Büchi automaton

Label denotation

Record whether labels denote individual letters or sets of letters through Boolean guards, including the atomic-proposition interpretation.

1. What is Σ, and what concrete input observation does each letter denote? definition
2. How are symbolic guards interpreted, and which guards denote no possible input? boundary

Legal run

For w = a0a1…, a run q0q1… starts in I and satisfies (qi, ai, qi+1) ∈ Δ at every position; a finite dead end is not an accepting infinite run.

1. Which states are initial, and which labeled successors are permitted from each state? definition
2. Do any states lack successors for some letters in the declared alphabet? measurement

Accepting recurrence

For state-based acceptance, Inf(r) ∩ F must be nonempty. Transition-based acceptance instead requires infinitely many occurrences of marked transitions.

1. Which states or transitions carry the single Büchi acceptance mark? definition
2. Does the encoded condition require infinite recurrence, rather than one visit or eventual permanent residence? boundary

Existential word acceptance

An infinite word belongs to the language when at least one legal run satisfies the Büchi condition.

1. Is acceptance existential over runs, as required by the baseline nondeterministic word semantics? definition
2. Does an imported artifact introduce universal branching or another semantics requiring a neighboring model? boundary

Accepting cycle test

In a state-based automaton, nonemptiness requires a reachable strongly connected component containing an accepting state and a nonempty cycle through it, using feasible transitions.

1. Which reachable strongly connected components support a cycle through an accepting state? measurement
2. Have singleton components without self-loops and transitions with unsatisfiable guards been excluded? boundary

Prefix and loop witness

Record a finite path from an initial state followed by a nonempty repeatable accepting loop, together with an input word uv^ω where v is nonempty.

1. What prefix, loop, and corresponding input labels constitute the nonemptiness witness? measurement
2. Can each step and the accepting recurrence be independently replayed against this exact automaton? action

Verified structural properties

Record determinism, completeness, and acceptance placement as checked properties; for symbolic labels, determinism depends on guard overlap.

1. Can one input letter enable distinct successors from the same state, or can a word start from multiple initial states? measurement
2. Which structural properties were verified, by what procedure, and for which artifact revision? provenance

Operation preconditions

Require operation-specific acceptance handling: intersection must enforce both recurrence obligations, complementation is not simple accepting-set inversion, and determinization may require another acceptance formalism.

1. What construction establishes the intended language relation between the input and output automata? action
2. Does the result retain ordinary Büchi acceptance, or must it be represented under a different acceptance model? boundary

Translation contract

Record the source specification, proposition mapping, and whether accepted words satisfy the property or its negation.

1. Which formula or behavioral requirement produced this automaton, and with what polarity? provenance
2. How do source observations map to alphabet letters and positions in the infinite word? definition

Verification result contract

For a product representing system behaviors intersected with property violations, nonemptiness supplies a counterexample; emptiness supports the property only under the recorded modeling assumptions.

1. How are system labels, deadlocks, and fairness assumptions incorporated into the product? boundary
2. What conclusion does emptiness or a returned lasso justify for the original system and property? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Büchi automaton this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these kinds and varieties hold for the sense of Büchi automaton this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Büchi automaton this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Büchi automaton this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of Büchi automaton this model covers, and on what evidence? provenance