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diff --git a/docs/case-study-hash-shift-bug.md b/docs/case-study-hash-shift-bug.md new file mode 100644 index 0000000..6ec4f7d --- /dev/null +++ b/docs/case-study-hash-shift-bug.md @@ -0,0 +1,157 @@ +# Case study: a latent bug the verification harness caught + +This documents a real defect that the verification work found **on the very +first run** of the new `make verify` target — before any of the heavier layers +(TLA+, Gobra) were even involved. It is a good illustration of *why* wiring +these checks into a gate pays off: the bug had been sitting in the repository +undetected because the existing tests could never trigger it. + +## TL;DR + +- **Where:** `search/hash.go`, the `Hash.hash` method. +- **What:** `key << 10`, where `key` has a generic integer type, silently + evaluates to `0` for narrow key types (`int8`/`uint8`), discarding a whole + term of the hash mix. +- **Who found it:** `go vet`'s shift analyzer, run as part of `make verify`. +- **Why the tests missed it:** every test instantiates the hash with 64-bit + `int` keys, where the shift is perfectly fine — so the bug is *latent*. +- **Fix:** compute the mix in a full-width `int64`. + +## The offending code + +Before: + +```go +func (h *Hash[K,V]) hash(key K) int { + i := key + key*2 + key<<10 + key>>2 + if i < 0 { + i = -i + } + return int(i) % h.capacity +} +``` + +`K` is a type parameter constrained by `ds.Integer` (`ds/types.go`), which +embeds `constraints.Integer` — i.e. `K` may be **any** of `int, int8, int16, +int32, int64, uint, uint8, …`. The intent of `key<<10` is clearly to spread the +key's low bits up into the high bits so that keys differing only in their low +bits land in different buckets. + +## What the tool reported + +``` +$ make verify +go vet ./... +search/hash.go:29:21: key (may be 8 bits) too small for shift of 10 +``` + +`go vet` bundles a *shift* analyzer that is, in effect, a lightweight formal +check: for every shift expression it computes a conservative lower bound on the +bit width of the left operand and flags any shift whose count is `>=` that +width. Here the narrowest type `K` can take is `int8` (8 bits), and `10 >= 8`, +so the analyzer proves that *for at least one legal instantiation* the shift is +degenerate. + +## Why it is genuinely a bug (Go shift semantics) + +This is not a false positive. The Go specification defines non-constant left +shifts operationally: + +> Shifts behave as if the left operand is shifted `n` times by 1 for a shift +> count of `n`. […] There is no upper limit on the shift count. + +For an 8-bit value, shifting "one bit at a time" ten times pushes **every** +original bit out of the value's width. The result is therefore always `0`. So +for `K = int8`/`uint8`: + +``` +key<<10 == 0 // always, for every key +``` + +and the hash silently collapses to `key + key*2 + key>>2` — the high-bit mixing +the author intended is simply gone. + +Two subtleties worth recording: + +1. **It is width-dependent, not universally broken.** For `int16` the count + `10 < 16`, so the term is fine; for 32-/64-bit types it is obviously fine. + `go vet` still (correctly) flags the expression because it must be sound for + *all* instantiations, and `int8` is in the constraint set. The narrowest + type is what governs safety. + +2. **It is a quality/portability bug, not a memory-safety or a Set-contract + violation.** The hash table stays *functionally correct* even for `int8` + keys: `Put`, `Get`, and `Del` all call the same `hash`, and collisions are + resolved by chaining in the per-bucket `Elementary` list. What degrades is + the *distribution* — more keys collide into the same bucket, turning the + intended O(1) operations toward O(n). So the failure mode is silent + performance rot for narrow-key instantiations, exactly the kind of thing that + never shows up as a failing assertion. + +## Why no test caught it + +Every instantiation in the test suite uses `int` keys: + +```go +test[int,int](NewHash[int,int](i*2), i, t) // search/search_test.go +``` + +`int` is 64 bits on