---
title: "algorithm-forward-backward-tests — Mova code model"
description: "Mova code model of algorithms/src/forward_backward/tests.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/forward_backward/tests.rs.groupant/"
---
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# algorithm-forward-backward-tests

The groupant beside algorithms/src/forward_backward/tests.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 descriptive instance
- **source:** [`algorithms/src/forward_backward/tests.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/forward_backward/tests.rs)
- **source SHA-256:** `7a9a64a47411d70b…`
- **repository folder:** [`algorithms/src/forward_backward/tests.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/forward_backward/tests.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** | partial | [approximant](#approximant) |
| **ast** | partial | `syntax.json` |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | partial | [model](#model) |
| **tests** | existing checks referenced | [tests](#tests) |

## 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 |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial source derived model | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial source derived model | [reverse_reconstruction](#reverse-reconstruction) |
| `independent_oracle` | independent_oracle | partial source derived model | [independent_oracle](#independent-oracle) |
| `boundary_partition` | boundary_partition | partial source derived model | [boundary_partition](#boundary-partition) |
| `compiled_mutation` | compiled_mutation | partial source derived model | [compiled_mutation](#compiled-mutation) |

## Model text


### descriptant

#### forward_backward checks

Original std-only implementation and fixtures; source citations appear in module documentation.

Independent finite oracle and boundary tests for forward_backward. Production: n1..64,T0..4096; normalized finite initial/transition rows and time-major emissions in[0,1]. Shape then values then row normalization. Alpha log-sum-exp over incoming paths; beta log-sum-exp over outgoing paths. exp(alpha+beta-logZ), normalized per posterior row. Empty returns logZ0/no rows; zero evidence returns None.

Epoch 2026-10-30: independent tiny-domain oracle, repeated debug/release and compiled production mutation. No timing or end-to-end Mova claims.

### model

#### Invariants and state

Independent finite oracle and boundary tests for forward_backward. Production: n1..64,T0..4096; normalized finite initial/transition rows and time-major emissions in[0,1]. Shape then values then row normalization. Alpha log-sum-exp over incoming paths; beta log-sum-exp over outgoing paths. exp(alpha+beta-logZ), normalized per posterior row. Empty returns logZ0/no rows; zero evidence returns None.

All-pass excludes represented oracle mismatches and rejected boundary violations in these finite fixtures only. Production excluded-output relation: Negative or nonnormalized posterior rows are excluded by the real model; represented rows are normalized by their computed sum. Impossible evidence excludes a posterior. Validation errors take precedence over empty-sequence success.

O(T*n*n) arithmetic, O(T*n+n) space; log-sum-exp allocates a temporary bounded n-vector per recurrence.

All allocations follow standard Rust process policy, not fallible allocation APIs. No callbacks or external effects. f64 roundoff and underflow remain explicit limitations.

### approximant

#### Forward and cost approximant

Independent finite oracle and boundary tests for forward_backward. Production: n1..64,T0..4096; normalized finite initial/transition rows and time-major emissions in[0,1]. Shape then values then row normalization. Alpha log-sum-exp over incoming paths; beta log-sum-exp over outgoing paths. exp(alpha+beta-logZ), normalized per posterior row. Empty returns logZ0/no rows; zero evidence returns None.

O(T*n*n) arithmetic, O(T*n+n) space; log-sum-exp allocates a temporary bounded n-vector per recurrence.

No executable cheaper surrogate or supported generic 1/2/5/10% profile is claimed. Residuals and floating estimates are not solution-error certificates.

### models

### behavior_complement

All-pass excludes represented oracle mismatches and rejected boundary violations in these finite fixtures only. Production excluded-output relation: Negative or nonnormalized posterior rows are excluded by the real model; represented rows are normalized by their computed sum. Impossible evidence excludes a posterior. Validation errors take precedence over empty-sequence success.

