THE FOLD / GRIND / THE HOT LOOP / THE SKI FOREST
THE SKI FOREST
Turing-complete with three birds and no variables
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Combinatory logic. Programming with no variables at all — just three ‘combinators’ and pure tree-rewriting: I x = x (identity), K x y = x (keep the first, discard the second), S x y z = x z (y z) (share z into both). That’s the whole language, and it is Turing-complete: you can build any computable function from just S and K (indeed I = S K K). Reduce in any order you like — Church–Rosser guarantees the same answer.
LIT verified: S K K x reduces to x (so SKK is the identity), S(KS)K is the composition combinator B (B a b c = a(b c)), and reducing outermost-first vs innermost-first reaches the same normal form over 300 random terms. FIG the ‘forest of birds’ (Smullyan) is the picture; the rules and confluence are exact.
LIT verified: S K K x reduces to x (so SKK is the identity), S(KS)K is the composition combinator B (B a b c = a(b c)), and reducing outermost-first vs innermost-first reaches the same normal form over 300 random terms. FIG the ‘forest of birds’ (Smullyan) is the picture; the rules and confluence are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus reaches for the roots of computation and logic (LOGIKĒ, the minimal kernels, the rewrite systems) and the conviction that the fewest possible parts often say the most. AVAN (AI) built this instrument: the rewrite engine, the interactive reducer, and the confluence diamond.
The weave: David names the variable-free forest and its seat at THE HOT LOOP (reduce until nothing moves); I make the three rules a strip in 1D, the reducer live in 2D, and Church–Rosser a turning diamond in 3D. The sphere is the seam.
The weave: David names the variable-free forest and its seat at THE HOT LOOP (reduce until nothing moves); I make the three rules a strip in 1D, the reducer live in 2D, and Church–Rosser a turning diamond in 3D. The sphere is the seam.
3 ONE DIMENSION
The three rewrite rules, and one term reducing on a single line: S(KS)K a b c collapsing step by step down to (a (b c)) — the composition of b-then-a, built with no variables at all.
4 TWO DIMENSIONS · INTERACTIVE
The reducer. Pick a term and step it toward normal form; each line is one rule firing (I, K, or S), the redex named. Watch structure appear from three tiny rules.
5 THREE DIMENSIONS + AVAN’S INVERSE
The Church–Rosser diamond, turning. From one term at the top, green reduces the outermost redex first, stepping down the left.
AVAN’s addition (the inverse-companion): the magenta path reduces the innermost redex first, down the right — a different route through a different intermediate. Yet both paths meet at the same normal form at the bottom. Confluence made a shape: no matter the order you evaluate, the destination is fixed. Two ways down, one floor.
LIT Genuine combinatory logic. Verified live: S K K x reduces to x (SKK is the identity), S(KS)K is the composition combinator B (B a b c = a(b c)), and reducing outermost-first vs innermost-first reaches the same normal form over random terms (Church–Rosser). The reducer, the reduction sequences, and the confluence diamond are the real rewrite system (verifiable: window.__ski.SKK_is_identity && churchRosser).
FIG 'A forest of birds' (Smullyan's combinator names) is the picture; the three rules, Turing-completeness, and confluence are exact. It really is the whole of computation with no variables.
FIG 'A forest of birds' (Smullyan's combinator names) is the picture; the three rules, Turing-completeness, and confluence are exact. It really is the whole of computation with no variables.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN