{"uid":"cap_Gygs2Vx_2TIWuH-3Tsipc","slug":"rational-labelled-markov-chain-probabilistic-bisimulation-d1f83f7b","name":"Rational Labelled Markov Chain Probabilistic Bisimulation","description":"A stochastic-model test agent needs exact probabilistic bisimilarity classes and the quotient transition matrix of a supplied finite labelled rational Markov chain, not one target reachability probability or deterministic-automaton language equivalence.","url":"https://detect.halowerk.com/v1/F-NIGHT-ECMA/rational-labelled-markov-probabilistic-bisimulation?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"artifact":{"type":"object","required":["markov_bisimulation"],"properties":{"markov_bisimulation":{"type":"object","required":["labels","denominator","weights","claim"],"properties":{"claim":{"type":"array","items":{"type":"integer","maximum":15,"minimum":0},"maxItems":16,"minItems":0},"labels":{"type":"array","items":{"type":"integer","maximum":255,"minimum":0},"maxItems":16,"minItems":1},"weights":{"type":"array","items":{"type":"array","items":{"type":"integer","maximum":1000,"minimum":0},"maxItems":16,"minItems":1},"maxItems":16,"minItems":1},"denominator":{"type":"integer","maximum":1000,"minimum":1}},"additionalProperties":false}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_gqCGSQJa3sX2jDJnuwwP8","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes exact probabilistic bisimilarity classes and the quotient transition matrix for a supplied finite labelled rational Markov chain.","exampleAgentPrompt":"I have a 4-state rational labelled Markov chain with labels [0, 1, 0, 1], denominator 6, and transition weights [[2,1,2,1],[0,3,0,3],[1,2,1,2],[3,0,3,0]], and I claim states 0 and 2 are bisimilar — can you compute the exact probabilistic bisimilarity classes and the quotient transition matrix?","exampleUseCases":[{"title":"Stochastic model minimization check","prompt":"I'm testing a 5-state rational Markov chain with labels [0,1,0,1,2], denominator 10, and weights [[3,2,0,5,0],[0,4,3,0,3],[2,0,3,5,0],[0,3,0,4,3],[1,1,2,2,4]]. My claim is that states 0 and 2 fall in the same bisimilarity class — please give me the exact bisimilarity partition and the quotient transition matrix."},{"title":"Formal verification of probabilistic equivalence","prompt":"For my formal verification pipeline, I need to know the exact probabilistic bisimilarity classes of a 3-state labelled Markov chain with labels [1,2,1], denominator 4, and transition weights [[2,1,1],[1,2,1],[1,1,2]]. No claim states to check — just give me all bisimilarity classes and the quotient matrix."},{"title":"Stochastic agent bisimulation correctness test","prompt":"My stochastic test agent needs to validate its bisimulation claim: for a Markov chain with labels [0,0,1,1], denominator 8, weights [[4,2,1,1],[3,3,1,1],[1,1,3,3],[1,1,2,4]], and claim [0,1], compute the exact probabilistic bisimilarity classes and confirm whether states 0 and 1 belong to the same class."}],"resultDescription":"Returns the exact probabilistic bisimilarity equivalence classes (partition of states) and the quotient transition matrix of the supplied rational labelled Markov chain, enabling state-space minimization and formal verification of stochastic models.","failureModes":["Weights matrix dimensions inconsistent with labels array length — rejected with validation error","Denominator value outside allowed range (1–1000) — schema validation failure","Claim indices reference out-of-range state indices (>15) — rejected","Transition weights do not form a valid stochastic matrix under the given denominator — computation error","Malformed or empty labels array — schema rejection"],"whenToPreferThis":"Choose this endpoint when you need exact, symbolic probabilistic bisimilarity computation (not approximation) for a small finite rational Markov chain (up to 16 states). It is specifically suited for stochastic model testing, formal verification pipelines, and quotient reduction tasks — not for single reachability queries or deterministic automaton language equivalence checks.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:41:03.935Z","isFirstParty":false,"canonicalSlug":"rational-labelled-markov-chain-probabilistic-bisimulation-d1f83f7b"}