### reverse_reconstruction

All-pass observations admit implementations differing on unsampled inputs. This finite test fiber does not uniquely identify source or prove universality. Production fiber: Posterior and likelihood admit many hidden models, e.g. a one-state model with different emission factorizations sharing the same product. Observed marginals do not recover joint state paths or transitions uniquely.

### independent_oracle

Tiny path enumeration, analytical rank-one matrices or independent weighted moments reconstruct expectations without calling the production recurrence; scope is finite.

### boundary_partition

Shape, zero/empty, invalid finite values, ties or budget boundaries are explicitly asserted. Numeric tolerance is fixture-specific.

### compiled_mutation

A compiled changed production recurrence must fail the named oracle assertion. Restoration plus positive rerun checks that the gate can distinguish this wrong behavior.

### tests

#### Validation

Run cargo test -p mova-algorithms forward_backward::tests. Evidence in $MOVA_DATA/runs/octopus-wave26-probability-2026-10-30: repeated profiles, mutation logs and restored positive runs. Three named tests per module. See receipt.json for evidence hashes. No allocator fault injection, timing or full-school measurement.

### prospective

#### Probability batch prospective math

Written before implementation. All code original, std-only. Bounded allocations use standard Vec; allocator failure follows Rust process policy, no callbacks or RNG.

Alias: Walker (1977), An efficient method for generating discrete random variables with general distributions; Vose (1991), A linear algorithm for generating random numbers with a given distribution. n 1..4096, finite nonnegative weights, positive maximum. Scale by maximum then total to avoid overflow. Small/large work lists conserve total scaled mass. Column uniform and coin uniform [0,1) produce the specified categorical distribution in real arithmetic; caller explicitly supplies coordinates. O(n) build/storage, O(1) draw. No probabilistic draw generation or numeric percentage guarantee.

HMM: Rabiner (1989), A tutorial on hidden Markov models and selected applications in speech recognition; Viterbi (1967), Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. n1..64, T0..4096; finite probabilities in [0,1], initial/transition rows sum to one within 1e-10. Emissions time-major likelihoods in [0,1], need not sum across states. Validate structure then numeric values before empty sequence. Empty log likelihood0/path empty. Viterbi log max-plus recurrence, predecessor ties lowest index and terminal ties lowest index (not global lexicographic). No positive path =>None. Forward/backward log-sum-exp, posterior proportional exp(alpha+beta-logZ); per-row normalization corrects rounding. O(Tn²) time/O(Tn) space. Tiny exhaustive state-sequence enumeration is independent oracle. Log scales avoid direct-product underflow; no general floating error bound.

Sinkhorn: Sinkhorn and Knopp (1967), Concerning nonnegative matrices and doubly stochastic matrices. Positive matrix entries and marginals restricted [1e-100,1e100], dims1..64, equal marginal totals within tolerance1e-12 relative. Alternate row and column multiplication to target marginals; residual max absolute marginal discrepancy. Budget0..10000, finite tolerance>0. Stop on residual<=tolerance or budget; overflow/underflow to zero explicit arithmetic error. O(iterations*r*c) time/O(rc) space. Residual is feasibility evidence, not solution distance/optimality guarantee. Scale-invariant input fibers and budget status explicit.

Weighted covariance: West (1979), Updating mean and variance estimates: an improved method; Chan, Golub, LeVeque (1983), Algorithms for computing the sample variance. dimension1..64; finite coordinates |x|<=1e100, weights[0,1e100]. Population covariance M2/W, not unbiased sample covariance. Zero weight validates coordinates then does nothing. Merge dimensions must match, empty neutral. Merge W=A+B; delta=muB-muA; mu=muA+delta*B/W; M2=M2A+M2B+delta outer delta*A*(B/W). Atomic error updates, finite intermediates required. O(d²) push/merge/storage. Independent batch weighted centered moments oracle, partition merge equivalence within tolerance. No order-bitwise identity or universal covariance error certificate.