---
title: "algorithm-transitive-reduction-dag — Mova code model"
description: "Mova code model of algorithms/src/transitive_reduction_dag/mod.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/transitive_reduction_dag/mod.rs.groupant/"
---
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# algorithm-transitive-reduction-dag

The groupant beside algorithms/src/transitive_reduction_dag/mod.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

## Facts

- **kind:** groupant
- **status:** partial source derived instance
- **source:** [`algorithms/src/transitive_reduction_dag/mod.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/transitive_reduction_dag/mod.rs)
- **source SHA-256:** `58125a7da360e46f…`
- **repository folder:** [`algorithms/src/transitive_reduction_dag/mod.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/transitive_reduction_dag/mod.rs.groupant)
- **repository commit:** `855da7bd4a7b`

> **Implementation material.** A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.

## Components

| component | label | where |
|---|---|---|
| **approximant** | bounded checked | [Forward](#forward) |
| **ast** | partial | `syntax.json` |
| **cost** | symbolic unmeasured | [Cost](#cost) |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | source derived | [Invariant](#invariant) |
| **tests** | bounded executed | [evidence](#evidence) |

## Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

| id | type | label | where |
|---|---|---|---|
| `nonbehavior` | behavior_complement | partial source derived | [nonbehavior](#nonbehavior) |
| `compatible_inputs` | reverse_reconstruction | partial source derived | [compatible_inputs](#compatible-inputs) |
| `essential_edge` | deletion_certificate | partial source derived | [essential_edge](#essential-edge) |
| `closure_fiber` | constructive_preimage | partial source derived | [closure_fiber](#closure-fiber) |
| `idempotence` | fixed_point | partial source derived | [idempotence](#idempotence) |
| `representation` | representation_invariance | partial source derived | [representation](#representation) |

## Model text


Migration: consolidated into model.md, 2026-11-04, operator decision; fresh hash-bound references to the former files were rebound to the sections below.

### descriptant

#### transitive_reduction_dag mod

Original std-only Rust. Prospective model is retained beside implementation. Source citations identify algorithm provenance; no copied implementation. Exact bounded graph semantics are useful independently of any unmeasured optimization. Finite oracle checks and mutant discrimination improve confidence; groupant count alone proves no improvement.

### model

## transitive_reduction_dag mod model

### Forward

Directed DAG n<=256,m<=65536, normalize duplicate arcs; reject cycles incl loops. Compute Boolean Warshall reachability, retain edge u->v iff no other direct successor w reaches v. Ascending rows; closure invariant, unique minimal reachability-equivalent DAG. O(V^3+E) time O(V^2) memory. Aho, Garey, Ullman (1972).

### Invariant

Warshall after k permits intermediate vertices through k. Reduction keeps precisely original edges with no alternate path; all original reachability remains.

### Cost

Directed DAG n<=256,m<=65536, normalize duplicate arcs; reject cycles incl loops. Compute Boolean Warshall reachability, retain edge u->v iff no other direct successor w reaches v. Ascending rows; closure invariant, unique minimal reachability-equivalent DAG. O(V^3+E) time O(V^2) memory. Aho, Garey, Ullman (1972).

### nonbehavior

Type: behavior_complement. Let R be the deterministic relation specified by Forward including validation precedence and tie rules, X the declared input type and Y its Result type. Behavior complement is (X times Y) minus R. Specifically outputs violating Warshall after k permits intermediate vertices through k. Reduction keeps precisely original edges with no alternate path; all original reachability remains. are excluded; a valid-input Err or a wrong successful witness is excluded. Resource, cycle and structural rejections are determined before algorithmic output.

### compatible_inputs

Type: reverse_reconstruction. Given DAG H already transitively reduced, every graph formed by adding any subset of H's reachable pairs reduces to H. Duplicate input edges also collapse. Cyclic Error recovers any directed cycle, without a unique witness.

### essential_edge

Type: deletion_certificate. Edge (u,v) is retained iff deleting it from the normalized input destroys u-to-v reachability. This characterizes unique DAG reduction.

### closure_fiber

Type: constructive_preimage. For fixed reduced H, H and its transitive closure and every intermediate graph share the same reduction. On chain 0->1->2, adding 0->2 illustrates the nonunique fiber.

### idempotence

Type: fixed_point. Reduction R satisfies reduction(R)=R. Every retained edge is essential in R.

### representation

Type: representation_invariance. Input row order and duplicate count do not affect sorted output. A self-loop always causes Cycle; empty graph returns empty.

### Limits

Partial syntax omits macro expansion and type resolution. No executable approximate profile: categorical graph semantics and feasibility have no justified percentage relaxation here. Integer operations are bounded by caps; flow uses checked i128/u64 arithmetic, with Overflow defensive under current caps. No callbacks. Allocator failure is outside typed status.

### evidence

#### Evidence — 2026-10-30

Eleven tests across five algorithms, including independent exhaustive small-domain characterization. This file belongs to transitive_reduction_dag. Two debug and two optimized runs matched normalized statuses. Strict clippy-driver --test -D warnings passed. A compiled production mutant failed a named assertion; exact bytes restored and positive rerun passed. Raw receipts: $MOVA_DATA/runs/octopus-wave25-graphs-2026-10-30/transitive_reduction_dag-*.txt. No timing or whole-system improvement claim. Source SHA-256: 58125a7da360e46fdcf8bb5b9f613f5c24ea808e44c576488abbfacebb512da7.

### prospective

#### Prospective model — 2026-10-30

Directed DAG n<=256,m<=65536, normalize duplicate arcs; reject cycles incl loops. Compute Boolean Warshall reachability, retain edge u->v iff no other direct successor w reaches v. Ascending rows; closure invariant, unique minimal reachability-equivalent DAG. O(V^3+E) time O(V^2) memory. Aho, Garey, Ullman (1972).
Allocation failure is outside typed errors. Resource caps precede vertex/edge validation. No callbacks. Exact categorical output has no justified percentage surrogate; executable approximation profiles unsupported.