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| 1 | +// SPDX-License-Identifier: MPL-2.0 |
| 2 | +// Copyright (c) Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk> |
| 3 | +// |
| 4 | +// Ephapax stdlib — Allen's interval algebra (1983). |
| 5 | +// |
| 6 | +// Citation: |
| 7 | +// Allen, J. F. "Maintaining knowledge about temporal intervals." |
| 8 | +// Communications of the ACM 26 (11): 832-843. November 1983. |
| 9 | +// |
| 10 | +// Allen's interval algebra is the canonical qualitative-temporal- |
| 11 | +// reasoning framework. Given two intervals A and B, there are exactly |
| 12 | +// 13 mutually-exclusive base relations describing their possible |
| 13 | +// configurations (7 plus 6 inverses, with `equals` being self- |
| 14 | +// inverse). |
| 15 | +// |
| 16 | +// Pedigree: |
| 17 | +// Snodgrass (1995) "The TSQL2 Temporal Query Language" applied this |
| 18 | +// algebra to relational databases, motivating the SQL:2011 valid- |
| 19 | +// time / transaction-time period support. |
| 20 | +// |
| 21 | +// Fit in ephapax: |
| 22 | +// This is a fully self-contained foundation primitive — no L1/L2/ |
| 23 | +// L3/L4 dependency, no proof obligations attached at landing time. |
| 24 | +// Composition table proofs can follow as a separate Coq file in |
| 25 | +// `formal/Allen.v` if/when bitemporal types (D12) land. |
| 26 | +// |
| 27 | +// Companion stdlib modules to expect later: |
| 28 | +// - `stdlib/Bitemporal.eph` (D12 — valid-time × transaction-time) |
| 29 | +// - `stdlib/Provenance.eph` (D09 — why/where/how provenance) |
| 30 | + |
| 31 | +module Allen |
| 32 | + |
| 33 | +// --- The 13 base relations --- |
| 34 | +// |
| 35 | +// Every pair of intervals (A, B) on a totally-ordered time domain |
| 36 | +// stands in exactly one of these relations. Mutual exclusion + |
| 37 | +// joint exhaustiveness is the canonical Allen result. |
| 38 | +// |
| 39 | +// Pictorial cheat-sheet (A = top line, B = bottom line): |
| 40 | +// |
| 41 | +// Before(A, B) A: === B: === |
| 42 | +// Meets(A, B) A: === B: === (A.end == B.start) |
| 43 | +// Overlaps(A, B) A: ===== B: ===== (A.end strictly inside B) |
| 44 | +// Starts(A, B) A: == B: ===== (A.start == B.start, A.end < B.end) |
| 45 | +// During(A, B) A: == B: ===== (strictly inside) |
| 46 | +// Finishes(A, B) A: == B: ===== (A.end == B.end, A.start > B.start) |
| 47 | +// Equals(A, B) A: ===== B: ===== (identical) |
| 48 | +// (and their 6 inverses) |
| 49 | +// |
| 50 | +type Relation = |
| 51 | + | Before |
| 52 | + | After |
| 53 | + | Meets |
| 54 | + | MetBy |
| 55 | + | Overlaps |
| 56 | + | OverlappedBy |
| 57 | + | Starts |
| 58 | + | StartedBy |
| 59 | + | During |
| 60 | + | Contains |
| 61 | + | Finishes |
| 62 | + | FinishedBy |
| 63 | + | Equals |
| 64 | + |
| 65 | +// --- The interval type --- |
| 66 | +// |
| 67 | +// An interval is a half-open `[start, end)` pair on i64. We use i64 |
| 68 | +// (not i32) to give 64-bit time stamps room — enough for |
| 69 | +// nanosecond-resolution UNIX time well past year 2200. |
| 70 | +// |
| 71 | +// Affinely consumable: an `Interval` carries no resource obligation |
| 72 | +// at L1 (no region capability) and is dropped freely at L2 (Affine |
| 73 | +// or Linear). |
| 74 | +// |
| 75 | +// Invariant: callers should maintain `start <= end`. The smart |
| 76 | +// constructor `make` enforces it. |
| 77 | +type Interval = Interval(i64, i64) |
| 78 | + |
| 79 | +// --- Smart constructor --- |
| 80 | +// |
| 81 | +// Constructs an interval, returning `Err` if `start > end`. The |
| 82 | +// degenerate point-interval `start == end` is permitted; the |
| 83 | +// half-open convention means it represents the empty interval. |
| 84 | +fn make(start: i64, end: i64) -> Result<Interval, String> { |
| 85 | + if start > end then |
| 86 | + Err("Allen.make: start > end") |
| 87 | + else |
| 88 | + Ok(Interval(start, end)) |
| 89 | +} |
| 90 | + |
| 91 | +// --- Accessors --- |
| 92 | + |
| 93 | +fn start(i: &Interval) -> i64 { |
| 94 | + let Interval(s, _) = i in s |
| 95 | +} |
| 96 | + |
| 97 | +fn end(i: &Interval) -> i64 { |
| 98 | + let Interval(_, e) = i in e |
| 99 | +} |
| 100 | + |
| 101 | +fn duration(i: &Interval) -> i64 { |
| 102 | + end(i) - start(i) |
| 103 | +} |
| 104 | + |
| 105 | +// --- The 13-way classifier --- |
