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//! [Standard term]s and [combinator]s //! //! This module defines [standard term]s and [combinator]s with commonly //! accepted names. A combinator is a closed lambda expression, meaning that it //! has no free variables. //! //! The combinators are collected from these sources: //! //! * [SKI] system by Moses Schönfinkel and Haskell Curry //! * [BCKW] system by Haskell Curry //! * [fixed-point combinator]s by Haskell Curry //! * the self-application combinator ω //! * the divergent combinator Ω //! * the reverse application (thrush) combinator //! //! The standard terms and combinators defined in this module are also //! predefined named constants in the default environment which can be created //! by calling `Environment::default()`. //! //! [standard term]: https://en.wikipedia.org/w/index.php?title=Lambda_calculus#Standard_terms //! [combinator]: https://en.wikipedia.org/wiki/Combinatory_logic#Combinatory_calculi //! [fixed-point combinator]: https://en.wikipedia.org/wiki/Fixed-point_combinator //! [SKI]: https://en.wikipedia.org/wiki/SKI_combinator_calculus //! [BCKW]: https://en.wikipedia.org/wiki/B,_C,_K,_W_system //! [Iota]: https://en.wikipedia.org/wiki/Iota_and_Jot #![allow(non_snake_case)] use std::collections::HashSet; use environment::Binding; use term::{app, lam, var, Term}; /// Creates a set of bindings for all combinators implemented in this module. pub fn default_bindings() -> HashSet<Binding> { binds! { I => I(), K => K(), S => S(), B => B(), C => C(), W => W(), ω => M(), M => M(), Ω => MM(), MM => MM(), T => T(), U => U(), V => V(), Y => Y(), Z => Z(), Θ => UU(), UU => UU(), } } /// Creates a set of bindings for all combinators of the [SKI] system. /// /// The returned `HashSet` contains a binding for the combinators: /// /// * S - Starling - Substitution /// * K - Kestrel - Constant /// * I - Idiot - Identity /// /// [SKI]: https://en.wikipedia.org/wiki/SKI_combinator_calculus pub fn ski_set() -> HashSet<Binding> { binds! { S => S(), K => K(), I => I(), } } /// I - Idiot - Identity combinator /// /// I ≡ λa.a ≡ S K K pub fn I() -> Term { lam("a", var("a")) } /// K - Kestrel - Constant combinator (TRUE) /// /// K ≡ λab.a pub fn K() -> Term { lam("a", lam("b", var("a"))) } /// S - Starling - Substitution combinator /// /// S ≡ λabc.ac(bc) pub fn S() -> Term { lam( "a", lam( "b", lam("c", app![var("a"), var("c"), app(var("b"), var("c"))]), ), ) } /// Creates a set of bindings for all combinators of the [BCKW] system. /// /// The returned `HashSet` contains a binding for the combinators: /// /// * B - Bluebird - Composition /// * C - Cardinal - Swapping /// * K - Kestrel - Constant /// * W - Warbler - Duplication /// /// [BCKW]: https://en.wikipedia.org/wiki/B,_C,_K,_W_system pub fn bckw_set() -> HashSet<Binding> { binds! { B => B(), C => C(), K => K(), W => W(), } } /// B - Bluebird - Composition combinator /// /// B ≡ λabc.a(bc) ≡ S (K S) K pub fn B() -> Term { lam( "a", lam("b", lam("c", app(var("a"), app(var("b"), var("c"))))), ) } /// C - Cardinal - Swapping combinator /// /// C ≡ λabc.acb ≡ S (B B S) (K K) pub fn C() -> Term { lam("a", lam("b", lam("c", app![var("a"), var("c"), var("b")]))) } /// W - Warbler - Duplication combinator /// /// W ≡ λab.abb ≡ C (B M R) pub fn W() -> Term { lam("a", lam("b", app![var("a"), var("b"), var("b")])) } /// ω - M - Mockingbird - Self-application combinator /// /// ω ≡ M ≡ λa.aa ≡ S I I pub fn M() -> Term { lam("a", app![var("a"), var("a")]) } /// Ω - Omega - Divergent combinator /// /// Ω ≡ ω ω ≡ M M pub fn MM() -> Term { app( lam("a", app![var("a"), var("a")]), lam("a", app![var("a"), var("a")]), ) } /// T - Thrush - Reverse application combinator /// /// T ≡ λab.ba ≡ C I pub fn T() -> Term { lam("a", lam("b", app![var("b"), var("a")])) } /// U - Turing /// /// U ≡ λab.b(aab) ≡ L O pub fn U() -> Term { lam( "a", lam("b", app(var("b"), app![var("a"), var("a"), var("b")])), ) } /// V - Vireo - Pairing combinator (PAIR) /// /// V ≡ λabc.cab ≡ B C T pub fn V() -> Term { lam("a", lam("b", lam("c", app![var("c"), var("a"), var("b")]))) } /// Y - lazy fixed-point combinator /// /// Y ≡ λf.(λa.f(aa))(λa.f(aa)) /// /// discovered by Haskell Curry. pub fn Y() -> Term { lam( "f", app( lam("a", app(var("f"), app(var("a"), var("a")))), lam("a", app(var("f"), app(var("a"), var("a")))), ), ) } /// Z - strict fixed-point combinator /// /// Z ≡ λf.(λa.f(λb.aab))(λa.f(λb.aab)) pub fn Z() -> Term { lam( "f", app( lam( "a", app(var("f"), lam("b", app![var("a"), var("a"), var("b")])), ), lam( "a", app(var("f"), lam("b", app![var("a"), var("a"), var("b")])), ), ), ) } /// Θ - Turing fixed-point combinator /// /// Θ ≡ (λab.b(aab))(λab.b(aab)) ≡ U U /// /// discovered by Alan Turing. pub fn UU() -> Term { app( lam( "a", lam("b", app(var("b"), app![var("a"), var("a"), var("b")])), ), lam( "a", lam("b", app(var("b"), app![var("a"), var("a"), var("b")])), ), ) }