Scope and Manufacture

The region is already built. Chapter 18 made a scope a predicate rather than a list, showed that disjointness makes a composite scope parse uniquely, and put fixed points into the name-predicate language so that a self-similar region can be finitely described and unboundedly large. What remains is to count in it, to ask what the counting costs, and to ask what had to be manufactured before there was anything there to count.

53.1 Copy number, graded

53.1.1 Well posed is not effective

The identity criterion is easy to state once the region is fixed. Two processes are copies when no experiment distinguishes them, and the relation that says so is bisimilarity. Definition 48.4 already takes this line, and it is the right one: structural identity is at once too strict, since two conformers of a molecule are the same reagent, and too permissive, since two graphs may agree while the things they denote behave differently under interaction. Behavioral identity is what a count of copies was always trying to track.

What Definition 48.4 gives is therefore well posed. It is not effective. Bisimilarity is undecidable for calculi of this expressiveness [120], so no learner computes it, and in particular no learner computes that count. The gap is the one \(\chSci\) meets in the assay, where a hypothesis that is true but unaffordable is refuted relative to a budget, and the remedy is the same.

The resolution \(n\) is not a free parameter. It is the modal depth a learner can afford to evaluate, and Chapter 52 bounds that: by integration depth, hence by tower height, hence by assembly index. So a count of copies has three arguments, and the third belongs to whoever is counting.

53.1.2 The observer enters the count

The direction is the opposite of the one intuition supplies. A weak instrument does not miss copies. It manufactures them, by failing to separate things that differ below its resolution, and the manufacture is not a small correction.

53.1.3 The count has a grading it should have kept

In a generated scope there is a second thing wrong with reporting an integer.

53.1.4 A worked series

Take the two-atom generated scope of Definition 18.2 and populate it: let \(\phi\)-atoms be sources and \(\psi\)-atoms be foragers, and let the population instantiate every shape at strata \(0\) through \(2\) with multiplicity \(12\). By Table 18.1 there are \(17\) shapes, distributed \(2 : 3 : 12\) over the strata, so the populated scope has \(204\) channels.

Ask for the copy number of a stratum-\(1\) shape, say \(\quo{(\phi \mid \psi)}\). The truth is \(12\). A learner of resolution \(2\) separates all three stratum-\(1\) shapes and reports \(12\). A learner of resolution \(1\) separates \(\quo{(\phi\mid\psi)}\) from \(\quo{(\phi\mid\phi)}\) but not from \(\quo{(\psi\mid\psi)}\), if the two atoms agree on their first step, and reports \(24\). A learner of resolution \(0\) reports \(36\); and an instrument that counts occurrences rather than channels, and so fails to separate strata at all, reports \(204\).

The series makes legible what the integer does not: \[\CN_2^{\bullet} = (0,\,12,\,0), \qquad \CN_1^{\bullet} = (0,\,24,\,0), \qquad \CN_0^{\bullet} = (0,\,36,\,0), \qquad \text{flat} = 204 .\] Every one of these is a defensible answer to “how many copies are there”. Only the first is an answer to “how many copies are there, at the resolution at which copies are a kind”.

53.2 Some cheap lower bounds

53.2.1 What a lower bound would have to be

Upper bounds on what an assembled thing can do come easily here, and Chapter 52 has several: exhibit an invariant that cannot exceed the assembly index, then bound capability by the invariant. Lower bounds run the other way and are much harder, because \(\AI\) is a minimum over constructions, so a lower bound asserts that no shorter construction exists. That is an incompressibility statement, and incompressibility statements are notoriously resistant.

There is one tractable route. Find quantities \(\iota_1, \dots, \iota_m\) that are monotone along assembly edges, forced to a minimum value by the capability in question, and witnessed by disjoint sets of edges. Then \(\AI(E) \ge \sum_i \iota_i(E)\); without the third condition one gets only \(\AI(E) \ge \max_i \iota_i(E)\). The third is the hard one in general, and §53.2.4 observes that the disjointness of Chapter 18 certifies it in the cases that matter.

Six such quantities follow. They are all obvious once stated, none requires an argument longer than a paragraph, and we make no claim that their sum is anywhere near the truth.

53.2.2 Six floors

L1: having an inside.

An ecology-as-mind is a region distinguished from its environment, so there is at least one medium whose endpoints straddle the boundary. A medium is a process and must be constructed — by no-self-code it is not among the terms it carries — so its construction is a joining step.