| 106 | +// |
| 107 | +// Computes the unique Allen relation between two intervals. |
| 108 | +// Total: every (A, B) pair returns exactly one variant. |
| 109 | +// |
| 110 | +// Complexity: O(1) — at most six integer comparisons. |
| 111 | +// |
| 112 | +// The case-tree below follows Allen 1983 §2, exhaustively |
| 113 | +// partitioning the (start_a vs start_b, end_a vs end_b, end_a vs |
| 114 | +// start_b) lattice. |
| 115 | +fn relate(a: &Interval, b: &Interval) -> Relation { |
| 116 | + let sa = start(a) |
| 117 | + let ea = end(a) |
| 118 | + let sb = start(b) |
| 119 | + let eb = end(b) |
| 120 | + |
| 121 | + if ea < sb then |
| 122 | + Before |
| 123 | + else if sa > eb then |
| 124 | + After |
| 125 | + else if ea == sb then |
| 126 | + Meets |
| 127 | + else if sa == eb then |
| 128 | + MetBy |
| 129 | + else if sa == sb && ea == eb then |
| 130 | + Equals |
| 131 | + else if sa == sb && ea < eb then |
| 132 | + Starts |
| 133 | + else if sa == sb && ea > eb then |
| 134 | + StartedBy |
| 135 | + else if ea == eb && sa > sb then |
| 136 | + Finishes |
| 137 | + else if ea == eb && sa < sb then |
| 138 | + FinishedBy |
| 139 | + else if sa > sb && ea < eb then |
| 140 | + During |
| 141 | + else if sa < sb && ea > eb then |
| 142 | + Contains |
| 143 | + else if sa < sb && ea < eb then |
| 144 | + Overlaps |
| 145 | + else |
| 146 | + OverlappedBy |
| 147 | +} |
| 148 | + |
| 149 | +// --- Inverse --- |
| 150 | +// |
| 151 | +// `relate(a, b)` and `relate(b, a)` are related by `inverse`. |
| 152 | +// This is one of the two foundational Allen properties (the other |
| 153 | +// being composition — see Future Work). |
| 154 | +fn inverse(r: Relation) -> Relation { |
| 155 | + match r of |
| 156 | + | Before => After |
| 157 | + | After => Before |
| 158 | + | Meets => MetBy |
| 159 | + | MetBy => Meets |
| 160 | + | Overlaps => OverlappedBy |
| 161 | + | OverlappedBy => Overlaps |
| 162 | + | Starts => StartedBy |
| 163 | + | StartedBy => Starts |
| 164 | + | During => Contains |
| 165 | + | Contains => During |
| 166 | + | Finishes => FinishedBy |
| 167 | + | FinishedBy => Finishes |
| 168 | + | Equals => Equals |
| 169 | + end |
| 170 | +} |
| 171 | + |
| 172 | +// --- Convex / disjoint predicates --- |
| 173 | + |
| 174 | +// Two intervals are disjoint iff one is entirely before/after the |
| 175 | +// other (no contact AT ALL — meets/met-by share an endpoint). |
| 176 | +fn is_disjoint(a: &Interval, b: &Interval) -> bool { |
| 177 | + match relate(a, b) of |
| 178 | + | Before => true |
| 179 | + | After => true |
| 180 | + | _ => false |
| 181 | + end |
| 182 | +} |
| 183 | + |
| 184 | +// Two intervals overlap (in the inclusive sense — share at least one |
| 185 | +// instant) iff they are NOT disjoint AND NOT merely touching at an |
| 186 | +// endpoint. |
| 187 | +fn has_proper_overlap(a: &Interval, b: &Interval) -> bool { |
| 188 | + match relate(a, b) of |
| 189 | + | Before => false |
| 190 | + | After => false |
| 191 | + | Meets => false |
| 192 | + | MetBy => false |
| 193 | + | _ => true |
| 194 | + end |
| 195 | +} |
| 196 | + |
| 197 | +// --- Future work (not in this slice) --- |
| 198 | +// |
| 199 | +// 1. Composition table: `compose : Relation -> Relation -> Set<Relation>`. |
| 200 | +// Allen 1983 §3 specifies the 13×13 = 169-entry table. The table |
| 201 | +// is canonical; encoding it requires a set type — landing this |
| 202 | +// is a follow-up once `stdlib/Set.eph` exists. |
| 203 | +// |
| 204 | +// 2. Coq mechanisation: prove `inverse(inverse(r)) = r` (self- |
| 205 | +// inverse property) and `relate(a, b) = inverse(relate(b, a))` |
| 206 | +// (the foundational orientation lemma). Target file: |
| 207 | +// `formal/Allen.v`. Both proofs are case-analysis on the 13 |
| 208 | +// variants; trivial. |
| 209 | +// |
| 210 | +// 3. Bitemporal extension (D12): combine valid-time interval + |
| 211 | +// transaction-time interval into a 2D Allen lattice. Snodgrass |
| 212 | +// 1995 §4. |
| 213 | +// |
| 214 | +// 4. Point intervals: half-open semantics already model the empty |
| 215 | +// interval at `start == end`. A future sibling type |
| 216 | +// `PointInstant = i64` would distinguish a single-instant event |
| 217 | +// from any interval. |
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