10 The meme as an abstract constructor
(ns meme-constructor.posts.post-01-meme-as-constructor
(:require [scicloj.kindly.v4.kind :as kind]
[meme-constructor.tasks :as t]
[meme-constructor.figures :as fig]))Post 1 of Part II, “Memes as Abstract Constructors”.
A meme is what a brain can do with an idea, not the brain state that encodes it. Copying happens on two levels. The channel (sounds, words, writing, a recited text, a fixed set of behaviours) is a closed alphabet that is fixed in the receiver at the time of copying. Copying on it is digital, and the copy can be made arbitrarily accurate. The function is not a string: the receiver rebuilds it from the evidence, from reasons. Two heads share the signal alphabet but not the concepts. An exact copy of the function can happen. It can never be guaranteed, and it can never be confirmed. So no such copy is ever treated as final: it stays open to criticism even when it is exact. An ape meme is the case where a string on the closed level fixes the whole repertoire. A culture meme is carried by a closed channel, but its function lives on the open level. Sperber’s attractors explain why copying is faithful on both levels: the symbols of the fixed alphabet on the closed level, hard-to-vary content on the open level.
That stance is fallibilism (C10). Part I defines it as a property of a process (post I.6). This post shows why copying the function of a meme needs it. Two further points come out of the same analysis. For an open receiver the evidence is a criterion, not a source. And whether a meme is hard to vary depends on the receiver’s conjecture space, not on the meme alone.
The post first imports the constructor theory the book needs. It then defines a meme and its copy task, proves what can be said about the copy error ε, and checks it in a small toy model. The toy tests the definitions. It does not model brains.
10.1 1. Imported: constructor theory in five definitions
Everything in this section is taken from Deutsch (2013), Deutsch & Marletto (2015) and Marletto (2015). We only restate it.
Definition 1.1 (Substrate, attribute, task; Deutsch 2013). A substrate is a physical system that gets transformed. An attribute is a set of states of a substrate, for example “red” or “contains the sequence ACGT”. A task is a set of pairs of input and output attributes, written \(\{a \to b\}\).
Definition 1.2 (Constructor, accuracy, reliability, possible; Deutsch 2013 §1.1; Deutsch & Marletto 2025 §2–3). A constructor for a task performs the task and afterwards keeps the ability to perform it again: a catalyst, a machine, a program running on a computer. Real devices are approximate constructors. Their accuracy is how close one run comes to the output the task asks for. Their reliability is how well they keep that accuracy over repeated use (Deutsch & Marletto 2025, §2). Write \(\alpha(K, T)\) for the inaccuracy and \(\rho(K, T)\) for the unreliability of \(K\) on \(T\).
A task is possible, written \(T^{\checkmark}\), if “the laws of nature impose no limit, short of perfection, on how accurately A could be performed, nor on how well things that are capable of approximately performing it could retain their ability to do so” (Deutsch 2013, §1.1). Otherwise it is impossible, \(T^{\text{✘}}\) (Deutsch & Marletto 2025, eqs. 2–3). In our paraphrase:
\[\begin{aligned} T^{\checkmark} &\iff \forall \varepsilon > 0\ \exists K_\varepsilon:\ \alpha(K_\varepsilon, T) \le \varepsilon \ \text{and}\ \rho(K_\varepsilon, T) \le \varepsilon \\ T^{\text{✘}} &\iff \neg\,\big(T^{\checkmark}\big) \end{aligned}\]
The \(\neg\) negates the statement “\(T\) is possible”, not the task. So “possible” means “arbitrarily accurate and arbitrarily reliable”, never “exact”. Deutsch adds: “Presumably no perfect constructors can exist in nature.”
Definition 1.3 (Information variable; Deutsch & Marletto 2015). A set of attributes is an information variable if permuting its elements and copying them are possible tasks. A substrate whose attributes form one is an information medium. The interoperability principle says that information can be copied between physically different media. So information is abstract, not tied to one substrate, and physical, defined by possible tasks on physical substrates.
Definition 1.4 (Abstract constructor, knowledge; Deutsch 2013 §2.14–2.15). Athens restores the Ship of Theseus plank by plank over generations. Neither the ship, nor the scroll with its plan, nor the Athenians persist. What persists is “the abstract information expressed in the scroll and the ideas”, which “meets the definition of a constructor more closely than anything else in the situation. It is an abstract constructor.” And: “The most important kind of abstract constructor is knowledge”: information that, once instantiated in a suitable environment, tends to remain instantiated. The Athenians’ ideas must contain “knowledge of how to cause Athenians to preserve those ideas themselves through the generations, and to reject rival ideas”. That sentence is already about memes. Part I makes the abstract constructor precise as the class \(K_\sigma\) of all programs that perform a task \(\sigma\), whatever their text (post I.1, Definition I.1.6).
Definition 1.5 (Self-reproducer architecture; Marletto 2015). Accurate self-reproduction under laws without design needs digital information media and von Neumann’s (1966) architecture:
\[\begin{aligned} S &= R \oplus V \\ V &= C \oplus P \end{aligned}\]
with recipe R (the replicator), vehicle V, copier C and programmable constructor P. The two tasks are
\[\begin{aligned} \text{replication:}\quad & R \oplus N \to R \oplus R \\ \text{construction:}\quad & R \oplus N \to R \oplus V \end{aligned}\]
from generic resources N. The recipe is digital and modular, and V is rebuilt from R each time, so errors in V are not inherited. Marletto calls the information in the recipe “an abstract constructor that I shall call knowledge (without a knowing subject)” (§3.1).
