{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.b-chi-automaton","code":"thing-q1020380","url":"https://ver.cy/models/thing/q1020380/","name":"Büchi automaton","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Information and virtual systems","entryKind":"thing","plane":"INF","domain":["INF.KNW"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A Büchi automaton is a finite-state automaton on infinite words that accepts a word if at least one run over that word visits designated accepting states infinitely often.","purpose":"Enable an agent to identify a Büchi automaton, interpret its infinite-word language, assess its acceptance behavior, and select sound verification or transformation operations.","scope":{"in":["Finite states, input alphabet, initial states, transition relation, and accepting states or transitions","Infinite runs and existential Büchi acceptance of infinite input words","Determinism, completeness, reachability, and recurrent accepting behavior","Language emptiness, ultimately periodic witnesses, and language comparison","Acceptance-preserving transformations and use in automata-based verification"],"out":["Automata theory as a discipline and its institutional classifications","Finite-word automata whose acceptance depends on the state at termination","Generalized Büchi, co-Büchi, parity, Rabin, and Streett automata as independent acceptance formalisms","Infinite-tree, alternating, probabilistic, and timed automata beyond explicitly recorded conversions","Temporal-logic specifications and transition systems independently of their mappings to this automaton","Model-checking software implementation and execution infrastructure"],"boundaries":[]},"distinguishingFeatures":["Its inputs are infinite sequences over an alphabet, so acceptance does not depend on reaching an end-of-input state.","In the ordinary state-based convention, a run accepts exactly when it visits the accepting set infinitely often; reaching it once is insufficient. See [Automata and Infinite Words](https://www.sciencedirect.com/science/article/pii/S0079816904800023).","For nondeterministic word automata, one accepting run suffices; acceptance does not require every possible run to accept.","Ordinary Büchi acceptance has one recurring acceptance obligation; generalized Büchi acceptance requires recurrence for each designated acceptance set.","A transition-based representation marks recurring transitions instead of states and must declare that convention. See [Spot acceptance concepts](https://spot.lre.epita.fr/concepts.html)."],"structure":{"bundles":[{"id":"word-and-run-semantics","name":"Word and run semantics","description":"The finite structure and interpretation that turn infinite input sequences into runs.","layers":[{"id":"alphabet-and-labels","name":"Alphabet and labels","description":"The input universe and the meaning of transition labels.","findings":[{"id":"label-denotation","name":"Label denotation","description":"Record whether labels denote individual letters or sets of letters through Boolean guards, including the atomic-proposition interpretation.","questions":[{"text":"What is Σ, and what concrete input observation does each letter denote?","id":"label-denotation-q01","kind":"definition"},{"text":"How are symbolic guards interpreted, and which guards denote no possible input?","id":"label-denotation-q02","kind":"boundary"}]}]},{"id":"initialization-and-successors","name":"Initialization and successors","description":"How a run starts and advances through the finite state space.","findings":[{"id":"legal-run","name":"Legal run","description":"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.","questions":[{"text":"Which states are initial, and which labeled successors are permitted from each state?","id":"legal-run-q01","kind":"definition"},{"text":"Do any states lack successors for some letters in the declared alphabet?","id":"legal-run-q02","kind":"measurement"}]}]}]},{"id":"buchi-acceptance","name":"Büchi acceptance","description":"The recurring obligation that distinguishes accepting infinite runs.","layers":[{"id":"recurrence-obligation","name":"Recurrence obligation","description":"The designated accepting set and the event required infinitely often.","findings":[{"id":"accepting-recurrence","name":"Accepting recurrence","description":"For state-based acceptance, Inf(r) ∩ F must be nonempty. Transition-based acceptance instead requires infinitely many occurrences of marked transitions.","questions":[{"text":"Which states or transitions carry the single Büchi acceptance mark?","id":"accepting-recurrence-q01","kind":"definition"},{"text":"Does the encoded condition require infinite recurrence, rather than one visit or eventual permanent residence?","id":"accepting-recurrence-q02","kind":"boundary"}]}]},{"id":"run-quantification","name":"Run quantification","description":"How individual accepting runs determine word acceptance.","findings":[{"id":"existential-word-acceptance","name":"Existential word acceptance","description":"An infinite word belongs to the language when at least one legal run satisfies