Mechanisms follow sequences

The historical question

Frequency analysis inferred hidden symbols from statistical regularity; programmable automata encoded changing behavior into cams, floats, valves, and timed releases. Both reveal state through consequences.

A system becomes computationally legible when we can name its state, transition rule, input, and output.

Hidden state and observable events

Model a simplified water-driven automaton using three states: filling, released, and resetting. A threshold event takes the machine from filling to released; the released state produces an output, such as moving a figure; reset restores the initial configuration. This is a modern teaching model, not a detailed reconstruction of al-Jazarī's elephant clock. It makes the distinction between internal state, event, and visible output explicit.

A transition table must say what happens for unexpected events. If a second threshold event arrives while resetting, should it be ignored, remembered, or treated as a fault? A picture of the mechanism alone may hide this question. A state machine forces it into the specification. Physical implementations add timing variation, friction, leakage, and failure modes absent from the abstract table.

Frequency analysis poses a related observation problem with different mathematics. In a monoalphabetic substitution, a fixed permutation renames symbols but preserves their counts. A frequent ciphertext symbol is therefore evidence about a frequent plaintext symbol, not proof of its identity. A short or unusual text can defeat the usual language-frequency guess. Multiple words and structural constraints provide stronger evidence than a single count. Neither statistical inference nor a cyclic automaton, by itself, establishes general-purpose stored-program computation.

What the evidence does not establish

Do not call every automaton a computer. Make the specific control analogy and its expressive limits explicit.

Reconstruction exercise

Specify the three-state automaton as a transition table. Then explain why the most frequent ciphertext symbol need not decode to the most frequent letter in ordinary prose.

Worked solution

Use filling + threshold → released; released + output-finished → resetting; resetting + reset-finished → filling. Define other events as ignored or faults, rather than leaving them implicit. The cipher preserves counts within this particular message, but a short message or specialized vocabulary need not have typical language frequencies. A guessed mapping must be checked against other letters and word patterns.

Test the limits and transfer the idea

1. State-machine design · allow 30 minutes

A delayed reset exposes missing state

A toy automaton has Ready, Running and Resetting states. A trigger in Ready starts one output cycle; finish moves Running to Resetting; reset_done returns to Ready. A new requirement says exactly one trigger arriving during Resetting must be remembered and started immediately after reset. Additional triggers while one is pending should be rejected. Extend the state machine and trace trigger, finish, trigger, trigger, reset_done. This is not a reconstruction of the Elephant Clock.

Get a starting hint

Resetting without a pending trigger and Resetting with one must lead to different behavior on reset_done.

Compare your reasoning with the worked solution

Split Resetting into ResettingEmpty and ResettingPending. Starting from Ready, the first trigger moves to Running and starts cycle 1. Finish moves to ResettingEmpty. The next trigger moves to ResettingPending without starting output yet. The following trigger is rejected and the state remains ResettingPending. On reset_done, start cycle 2 and enter Running. In ResettingEmpty, reset_done instead returns to Ready. The two reset states cannot be merged because the same next event requires different actions. Define unexpected events explicitly, for example reject finish outside Running. The Met folio is a primary visual artifact with later catalog interpretation; it does not establish this invented trigger queue or its timing behavior in an historical machine.

Check your reasoning

• Introduces state that distinguishes pending from absent work.
• Traces the five events and exactly two starts.
• Separates original model from artifact-based historical reconstruction.

2. Probability and evidence · allow 25 minutes

A frequency clue is likelihood rather than certainty

In an invented independent-symbol model, hypothesis H1 gives ciphertext symbol X probability 0.6 and H2 gives it probability 0.4. Observe the exact sequence XXXY, where Y means not-X. Both hypotheses initially have probability 1/2. Compute the likelihood ratio and posterior probability of H1. Explain why this does not identify a real substitution key, and how the independence and prior assumptions could fail for language.

Get a starting hint

Multiply symbol probabilities within each hypothesis, then normalize the two weighted likelihoods.

Compare your reasoning with the worked solution

The likelihoods are 0.6³×0.4=0.0864 and 0.4³×0.6=0.0384. Their ratio is 2.25. With equal priors, the posterior is 0.0864/(0.0864+0.0384)=0.6923077, or 9/13. This favors H1 without proving it. Real plaintext symbols have context, genre and word dependencies, and many candidate keys affect more than the frequency of one symbol. Unequal priors would change the posterior. Longer texts, repeated-pattern constraints and independently justified language models can provide additional evidence. The retrospective Arabic cryptology article supplies historical interpretation; the numerical inference here is a modern teaching construction and must not be passed off as a surviving calculation by al-Kindī.

Check your reasoning

• Computes both likelihoods and posterior 9/13.
• Avoids confusing a favored hypothesis with certainty.
• Identifies model and historical-attribution limits.

Check your understanding

What survives a fixed monoalphabetic substitution?

• The original symbol names
• Every word meaning as readable text
• The frequency counts, with symbols renamed

Explanation

Use filling + threshold → released; released + output-finished → resetting; resetting + reset-finished → filling. Define other events as ignored or faults, rather than leaving them implicit. The cipher preserves counts within this particular message, but a short message or specialized vocabulary need not have typical language frequencies. A guessed mapping must be checked against other letters and word patterns.

Sources and further reading

• Elephant Clock folio: The Metropolitan Museum of Art · public domain
• Arabic cryptology: Muslim Heritage · retrospective historical article