📐algorithm-knuth-optimal-bst

The groupant beside algorithms/src/knuth_optimal_bst/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

Components

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.

Model text

descriptant

knuth_optimal_bst mod.rs

Original std-only implementation and independent fixtures. Knuth (1971), Optimum binary search trees; successful-search-only specialization.

Prospective model saved before changes in $MOVA_DATA/runs/octopus-wave20-dp-2026-10-30/prospective.md. Current epoch 2026-10-30: exact optimization with explicit bounded-resource outcomes. Finite oracle agreement does not establish the caller's monotonicity premises. No timing or whole-Mova speed claim.

approximant

Forward model

Successful-search BST recurrence C[i,j]=sum(w[i..j])+min_{i<=r<j}(C[i,r]+C[r+1,j]), C[i,i]=0. Nonnegative weights give root[i,j-1]<=root[i,j]<=root[i+1,j]. Scan only that interval, choosing first equal optimum.

Cost

O(n^2) candidate scans by telescoping Knuth ranges and O(n^2) memory. max_cells bounds each (n+1)^2 table. Empty input returns before allocation; dimensions then budget then fallible allocations then checked u128 sums. Prefix array adds O(n) storage.

Invariants and limits

Owned input slices remain unchanged. Deterministic ties and traversal; callback panics propagate and purity/monotonicity remain caller obligations. No allocator-refusal injection, no semantic AST proof, no universal finite-test proof. Checked arithmetic reports errors rather than wrapping.

Executable approximant gap

No executable surrogate is attached: categorical reachability/argmin has no justified relaxed metric here. Objective-only surrogate speed/error has not been measured. Existing knapsack surrogate is not automatically transferable. No 1/2/5/10% profile or approximate substitution is claimed.

models

behavior_complement

A singleton of weight 9 excludes costs other than 9, since root depth is one. Cost below sum(weights) is impossible; all nodes have positive depth.

reverse_reconstruction

Cost 0 admits any length all-zero weight vector, including empty. Cost 2 admits [2] and [0,1]. Output alone determines neither key count nor optimal tree.

positive_scale

Multiplying weights by integer k>=0 scales optimum by k when input and u128 sums fit; k=0 may change chosen roots but not cost law.

reversal

Reversing weights mirrors each feasible BST and preserves optimum, although left-tie chosen roots need not mirror.

weight_monotonicity

Increasing one weight by delta cannot decrease optimum; for nonempty n the increase lies between delta and n*delta when arithmetic fits.

root_interval

For nonnegative weights, neighboring optimal roots bound the current leftmost optimum; the sum of candidate interval lengths per diagonal is O(n).

tests

Checks

cargo test -p mova-algorithms knuth_optimal_bst::tests

Independent small oracle and resource boundary fixtures live in ../tests.rs. Repeated debug/release and compiled mutation receipts: $MOVA_DATA/runs/octopus-wave20-dp-2026-10-30.

combiner

Composition

Raw mod.rs syntax and mathematical recurrence are separate evidence. The partial syntax projection binds the exact source bytes but does not certify the recurrence. Oracle tests compare observable outputs; typed relations constrain lawful transformations and compatible preimages. For knuth_optimal_bst, resource caps and checked failures remain separate from mathematical objective equivalence. Prospective expectations remain unchanged; finite validation cannot turn monotonicity assumptions into runtime validation.

validation

Validation — 2026-10-30

Ten new native tests cover five exact kernels. Full mova-algorithms library: 557 tests pass in two debug and two release executions; sorted native test outcomes agree byte-for-byte across all four. Strict all-target Clippy and focused rustfmt pass. External receipts: $MOVA_DATA/runs/octopus-wave20-dp-2026-10-30/{debug1,debug2,release1,release2,clippy}.log.

Compiled production controls: zero initial subset bit, forced zero MITM objective, right-tie SMAWK reduction, forced zero layered-DP objective and forced zero BST objective all fail named equality assertions. Byte restoration followed by full positive library rerun passes. A separate SMAWK interpolation tie mutation fails lawful_matrices after adding half-integer squared-distance centers; the restored focused test passes. Receipts: controls.log, mutant-*.log, restored.log, smawk-restored.log in the same external run directory. The original constant-matrix fixture alone did not distinguish interpolation ties; the stronger fixture closes that observed gap.

Every source/test file has two independently generated identical syntax projections and current source SHA. An isolated ten-leaf modelcheck groupants run reports valid=true, diagnostics=[], ten sidecars, zero missing source sidecars; the temporary collection parent has no sidecar and makes no composition claim. Root integration owns global registry and ancestor binding checks.

No timing speedup, machine-independent resource guarantee, allocation-failure injection, monotonicity validation, or end-to-end Mova improvement is inferred. Successful finite tests certify the listed observations only. No teacher, model, corpus or live database operations were used.

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