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Concatenative Programming Concatenative Programming From Ivory to Metal Jon Purdy ● Why Concatenative Programming Matters (2012) ● Spaceport (2012–2013) Compiler engineering ● Facebook (2013–2014) Site integrity infrastructure (Haxl) ● There Is No Fork: An Abstraction for Efficient, Concurrent, and Concise Data Access (ICFP 2014) ● Xamarin/Microsoft (2014–2017) Mono runtime (performance, GC) What I Want in a ● Prioritize reading & modifying code over writing it Programming ● Be expressive—syntax closely Language mirroring high-level semantics ● Encourage “good” code (reusable, refactorable, testable, &c.) ● “Make me do what I want anyway” ● Have an “obvious” efficient mapping to real hardware (C) ● Be small—easy to understand & implement tools for ● Be a good citizen—FFI, embedding ● Don’t “assume you’re the world” ● Forth (1970) Notable Chuck Moore Concatenative ● PostScript (1982) Warnock, Geschke, & Paxton Programming ● Joy (2001) Languages Manfred von Thun ● Factor (2003) Slava Pestov &al. ● Cat (2006) Christopher Diggins ● Kitten (2011) Jon Purdy ● Popr (2012) Dustin DeWeese ● … History Three ● Lambda Calculus (1930s) Alonzo Church Formal Systems of ● Turing Machine (1930s) Computation Alan Turing ● Recursive Functions (1930s) Kurt Gödel Church’s Lambdas e ::= x Variables λx.x ≅ λy.y | λx. e Functions λx.(λy.x) ≅ λy.(λz.y) | e1 e2 Applications λx.M[x] ⇒ λy.M[y] α-conversion (λx.λy.λz.xz(yz))(λx.λy.x)(λx.λy.x) ≅ (λy.λz.(λx.λy.x)z(yz))(λx.λy.x) (λx.M)E ⇒ M[E/x] β-reduction ≅ λz.(λx.λy.x)z((λx.λy.x)z) ≅ λz.(λx.λy.x)z((λx.λy.x)z) ≅ λz.z Turing’s Machines ⟨ ⟩ M = Q, Γ, b, Σ, δ, q0, F ● Begin with initial state & tape ● Repeat: Q Set of states ○ If final state, then halt Γ Alphabet of symbols ○ Apply transition function ∈ b Γ Blank symbol ○ Modify tape ⊆ ∖ Σ Γ {b} Input symbols ○ Move left or right ∈ ⊆ q0 Q, F Q Initial & final states δ State transition function δ : (Q ∖ F) × Γ → Q × Γ × {L, R} Gödel’s Functions f(x1, x2, …, xk) = n Constant S(x) = x + 1 Successor k Pi (x1, x2, …, xk) = xi Projection f ∘ g Composition ρ(f, g) Primitive recursion μ(f) Minimization Three Four ● Lambda Calculus (1930s) Alonzo Church Formal Systems of ● Turing Machine (1930s) Computation Alan Turing ● Recursive Functions (1930s) Kurt Gödel ● Combinatory Logic (1950s) Moses Schönfinkel, Haskell Curry Combinatory Logic (SKI, BCKW) Just combinators and applications! Bxyz = x(yz) Compose Cxyz = xzy Flip Sxyz = xz(yz) Application Kxy = x Constant S = λx.λy.λz.xz(yz) “Starling” Wxy = xyy Duplicate Kxy = x Constant SKKx = Kx(Kx) = x K = λx.λy.x “Kestrel” M = SII = λx.xx Ix = x Identity L = CBM = λf.λx.f(xx) I = λx.x “Idiot” Y = SLL = λf.(λx.f(xx))(λx.f(xx)) What is Turing machines → imperative Lambda calculus → functional concatenative Combinatory logic →* concatenative programming? “A concatenative programming language is a point-free computer programming language in which all expressions denote functions, and the juxtaposition of expressions denotes function composition.” — Wikipedia, Concatenative Programming Language “…a point-free computer programming language…” Point-Free Programming find . -name '*.txt' define hist (List<Char> | awk '{print