For memes the architecture holds in full only where a text is copied from the text. Elsewhere on the closed level the fixed alphabet gives high fidelity by another route (Conjecture 1.20, Section 3.1). The function of a culture meme is not a recipe.
For Marletto the abstract constructor is the genetic information, not the DNA molecule. Calling a meme an abstract constructor applies her term to another substrate. What is new, if anything, is how the meme is individuated and who performs its copy task. The recipe and vehicle architecture itself carries over to the closed level of every meme (Conjecture 1.20, Section 3.1). The function of a culture meme is not a recipe.
10.2 2. The meme as a class of brain states
The standard copy task on two substrates,
\[C = \{ (x, x_0) \to (x, x) \mid x \in X \},\]
assumes that \(x\) is the same attribute in both. For DNA it is: the same base sequence is the same attribute in every cell. For brains it is not. Two brains encode the same idea in physically different states, and no brain state is the idea. So the attribute must be defined without referring to the encoding.
We define it by what the brain can do. Part I makes the same move for programs. A brain is a computing system (C1, Definition I.1.1) whose output medium is the muscles, and a meme is one of its patterns.
Definition 1.6 (Task universe, repertoire). Fix a context and a set \(\mathcal{T}\) of tasks, the task universe: explaining something, predicting it, doing it, justifying it, passing it on. The repertoire \(T(b) \subseteq \mathcal{T}\) of a brain state \(b\) is the set of tasks in \(\mathcal{T}\) for which \(b\) is a constructor.
Definition 1.7 (Meme, carrying). A meme \(m\) is given by a repertoire \(T_m \subseteq \mathcal{T}\). A brain state \(b\) carries \(m\) iff \(T_m \subseteq T(b)\). The attribute “carries m” is
\[x_m = \{\, b \mid T_m \subseteq T(b) \,\}.\]
So \(x_m\) is Part I’s abstract constructor \(K_{T_m}\) with brain states in place of programs. It groups physically different states by one functional property, and any substrate whose states have repertoires can carry \(m\). That puts memes in the same category as Deutsch’s and Marletto’s abstract genes. For genes the functional class and the physical attribute coincide. For memes they do not.
Three consequences are easy to miss.
From outside, membership in \(x_m\) is known only from above. An observer can try finitely many tasks. A failed task rules membership out. A passed test does not confirm it, because two programs can agree on every tested input and differ on an untested one (post I.1, Observation I.1.7). Finite tests suffice only where a bound is known, as for polynomials of bounded degree (Proposition I.1.9). For brains no such bound is known. Speed is not part of the class either: members of one abstract constructor can differ about thirtyfold in time (Proposition I.1.12).
Definition 1.7 says what it is to carry a given \(T_m\). It does not say which subsets of \(\mathcal{T}\) are memes. Repertoires overlap (“explain the tides” belongs to Newtonian gravity and to a theory of the moon’s orbit), and they are holistic: whether a brain can perform a task of one meme depends on the others it carries (Quine 1951 made this point for the meaning of sentences). A candidate unit is the smallest repertoire that can be copied as a whole. It depends on the receiver and need not be unique. The toy model below fixes the unit by fiat: a meme is the repertoire of one hypothesis from a fixed class.
10.3 3. The copy task and its error
10.3.1 3.1 Copying is rebuilding from evidence
The receiver never gets the meme. It gets evidence \(e_m\): the source’s behaviour, words and artefacts.
Definition 1.8 (Copy task of a meme). The meme copy task is Part I’s copy task (Definition I.3.7) with the meme \(m\) as original:
\[C_m = \{ (e_m, x_0) \to (e_m, x_m) \} \quad \text{on } E \oplus B .\]
\(E\) is the evidence medium, \(B\) the receiving brain, and \(x_0\) the receiver’s state before, in which it does not carry \(m\). \(x_0\) is not a blank slate: the receiver is an explainer with memes of its own (Deutsch 2011). The source supplies hints, and the idea is rebuilt in the receiver.
Like every copy task, \(C_m\) is synthesis from examples (C4, post I.3, Definition I.3.2). The evidence, a finite set of the source’s answers, is the specification, and the receiving brain is the program medium. \(C_m\) asks for a state that performs all of \(T_m\), and \(T_m\) is in general much larger than what \(e_m\) shows. The receiver is handed a weaker specification than the one it has to meet. The gap is filled from its own conjecture space.
For an open receiver, evidence is a criterion, not a source. If the receiver is self-reaching (C3, Definition I.2.6), every computable extension of the evidence is a conjecture it can form, and no finite evidence picks one of them (Proposition I.3.8). The evidence rejects reconstructions. It does not produce them. So the constructor for \(C_m\) must propose conjectures of its own: it is the receiver’s creativity, together with the source and the channel.
The contrast case is a closed receiver. If its conjectures are integer polynomials of degree at most \(d\), then \(d+1\) answers fix the meme and interpolation computes the copy with no search and no test (Proposition I.3.9). There the evidence is the source, because the knowledge that closes the gap sits in the conjecture space.
