IKI to find the symbol duration time t. Studies have reported an average IKI of 0.238 seconds [26], yielding a maximum human typing rate of approximately N=1/t=1/0,238=4.2 symbols per second

A similar approach can be applied to tasks such as furniture assembly. In this case, a plausible quantum is a single screw tightened, since it represents a minimal, repeatable unit of outcome. We then identify Δt as the average time required to tighten one screw. Empirical studies report that this task typically takes between 5 and 10 seconds[34]. Using the upper bound, we estimate the maximum screw-tightening rate as N=1/t=1/10=0.1 screws per second.

This methodology offers a principled way to estimate N using domain-specific quanta and empirically grounded time durations, enabling the application of our model to a broad range of human tasks.

The next question concerns the appropriate definition of outcome for measuring S and N.

Both N and S can always be discretized—or “binned”—in a way that preserves the total information rate, regardless of whether the outcome arises from natural processes, human behavior, or machines. By choosing a bin width Δt small enough (e.g., milliseconds), the range of possible tangible outcomes within each bin shrinks dramatically. This reduced range leads to less uncertainty in each bin, which compensates for the smaller time interval. Yet the ratio

remains an accurate measure of information rate.

As Δt becomes smaller, the measurements of S and N become more precise, as they reflect outcome over finer time intervals. But how small should Δt be? This dilemma is resolved by considering the granularity of outcomes associated with the outcome. The set E of outcomes can be thought of as the effects of the regulation process — the resulting states after the regulator responds to disturbances. In our model E is a sequence of {0,1}, where 0 = wrong outcome(failure to regulate) and 1 = acceptable outcome. So the presence of a concrete outcome leads to a natural binning of the outcomes, It also enables a clear distinction between signal (the entropy associated with producing the outcome) and noise (the residual variability unrelated to success or failure).

For example, two distinct symbols typed (e.g., ‘a' vs. ‘b') are clearly different outcomes. However, if one symbol is typed in 91 milliseconds and another in 92 milliseconds, this minute variation is inconsequential to the outcome. Such timing fluctuations are typically unintentional, irrelevant to task performance, and should not be considered part of the outcome. In practical terms, if the theoretical upper bound N is known—for instance, 4.2 symbols per second as derived from human typing speed, and the observed rate is S=1 symbol per second, then time should be partitioned into one-second bins. Each bin then yields a single outcome: either 1 (a symbol was successfully typed) or 0 (no symbol typed or incorrect input).

This binning principle generalizes beyond typing. Whether analyzing foot strikes in trail running (where negligible spatial change occurs over milliseconds) or the discrete moves in solving a Rubik's cube (where each turn resolves multiple potential states into a single action), binning ensures that no intermediate state need be modeled explicitly.

Tightening screws

We can apply our model to motor tasks such as furniture assembly. In this context, a natural unit of outcome — or “quantum” — is the tightening of a single screw.

Skilled workers engaged in manual assembly tasks can typically insert and tighten standard screws at a rate of 6'-12 screws per minute under optimal, repetitive conditions — such as those found in furniture construction or industrial assembly lines. In contrast, automated screw-tightening machines can achieve significantly higher rates, often between 30 and 60 screws per minute [34] More complex manual tasks, such as high-torque applications involving ratchets or Allen keys, typically reduce the rate to 2'-4 screws per minute due to the increased effort and precision required. In surgical or medical contexts, such as orthopedic screw insertion, accuracy and the avoidance of overtightening are paramount; here, rates often fall to 1'-2 screws per minute, or approximately one screw every 30'-60 seconds [46].

The key observation is that rates decrease as torque, task complexity, or required precision increases. If we take the machine rate as the maximum possible outcome N and the observed human rate as S, we can estimate the average number of bits of information H(X|Y) that the human operator must process per action.

This implies that the human must absorb approximately 4 bits of information, on average, to tighten a single screw under typical conditions.