the Büchi condition.","questions":[{"text":"Is acceptance existential over runs, as required by the baseline nondeterministic word semantics?","id":"existential-word-acceptance-q01","kind":"definition"},{"text":"Does an imported artifact introduce universal branching or another semantics requiring a neighboring model?","id":"existential-word-acceptance-q02","kind":"boundary"}]}]}]},{"id":"recurrent-structure-and-witnesses","name":"Recurrent structure and witnesses","description":"Finite graph evidence for infinite accepting behavior.","layers":[{"id":"accepting-components","name":"Accepting components","description":"Reachable strongly connected structure capable of sustaining acceptance.","findings":[{"id":"accepting-cycle-test","name":"Accepting cycle test","description":"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.","questions":[{"text":"Which reachable strongly connected components support a cycle through an accepting state?","id":"accepting-cycle-test-q01","kind":"measurement"},{"text":"Have singleton components without self-loops and transitions with unsatisfiable guards been excluded?","id":"accepting-cycle-test-q02","kind":"boundary"}]}]},{"id":"lasso-evidence","name":"Lasso evidence","description":"Finite certificates describing accepting infinite repetition.","findings":[{"id":"prefix-and-loop-witness","name":"Prefix and loop witness","description":"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.","questions":[{"text":"What prefix, loop, and corresponding input labels constitute the nonemptiness witness?","id":"prefix-and-loop-witness-q01","kind":"measurement"},{"text":"Can each step and the accepting recurrence be independently replayed against this exact automaton?","id":"prefix-and-loop-witness-q02","kind":"action"}]}]}]},{"id":"representation-and-transformation","name":"Representation and transformation","description":"Structural classifications and constraints on language-preserving changes.","layers":[{"id":"determinism-and-encoding","name":"Determinism and encoding","description":"Properties of the actual representation that govern available algorithms.","findings":[{"id":"verified-structural-properties","name":"Verified structural properties","description":"Record determinism, completeness, and acceptance placement as checked properties; for symbolic labels, determinism depends on guard overlap.","questions":[{"text":"Can one input letter enable distinct successors from the same state, or can a word start from multiple initial states?","id":"verified-structural-properties-q01","kind":"measurement"},{"text":"Which structural properties were verified, by what procedure, and for which artifact revision?","id":"verified-structural-properties-q02","kind":"provenance"}]}]},{"id":"language-preservation","name":"Language preservation","description":"Proof obligations for reduction, composition, complementation, and conversion.","findings":[{"id":"operation-preconditions","name":"Operation preconditions","description":"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.","questions":[{"text":"What construction establishes the intended language relation between the input and output automata?","id":"operation-preconditions-q01","kind":"action"},{"text":"Does the result retain ordinary Büchi acceptance, or must it be represented under a different acceptance model?","id":"operation-preconditions-q02","kind":"boundary"}]}]}]},{"id":"verification-meaning","name":"Verification meaning","description":"How the accepted language connects to specifications and verification conclusions.","layers":[{"id":"specification-origin","name":"Specification origin","description":"The semantic contract of a generated or supplied automaton.","findings":[{"id":"translation-contract","name":"Translation contract","description":"Record the source specification, proposition mapping, and whether accepted words satisfy the property or its negation.","questions":[{"text":"Which formula or behavioral requirement produced this automaton, and with what polarity?","id":"translation-contract-q01","kind":"provenance"},{"text":"How do source observations map to alphabet letters and positions in the infinite word?","id":"translation-contract-q02","kind":"definition"}]}]},{"id":"product-and-conclusion","name":"Product and conclusion","description":"The assumptions needed to interpret verification through a product automaton.","findings":[{"id":"verification-result-contract","name":"Verification result contract","description":"For a product representing system behaviors intersected with property violations, nonemptiness supplies a counterexample; emptiness supports the property only under the recorded modeling assumptions.","questions":[{"text":"How are system labels, deadlocks, and fairness assumptions incorporated into the product?","id":"verification-result-contract-q01","kind":"boundary"},{"text":"What conclusion does emptiness or a returned lasso justify for the original system and property?","id":"verification-result-contract-q02","kind":"action"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of Büchi automaton this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of Büchi automaton this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of Büchi automaton this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of Büchi automaton this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"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?