- no finite evidence picks one of the computable extensions
- the evidence rejects reconstructions; creativity proposes them
- polynomials of degree ≤ d: d+1 answers fix the meme
- interpolation computes the copy, with no search and no test
Two levels of copying.
The closed receiver of the polynomial example is a formal contrast case. Transmission between people and animals has the same two cases, and they are two levels of one copy task, not two kinds of meme. The channel carries the evidence, and sender and receiver share its alphabet. The function is what the receiver builds from that evidence, and there they share no alphabet of concepts.
The closed level is the channel: the steps of an ape’s technique, the sounds of a recited text, the symbols of a checklist or of source code, the phonemes, words and letters of a language. It is a closed alphabet that is fixed in the receiver at the time of copying. It does not matter where the alphabet came from. It may be innate, learned in early childhood or invented. That it is fixed is the assumption of this level, and the fixed alphabet is its central property. A meme whose repertoire is fixed by a string on the closed level is an ape meme. The receiver parses what it observes into the alphabet, with errors. A young gorilla watching an older one prepare nettles does not copy the movement. Gorillas prepare nettles in a standard sequence of steps, while the hand movements within the steps differ from animal to animal (Byrne & Byrne 1993). Byrne and Russon (1998) call the copying of such a sequence program-level imitation, and Byrne (1999) proposes that it works by behaviour parsing: the observer picks out recurring patterns in the stream of behaviour and copies its organisation, without understanding why it works. We read this as parsing into a fixed alphabet: the young gorilla parses the movement into steps it already has. Because the alphabet is fixed, a deviation smaller than the distance between two symbols snaps to the right symbol and never reaches the copy of the string. (It can still travel on a separate, analog path that is not part of the string: children pick up an accent.) What remains are errors of whole symbols, and watching more often or taking the majority over several sources makes those as rare as one likes. That deviations snap to the right symbol, and that the errors left can be made as rare as one likes, is our argument from the digital alphabet, not a finding of Byrne’s. So the copy of the string is arbitrarily accurate, and by Definition 1.2 the copy task is possible. We conjecture that the symbols of the alphabet are attractors in Sperber’s (1996) sense: every rendition is pulled to one of them. Its limits lie elsewhere: a closed alphabet is expensive to pass on, and it can only express what the alphabet composes (Deutsch 2011, ch. 16).
The open level is the function of the meme (Definition 1.7), and a meme whose function lives on this level, carried by a closed channel, is a culture meme. The function is not a string. Two heads share the signal alphabet but not the concepts, since each builds its own concept for the idea. The receiver does not execute the signal, it rebuilds the function, in the open mode, from the reasons the evidence gives, with its own creativity. An utterance, a text or a behaviour is a recipe-like signal that is read as evidence that lasts. The word dog shows both levels: the phonemes and the word are copied digitally, and what the word means to the child is built by the child. This is the case of the rest of this section: the evidence is a criterion, and no exact copy of the function can be guaranteed or confirmed (Proposition 1.11).
Conjecture 1.20 (Two levels of copying). Copying a meme happens on two levels. The closed level is the channel, a closed alphabet fixed in the receiver at the time of copying, whether innate, learned or invented. Every meme has a closed level, and on it the meme is a string over the alphabet, which the receiver parses with error. The fixed alphabet is what makes its copies arbitrarily accurate. It gives the closed level high fidelity by another route than Marletto’s separation of recipe and vehicle (Definition 1.5), but not all of what the separation gives: the string is mostly read back from a performance, and redundancy (watching often, majority over sources) makes the errors of single renditions as rare as one likes, though at any finite redundancy a small error still enters the string each generation (post 6, Proposition 6.3); an error that every rendition shares stays. A recipe is copied from a recipe only where a text is copied from the text. The open level is the function. The receiver rebuilds it from the closed-level evidence: an utterance is a recipe-like signal that the receiver reads as evidence and does not execute. An ape meme is the case where a string on the closed level fixes the whole repertoire, and a culture meme is the case where the function lives on the open level. The prediction: variants in ape traditions come in whole elements, a step swapped, inserted or missing, with no gradual drift of one step into another, and fidelity rises with the number of observations. The conjecture is refuted if, in an ape tradition, a step of the transmitted sequence drifts gradually into another over generations.
Behaviour parsing is Byrne’s. That its fixed alphabet makes the copy arbitrarily accurate is ours.
So the answer to Sperber (1996) takes his own terms. He holds that transmission is mostly transformation, and that traditions stay stable because reconstructions are pulled towards attractors. We read his attractors as what makes copying faithful, not as its rival. On the closed level, we conjecture, they are the symbols of the fixed alphabet, the central property of that level: the factor of attraction is the receiver’s parsing, which snaps every rendition to a symbol, so copying is digital. On the open level the attractor is the meme’s hard-to-vary content (Conjecture 1.19), and the factor is the receiver’s test: most reconstructions that differ from the idea fail to explain what it explains and are thrown out, so reconstruction converges, and it is never confirmed exact. The open-level attractors are rational; the psychological attractors of post 6 pull towards what the hosts’ psychology rewards. What has no reason at all, not even an internal anchor (post 4), such as a name or a wording, has no attractor on the open level and is kept only on the closed level.
