ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Adding a sink turns the published prefix automaton for avoiding `11` into a DFA

Example

Let $B = {11}$ over the alphabet ${0, 1}$ . The published proper-prefix state set is $S_{B} = {ε, 1} .$ Adjoin a sink state $⊥$ , keep $ε$ and $1$ accepting, and define the transitions by $δ (ε, 0) = ε, δ (ε, 1) = 1,$ $δ (1, 0) = ε, δ (1, 1) = ⊥, δ (⊥, 0) = ⊥, δ (⊥, 1) = ⊥ .$

This DFA accepts exactly the binary words with no factor 11.

Facts & Assumptions

Given: The forbidden-factor set $B = {11}$ over ${0, 1}$ .

[L1]

The published prefix automaton extends canonically to a DFA for the factor-avoidance language by adjoining a sink and declaring the states in $S_{B}$ accepting, by The published prefix automata extend canonically to DFAs for factor-avoidance languages.

Verification

technique · direct

1.1given

The only proper prefixes of the forbidden word 11 are $ε$ and 1, so these are exactly the nonsink states. The missing transition is from 1 on input 1, and it must go to the sink because the extended word 11 already contains the forbidden factor.

1.2given

Once the machine reaches $⊥$ , the forbidden factor has already appeared, so looping at $⊥$ on both letters is the correct total completion. The words $ε$ , 1, 10, and 1010 stay in the accepting states, while 11 and 1011 reach $⊥$ .

2.1L1step 1.1step 1.2∎

This is exactly the sink-state completion described by [L1], so the resulting DFA recognizes the binary words avoiding 11.

Depends on

The published prefix automata extend canonically to DFAs for factor-avoidance languages

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Example 4.7.7 and Proposition 4.7.8 (standard reference, not scraped)

Adding a sink turns the published prefix automaton for avoiding 11 into a DFA