this platform, so `key<<10` behaves as intended and all +tests pass. There is no `Hash[int8, …]` anywhere, so the degenerate path is +never exercised. A property test or a fuzz run over `int` keys would *also* miss +it — the bug lives in the *type dimension*, not the value dimension, and only a +tool that reasons about the type (like `go vet`) or an actual narrow-type +instantiation can surface it. This is precisely the class of latent defect that +static analysis is good at and dynamic testing is blind to. + +## The fix + +Perform the mixing in a full-width `int64`, then reduce: + +```go +func (h *Hash[K,V]) hash(key K) int { + // Mix the key in a full-width int64 rather than in K. K is any ds.Integer, + // so for a narrow type (e.g. int8) the "key<<10" term would shift past the + // type width and vanish to 0, destroying the intended high-bit mixing (and + // go vet rightly flags it). Widening to int64 first keeps the result + // identical for 64-bit int keys while making the mix well-defined for every + // integer width. + i := int64(key) + i = i + i*2 + i<<10 + i>>2 + if i < 0 { + i = -i + } + return int(i) % h.capacity +} +``` + +Why this is the right fix: + +- **Behavior-preserving for the code that exists.** For `K = int` (64-bit), the + arithmetic is byte-for-byte identical to before — `int64(key)` is a no-op + widening, and every operation stays in 64 bits — so every existing test still + passes unchanged. +- **Correct for the code that might exist.** For narrow `K`, the key is widened + *before* the shift, so `i<<10` now mixes real bits instead of vanishing. The + hash finally does for `int8` keys what it always did for `int` keys. +- **It silences the analyzer for the right reason.** `int64` is 64 bits, `10 < + 64`, so the shift is provably well-defined for the actual operand type. We are + not suppressing the warning; we are removing the condition that made it true. + +An `int64` cast rather than the value's own width also documents intent: "this +mixing is meant to happen in a wide register, independent of the key type." + +> Residual note, left as-is: `if i < 0 { i = -i }` still has the classic +> `-math.MinInt64` overflow corner. It predates this change, is astronomically +> unlikely for these inputs, and is out of scope here — recorded for honesty. + +## How this maps to the verification layers + +This defect was caught by **Layer 2** of the harness (see +[`verification.md`](verification.md)) — `go vet` inside `make verify`. It is the +cheapest layer, and it found a bug that the paper proofs (Layer 0, which focus +on the sorts) and the property tests (Layer 1, which only ever run `int`) did +not. The lesson is the ordering of the layers is not the ordering of their +value: a one-line static check surfaced a real, shipped-in latent bug that no +amount of value-space testing would have. Cheap, broad checks first; deep proofs +where they earn their keep. diff --git a/docs/verification.md b/docs/verification.md new file mode 100644 index 0000000..3959e5b --- /dev/null +++ b/docs/verification.md @@ -0,0 +1,255 @@ +# Hand-written correctness proofs + +This document contains human-written (paper) correctness proofs for the +algorithms in this repository, derived by reading the actual source. Each proof +is a Hoare-style argument: a **precondition**, a **postcondition**, one **loop +invariant** per loop, a **termination measure**, and — for the sorts — a +**permutation argument**. + +These proofs are the human-readable source of truth. They are *human-checked*, +not machine-checked; the automated layers corroborate them: + +- `sort/property_test.go` checks *ordering* **and** *permutation* on thousands + of random inputs (empirical corroboration — see the "permutation" note below). +- `make verify` runs `go vet`, `staticcheck`, and the race detector. This layer + already paid off: it caught a latent bug in `search/hash.go` on its first run + — see [`case-study-hash-shift-bug.md`](case-study-hash-shift-bug.md). +- `formal/tla/SleepSort.tla` exhaustively model-checks the concurrent sleep sort. +- `formal/insertion.go` is