Every culture meme mixes both levels. A tradition that hands on its wording by rote runs on the closed level, and the reasons that make the wording worth keeping are rebuilt on the open level. The same holds for software: a fork copies the source as a string, a rewrite rebuilds the program from what it is for.
10.3.2 3.2 Loss and gain
Let \(b'_0\) be the receiver’s state before and \(b'\) its state after the transmission.
Definition 1.9 (Loss, gain).
\[\begin{aligned} \varepsilon^- &= T_m \setminus T(b') \\ \varepsilon^+ &= \big(T(b') \setminus T(b'_0)\big) \setminus T_m \end{aligned}\]
Gain is measured against \(T_m\) and against the receiver’s prior repertoire. Measured against the source’s whole repertoire, it would count everything the receiver knew before. A third term, the collateral loss \(T(b'_0) \setminus T(b')\) of prior tasks, is empty in every model of this book, so we leave it aside.
The gain holds errors and improvements alike. If the source answers \(q\) with \(a\) and the receiver with \(a'\), then \((q,a) \in \varepsilon^-\) and \((q,a') \in \varepsilon^+\). Whether a gain is an improvement takes a criterion, the subject of post 4. Below we split \(\varepsilon^+\) into conflicts, other answers to questions \(m\) answers, and extensions, answers to questions \(m\) leaves open.
A tiny example. The third argument is the receiver’s prior repertoire.
(t/epsilon #{:explain-tides :predict-eclipse :teach-it}
#{:explain-tides :teach-it :predict-comet :old-skill}
#{:old-skill}){:loss #{:predict-eclipse}, :gain #{:predict-comet}, :collateral #{}}Definition 1.10 (Measure, size of ε, fidelity). Let \(\mu\) be a measure on \(\mathcal{T}\) that weights tasks by importance, for example by how often the environment demands them. Then
\[\begin{aligned} |\varepsilon| &= \mu(\varepsilon^-) + \mu(\varepsilon^+) \\ \varphi &= \frac{\mu(T_m \cap T(b'))}{\mu(T_m)} \end{aligned}\]
\(\varphi\) is the fidelity of Definition I.3.7 with the meme \(m\) as original. For a \(\mu\) with full support, \(\varphi = 1\) iff the receiver carries \(m\). Summing loss and gain into one number is a choice, not a fact, so we report both parts where it matters.
10.3.3 3.3 No copy can be guaranteed or confirmed exact
Let \(M\) be the memes a source could carry, and \(M_e \subseteq M\) those that could have produced the evidence \(e\). A reconstruction \(\rho\) maps evidence to a receiver repertoire, possibly at random.
Proposition 1.11 (No guaranteed exact copy).
The proposition is about the open level, the function. On the closed level the receiver has a fixed alphabet, and there \(M_e\) is a single meme once the evidence is long enough (Section 3.1).
No guarantee. If \(M_e\) contains two memes \(m_1, m_2\) with \(T_{m_1} \neq T_{m_2}\), then every reconstruction \(\rho\) has an \(i \in \{1,2\}\) with
\[\Pr\big[\varepsilon(m_i, \rho(e)) = \emptyset\big] \le 1/2 ,\]
and a deterministic \(\rho\) has an \(i\) with \(\varepsilon(m_i, \rho(e)) \neq \emptyset\).
Open sources make this the rule. Let the task universe contain infinitely many questions, and let \(M\) contain every meme whose repertoire is the graph of a partial computable answer function, as it does when sources are self-reaching (C3). Then for every finite evidence \(e\), \(M_e\) contains infinitely many memes with pairwise different repertoires.
No confirmation. Let the receiver be effectively self-reaching (Definition I.2.6) and \(T_m\) infinite. Then no procedure, the receiver’s own included, confirms that a copy carries \(m\). Checks against further evidence can refute a copy, never certify it.
Proof. (1) Take the prior repertoire disjoint from \(T_{m_1} \cup T_{m_2}\). An empty \(\varepsilon\) for \(m_i\) means that the newly acquired repertoire is exactly \(T_{m_i}\). So “exact for \(m_1\)” and “exact for \(m_2\)” are disjoint events, their probabilities sum to at most 1, and one is at most 1/2. For deterministic \(\rho\) one has probability 0. (2) This is Proposition I.3.8 read for memes: a finite table has infinitely many partial computable extensions, with different graphs, and each fits \(e\). (3) The receiver’s reachable states form an acceptable language (post I.3), and “carries \(m\)” holds of an answer function without holding of any finite part of it. By the Rice–Shapiro theorem (Proposition I.3.4) it is not even semi-decidable. \(\square\)
The proof is trivial. The content lies in the premises. Part 2 turns the informal argument, that meme repertoires reach unboundedly many situations while evidence is finite, into a proof for self-reaching sources. It does not prove that human sources are self-reaching. That is Deutsch’s thesis (2011, chs. 3 and 6), and Part I defines the property without establishing it for brains (post I.2). Nor does the proposition give Popper’s stronger claim that there is no inductive justification at all. A receiver whose prior matches how sources are distributed can be right with high probability. What is ruled out is certainty, not reliability.
One might conclude that the copy error is never zero. It does not hold, because single copies are exact: the toy model below produces exact copies from finite evidence (Observation 1.12). What holds is weaker and more useful.
