DFA Dead State Partitioner

Description

Partition declared states into states with a path to acceptance and dead states without such a path.

A deterministic transition maps a state and one literal alphabet symbol to at most one next state. A partial table may leave some pairs undefined; a complete table defines every state/symbol pair. Tracing one word, exploring all graph paths and changing the accepting set are distinct operations. IDs and symbols use exact string identity.

Partition declared states into states with a path to acceptance and dead states without such a path.

When to use this tool

Use the dead-state partition to distinguish states with any path to acceptance from those without one.

How to use the tool and how it works

Enter transitionsJson, states, alphabet, accepting using the definitions below. Serialize JSON text fields in the decoded shape shown below.
Run the operation. Resolve input-format errors before interpreting an output.
Read result.coaccessible, result.dead, using the stated boundary, ordering, sign and absence conventions.

Validate the deterministic partial table and accepting-state membership. Traverse transitions in reverse from the accepting states to find coaccessible states, then report all other declared states as dead.^[1]

Input field	Meaning and units	Accepted input
transitionsJson	JSON array of {from,symbol,to} literal string records, preserving transition indexes. Determinism is checked or audited as described above.	Text, at most 8,192 UTF-16 code units; narrower grammar/range rules also apply.
states	Distinct literal declared state IDs in bank order; full automaton operations require at least one.	Dense list of at most 128 literal strings, each at most 8,192 UTF-16 units.
alphabet	Distinct literal alphabet symbols in bank order; an empty alphabet is allowed.	Dense list of at most 128 literal strings, each at most 8,192 UTF-16 units.
accepting	Explicit literal accepting-state IDs; full automaton operations require membership in states.	Dense list of at most 128 literal strings, each at most 8,192 UTF-16 units.

These decoded structures show the examples inside JSON text fields. Supply their serialized JSON strings as input.

json

{
  "transitionsJson": [
    {
      "from": "a",
      "symbol": "x",
      "to": "b"
    },
    {
      "from": "b",
      "symbol": "x",
      "to": "b"
    }
  ]
}

Output field	Interpretation
result	Object containing the fields below.
result.coaccessible	States with a graph path to an accepting state, in reverse breadth-first discovery order.
result.dead	Declared states with no graph path to acceptance, in bank order.

json

{
  "input": {
    "transitionsJson": "[{\"from\": \"a\", \"symbol\": \"x\", \"to\": \"b\"}, {\"from\": \"b\", \"symbol\": \"x\", \"to\": \"b\"}]",
    "states": [
      "a",
      "b",
      "c"
    ],
    "alphabet": [
      "x"
    ],
    "accepting": [
      "b"
    ]
  },
  "output": {
    "result": {
      "coaccessible": [
        "b",
        "a"
      ],
      "dead": [
        "c"
      ]
    }
  }
}

Limitations and assumptions

Dead means no graph path to a supplied accepting state. Coaccessible states follow reverse breadth-first discovery; dead states retain declared bank order. This is not a minimal-state partition.
Transition JSON is an array of {from,symbol,to} string records. Tools requiring a declared automaton use a nonempty distinct state bank and distinct alphabet, validate endpoints and symbols, and reject duplicate [from,symbol] keys. Trace-only tools validate keys without inferring a declared universe. No normalization or implicit sink is provided.
Every declared outer field is required and unknown outer fields are rejected. Strings and JSON text have size limits, lists are bounded to 128 entries, and JSON nesting/node counts are bounded. Tighter operation-specific bounds stated above take precedence.
Numeric outputs must remain finite and within safe-integer magnitude. Original record indexes and cell coordinates are zero-based; stated half-open ranges exclude their end.

Alternative or Complementary methods

Use endpoint and duplicate-key audits before transformations. Check completeness before complementing an accepting set, and preserve the declared alphabet, start-state convention and acceptance policy with exported tables.

References

Tool behavior and example: DFA Dead State Partitioner — tool.io

Similar or alternative tools

DFA Coaccessible State Inventory
List automaton states with a graph path to a supplied accepting state.
DFA Unreachable State Partitioner
Partition declared automaton states into states reachable from start and the remaining unreachable states.