The questions: Id(s, t) and Q1–Q11

Besides the central question Id(s, t) (how many steps from s to t), Structure Computer answers eleven questions about the structure of a system (Q1–Q11). Every answer carries a certificate, and only answers that pass the independent checker are returned. Each answer has a kind that says how certain it is (certain, probably certain, reconstructed, statistical, no result); the kinds are explained in “Reading the results, and a glossary”.

How the Try page computes them

• “Compute everything” computes Id(s, t), Q1–Q7 and Q11 together. Q5 checks the first conserved quantity that Q4 found.
• Q8–Q10 need more input: enter it in their own cards and compute them there.
• The questions differ in which number system they answer about.

Overview

Id(s, t): how many steps from s to t

Writing σ for one step, the answer is one of these:

• t = σⁿ s ⇔ n ≡ r (mod M): n steps reach t exactly when n leaves remainder r on division by M (a negative n means running backwards).
• no path: no number of steps, forwards or backwards, reaches t.
• s is not regular: going forwards or backwards from s runs into a state where the formulas are undefined (a division by zero); if t is reached before that, the step count is given.
• Over ℚ, when t is reached, the answer gives the step count n and the form n ≡ r (mod M) any other solution would have to take.

Input on the Try page

Advanced options (usually left as they are)

Q1 Invertibility

Whether a step can be run backwards (whether it is birational).

• If it can, the answer gives the formulas of the backward step, confirmed as identities of formulas over ℚ (going forwards then backwards returns where you started). It also gives where the formulas are undefined (where a denominator vanishes). If the system file gives backward formulas (inverse:), it also says whether they really are the inverse.
• If it cannot, the answer gives the reason, which is that different states are sent to the same place (technically: one variable with degree 2 or more, a Jacobian that vanishes identically, or two different points sent to the same point).
• When neither can be confirmed, the answer is a statistic: how often states collide over small primes.

Q2 Period structure

How the states over a finite field split into cycles with a period and chains that break off.

• When there are few states, every state is counted (certain): the number of cycles of each period, and the lengths of the chains.
• When there are many, orbits are sampled to estimate the distribution of periods (statistical). The period of each sampled orbit that could be confirmed is recorded as a certain fact.
• When ℚ is chosen, the counting is over the primes 11 and 13.

Q3 Integrability

Whether the motion is regular, i.e. the system is “integrable”.

• If there are as many independent conserved quantities as the number of variables minus one, every orbit lies on a curve, and the answer is “integrable” (certain), with the formulas of the conserved quantities.
• Otherwise the verdict is read from features of the distribution of periods over finite fields (statistical): “integrable signs”, “non-integrable (reversible)”, “non-integrable (random)” or “undecided”.

Q4 Conserved quantities

Finds the formulas, up to the maximum degree (the Try page's “Maximum degree of conserved quantities”), whose values do not change along an orbit.

• For polynomial conserved quantities, the answer gives a basis and says, as a certain statement, whether it is all of them (completeness).
• Conserved quantities in the form of fractions (rational functions) are listed as found; the list need not be complete.
• It also gives how many of the listed formulas are independent of each other.
• Every listed formula is proved as an identity over ℚ.

Q5 Verify a conserved quantity

Whether a given formula I is conserved. The Try page checks the first formula Q4 found.

• If it is conserved, the answer proves, as an identity over ℚ, that one step leaves I unchanged.
• If it is not, the answer gives a concrete point where one step changes the value of I.
• For a system whose step is not a rational formula (given by a Lagrangian, where the next point is not fixed by a linear equation), the answer is a fact about a finite field, or a statistic from samples.

Q6 Holonomy

Answered only when the system file declares a symmetry (symmetry: with so: and a quadratic form). For a system without one, the request is refused as an input error.

• Per prime, the answer gives the number of steps until the orbit comes back to s up to the symmetry, and how far the state has then been turned (an element of the symmetry group).
• It also gives how many times that turn must be repeated to come back (its order), and what kind of turn it is.
• How to declare it is in “Writing system files”.

Q7 Integer invariant (steps over ℚ)

The integer n with t = σⁿ s over ℚ. The Try page always asks this over ℚ, whatever field is chosen.

• Under the assumption that a solution with |n| ≤ B (the step bound) exists, n is determined uniquely (reconstructed).
• When there is no solution in that range, or t is never reached at all, the answer is certain: “no such n”.

Q8 Path equivalence

Whether two paths from s (words in the generators: r s^-1 r^2 means r, then the inverse of s, then r twice, read from the left) are the same.

• For a system with a single step, two paths are the same when their total step counts agree (certain).
• For a system of several generators, the maps the two words compose to are compared: shown equal by an identity of formulas, or different at a concrete point (certain). When the formulas are too large, the answer is a statistic from samples.

Q9 System equivalence

Whether another system is the same as this one up to an affine change of coordinates (x ↦ Mx + c).

• If so, the answer gives the change of coordinates and proves, by identities of formulas, that the two steps correspond. It also says whether the places where the formulas are undefined correspond too.
• If they can be shown not to be the same, the answer is certain; if neither can be shown, it is statistical.
• You can give a change of coordinates φ yourself, and it is checked (even one not of the form x ↦ Mx + c, as long as it has backward formulas).

Q10 Transport

Between two systems Q9 has shown to be equivalent, moves a result (with its certificate) about one system to the corresponding result about the other. The checker confirms everything again: the original result, the change of coordinates, and the moved result. The moved result has the kind of the original.

Q11 Closed term

Treats Id(s, t) as a term of type theory (homotopy type theory) and answers two things:

• Whether it can be proved from the syntax alone. Since s and t are separate constants, it cannot. This is a general fact, the same for every system, and carries almost no information.
• Its value in this system taken as a model: the step count over ℚ, or no path. This is the part that carries information.