L2: owning a boundary.

Definition 24.6 asks that internal media be closed and boundary media not, which requires the two to be distinguishable and hence separately constructed: a single medium serving both roles would put the interior under external naming and dissolve the closure that makes repair possible. And by Proposition 24.7 a mind that has paid for the boundary must also have paid for at least one redex pathway across it, since perfect closure is death. We fold the pathway into L2 rather than counting it separately.

L3: carrying a germ.

The germ of §21.12.2 is a quotation, transmissible and inert until dropped. By no-self-code it is not among the names occurring in what it encodes: the germ sits strictly above what it quotes. So the machinery that manipulates germ material occupies a level of the tower above the machinery it manipulates, and by Proposition 52.1 each level costs at least one edge of a minimum path. Reproduction is \(+1\), unconditionally, over the floor of a thing that does not reproduce.

L4: resolving at depth \(n\).

This one is free: it is the contrapositive of a theorem already proved. If an ecology of assembly index \(\AI\) cannot distinguish environments that are \(\AI\)-step bisimilar, then an ecology that does distinguish environments differing only at depth \(n\) has \(\AI \ge n\). Every upper bound on capability is a lower bound on manufacture read backwards, and this is the cleanest instance.

L5: escaping a basin.

By Proposition 21.4 a walk whose steps are all shorter than \(2^{-n}\) never leaves the ball of that radius. Escape requires a revision move of distance at least \(2^{-n}\), which by Definition 21.6 means a locus at term-depth at most \(n\) together with an operation applied there. Addressing a locus at depth \(n\) requires a one-hole context of that depth, and that must be built. So a learner able to leave its initial basin pays on the order of \(n\) steps for the addressing alone.

L6: repairing itself.

By Theorem 24.2 the total detection strength of the price code is exactly \(\bt\), and an edge lying on no cycle carries no discipline at all. So a region that can detect a single-edge fault has \(\bt \ge 1\) and one detecting \(m\) independent faults has \(\bt \ge m\). Each independent cycle requires an edge that closes it, and those edges are joining steps.

53.2.3 Genetic exploration, and the difference between a learner and a scorer

L3 and L5 combine into the one bound here we find genuinely interesting, because it separates two things the framework has otherwise treated together.

53.2.4 Disjointness is the additivity certificate

53.2.5 Why we stop here

The sum of Table 53.1 for a reproducing, repairing, depth-\(n\)-resolving individual is something like \(n + \bt + 4\), and §53.4.5 evaluates it at \(28\). The true minimum is presumably very much larger, for the reason Remark 52.1 gives: the assembly index counts everything built, including redundancy, repair machinery and bulk, while these floors count only the building that buys a named capability. The ratio between them is the interesting quantity and we cannot compute it.

That is a reasonable place to leave it. A framework earns attention by making quantities visible before it makes them sharp, and a floor anyone can check and improve is a better invitation than a theorem that closes the question. The lower-bound problem for minds is at about the stage the lower-bound problem for circuits was before anyone had a technique: everybody can see what a bound would have to look like and nobody can produce one. We would be pleased to be scooped on this.

53.3 Generator and unfolding

Table 18.1 showed a generator of fixed size whose extension grows doubly exponentially. With the assembly index in hand there are three quantities to relate, not two.

53.4 The pressure on copy number

The two parameters of the biosignature look independent. One is a property of an object and the other of a population, and nothing in assembly theory relates them beyond the observation that in living matter both are large. Here they are related, by two mechanisms pushing the same way: an architecture that is deep requires copies before anyone wants them for statistics.

53.4.1 Detection forces copies

By Theorem 24.2 the total detection strength of the price code is \(\bt\), distributed over edges by effective resistance, and an edge on no cycle is undetectable. A region whose internal media form a tree therefore has no error correction at all: every fault is silent, and by Corollary 24.10 a bridge fails twice, having neither a cycle to check it nor an owner to repair it.

Repairability requires cycles; a cycle requires a redundant pathway; and a redundant pathway between two functional positions is two copies of a relation where one would carry the traffic. The requirement recurs per stratum, since each level of the tower has its own media and its own faults.

This is a floor on copies of media, and it is small. The second mechanism is not.