- an exact copy cannot be told apart, from inside or outside, from one that fails on a question nobody has asked
- so each copy stays open to criticism, exact or not
That is fallibilism (C10), and it is a property of the process, not of the accuracy of a copy. Post I.2 gives it its structure: a persistently self-reaching system has no state that closes off the alternatives. Post I.6 defines it (Definition I.6.12) and finds two limits. Fallibilism creates knowledge only if criticism follows an anchor outside the process (Conjecture I.6.6). And it has a price: a correct idea that stays open to variation can only get worse, so how a process that never makes an idea final keeps a correct one is the keeping question of post I.5.
The result fits Definition 1.2. \(C_m\) can be possible, arbitrarily accurate as evidence and criticism accumulate, although no finite evidence guarantees exactness.
The closed case is the contrast, and it is the formal counterpart of the closed level (Conjecture 1.20). When source and receiver share a closed conjecture space, part 1’s premise can fail: \(d+1\) answers leave one polynomial of degree at most \(d\) (Proposition I.3.9), and a fixed procedure copies exactly every time. In a finite task universe, exhaustive evidence does the same. The toy model shows both.
10.4 4. A toy model
10.4.1 4.1 The world
The task universe has 24 questions \(x \in \{0,\dots,23\}\) and 5 possible answers \(y \in \{0,\dots,4\}\). A task is a pair \([x\ y]\), “answer \(x\) with \(y\)”. A hypothesis is a rule together with a domain interval \([lo, hi]\). Inside the domain it answers by the rule, and outside it gives no answer. The conjecture space has 505 rules in four families: constants, affine rules \(y = (ax + b) \bmod 5\), thresholds and parity rules. A meme is the repertoire of one hypothesis. Sources are drawn at random from the same space, with domains of at least 8 questions.
The conjecture space is finite, so every host in the toy is closed. The toy can show part 1 of Proposition 1.11 and the closed contrast. It cannot show an open receiver.
(def world {:n-questions 24 :n-answers 5})(def rules (t/all-rules world [:const :affine :threshold :parity]))(count rules)505Two hypotheses can have the same repertoire. On the domain \([3,3]\) the affine rule \(y = x\) and the constant rule \(y = 3\) both give the single task \([3\ 3]\). The meme is the repertoire, not the representation.
(= (t/repertoire world {:rule {:family :affine :a 1 :b 0} :domain [3 3]})
(t/repertoire world {:rule {:family :const :c 3} :domain [3 3]}))true10.4.2 4.2 The receiver: conjecture and refutation
Evidence is a set of distinct questions drawn uniformly, each with the source’s response (an answer, or nil for none). The receiver draws a rule from a uniform prior, counts how many evidence items refute it, and repeats until a rule survives or a budget of attempts is spent. It does the same for the domain. With a large budget this returns a uniform draw from all hypotheses consistent with the evidence. That is a deliberately weak receiver with no taste for good explanations. It shows what evidence and criticism alone achieve. In the terms of post I.3 it is generate-and-test: the generator never sees the evidence, and the evidence only rejects.
This is Popper’s (1972) schema of tentative theory and error elimination, run inside the receiver. For genes an error is eliminated by the death of the vehicle. For memes criticism eliminates the variant and the host survives: “Let our conjectures, our theories, die in our stead!” (Popper 1972, ch. 7).
(loop [k 1 best nil best-r Long/MAX_VALUE]
(let [h (conjecture rng)
r (long (criticise h))
[best best-r] (if (< r best-r) [h r] [best best-r])]
(if (or (zero? best-r) (>= k budget))
{:hypothesis best :refutations best-r :attempts k}
(recur (inc k) best best-r))))One transmission, with seed 42 and 6 evidence items:
(t/copy-run world {:rules rules :evidence-size 6} 42){:gain-extension 0,
:evidence-size 6,
:gain-conflict 1,
:fidelity 0.1,
:attempts 4,
:exact? false,
:source-size 10,
:gain 1,
:epsilon 10,
:loss 9,
:consistent? true}10.4.3 4.3 Error against evidence
200 transmissions for each evidence size from 0 to 24. Each size uses its own seeds, derived from master seed 1 plus the size.
