# Finite Kripke CTL Branching-Time Satisfaction Sets Verifier

> Finite Kripke CTL Branching-Time Satisfaction Sets Verifier is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-01).

Computes the satisfaction set of every CTL formula over a supplied finite total Kripke structure, verifying all branching-time temporal logic claims.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-REST-B20/finite-total-kripke-ctl-branching-time-satisfaction-sets?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-01
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/finite-kripke-ctl-branching-time-satisfaction-sets-verifier-225577d8
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_Oor10_l5LBZvWu-aRY-7Z

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 finite-kripke-ctl-branching-time-satisfaction-sets-verifier-225577d8 -d '<json body>'
```

Example prompt: I have a finite Kripke structure with 3 states, edges [[0,1],[1,2],[2,1]], per-state label sets [[0],[1],[0,1]], and two CTL formulas — atom(0) and EG(atom(1)) — can you compute the satisfaction set for each formula over this model?

## When to prefer this

Use this endpoint when you need to formally verify CTL (Computation Tree Logic) satisfaction sets over a small finite total Kripke structure (up to 8 states, 64 edges, 40 formulas). It is ideal for model checking tasks involving branching-time temporal operators (EX, AX, EG, AG, EF, AF, EU, AU) on finite automata, protocol verification, or academic CTL exercises. Prefer this over custom implementations when you need a reliable, hosted model checker for bounded-size models without writing your own fixpoint algorithms.

## Known failure modes

- Invalid graph: edges reference state indices >= n
- Non-total structure: some state has no outgoing edge (violates totality constraint)
- Formula references undefined sub-formula index (out of bounds)
- n exceeds maximum of 8 or is below minimum of 1
- Formula count exceeds 40 or edges exceed 64
- Malformed formula operator not in allowed enum
- Cycle or formula index self-reference causing invalid evaluation order

## How this service works

Verify every CTL state satisfaction set of a supplied total finite branching-time model.

## Output

Returns the satisfaction set for every CTL formula provided — i.e., the set of state indices (from 0 to n-1) that satisfy each formula in the given finite total Kripke structure. Results allow verification of branching-time temporal logic claims against the model.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "ctl"
   ],
   "properties": {
    "ctl": {
     "type": "object",
     "required": [
      "data",
      "claim"
     ],
     "properties": {
      "data": {
       "type": "object",
       "required": [
        "n",
        "edges",
        "labels",
        "formulas"
       ],
       "properties": {
        "n": {
         "type": "integer",
         "maximum": 8,
         "minimum": 1
        },
        "edges": {
         "type": "array",
         "items": {
          "type": "array",
          "items": {
           "type": "integer",
           "maximum": 255,
           "minimum": 0
          },
          "maxItems": 2,
          "minItems": 2
         },
         "maxItems": 64,
         "minItems": 0
        },
        "labels": {
         "type": "array",
         "items": {
          "type": "array",
          "items": {
           "type": "integer",
           "maximum": 255,
           "minimum": 0
          },
          "maxItems": 16,
          "minItems": 0
         },
         "maxItems": 8,
         "minItems": 1
        },
        "formulas": {
         "type": "array",
         "items": {
          "anyOf": [
           {
            "type": "object",
            "required": [
             "op",
             "proposition"
            ],
            "properties": {
             "op": {
              "const": "atom"
             },
             "proposition": {
              "type": "integer",
              "maximum": 255,
              "minimum": 0
             }
            },
            "additionalProperties": false
           },
           {
            "type": "object",
            "required": [
             "op",
             "body"
            ],
            "properties": {
             "op": {
              "enum": [
               "not",
               "EX",
               "AX",
               "EG",
               "AG",
               "EF",
               "AF"
              ]
             },
             "body": {
              "type": "integer",
              "maximum": 255,
              "minimum": 0
             }
            },
            "additionalProperties": false
           },
           {
            "type": "object",
            "required": [
             "op",
             "left",
             "right"
            ],
            "properties": {
             "op": {
              "enum": [
               "and",
           
… (truncated)
```

## More

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