53.4.2 Sampling forces copies, exponentially

By Proposition 24.9 a percept of integration depth \(d\) traverses \(d\) hops of internal medium and survives with probability \(p^{d}\). A hypothesis at that depth needs \(m\) independent confirmations before a learner will fund a revision on it, and confirmations attempted in parallel are attempted by copies.

53.4.3 The two axes are coupled

53.4.4 A prediction about scarcity

53.4.5 A worked composite: four skills, one individual, twelve of them

Now the numbers, carried through a mind whose skills are themselves ecologies-as-mind, in the manner of §23.11. The illustration is orca-shaped and the shape does no work beyond making the skills memorable: a hunting skill, a skill for one’s place in the pod, a mating skill, and a skill for rearing calves. Each is a population of mortal scientists in its own typed namespace with its own token flavor; the four are wired by enzymes; the whole closes over its namespace and is an individual; and twelve such individuals form a pod, which is a candidate individual one stratum up.

The generator.

The individual’s scope is \[\Nsp \;=\; \mu X.\; \quo{\bigl( (\phi_{\mathrm{for}} \lor X) \mid (\phi_{\mathrm{pod}} \lor X) \mid (\phi_{\mathrm{mat}} \lor X) \mid (\phi_{\mathrm{rea}} \lor X) \bigr)},\] the four-predicate version of Definition 18.2. The four predicates are pairwise disjoint — each skill’s population is rooted at its own quotation, by Proposition 23.9 — so the scope parses uniquely, membership is decidable by Proposition 18.2, and the floors may be added by Proposition 53.5.

Depths and populations.

Each skill sits on media of its own reliability: the pod and rearing skills run on internal, owned media (\(p = 0.99\)); hunting reaches across the boundary into a noisy world (\(p = 0.95\)); mating runs across another individual’s boundary and is worst (\(p = 0.90\)). Optimal integration depth follows Proposition 24.9, maximizing \(Y(d)p^{d} - cd\) with \(Y(d) = 100(1-2^{-d})\) and \(c = 2\); the sampling floor follows Proposition 53.8 with \(m = 20\).

Note which skill is most expensive in copies. It is not the deepest; it is the one on the worst medium. Mating costs \(28\) learners to sustain depth \(3\) while the pod skill costs \(22\) to sustain depth \(5\), because reliability enters the floor exponentially and depth only through the exponent. An architecture is expensive where it is unreliable, not where it is deep.

The copy series.

Twelve individuals, each with \(96\) learners in four skill-ecologies: \[\CN^{\bullet}(\text{pod}) \;=\; (1152,\; 48,\; 12,\; 1),\] being learners, skill-ecologies, individuals, and the pod itself. The flat collapse is \(1213\), a number that answers no question anyone would ask. The ratios \(1152 : 48 : 12 : 1\) are \(24\), \(4\) and \(12\), and those are respectively a sampling floor, a skill count and a group size — three different kinds of fact about the animal, which the collapse blends into one meaningless total.

The assembly floor.

Applying Table 53.1 with the additivity of Proposition 53.5:

Two numbers to take from this. Sharing the kernel across the four skills saves \(15\) of \(43\) steps, or \(34.9\%\); and of the \(28\) remaining, \(16\) — \(57.1\%\) — are content rather than architecture. The fractal is cheap and the niches are expensive.

Does the whole close?

By §24.8 a region may close over its namespace exactly when its isoperimetric ratio clears \((\varepsilon\theta/\rho)\,r\). Taking \(\theta = 0.2\), \(\rho = 5\) and \(\varepsilon = 1-p\) for the relevant media: the individual, with internal media at \(p = 0.99\) and two of its four skills facing outward so that \(h = 0.5\), has \(\varepsilon\theta/\rho = 4\times 10^{-4}\) and critical radius \(r^\ast = 1250\); it closes with enormous margin. The pod, whose media run across individual boundaries at \(p = 0.90\) and whose isoperimetric ratio we take as \(0.3\), has \(\varepsilon\theta/\rho = 4\times 10^{-3}\) and \(r^\ast = 75\). A pod of twelve closes. An aggregation of two hundred does not.

What a passing instrument sees.