(def sizes [0 1 2 4 6 8 12 16 20 24])(def sweep-rows (t/sweep world {:rules rules} sizes 200 1))(def colors ["#2a78d6" "#eb6834" "#1baf7a" "#eda100"])(defn line-chart [title x-title y-title series & {:keys [x-type]}]
(kind/plotly
{:data (map-indexed
(fn [i {:keys [name x y dash]}]
{:type "scatter" :mode "lines+markers" :name name :x x :y y
:line {:width 2 :color (colors i) :dash (or dash "solid")}
:marker {:size 8 :color (colors i)}})
series)
:layout {:title {:text title}
:xaxis (cond-> {:title {:text x-title} :zeroline false}
x-type (assoc :type x-type))
:yaxis {:title {:text y-title} :rangemode "tozero"}
:hovermode "x unified"
:legend {:orientation "h" :y -0.25}
:margin {:t 50 :r 20}}}))(line-chart "Exact vs merely consistent copies (200 runs per size, seed 1)"
"evidence items (distinct questions)" "fraction of runs"
[{:name "exact copy (ε = ∅)" :x sizes :y (map :exact sweep-rows)}
{:name "consistent with evidence but ε ≠ ∅" :x sizes
:y (map :consistent-but-wrong sweep-rows)}])(kind/table
(map #(update-vals % (fn [v] (if (double? v) (/ (Math/round (* 100 v)) 100.0) v)))
sweep-rows))| gain-extension | evidence-size | gain-conflict | fidelity | consistent-but-wrong | exact | gain | epsilon | consistent | loss |
|---|---|---|---|---|---|---|---|---|---|
| 2.84 | 0 | 4.69 | 0.05 | 1.0 | 0.0 | 7.53 | 20.38 | 1.0 | 12.85 |
| 2.57 | 1 | 3.9 | 0.27 | 1.0 | 0.0 | 6.46 | 16.02 | 1.0 | 9.56 |
| 2.15 | 2 | 2.9 | 0.42 | 1.0 | 0.01 | 5.05 | 12.81 | 1.0 | 7.76 |
| 1.58 | 4 | 1.95 | 0.6 | 0.99 | 0.01 | 3.53 | 8.63 | 1.0 | 5.1 |
| 1.33 | 6 | 1.28 | 0.75 | 0.97 | 0.04 | 2.61 | 5.86 | 1.0 | 3.26 |
| 0.89 | 8 | 0.95 | 0.81 | 0.91 | 0.09 | 1.83 | 4.28 | 1.0 | 2.44 |
| 0.73 | 12 | 0.46 | 0.91 | 0.83 | 0.18 | 1.19 | 2.37 | 1.0 | 1.18 |
| 0.33 | 16 | 0.28 | 0.95 | 0.62 | 0.39 | 0.62 | 1.3 | 1.0 | 0.68 |
| 0.18 | 20 | 0.1 | 0.98 | 0.38 | 0.63 | 0.28 | 0.56 | 1.0 | 0.28 |
| 0.0 | 24 | 0.01 | 1.0 | 0.0 | 1.0 | 0.01 | 0.01 | 1.0 | 0.01 |
Observation 1.12 (Exact copies occur, underdetermination remains; 200 runs per size, master seed 1). Mean \(|\varepsilon|\) falls monotonically with evidence and mean fidelity rises. Almost every receiver is consistent with all its evidence, yet at 8 items about nine in ten still hold a different meme than the source, and at 16 items more than half. Up to 8 items most of the gain is conflict, that is, wrong answers. From 12 items on, extension dominates, because the receiver’s domain guess overshoots. Exact copies are rare at first, one in 200 at 2 items, then 9 % at 8 items and 39 % at 16. At 24 items, the whole universe, 199 of 200 copies are exact. (The remaining run ran out of its budget of 3,000 attempts.)
So consistency with the evidence does not imply the same repertoire (part 1 of Proposition 1.11), exact copies from finite evidence do occur, and exhaustive evidence in a finite universe removes the underdetermination.
10.4.4 4.4 Criticism and dialogue
In the toy, criticism has a precise meaning: the number of conjectures the receiver may test. We fix 8 evidence items and vary the budget.
(def budgets [1 3 10 30 100 300 1000 3000])(def budget-rows
(mapv #(first (t/sweep world {:rules rules :budget %} [8] 200 2)) budgets))(line-chart "Criticism budget at 8 evidence items (200 runs, seed 2)"
"conjectures tested per part (log scale)" "mean |ε| (tasks)"
[{:name "|ε|" :x budgets :y (map :epsilon budget-rows)}]
:x-type "log")Observation 1.13 (Criticism helps up to the evidence limit; 200 runs per budget, master seed 2). With budget 1 the receiver keeps its first guess. \(|\varepsilon|\) falls fast up to about 30 attempts and then levels off at the value of Observation 1.12 for 8 items. Beyond that, all surviving conjectures fit the evidence equally well. More criticism then needs more evidence, or a better criterion than fit.
A dialogue gives better evidence. The receiver expresses its version, and the source answers a question on which the two disagree. We compare this with the source answering a random new question. Both start from 2 random evidence items.
(def dialogue-curves
(into {}
(for [mode [:dialogue :random]]
(let [runs (mapv #(t/dialogue-run world {:rules rules :start 2 :rounds 12 :mode mode} %)
(t/run-seeds 5 200))]
[mode (mapv (fn [i] (t/mean (map #(:epsilon (nth % i)) runs))) (range 13))]))))(line-chart "Dialogue vs random questions (200 runs, seed 5)"
"rounds after 2 initial items" "mean |ε| (tasks)"
[{:name "dialogue: source answers a disputed question" :x (range 13)
:y (:dialogue dialogue-curves)}
{:name "random new question" :x (range 13) :y (:random dialogue-curves)
:dash "dot"}])Observation 1.14 (Dialogue beats random evidence; 200 runs, master seed 5). With the same number of items, counterexamples chosen by the source lower \(|\varepsilon|\) much faster. After 12 rounds the dialogue curve is close to zero and the random curve is not. Every counterexample refutes the receiver’s current conjecture, while a random question often confirms it and teaches nothing. Teaching is joint error elimination, and it needs a source that can see the receiver’s version.
10.5 5. Hard to vary, relative to the receiver
DNA polymerase copies any sequence equally well. Whether \(C_m\) succeeds depends on what \(m\) says.