The rate at which a person tightens screws depends on various factors, including:

• Screw type and size
• Material being fastened
• Required torque
• Tool used (screwdriver, ratchet, etc.)
• Operator skill and fatigue

This interpretation aligns with existing research, which suggests that task difficulty directly influences the amount of information a task imparts [47, 48]. When difficulty is appropriately matched to the individual's skill level, the task yields maximal informational value [49], and the time required reflects the interaction between task complexity and the individual's regulatory capacity [50].

Using our model, we transform a sequence of real-world actions in furniture assembly into a granular, time-based measure of regulatory capacity. This enables us to quantify — in bits — how much variety the individual must absorb in order to successfully complete the task.

Typing the longest English word

Let's use an example scenario to see Ashby's law applied to human cognition and knowledge work.

For that we'll have myself executing the task of typing on a keyboard the word “Honorificabilitudinitatibus”. It means “the state of being able to achieve honours” and is mentioned by Costard in Act V, Scene I of William Shakespeare's “Love's Labour's Lost”. With its 27 letters “Honorificabilitudinitatibus” is the longest word in the English language featuring only alternating consonants and vowels.

The way I will execute this task is to go to the "play text" or "script" of “Love's Labour's Lost”, look up the word and type it down. The manual part of the task is to type 27 letters. The knowledge part of the task is to know which are those 27 letters.

In order to track the knowledge discovery process I will put "1" for each time interval when I have a letter typed and "0" for each time interval when I don't know what letter to type.

I start by taking a good look at the word “Honorificabilitudinitatibus” in the script of “Love's Labours' Lost”. That takes me two time intervals. Then I type the first letters “H”, “o”, and “n”.I continue typing letter after letter: “o”, “r”. At this point I cannot recall the next letter. What should I do? I am missing information so I go and open up the script of “Love's Labours Lost” and I look up the word again. Now I know what the next letter to type is but acquiring that information took me one time interval. This time I have remembered more letters so I am able to type “i”,”f”,”i”,”c”,”a”,”b”,”i”. Then again I cannot continue because I have forgotten what were the next letters of the word, so I have to look it up again.in the script. That takes two more time intervals. Now I can continue my typing of “l”,”i”,”t”. At this point I stop again because I am not sure what were the next letters to type, so I have to think about it. That takes one time interval. I continue my typing with “u”,”d”,”i”. Then I stop again because I have again forgotten what were the next letters to type, so I have to look it up again in the script of “Love's Labours Lost”. That takes two more time intervals. Now I know what the next letter to type is so I can continue typing “n”,”i”.At this point I cannot recall the next letter. so I have to look it up again in the script. That takes two more time intervals. After I know what the next letter to type is I can continue typing “t”,”a”,”t”,”i”,”b”,”u”,”s”. Eventually I am done!

At the end of the exercise I have the word “Honorificabilitudinitatibus” typed and along with it a sequence of zeros and ones.

In the table we have separated the manual work of typing from the knowledge work of thinking about what to type.

We made visible both the manual work and the knowledge discovery parts of a Knowledge Discovery process.

The first row of the table shows the knowledge I manually transformed into tangible outcome - in this case the longest English word. The second row of the table shows the way I discovered that knowledge. There is a "0" for each time interval when I was missing information about what to type next. There is "1" for each time interval when I had prior knowledge about what to type next. Each "0" represents a selection I needed to ask in order to acquire the missing information about what letter to type next. Each "1" represents prior knowledge.

We know that there is knowledge applied when we see the tangible outcome of the process. We know there was knowledge discovered when we see there was at least one selection made.

In the exercise above we witnessed the discovery and transformation of invisible knowledge into visible tangible outcome.

KEDE calculation

We can calculate the KEDE for this sequence of outcomes.

We can also calculate the knowledge discovered H(X|Y) in bits of information.

We've turned a real-world sequence of action and hesitation into a fine-grained, time-based measurement of regulatory capacity — effectively measuring how much variety I needed to absorb with external help i.e. my knowledge discovered.