Finally the point of §53.1, made concrete. The four skills’ learners share the kernel and separate at depths \(2, 3, 3, 4\). An observer surveying the pod’s namespace at resolution \(n\) reports:

An instrument at resolution \(0\) reports a copy number \(3.4\) times the truth and a monoculture where there are four kinds; at resolution \(2\) it is still wrong by a factor of \(2.6\) on the largest class. And note the small mercy at \(n = 3\): once every kind but one has been separated, the last comes free, because what remains in the pool is a kind by elimination. A survey therefore needs resolution equal to the second-deepest separation, not the deepest — a saving of exactly one kind’s worth of instrument, and the only place in this chapter where the accounting is generous.

53.5 The survey, in rholang

Two contracts. The first decides membership in a generated scope by descent, per Proposition 18.2. The second surveys a scope and returns a copy series. Both follow the conventions of §22.13: a bare parameter is a name, an @-pattern binds data, and a channel is passed as *c. Guards marked //! ideal stand for decision procedures assumed rather than implemented.

new InScope, Split, Survey, Series, BisimN, Atom, Stratum in {

  // ---- membership by descent -------------------------------------
  // mu X . @( (phi \/ X) | (psi \/ X) ), presented as a list of
  // atomic predicates and tested by structural descent on the quote.

  contract InScope(@nm, atoms, ret) = {
    new hit in {
      Atom!(nm, *atoms, *hit) |
      for (@isAtom <- hit) {
        if (isAtom) { ret!(true) }
        else {
          // By "Unique decomposition" the split is unique, so one
          // split and two recursive calls -- no search.
          new sp in {
            Split!(nm, *sp) |
            for (@p, @q, @ok <- sp) {
              if (not ok) { ret!(false) }
              else {
                new l, r in {
                  InScope!(p, *atoms, *l) |
                  InScope!(q, *atoms, *r) |
                  for (@a <- l & @b <- r) { ret!(a and b) }
                }
              }
            }
          }
        }
      }
    }
  } |

  // Split returns the unique decomposition of a quoted parallel
  // composition.  Termination of the recursion above is exactly the
  // no-self-code theorem: p and q code strictly smaller terms.
  contract Split(@nm, ret) = {
    match *nm {
      { P | Q } => { ret!(@P, @Q, true) }        //! ideal
      _         => { ret!(Nil, Nil, false) }
    }
  } |

  // ---- the survey ------------------------------------------------
  // Visit each channel of the scope, decide n-step bisimilarity with
  // the target, and bin by stratum.  One metered rendezvous per
  // visit: this contract is where a copy number is paid for.

  contract Survey(@target, chans, @n, ret) = {
    new tally in {
      tally!({}) |
      for (@c <= chans) {                     // one visit per channel
        new b, s in {
          BisimN!(c, target, n, *b) |         //! ideal, cost kappa(n)
          Stratum!(c, *s) |
          for (@same <- b & @j <- s) {
            for (@acc <- tally) {
              if (same) { tally!(acc.set(j, acc.getOrElse(j,0) + 1)) }
              else      { tally!(acc) }
            }
          }
        }
      } |
      for (@acc <- tally) { ret!(acc) }       // the copy series
    }
  } |

  // The flat collapse, for comparison with the received definition.
  contract Series(@series, ret) = {
    ret!( series.values().fold(0, (a, b) => a + b) )
  }
}

Three things the listing makes visible that the prose does not. The recursion in InScope has no depth bound in its own code — it terminates because of a theorem about the calculus, not because of a counter. Survey charges once per channel, which is what makes affordable scope a budget question rather than a definitional one. And Series exists only to throw information away; it is the received copy number, written as the lossy projection it is.

53.6 What this leaves

Four things, and they belong with the gaps of Chapter 54 rather than here, so they are recorded there. The shortest statement of each: whether a non-well-founded scope could be surveyed stratum by stratum; whether the copy series can tell a merged composite from a merely composed one; whether affordable scope and the individuation fixed point of Remark 24.11 have a common solution; and what a copy number looks like once one admits that a survey is an assay, and so has a third outcome.

There is also a connection that runs forward rather than sideways. A learner’s region is bounded by what it can pay to survey, so the unsurveyable is not the unknowable in principle but the unaffordable in fact — and that is precisely the distinction §68.5.1 builds on when it argues that the noumenal survives in this framework as a budget constraint rather than as a wall. Copy number turns out to be a place where that abstraction has a number attached: what a mind can count is bounded by what it was assembled to resolve, and what it can survey is bounded by what it can afford to visit. Both bounds are economic, both are movable, and neither is a limit on what is there.