Definition 1.15 (Generic and content-specific copiers). Let \(\varphi_K(m)\) be the distribution of fidelity when copier \(K\) copies meme \(m\) under a fixed channel and amount of evidence. \(K\) is generic on a set of memes \(M\) if \(\varphi_K(m)\) is the same for all \(m \in M\), and content-specific otherwise.
Conjecture 1.16 (Meme copiers are content-specific). Human receivers are content-specific copiers on any set of memes that contains both explanations and arbitrary lists. The function of an explanation is copied on the open level, an arbitrary list on the closed level. Test: hold channel and evidence fixed (say, the same number of worked examples) and measure fidelity across memes. A generic copier predicts equal fidelity. Transmission-chain experiments in the tradition of Bartlett (1932) are of this kind. We have not checked them for this purpose.
In the toy, content specificity comes only from the receiver’s conjecture space. A stronger idea, that the meme is part of its copier’s program and demands to be passed on, is not modelled in this book.
Deutsch (2011, ch. 1) calls a good explanation hard to vary. Following De la Guardia (2020, 2025c) we picture it as a small point in idea space: almost every change does not make it slightly worse but functionless. Rules of thumb are clouds, whose neighbouring variants are about equally good.
We use a second world with more answers, so that chance agreement is low: 24 questions, 11 answers, total domain. The small point is the affine rule \(y = (3x + 7) \bmod 11\). The cloud is a random lookup table (seed 17). A variation is one edit of the representation: change \(a\) or \(b\) of the rule, or one entry of the table.
(def world-11 {:n-questions 24 :n-answers 11})(def affine-meme {:family :affine :a 3 :b 7})(def cloud-meme (t/random-table world-11 (java.util.Random. 17)))(defn rep-11 [rule] (t/repertoire world-11 {:rule rule :domain [0 23]}))(def edits (range 0 13))(defn mean-overlap [rule d seed]
(let [rng (java.util.Random. (long seed))]
(t/mean (repeatedly 200 #(t/fidelity (rep-11 rule) (rep-11 (t/vary world-11 rule d rng)))))))(line-chart "Overlap with the original after d edits (200 variants, seed 23)"
"representational edits d" "fidelity of variant"
[{:name "hard to vary: affine rule" :x edits :y (map #(mean-overlap affine-meme % 23) edits)}
{:name "cloud: lookup table" :x edits :y (map #(mean-overlap cloud-meme % 23) edits)}])Observation 1.17 (A jump vs a slope; 200 variants per \(d\), seed 23). One edit of the affine rule drops the overlap from 1 to about 0.06, near chance agreement (1/11 ≈ 0.09), and further edits keep it there. Each edit of the table removes one of 24 tasks. The drop is to chance level, not to nothing.
Now copying. Receiver A conjectures affine rules: it looks for an explanation. Receiver B conjectures lookup tables: it memorises and guesses the rest. Both know the domain is total. 300 runs per evidence size, with master seed equal to the size.
(def copy-sizes [1 2 3 4 8 16])(def affine-rules (t/all-rules world-11 [:affine]))(defn fidelity-runs [source receiver budget s]
(mapv #(:fidelity (t/copy-run world-11 {:rules affine-rules :domain :total
:source {:rule source :domain [0 23]}
:receiver receiver :budget budget
:evidence-size s}
%))
(t/run-seeds s 300)))(def copier-series
[["A (affine) copies affine meme" affine-meme :rules 3000]
["B (table) copies affine meme" affine-meme :table 1]
["B (table) copies cloud meme" cloud-meme :table 1]
["A (affine) copies cloud meme" cloud-meme :rules 300]])(def copier-results
(into {}
(for [[name src rcv budget] copier-series]
[name (mapv #(fidelity-runs src rcv budget %) copy-sizes)])))(line-chart "Mean fidelity by copier and meme (24 questions, 11 answers)"
"evidence items" "mean fidelity"
(for [[name] copier-series]
{:name name :x copy-sizes :y (map t/mean (copier-results name))}))Observation 1.18 (Bimodal vs graded fidelity; 300 runs per size, master seed = size).
- Receiver A on the affine meme is bimodal. At one evidence item about one copy in ten is exact and the rest sit near chance. From two items on almost every copy is exact, since two points fix an affine rule when their questions differ mod 11. A wrong affine conjecture is refuted by almost any further item, so being hard to vary also works as error detection. For A the evidence could be the source: two linear equations mod 11 give the rule without search, as in Proposition I.3.9.
- Receiver B is graded on either meme: fidelity is the share of questions seen plus chance on the rest. B is a generic copier. It copies the explanation exactly as badly as the list.
- Receiver A is content-specific. It copies the affine meme almost perfectly from two items and the cloud badly at any size.
The same task set, the graph of \(y = (3x+7) \bmod 11\), is a small point for receiver A and a cloud for receiver B. So being hard to vary is not a property of the task set alone.
Conjecture 1.19 (Hard to vary is relative to a conjecture space). For copy task and ε (F3), the bearer of hard-to-varyness is the pair (\(T_m\), conjecture space of the hosts): \(m\) is hard to vary for hosts \(H\) if, in \(H\)’s conjecture space, almost every neighbour of the simplest representation of \(T_m\) has near-chance overlap with \(T_m\). It does not depend on the measure \(\mu\), as long as \(\mu\) has full support, since the jump is in the overlap set. Test: compute the overlap curve of Observation 1.17 for the same meme under two different host conjecture spaces. The conjecture predicts different curves.