Measuring software development

In order to use the KEDE formula (6) in practice we need to know both S and N. We can count the actual number of symbols of source code contributed straight from the source code files. For N we want to use some naturally constrained value.

N is the maximum number of symbols that could be contributed for a time interval by a single human being.

To achieve this, the following estimation is performed. We pick T = 8 hours of work because that is the standard length of a work day for a software developer.

To calculate the value of r we need to pick the symbol duration t.

The value of the symbol duration time t is determined by two natural constraints:

1. the maximum typing speed of human beings
2. the capacity of the cognitive control of the human brain

Typing speed has been subject to considerable research. One of the metrics used for analyzing typing speed is inter-key interval (IKI), which is the difference in timestamps between two keypress events. We see that IKI is defined equal to the symbol duration time t. Hence we can use the research of IKI to find the symbol duration time t. It was found that the average IKI is 0.238s [26]. There are many factors that affect IKI [6]. It was also found that proficient typing is dependent on the ability to view characters in advance of the one currently being typed. The median IKI was 0.101s for typing with unlimited preview and for typing with 8 characters visible to the right of the to-be-typed character but was 0.446s with only 1 character visible prior to each keystroke [7]. Another well-documented finding is that familiar, meaningful material is typed faster than unfamiliar, nonsense material[8]. Another finding that may account for some of the IKI variability is what may be called the “word initiation effect”. If words are stored in memory as integral units, one may expect the latency of the first keystroke in the word to reflect the time required to retrieve the word from memory[55].

Cognitive control, also known as executive function, is a higher-level cognitive process that involves the ability to control and manage other cognitive processes that permit selection and prioritization of information processing in different cognitive domains to reach the capacity-limited conscious mind. Cognitive control coordinates thoughts and actions under uncertainty. It's like the "conductor" of the cognitive processes, orchestrating and managing how they work together. Information theory has been applied to cognitive control by studying the capacity of cognitive control in terms of the amount of information that can be processed or manipulated at any given time. Researchers found that the capacity of cognitive control is approximately 3 to 4 bits per second[32][33], That means cognitive control as a higher-level function has a remarkably low capacity.

Based on the above research we get:

1. Maximum typing speed of human beings to be r=1/t=1/0,238=4.2 symbols per second
2. Capacity of the cognitive control of the human brain to be approximately 3 to 4 bits per second. Since we assume one question equals one bit of information we get 3 to 4 questions per second.
3. Asking questions is an effortful task and humans cannot type at the same time. If there was a symbol NOT typed then there was a question asked. That means the question rate equals the symbol rate, as explained here.

In order to get a round value of maximum symbol rate N of 100 000 symbols per 8 hours of work we pick symbol duration time t to be 0.288 seconds. That is a bit larger than what the IKI research found but makes sense when we think of 8 hours of typing. Having t of 0.288 seconds makes a symbol rate r of 3.47 symbols per second. That is between 3 and 4 and matches the capacity of the cognitive control of the human brain.

We define CPH as the maximum rate of characters that could be contributed per hour. Since r is 3.47 symbols per second we get CPH of 12 500 symbols per hour. We substitute T = h and r=CPH and the formula for N becomes:

where h is the number of working hours in a day and CPH is the maximum number of characters that could be contributed per hour. We define h to be eight hours and get N to be 100 000 symbols per eight hours of work.

Total working time consist of four components:

• Time spent typing (coding)
• Time spent figuring out WHAT to develop
• Time spent figuring out HOW to code the WHAT
• Time doing something else (NW)

Let us assume an ideal system where the time spent doing something else T_NW is zero. Using the new formula for N the formula for H becomes

Note, that since N is calculated per hour so S also needs to be counted in an hour.

We see that the more symbols of source code contributed during a time interval the less missing information was there to be acquired. We want to compare the performance of different software development processes in terms of the efficiency of their knowledge discovery processes. Hence we rearrange the formula to emphasize that.