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain Büchi automata and their role in model checking of infinite behaviours.","Construct or check automata for properties in systems it is authorized to verify."],"mustNot":["Present a system as verified when the automaton or property does not match the specification.","Hide state-space limits or timeouts that left a check incomplete.","Copy proofs or constructions from published works without credit."],"requiresHuman":["Signing off safety-critical systems based on model-checking results."]},"ethics":{"considerations":["Verification results are trusted for safety-critical systems, so false assurance can harm people."],"affectedParties":["Users of verified systems","Engineers","Students"]},"owners":{"steward":"Nobody owns it; it is a concept of automata theory.","roles":[],"masterSystems":[]},"relations":[{"target":"Q291256","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.b-chi-automaton","Wikidata: Q1020380 (https://www.wikidata.org/wiki/Q1020380)"]},"properties":{"applicability":"not-applicable","items":["Automaton signature: A = (Q, Σ, Δ, I, F), with finite Q and Σ, Δ ⊆ Q × Σ × Q, I ⊆ Q, and F ⊆ Q for the baseline convention","Alphabet interpretation: Explicit symbols or symbolic encodings, including valuations of a finite atomic-proposition set","Acceptance placement: State-based or transition-based; one Büchi acceptance set","Branching discipline: Deterministic or nondeterministic, including the declared initial-state convention","Transition completeness: Complete, incomplete, or unchecked relative to the declared alphabet","Representation size: State count, transition-record count, and symbolic-label size under a named encoding","Language emptiness: Empty, nonempty, or unchecked, with supporting certificate or witness","Specification correspondence: Source formula or system, alphabet mapping, translation convention, and preservation claim"]},"recognition":{"applicability":"optional","items":["Drawn as a state diagram with accepting states marked by double circles, read over infinite input."]},"capabilities":{"applicability":"required","items":["Validate the transition structure, alphabet interpretation, and declared Büchi acceptance convention.","Check language emptiness and extract an accepting prefix-and-loop witness when one exists.","Decide acceptance for a supplied ultimately periodic word using a finite representation of its repeating suffix.","Compare languages through sound inclusion or equivalence procedures and retain counterexample words.","Compose compatible automata for verification using a construction that preserves all required acceptance obligations.","Simplify or convert the automaton while recording the preserved language and any change of acceptance formalism."]},"hazards":{"applicability":"required","items":["Mistranslated properties give false assurance that a system is safe.","State explosion can stop checks before completion."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["A Buchi automaton represents permitted or prohibited infinite sequences in formal language theory and automated verification.","It is used in model checking to reason about ongoing system behavior, including properties that require events to recur indefinitely."]}},"sources":[{"title":"Wikidata item Q1020380: Büchi automaton","url":"https://www.wikidata.org/wiki/Q1020380","note":"identity and sense of the item"},{"title":"Wikipedia: Büchi automaton","url":"https://en.wikipedia.org/wiki/B%C3%BCchi_automaton","note":"general description of the item"}],"openQuestions":["Does the registry intend only the conventional state-based word automaton, or should transition-based Büchi automata be first-class variants within this entry?","Does an existing Vercy world model already own this concept or its finite-state omega-automaton foundation?","Which authoritative definition and notation should govern initial-state multiplicity, incomplete transitions, and symbolic alphabet encodings?","Which exchange format and independent validation procedure should support reproducible acceptance and language-preservation evidence?","Should specialized subclasses such as weak or limit-deterministic Büchi automata be recorded as characteristics here or linked to separately registered concepts?"],"resources":{"spec":"/models/things/publications/thing-q1020380/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1020380/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T08:19:01Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:25+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane INF: no invented physical properties.","interaction.interfaces":"No specific technical standard or regulation is intrinsic to this mathematical automaton type."},"score":1.0}}