The evidence showed the same relativity in Section 3.1. Whether it can produce a copy or only criticise one depends on the receiver’s conjecture space, and so does whether a meme is hard to vary. In both cases part of the knowledge that makes a copy exact lies in the receiver, not in the meme or the evidence.
10.6 Summary
- A meme is a class of brain states: a repertoire \(T_m\), carried iff \(T_m \subseteq T(b)\). It is Part I’s abstract constructor with brain states for programs, and from finite tests it is known only from above.
- Copying has two levels (Conjecture 1.20). The channel is a closed alphabet fixed in the receiver, which the receiver parses with error; because the alphabet is fixed, the copy on it is arbitrarily accurate. The function is rebuilt by the receiver from reasons. An ape meme is the case where a string on the closed level fixes the whole repertoire, a culture meme the case where the function lives on the open level. Sperber’s attractors explain fidelity: the symbols of the fixed alphabet, and hard-to-vary content.
- Copying is synthesis from examples with a weaker specification than the one to be met. For an open receiver the evidence is a criterion, not a source, and the copy needs the receiver’s creativity.
- The copy error splits into a loss \(\varepsilon^-\) and a gain \(\varepsilon^+\). The gain is mostly error in the toy.
- No exact copy can be guaranteed under underdetermination, open sources always leave it, and no exact copy of an infinite repertoire can be confirmed (Proposition 1.11). Single copies are exact all the same. So no copy is treated as final: fallibilism (C10).
- In the toy, error shrinks with evidence, with criticism and fastest in dialogue. Hard-to-vary memes give bimodal fidelity, but only relative to a conjecture space (Conjecture 1.19).
10.7 Open problems
- Individuating one meme among overlapping, holistic repertoires. “The smallest repertoire that can be copied as a whole” is a candidate, not a solution.
- Are human sources self-reaching? Part 2 of Proposition 1.11 needs it. Part I defines the property but does not establish it for brains.
- Conjecture 1.20 against ape traditions: whole-step variants and no gradual drift of one step into another.
- Conjecture 1.16 against real transmission chains, and Conjecture 1.19 under two host conjecture spaces.
- An open receiver in a model. The toy’s conjecture spaces are finite, and its receiver has no preference for good explanations.
10.8 References
- Bartlett, F. C. (1932). Remembering: A Study in Experimental and Social Psychology. Cambridge University Press. (not re-checked)
- Byrne, R. W. (1999). Imitation without intentionality. Using string parsing to copy the organization of behaviour. Animal Cognition 2(2), 63–72. doi:10.1007/s100710050025. (verified: bibliographic data and abstract; the claims about accuracy are ours)
- Byrne, R. W. & Byrne, J. M. E. (1993). Complex leaf-gathering skills of mountain gorillas (Gorilla g. beringei): variability and standardization. American Journal of Primatology 31(4), 241–261. (verified: bibliographic data and abstract)
- Byrne, R. W. & Russon, A. E. (1998). Learning by imitation: a hierarchical approach. Behavioral and Brain Sciences 21(5), 667–721. (verified: bibliographic data and abstract)
- De la Guardia, C. (2020). A few questions on AGI. Making Minds and Making Progress (Substack). https://carlosd.substack.com/p/6-a-few-questions-on-agi (not re-checked)
- De la Guardia, C. (2025c). Are good ideas rare? Making Minds and Making Progress (Substack). https://carlosd.substack.com/p/rare (not re-checked)
- Deutsch, D. (2011). The Beginning of Infinity. Allen Lane. Chs. 1, 3, 6, 16 (limits of ape memes). (not re-checked)
- Deutsch, D. (2013). Constructor theory. Synthese 190(18), 4331–4359. arXiv:1210.7439. §1.1, §2.14, §2.15. (not re-checked)
- Deutsch, D. & Marletto, C. (2015). Constructor theory of information. Proc. R. Soc. A 471: 20140540. arXiv:1405.5563. (not re-checked)
- Deutsch, D. & Marletto, C. (2025). Constructor theory of time. arXiv:2505.08692 (v3, June 2026). §2 (accuracy and reliability, after Violaris & Marletto 2022), §3 (eqs. 2–3). (verified against the v3 PDF)
- Marletto, C. (2015). Constructor theory of life. J. R. Soc. Interface 12(104): 20141226. arXiv:1407.0681. §2, §3.1. (not re-checked)
- Popper, K. R. (1972). Objective Knowledge: An Evolutionary Approach. Oxford University Press. Ch. 7. (not re-checked; the ch. 7 quote comes from secondary sources)
- Sperber, D. (1996). Explaining Culture: A Naturalistic Approach. Blackwell. (not re-checked)
- Quine, W. V. O. (1951). Two dogmas of empiricism. Philosophical Review 60, 20–43. (not re-checked)
- von Neumann, J. (1966). Theory of Self-Reproducing Automata. Ed. A. W. Burks. University of Illinois Press. (not re-checked)
source: book2/notebooks/meme_constructor/posts/post_01_meme_as_constructor.clj