# Two automata, one parity language

KL-FCS-002 · Automata · version 1.0.0

## Problem

Decide whether two complete deterministic automata accept the same binary strings.

## Context

Compare complete transition systems through their reachable product, rather than checking a short sample of strings.

## Definitions

- **DFA**: A deterministic transition function over a finite state set and alphabet.
- **Product state**: A pair of states reached by reading the same prefix in both automata.
- **Distinguishing word**: A finite input accepted by exactly one of the compared machines.

## Checked result

Equivalent: yes.

Starting from the initial pair, the checker explores every reachable pair of states and requires matching acceptance. No mismatch is reachable, so equivalence holds for every finite binary word. Both accept an even number of ones.

## Checker reasoning

1. Start from the pair of initial states.
2. Explore every symbol transition until no new pair is reachable.
3. Compare acceptance in every reachable pair.
4. For inequivalence, replay a distinguishing input from both initial states.

## Dataset construction

{
  "family": "automata",
  "task": "Decide whether two complete deterministic automata accept the same binary strings.",
  "input_encoding": "Structured JSON; field meanings are stated in the specification.",
  "coverage": "Complete reachable product · 3 state pairs",
  "acceptance": [
    "Start from the pair of initial states.",
    "Explore every symbol transition until no new pair is reachable.",
    "Compare acceptance in every reachable pair.",
    "For inequivalence, replay a distinguishing input from both initial states."
  ],
  "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
  "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
}

## Formal payload

```json
{
  "specification": {
    "alphabet": [
      "0",
      "1"
    ],
    "left": {
      "start": "E",
      "accepting": [
        "E"
      ],
      "transitions": {
        "E": {
          "0": "E",
          "1": "O"
        },
        "O": {
          "0": "O",
          "1": "E"
        }
      }
    }
  },
  "claim": {
    "equivalent": true
  },
  "witness": {
    "right": {
      "start": "A",
      "accepting": [
        "A",
        "B"
      ],
      "transitions": {
        "A": {
          "0": "B",
          "1": "C"
        },
        "B": {
          "0": "A",
          "1": "C"
        },
        "C": {
          "0": "C",
          "1": "A"
        }
      }
    }
  }
}
```

## Complexity

At most |Q₁|·|Q₂| product states are visited, with one outgoing edge per alphabet symbol.

## Limits

Applies only to these two specified complete deterministic automata; no claim of automaton minimality.

## Common error and further work

Bounded string testing is weaker than complete DFA product exploration.

Extend the checker to emit shortest distinguishing words and state-minimization partitions.

## Verification

Complete reachable product · 3 state pairs. 3 checker units.
Replay with `python3 tools/verify.py`. Mechanical status: checked; human review
has not yet been recorded. Custom Python verification, not a proof-assistant
claim. Checker 1.0.0 and exact source hashes are in `verification.json`.

## Provenance and references

Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.
No external dataset or model-generation experiment. Reuse-license selection
remains pending.

- [Conceptual reference](https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/download/)