length($1),$1}' → List<Pair<Char, Int>>): | sort -rn | head { is_space not } filter sort group hist ∷ String → [(Char, Int)] { \head \length both_to hist = map (head &&& length) pair } map . group . sort . filter (not . isSpace) Point-Free ● Programming: dataflow style using combinators to avoid (Pointless, Tacit) references to variables or Programming arguments ● Topology/geometry: abstract reasoning about spaces & regions without reference to any specific set of “points” ● Variables are “goto for data”: unstructured, sometimes needed, but structured programming is a better default ● “Name code, not data” Value-Level Programming Can Programming Be Liberated int inner_product from the Von Neumann Style? (int n, int a[], int b[]) (1977) John Backus { int p = 0; CPU & memory connected by “von for (int i = 0; i < n; ++i) Neumann bottleneck” via primitive p += a[i] * b[i]; “word-at-a-time” style; programming return p; languages reflect that } Value-Level Programming int inner_product n=3; a={1, 2, 3}; b={6, 5, 4}; (int n, int a[], int b[]) p ← 0; { i ← 0; int p = 0; p ← 0 + 1 * 6 = 6; for (int i = 0; i < n; ++i) i ← 0 + 1 = 1; p += a[i] * b[i]; p ← 6 + 2 * 5 = 16; return p; i ← 1 + 1 = 2; } p ← 16 + 3 * 4 = 28; 28 Value-Level Programming ● Semantics & state closely ● No high-level combining forms: coupled: values depend on all everything built from primitives previous states ● No useful algebraic properties: ● Too low-level: ○ Can’t easily factor out ○ Compiler infers structure to subexpressions without optimize (e.g. vectorization) writing “wrapper” code ○ Programmer mentally ○ Can’t reason about subparts executes program or steps of programs without context through it in a debugger (state, history) FP Def InnerProd ≡ (Insert +) ∘ (ApplyToAll ×) ∘ Transpose Def InnerProd ≡ (/ +) ∘ (α ×) ∘ Trans innerProd ∷ Num a ⇒ [[a]] → a innerProd = sum . map product . transpose FP Def InnerProd ≡ InnerProd:⟨⟨1, 2, 3⟩, ⟨6, 5, 4⟩⟩ (Insert +) ∘ (ApplyToAll ×) ∘ ((/ +) ∘ (α ×) ∘ Trans):⟨⟨1,2,3⟩, ⟨6,5,4⟩⟩ Transpose (/ +):((α ×):(Trans:⟨⟨1,2,3⟩, ⟨6,5,4⟩⟩)) (/ +):((α ×):⟨⟨1,6⟩, ⟨2,5⟩, ⟨3,4⟩⟩) ≡ Def InnerProd (/ +):(⟨×:⟨1,6⟩, ×:⟨2,5⟩, ×:⟨3,4⟩⟩) ∘ ∘ (/ +) (α ×) Trans (/ +):⟨6,10,12⟩ +:⟨6, +:⟨10,12⟩⟩ +:⟨6,22⟩ 28 Function-Level Programming ● Stateless: values have no ● Made by only combining forms dependencies over time; all data ● Useful algebraic properties dependencies are explicit ● Easily factor out subexpressions: ● High-level: Def SumProd ≡ (+ /) ∘ (α ×) ○ Expresses intent Def ProdTrans ≡ (α ×) ∘ Trans ○ Compiler knows structure ● Subprograms are all pure ○ Programmer reasons about functions—all context explicit large conceptual units J innerProd =: +/@:(*/"1@:|:) innerProd >1 2 3; 6 5 4 (+/ @: (*/"1 @: |:)) >1 2 3; 6 5 4 +/ (*/"1 (|: >1 2 3; 6 5 4)) +/ (*/"1 >1 6; 2 5; 3 4) +/ 6 10 12 28 J You can give verbose names to things: sum =: +/ of =: @: products =: */"1 transpose =: |: innerProduct =: sum of products of transpose (J programmers don’t.) Function-Level ● Primitive pure functions ● Combining forms: combinators, Programming: HoFs, “forks” & “hooks” Summary ● Semantics defined by rewriting, not state transitions ● Enables purely algebraic reasoning about programs (“plug & chug”) ● Reuse mathematical intuitions from