a machine-checked (Gobra) proof of insertion sort. + +## Common notation and lemmas + +- `a[p..q]` denotes the inclusive slice of indices `p, p+1, …, q`. An empty + range (`p > q`) is vacuously sorted and vacuously a permutation of itself. +- **sorted(a[p..q])** ≡ `∀ p ≤ r < q : a[r] ≤ a[r+1]`. +- **perm(a, a₀)** ≡ the multiset `{a[0], …, a[n-1]}` equals the multiset of the + original contents `a₀`. + +### Lemma S (Swap preserves the multiset) + +The **only** operation any sort here uses to mutate the backing array is +`ArrayList.Swap` (`ds/arraylist.go:74`), which exchanges two elements. Swapping +two positions leaves the multiset of stored values unchanged. By induction over +the sequence of swaps performed by any algorithm below, **perm(a, a₀) holds at +every point** — the output is always a permutation of the input. This single +lemma discharges the permutation half of every sort's postcondition, so the +per-algorithm proofs below focus on *ordering* and *termination*. + +> The exceptions are `Merge`/`BottomUpMerge`, which write elements via `aux` +> rather than `Swap`; their permutation argument is given inline (Lemma M). + +--- + +## Selection sort — `sort/selection.go:7` + +- **Pre:** `a` holds arbitrary values; `l = len(a)`. +- **Post:** `sorted(a[0..l-1]) ∧ perm(a, a₀)`. + +**Outer invariant** (before iteration `i`, `0 ≤ i ≤ l`): +`sorted(a[0..i-1]) ∧ ∀ p < i ≤ q : a[p] ≤ a[q]` — i.e. the prefix `a[0..i-1]` +is sorted and every prefix element is ≤ every suffix element. + +**Inner loop** (`j := i+1 … l-1`) computes `min` = index of the smallest element +in `a[i..l-1]` (invariant: `a[min]` is the minimum of `a[i..j-1]`). After the +loop, `a.Swap(i, min)` moves that minimum to position `i`. This element is ≥ all +of `a[0..i-1]` (by the outer invariant, everything in `a[i..]` is ≥ the prefix) +and ≤ everything remaining in `a[i+1..]`, so the outer invariant re-establishes +for `i+1`. + +**Termination:** outer `i` and inner `j` each range over a fixed finite index +set and strictly increase. **Permutation:** Lemma S. At `i = l` the invariant +gives `sorted(a[0..l-1])`. ∎ + +## Insertion sort — `sort/insertion.go:7` + +- **Post:** `sorted(a[0..l-1]) ∧ perm(a, a₀)`. + +**Outer invariant** (before iteration `i`): `sorted(a[0..i-1])`. + +**Inner loop** (`j := i; j > 0; j--`) bubbles `a[i]` left, stopping via `break` +as soon as `a[j] > a[j-1]` (the pair is already in order) or when `j = 0`. +**Inner invariant** (at each test): `a[0..i]` is a permutation of its original +prefix contents; `sorted(a[j..i])`; and `∀ j < r ≤ i : a[r] ≥ a[j]`. When the +loop stops, the entire prefix `a[0..i]` is sorted, re-establishing the outer +invariant for `i+1`. + +Note the loop swaps on *equality* too (it breaks only on strict `a[j] > a[j-1]`), +which is harmless — it performs a few extra swaps but preserves both sortedness +and (by Lemma S) the multiset. + +**Termination:** inner measure `j` strictly decreases and is bounded below by 0; +outer `i` ranges over `range a`. ∎ + +## Shell sort — `sort/shell.go:7` + +Shell sort is insertion sort applied on a decreasing sequence of gaps +`h ∈ {…, 40, 13, 4, 1}` (built by `h = 3h+1`, then `h /= 3`). + +- **Post:** `sorted(a[0..l-1]) ∧ perm(a, a₀)`. + +For a fixed gap `h`, the body is an *h-interleaved* insertion sort: the inner +loop (`j := i; j >= h; j -= h`) inserts `a[i]` into the sorted-by-`h` +subsequence `…, a[i-2h], a[i-h], a[i]`, breaking when `a[j-h] < a[j]`. By the +insertion-sort argument applied to each residue class mod `h`, after the `h` +pass every h-strided subsequence is sorted (the array is "h-sorted"). + +The final gap is always `h = 1` (the loop condition is `h >= 1` and integer +division reaches 1). A 1-sorted array is fully `sorted(a[0..l-1])`. The earlier +larger-gap passes only reorder via `Swap`, so they neither break the final +1-sort's correctness nor the multiset. + +**Termination:** the gap loop strictly decreases `h` via `h /= 3` until `h < 1`; +each inner loop terminates as in insertion sort. **Permutation:** Lemma S. ∎ + +## Merge sort — `sort/merge.go:7` + +Recursive top-down merge sort; base case (`l ≤ 10`) delegates to insertion sort. + +- **Post of `mergeSort(a, aux)`:** `sorted(a) ∧ perm(a, a₀)`. + +**Induction on `l = len(a)`.