Finite Relation Transitive Triple Audit

Audit finite relation transitivity and return explicit failing witnesses; unknown universe elements are rejected.

Description

Audit finite relation transitivity and return explicit failing witnesses; unknown universe elements are rejected.

Transitivity requires every two-step relation chain to have its direct endpoint pair. Missing shortcuts provide concrete witnesses.

When to use this tool

Audit finite relation transitivity and return explicit failing witnesses; unknown universe elements are rejected.

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.holds, result.witnesses. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.

Validate and deduplicate pairs. For each [a,b] and [b,d], report [a,b,d] when [a,d] is absent. holds means no witness triples.^[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.holds	True exactly when the stated property passes; it does not establish other properties.
result.witnesses	Concrete missing or conflicting relation evidence; inspect holds alongside the list.

json

{
  "input": {
    "pairsJson": "[[\"a\",\"b\"],[\"b\",\"a\"]]",
    "universe": [
      "a",
      "b"
    ]
  },
  "output": {
    "result": {
      "holds": false,
      "witnesses": [
        [
          "a",
          "b",
          "a"
        ],
        [
          "b",
          "a",
          "b"
        ]
      ]
    }
  }
}

Limitations and assumptions

The tool does not add closure pairs. Diagonal shortcuts from two-way chains are required by the same rule.
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 Transitive Triple Audit — tool.io

Similar or alternative tools

Finite Relation Antisymmetric Pair Audit
Audit finite relation antisymmetry and return explicit failing witnesses; unknown universe elements are rejected.
Finite Relation Asymmetric Pair Audit
Audit finite relation asymmetry and return explicit failing witnesses; unknown universe elements are rejected.