non-programming education ● Simple factoring of subprograms: “extract method” is cut & paste Three Four Five ● Lambda Calculus (1930s) Alonzo Church Formal Systems of ● Turing Machine (1930s) Computation Alan Turing ● Recursive Functions (1930s) Kurt Gödel ● Combinatory Logic (1950s) Moses Schönfinkel, Haskell Curry ● Concatenative Calculus (~2000s) Manfred von Thun, Brent Kirby Concatenative Calculus The Theory of Concatenative [ A ] dup = [ A ] [ A ] Combinators (2002) Brent Kirby [ A ] [ B ] swap = [ B ] [ A ] E ::= C Combinator [ A ] drop = | [ E ] Quotation | E1 E2 Composition [ A ] quote = [ [ A ] ] ∘ (E2 E1) [ A ] [ B ] cat = [ A B ] [ A ] call = A Concatenative Calculus { dup, swap, drop, quote, cat, call } is Turing-complete! Smaller basis: [ B ] [ A ] k = A [ B ] [ A ] cake = [ [ B ] A ] [ A [ B ] ] [ B ] [ A ] cons = [ [ B ] A ] [ B ] [ A ] take = [ A [ B ] ] Combinatory Logic (BCKW) ● B — apply functions Bkab = k(ab) Compose/apply ● C — reorder values Ckab = kba Flip ● K — delete values Kka = k Constant ● W — duplicate values Wka = kaa Duplicate Connection to logic: substructure! ● W — contraction ● C — exchange ● K — weakening Combinatory Logic ● BCKW = SKI ● BI = ordered + linear ○ S = B(BW)(BBC) “Exactly once, in order” ○ K = K (Works in any category!) ○ I = WK ● BCI = linear ● SKI → LC (expand combinators) “Exactly once” ● LC → SKI (abstraction algorithm) ● BCKI = affine ● { B, C, K, W } = LC “At most once” ● BCWI = relevant “At least once” Substructural Type Systems ● Rust, ATS, Clean, Haskell (soon) ● Reason about any resource: ● Rust (affine): if a mutable memory, file handles, locks, reference exists, it must be sockets… unique—eliminate data races & ● Enforce protocols: “consume” synchronization overhead objects that are no longer valid ● Avoid garbage collection: ● Prevent invalid state transitions precisely track lifetimes of ● Reversible computing objects to make memory usage ● Quantum computing deterministic (predictable perf.) Substructural Rules in Concatenative Calculus [ A ] dup k = [ A ] [ A ] k ● Continuations are no longer Wka = kaa scary or confusing ● “Current continuation” (call/cc) [ A ] [ B ] swap k = [ B ] [ A ] k is simply the remainder of the Ckab = kba program [ A ] drop k = k ● Saving a continuation is as easy Kka = k as saving the stacks and instruction pointer Concatenative Calculus ≈ Combinatory Logic + Continuation-Passing Style “…all expressions denote functions […] juxtaposition…denotes function composition.” Stacks ● Composition is the main way to build programs, but what are we composing functions of? ● We need a convenient data structure to store the program state and allow passing multiple values between functions ● Most concatenative languages use a heterogeneous stack, separate from the call stack, accessible to the programmer ● Other models proposed; stack is convenient & efficient in practice “Everything is an object a list a function” Literals (“nouns”) take stack & return Term 2 is a function, pushes value 2. it with corresponding value on top. 2 3 + is a function, equal to 5. Can be split into 2 3 and + or 2 and 3 +. 2 : ∀s. s → s × ℤ "hello" : ∀s. s → s × string Higher-order
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