** *Base* (`l ≤ 10`): insertion sort, proven above. +*Step*: `mi = l/2`; the two recursive calls sort the disjoint halves `a[0..mi-1]` +and `a[mi..l-1]` (IH). `merge(a, aux, 0, mi, l-1)` then combines them. + +### Lemma M (merge is correct and permutation-preserving) — `sort/merge.go:27` + +`merge` first copies `a[lo..hi]` into `aux[lo..hi]`, then walks `k = lo … hi` +with two read cursors `i` (into the left run, starting `lo`) and `j` (into the +right run, starting `mi`). **Invariant** at each `k`: `a[lo..k-1]` is sorted and +is exactly the `k-lo` smallest elements of `aux[lo..hi]`, with `i`, `j` pointing +at the unconsumed heads of the two (individually sorted) runs. The 4-way +`switch` picks the smaller available head (`aux[i] > aux[j]` → take right, else +take left; boundary cases when a run is exhausted: `i >= mi` or `j > hi`). Each +step consumes exactly one source element and advances exactly one cursor, so +after `hi-lo+1` steps every element of `aux[lo..hi]` has been written back once +→ `perm` holds and `a[lo..hi]` is sorted. ∎ + +Because the merge preserves the multiset and produces a sorted whole from two +sorted halves, the step re-establishes the postcondition. + +**Termination:** each recursion halves the length, bottoming out at `l ≤ 10`. ∎ + +## Bottom-up merge sort — `sort/bottomupmerge.go:7` + +Iterative merge sort. **Outer invariant** (before the pass with subarray size +`sz`, a power of two): every aligned block `a[k·sz .. (k+1)·sz - 1]` is sorted. +The inner loop merges adjacent pairs of `sz`-blocks via the same `merge` +(Lemma M), using `min(lo+sz+sz-1, l-1)` (`sort/bottomupmerge.go:20`) to clamp +the final, possibly short, block to the array end. After the pass, every block +of size `2·sz` is sorted — the invariant for the next pass. + +**Termination:** `sz` doubles (`sz = sz + sz`) until `sz ≥ l`; the loop then +stops with the whole array as one sorted block. **Permutation:** Lemma M applied +to each merge. ∎ + +## Quick sort — `sort/quick.go:9` (highest scrutiny) + +Recursive quicksort; base case (`l ≤ 10`) delegates to insertion sort. The +interesting part is `quickPartition` (`sort/quick.go:25`), examined line by line +because its index bounds are the most error-prone code in the repo. + +Setup for an array of length `l ≥ 11` (partition is only reached from +`quick` when `l > 10`, so `hi = l-1 ≥ 10`): + +``` +i := 0; j := l; hi := l-1 +a.Swap(0, median(a, l)); v := a[0] // pivot chosen by median-of-3, parked at index 0 +``` + +**Left scan** `for i++; a[i] < v && i < hi; i++`: +`i` starts at 1. Because the test `i < hi` is ANDed in, `i` can advance at most +to `hi`; when `i == hi` the guard `i < hi` is false and the loop stops. The +array access `a[i]` therefore uses indices in `[1, hi] = [1, l-1]` — **always in +bounds**. The scan stops at the first index with `a[i] ≥ v` (or at `hi`), so on +exit `∀ 1 ≤ r < i : a[r] < v`. + +**Right scan** `for j--; v < a[j] && j > 0; j--`: +`j` starts at `l`, immediately decremented to `l-1 = hi`. The guard `j > 0` +caps it at 0; and since `a[0] == v`, the head test `v < a[0]` is false, so the +scan halts at `j = 0` at the latest — `j` **never goes negative**. Accesses use +`[0, hi]`. On exit `∀ j < r ≤ hi : a[r] > v`, and `a[j] ≤ v`. + +**Loop:** if `i ≥ j` the scans have crossed → `break`; otherwise `a.Swap(i, j)` +sends the `≥ v` element right and the `≤ v` element left, and the invariant +`a[1..i-1] < v ∧ a[j+1..hi] > v` is maintained across iterations. + +**Finalize** `a.Swap(0, j)`: at break, `a[j] ≤ v` (right scan stopped there), so +after the swap `a[j] = v` with `a[0..j-1] ≤ v ≤ a[j+1..hi]`. Return `j`. + +**Verdict:** the invariant *closes* — no out-of-bounds and no off-by-one. The +`i < hi` bound and the `a[0] == v` sentinel are exactly what keep the two scans +in range without relying on external sentinels. Duplicates equal to `v` are +handled correctly: strict inequalities make both scans stop on equal keys, which +is the standard technique to avoid quadratic blow-up on many duplicates and does +not violate the partition postcondition. On the all-equal input the scans meet +near the middle and the recursion still shrinks, so there is no infinite loop. + +`quick` then recurses on `a[0..j-1]` and `a[j+1..]