The right hand part is the KEDE we defined earlier. Thus, we define an instance of the metric KEDE - the general metric that we introduced earlier. This version of KEDE is for the case of knowledge workers that produce tangible outcome in the form of textual content:

KEDE from (8) contains only quantities we can measure in practice. KEDE also satisfies all properties we defined earlier. it has a maximum value of 1 and minimum value of 0; it equals 0 when H is infinite; it equals 1 when H is zero; it is anchored on a natural constraint—the maximum typing speed of a human being.

If we convert the KEDE formula into percentages then it becomes:

We can use KEDE to compare the knowledge discovery efficiency of software development organizations.

Testing Intelligence

Today all measure intelligence by the power of appropriate selection (of the right answers from the wrong). The tests thus use the same operation as is used in the theorem on requisite variety, and must therefore be subject to the same limitation. (D, of course, is here the set of possible questions, and R is the set of all possible answers). Thus what we understand as a man's “intelligence” is subject to the fundamental limitation: it cannot exceed his capacity as a transducer. (To be exact, “capacity” must here be defined on a per-second or a per-question basis, according to the type of test.)[3]

We can also use our model to the testing of human and AI intelligence. We infer this capacity from performance under variety — i.e., how many different problems a system or a person can solve correctly.

The dominant mathematical models for testing intelligence by the number of answered problems are benchmark datasets like MMLU, GSM8K, MATH, and FrontierMath. These models measure intelligence by the raw count or percentage of correctly solved problems, with more advanced benchmarks designed to minimize guessing and require deep reasoning.

From the knowledge-centric perspective:

• The disturbances are the questions $Q=\{q_1,q_2,q_3,...q_n\}$
• The person gives responses $R=\{r_1,r_2,r_3,...r_n\}$
• The outcomes are $E=\{e_1,e_2,e_3,...e_n\} \in \{0,1\}$

Several mathematical models and benchmark datasets are used to evaluate intelligence—especially artificial intelligence (AI)—by measuring the number and complexity of math problems answered correctly. These models serve as standardized tests for both AI and, by analogy, human intelligence[52].

Massive Multitask Language Understanding (MMLU):

• MMLU is a widely used benchmark that tests AI models on a broad range of subjects, including mathematics at various levels (high school, college, abstract algebra, formal logic).
• The test is typically formatted as multiple-choice questions, and performance is measured by the percentage of correct answers out of the total number of questions
• For example, advanced AI models have achieved up to 98% accuracy on math sections of MMLU, indicating high proficiency in standard math tasks but not necessarily deep reasoning

Grade School Math 8K (GSM8K)

• GSM8K is a dataset of 8,500 high-quality, grade school-level word problems designed to test logical reasoning and basic arithmetic skills.
• Evaluation is based on exact match accuracy: the number of problems answered exactly correctly divided by the total number attempted
• This benchmark is used to assess step-by-step reasoning and the ability to handle linguistic diversity in problem statements.

MATH (Mathematics Competitions Dataset)

• MATH consists of problems from high-level math competitions (e.g., AMC 10, AMC 12, AIME), focusing on advanced reasoning rather than rote computation.
• Performance is measured by the percentage of correct answers, with human experts (e.g., IMO medalists) providing a reference for top-level performance
• The dataset is challenging for both humans and AI, with LLMs typically scoring much lower than expert humans.

FrontierMath[53]

• FrontierMath is a new benchmark featuring hundreds of original, expert-level math problems spanning major branches of modern mathematics.
• Problems are designed to be "guessproof" and require genuine mathematical understanding, with automatic verification of answers
• The benchmark is used to assess how well AI models can understand and solve complex mathematical problems, similar to human performance.

In human intelligence testing, Psychometric models such as IQ tests or psychometric approaches also use the number of correctly answered problems as a key metric. These tests are standardized, and the raw score (number of correct answers) is often converted into a scaled score or percentile.

