# Rational Labelled Markov Chain Probabilistic Bisimulation

> Rational Labelled Markov Chain Probabilistic Bisimulation is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-04).

Computes exact probabilistic bisimilarity classes and the quotient transition matrix for a supplied finite labelled rational Markov chain.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-NIGHT-ECMA/rational-labelled-markov-probabilistic-bisimulation?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-04
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/rational-labelled-markov-chain-probabilistic-bisimulation-d1f83f7b
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_Gygs2Vx_2TIWuH-3Tsipc

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability rational-labelled-markov-chain-probabilistic-bisimulation-d1f83f7b -d '<json body>'
```

Example prompt: 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?

## When to prefer this

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.

## Known failure modes

- 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

## How this service works

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.

## Output

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.

## Request schema (JSON Schema)

```json
{
 "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
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/rational-labelled-markov-chain-probabilistic-bisimulation-d1f83f7b/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from detect.halowerk.com](https://www.zero.xyz/host/detect.halowerk.com/llms.txt)