`, which by the partition +postcondition are correctly ordered relative to `v`; by induction on length the +whole array is sorted. **Termination:** each partition removes the pivot and +splits the rest into two strictly-smaller subranges. **Permutation:** Lemma S. ∎ + +## 3-way quicksort — `sort/quick3way.go:8` + +Dijkstra's 3-way (Dutch-national-flag) partition; shuffles first, base case +(`l ≤ 10`) insertion sort. Pivot `v = a[0]` (after `Swap(0, median)`). + +**Invariant** of the partition loop (`for i <= gt`), with `lt`, `i`, `gt`: +`a[0..lt-1] < v`, `a[lt..i-1] == v`, `a[gt+1..hi] > v`, and `a[i..gt]` unexamined. +The `switch` maintains it: `a[i] < v` → `Swap(lt, i); lt++; i++`; `a[i] > v` → +`Swap(i, gt); gt--` (leaves `i`, since the swapped-in element is unexamined); +`a[i] == v` → `i++`. When `i > gt` the middle band `a[lt..gt]` equals `v` and is +in final position, so only `a[0..lt-1]` and `a[gt+1..hi]` need recursion. + +**Termination:** each iteration either advances `i` or lowers `gt`, so the gap +`gt - i` strictly decreases; recursion shrinks the ranges. **Permutation:** +Lemma S. ∎ + +## Shuffle — `sort/shuffle.go:9` (NOT a sort) + +`Shuffle` produces a uniformly random permutation (used by `Quick3Way` and the +`TestShuffleSort` negative test). For each `i`, `r := l - rand.Intn(l-i) - 1`. +Since `rand.Intn(l-i) ∈ [0, l-i-1]`, we get `r ∈ [i, l-1]`, so each `Swap(i, r)` +exchanges `a[i]` with a uniformly chosen element of the unshuffled suffix — this +is the Fisher–Yates shuffle, yielding each of the `l!` permutations with equal +probability. **Post:** `perm(a, a₀)` (Lemma S); ordering is intentionally *not* +guaranteed. ∎ + +## Parallel merge / parallel quick — `sort/parallelmerge.go:9`, `sort/parallelquick.go:9` + +These reuse the sequential `mergeSort`/`quick`/`quickPartition` proven above and +parallelize the two recursive calls once the length crosses a threshold +(`< 1000` falls back to sequential). + +**Correctness reduces to data-race freedom.** In both, the two goroutines +operate on **disjoint** subranges: + +- `parallelMerge`: `a[0:mi]` / `a[mi:]` and, crucially, `aux[0:mi]` / `aux[mi:]` + are non-overlapping slices, so the two subtrees touch disjoint memory. The + `wg.Wait()` **happens-before** the top-level `merge`, so the merge observes + both halves fully sorted. No goroutine reads memory another writes + concurrently. +- `parallelQuick`: `quickPartition` runs *before* the goroutines are spawned and + fixes the pivot at index `j`; the children then own `a[0:j]` and `a[j+1:]` — + disjoint, and both exclude the settled pivot `a[j]`. `wg.Wait()` joins before + returning. + +Given disjointness + the `WaitGroup` join fence, the parallel executions compute +the same result as their sequential counterparts, whose correctness is proven +above. The **race detector** (`make verify`) corroborates the disjointness claim +dynamically. ∎ + +## Sleep sort — `sort/sleep.go:9` + +`Sleep` (integers only) spawns one goroutine per element that sleeps +`num` seconds, then sends `num` on a shared channel; a `WaitGroup` closes the +channel once all sends complete; the main goroutine appends received values. + +Correctness rests on the *timing assumption* that a larger value's sleep +finishes strictly later, so values arrive on the channel in non-decreasing +order. This assumption — and the concurrency safety (the closer goroutine's +`wg.Wait()` happening-after every `wg.Done()`, no send on a closed channel, and +termination without deadlock) — is **not** something a paper proof can settle +convincingly. It is instead model-checked exhaustively in +`formal/tla/SleepSort.tla`, which is the appropriate tool for this coordination +logic. See that model and its README for the machine-checked result. |