As an example we will use the Exact Match metric as the evaluation method[52]. Given that each question in our benchmark dataset has a single correct answer and the model produces a response per query, Exact Match ensures a rigorous evaluation by comparing the extracted answer to the ground truth.

Let ${ŷ}_i$ represent the extracted answer from the model's outcome for the $i^{\mathrm{th}}$ question, and let $y_i$ be the corresponding ground truth answer. The Exact Match accuracy is computed as:

where:

• $N$ is the total number of evaluated questions.
• $𝟙(\cdot )$ is the indicator function, returning 1 if the extracted model response matches the ground truth after preprocessing, and 0 otherwise.
• $\text{normalize}(\cdot )$ is a function that standardizes formatting, trims spaces, and normalizes numerical values.

The knowledge discovery efficiency of an LLM can be calculated as:

Let's pick the case of the performance of GPT-4o on the MATH benchmark, which achieved a significantly lower accuracy of 64.88%, lagging behind its peer models[52]. Now, we can calculate the average knowledge discovered H(X|Y).

Basketball Game

We can also use this model to assess the performance of a basketball player.

• Timeframe is a basketball game.
• We observe N total shot attempts.
• S of them are successful (shot made).
• We record a binary outcome sequence
  $E\in \{0,1{\}}^N$
• The empirical success rate:
  $\theta =\frac{S}{N}$
  is our observed probability of success.

Interpretation using Ashby's Law

The basketball shot is a regulation problem: the player must control their body and respond to the game environment to produce the desired outcome. The player is faced with a series of disturbances (D) in the form of different shots to make under different conditions. The player responds with a selection, drawn from their internal skills (regulatory variety R) in the form of different shooting techniques. Each shot is uncertain whether it will be successful. The outcome E is whether the shot is made (1) or missed (0).

Over N shots, the success rate

In this case, θ becomes a practical proxy for how often the regulator (player) has sufficient internal variety to absorb the disturbance presented by the game. However, it is important to note that this is a simplified model and does not account for all the complexities of basketball performance. For example, the player may have different success rates depending on the type of shot, the position on the court, or the level of defense. These factors can all affect the player's ability to regulate their performance and should be considered when interpreting the results. Thus, as explained here θ is a useful heuristics for P(E=1), but the full picture includes the quality of mapping, not just quantity.

Applying the Model

NBA keeps track of field goal attempts and makes for each player. The most field goal attempts by a player in a single NBA game is 63, achieved by Wilt Chamberlain during his legendary 100-point game against the New York Knicks on March 2, 1962 We take this as the natural constraint so N=63. We can also take the number of successful shots S=36, which is the most field goals made in a single game by a player[13].

We can calculate the KEDE for this sequence of outcomes.

We can also calculate the knowledge discovered H(X|Y) in bits of information.

That means that the player needed to absorb 0.75 bits of information on average to make the shot.

We've turned a real-world sequence of basketball shots into a fine-grained, time-based measurement of a regulatory capacity — effectively measuring how much variety the player needed to absorb.

We can also use this model to assess the performance of a basketball team. In this case the success rate coincides with the field goal percentage (FG%) of the team which is the percentage proportion of made shots over total shots that a player or a team takes in games. There is a statistical distribution for NBA field goal percentage (FG%) [10]. Analysts and researchers often study the distribution of FG% across players or teams to understand scoring efficiency and trends[11]. The NBA record for the highest FG% in a single game by a team is 69.3%, set by the Los Angeles Clippers on March 13, 1998, when they made 61 of 88 shots[12].

For example, in the 2023-24 season, team FG% ranged from about 43.5% (lowest) to 50.6% (highest), with the league average typically falling in the mid-to-high 40% range[11]. if we take the average FG% of 45% , we can calculate the average knowledge discovered H(X|Y).

That means that a team needed to absorb 1.22 bits of information on average to make a shot.

Assembly Line

We can also use this model to assess the knowledge discovery efficiency of an assembly line.

