Finite Relation Symmetric Completion

Construct the least symmetric completion of an explicit finite relation; no transitive closure is performed.

Description

Construct the least symmetric completion of an explicit finite relation; no transitive closure is performed.

Symmetric completion adds the reverse of every relation pair while retaining existing pairs.

When to use this tool

Construct the least symmetric completion of an explicit finite relation; no transitive closure is performed.

Prepare the supplied pairsJson, universe fields for this operation.
Compare the complete example with the representation used by your own input before carrying the result into another step.

How to use the tool and how it works

Enter pairsJson, universe. Use the exact text syntax, units and index conventions explained below.
Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
Read result.pairs. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.

Validate a unique universe and pair membership, deduplicate input pairs and append each missing reverse in original pair traversal order.^[1]

Input field	Meaning and representation	Accepted input
pairsJson	JSON text, entered as one text field. The decoded structure is illustrated below; it is not a separate file.	Text
universe	universe.	List of strings; at most 128 entries

For the JSON text fields, the following decoded structures show the example records before they are serialized into input strings. Keep field names, list nesting and value types as shown.

json

[
  {
    "field": "pairsJson",
    "decoded": [
      [
        "a",
        "b"
      ],
      [
        "b",
        "a"
      ]
    ]
  }
]

Output field	Interpretation
result	Object containing the named outputs below.
result.pairs	The pairs list in the ordering and representation described above.

json

{
  "input": {
    "pairsJson": "[[\"a\",\"b\"],[\"b\",\"a\"]]",
    "universe": [
      "a",
      "b"
    ]
  },
  "output": {
    "result": {
      "pairs": [
        [
          "a",
          "b"
        ],
        [
          "b",
          "a"
        ]
      ]
    }
  }
}

Limitations and assumptions

This does not add all diagonal or transitive pairs and does not prove an equivalence relation.
All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.

Alternative or Complementary methods

Declare the full universe and distinguish testing a property from adding its closure. Use separate functionality and totality checks before interpreting a relation as a function.

References

Tool behavior and example: Finite Relation Symmetric Completion — tool.io

Similar or alternative tools

Finite Relation Reflexive Completion
Construct the least reflexive completion of an explicit finite relation; no transitive closure is performed.
Finite Relation Composition Pairs
Compose finite relations by following first then second, coalescing duplicate witnesses into unique endpoint pairs; order convention is explicit.