The assembly line is a system that transforms raw materials into finished products. The assembly line has a set of disturbances (D) in the form of different raw materials, machines, and processes. The assembly line responds with a selection, drawn from its internal structure (R) in the form of different machines, processes, and workers.

From a knowledge-sentric perspective most of the knpwledge discovery happens in the design phase of the assembly line. This is the planning for design, fabrication and assembly. This activity has also been called design for manufacturing and assembly (DFM/A) or sometimes predictive engineering. It is essentially the selection of design features and options that promote cost-competitive manufacturing, assembly, and test practices[51]. Thus most of the disturbances D are already absorbed by the design of the assembly line. That means when the workers have most of the knowledge built into the assembly line and the operational procedures.

Assembly line efficiency (AE) is the ratio of the outcome to the maximum possible outcome, often expressed as a percentage.

The efficiency of the assembly line can be calculated as:

We can assume that an assembly line is designed to produce a certain number of successful products (S) with a maximum rate of N products per hour. So for example, a shoe manufacturer has an actual outcome of 100 shoes per day, and a maximum potential outcome of 120 shoes per day. Their production line efficiency would be 83%. Now, we can calculate the average knowledge discovered H(X|Y).

To optimize the AE, companies can apply DFA guidelines, such as minimizing the number and variety of parts, standardizing the fasteners and connectors, and simplifying the assembly sequence and orientation[51].

Interpreting the results involves a comprehensive analysis of the data to understand where and why inefficiencies occur. In general, the higher the AE, the better the design. On the other hand, AE close to 100% might indicate under-utilised capacity. It's essential to compare high efficiency with industry capacity standards to determine if an increase in production is feasible and beneficial.

If AE is consistently below industry benchmarks, this could highlight several potential issues:

• Machinery: It may indicate that machines are outdated, malfunctioning, or not suitable for the required tasks.
• Labour Skills: Low efficiency might be due to workforce training gaps.
• Process Design: Sometimes, the workflow or layout of the production line itself causes inefficiencies.

Speed of Light in Medium

We can also use this model to support an interpretation of Ashby's Law of Requisite Variety to assess the speed of light in a medium where the medium acts as a disturbance to photon flow. Here's how this perspective aligns with the physics of light-matter interactions:

• Disturbance: The medium's atomic/molecular structure introduces spatial and electromagnetic inhomogeneities (e.g., refractive index variations, turbulence).
• Control Mechanism: Photons' ability to "counteract" disturbances through wavelength compression and phase synchronization.
• Requisite Variety: Photons require sufficient adaptability (e.g., frequency range, polarization states) to navigate the medium's complexity without scattering or losing coherence.

The speed of light in a vacuum is 299,792,458 m/s. In a medium, the speed of light is reduced by a factor n, called the refractive index defined as:

The refractive index is a measure of how much the speed of light is reduced in the medium. The higher the refractive index, the more the speed of light is reduced.

For example, the refractive index of water is 1.33, which means that the speed of light in water is:

The knowledge discovery efficiency of the speed of light in a medium can be calculated as:

Now, we can calculate the average knowledge discovered H(X|Y) by a photon in water:

What learning could also do (but we are explicitly excluding)

Not every form of learning improves regulation H(E|Y) in Ashby's sense. Other possibilities include:

1. Expanding action variety without selectivity

   Learning might increase $H\left(X\right)$ (more possible actions, tools, behaviors) without reducing $H\left(X\right|Y)$.

   • The system becomes more capable in principle
   • But still does not know which action to take
   • Regulation does not improve

   This violates Ashby's requirement that variety must be constrained, not merely expanded.

2. Improving buffering instead of knowledge

   Learning might increase buffering capacity $K$ (delay, slack, tolerance), so disturbances are absorbed without better action selection.

   • Outcomes may improve
   • But $I(X:Y)$ does not increase
   • Regulation improves without learning the mapping

   This is explicitly separated from knowledge in Ashby's extended formulation.

3. Changing goals or success criteria

   Learning could redefine what counts as success $E$.

   • Apparent performance improves
   • But the structural coupling (mapping) is unchanged
   • Information-theoretically, nothing about $H\left(X\right|Y)$ need change

   This is semantic drift, not cybernetic learning.

4. One-off adaptation without structural retention

   The system may succeed through exploration $Z$ without storing the result.

   • Regulation succeeds this time
   • Next encounter repeats the same uncertainty
   • No accumulation of $I(X:Y)$

   This is regulation, not learning.

Cumulative Knowledge To Be Discovered

Using

Cumulative w.r.t. S
Choose a baseline $S_0$ > 0.
Define:

Key properties:

• $\frac{dC}{dS}=H\left(S\right)=\frac{N}{S}-1$
• $\frac{d^{2}C}{d^{2}S}=-\frac{N}{S^2}<0$ → $C$ is concave in $S$.

Domain: $S$ ∈ (0, N]. Since $H\left(S\right)$ > 0 for $S<N$, $C(S;S_0)$ increases with $S$ (for $S\ge S_0$) and is finite as long as $S_0>0$.

Useful normalizations:
Dimensionless form with $\widehat{S}=\frac{S}{N}$: $$\frac{C\left(\widehat{S};{\widehat{S}}_0\right)}{N}=ln\frac{\widehat{S}}{{\widehat{S}}_0}-\left(\widehat{S}-{\widehat{S}}_0\right).$$

Total cumulative up to completion S = N: $$C(N;S_0)=N\cdot ln\left(\frac{N}{S_0}\right)-(N-S_0).$$

This can be thought of as the total knowledge-effort curve or “cumulative residual variety as a function of performance level” i.e. how much “knowledge work” has been consumed to reach performance level S.

Residual Variety

In the context of Ashby's Law of Requisite Variety, residual variety refers to the variety that the regulator fails to absorb. In other words, the remaining uncertainty or uncontrolled states in a system after a regulator has applied its available counter-actions. Information-theoretically, this can be understood in two ways:

• The residual variety is the uncertainty H(X|Y) about the regulator's response given the disturbance i.e., the regulator's remaining uncertainty about what action to take given a known disturbance Y. It focuses on the input side - how much the regulator doesn't know about the disturbance that is hitting the system.
• The residual variety is the uncertainty H(E) in essential variables. It focuses on the outcome side - how much uncertainty persists in what we care about the essential variable.

Both interpretations quantify how much uncertainty is left once the regulator has made its move, because H(X|Y) upper-bounds the achievable reduction in H(E) given the fixed table T. If the regulator perfectly counters every disturbance (full requisite variety), the residual variety in both forms would be zero.

Observability of Residual Variety

Now we focus on one important consideration, which is that E (essential variable) values are observable. These are the outcomes we can measure and care about - system performance, outcome quality, stability measures, etc. Being observable, we can empirically estimate H(E) by collecting data on how E varies given different regulator states X This makes H(E) (residual variety) a measurable quantity in practice.

H(X|Y) presents observability challenges, as disturbances Y may not be directly observable i.e. they could be internal system dynamics, environmental factors we can't measure, or complex interactions we can't decompose. Even if some disturbances are observable, the full set Y might include hidden or latent factors that we cannot directly measure or quantify. This makes H(X|Y) potentially unobservable or only partially estimable and often theoretical or abstract.

This creates an important asymmetry in cybernetic analysis:

• H(E) is observable and measurable via H(B), allowing us to empirically assess how well the regulator is performing in controlling outcomes.
• H(X|Y) may not be fully observable, making it difficult to quantify the regulator's knowledge of disturbances it's regulating.

This is why H(E) is often the more practically useful measure - it tells us what we can observe about system performance. That is reflected in the literature as well, where work focuses on observable outcomes rather than unobservable disturbances[29,30,35,36,37,38,39,40,41,42,43,44,45].