# MIT License, Copyright (c) 2022 Marvin Borner # This file defines the most basic mathematical operations # → refer to std/Math for more advanced functions # Heavily inspired by the works of T.Æ. Mogensen (see refs in README) :import std/Combinator . :import std/Pair . :import std/Logic . # negative trit indicating coeffecient of (-1) t⁻ [[[2]]] # returns true if a trit is negative t⁻? [0 true false false] # positive trit indicating coeffecient of (+1) t⁺ [[[1]]] # returns true if a trit is positive t⁺? [0 false true false] # zero trit indicating coeffecient of 0 t⁰ [[[0]]] # returns true if a trit is zero t⁰? [0 false false true] :test (t⁻? t⁻) (true) :test (t⁻? t⁺) (false) :test (t⁻? t⁰) (false) :test (t⁺? t⁻) (false) :test (t⁺? t⁺) (true) :test (t⁺? t⁰) (false) :test (t⁰? t⁻) (false) :test (t⁰? t⁺) (false) :test (t⁰? t⁰) (true) # shifts a negative trit into a balanced ternary number ↑⁻‣ [[[[[2 (4 3 2 1 0)]]]]] :test (↑⁻(+0)) ((-1)) :test (↑⁻(-1)) ((-4)) :test (↑⁻(+42)) ((+125)) # shifts a positive trit into a balanced ternary number ↑⁺‣ [[[[[1 (4 3 2 1 0)]]]]] :test (↑⁺(+0)) ((+1)) :test (↑⁺(-1)) ((-2)) :test (↑⁺(+42)) ((+127)) # shifts a zero trit into a balanced ternary number ↑⁰‣ [[[[[0 (4 3 2 1 0)]]]]] :test (↑⁰(+0)) ([[[[0 3]]]]) :test (↑⁰(+1)) ((+3)) :test (↑⁰(+42)) ((+126)) # shifts a specified trit into a balanced ternary number up [[[[[[5 2 1 0 (4 3 2 1 0)]]]]]] :test (up t⁻ (+42)) (↑⁻(+42)) :test (up t⁺ (+42)) (↑⁺(+42)) :test (up t⁰ (+42)) (↑⁰(+42)) # shifts the least significant trit out - basically div by 3 down [~(0 z a⁻ a⁺ a⁰)] z (+0) : (+0) a⁻ [0 [[↑⁻1 : 1]]] a⁺ [0 [[↑⁺1 : 1]]] a⁰ [0 [[↑⁰1 : 1]]] # negates a balanced ternary number negate [[[[[4 3 1 2 0]]]]] -‣ negate :test (-(+0)) ((+0)) :test (-(-1)) ((+1)) :test (-(+42)) ((-42)) # converts a balanced ternary number to a list of trits list! [0 z a⁻ a⁺ a⁰] z [[0]] a⁻ [t⁻ : 0] a⁺ [t⁺ : 0] a⁰ [t⁰ : 0] # TODO: Tests! # strips leading 0s from balanced ternary number strip [^(0 z a⁻ a⁺ a⁰)] z (+0) : true a⁻ [0 [[↑⁻1 : false]]] a⁺ [0 [[↑⁺1 : false]]] a⁰ [0 [[(0 (+0) ↑⁰1) : 0]]] %‣ strip :test (%[[[[0 3]]]]) ((+0)) :test (%[[[[2 (0 (0 (0 (0 3))))]]]]) ((-1)) :test (%(+42)) ((+42)) # extracts least significant trit from balanced ternary numbers lst [0 t⁰ [t⁻] [t⁺] [t⁰]] :test (lst (-1)) (t⁻) :test (lst (+0)) (t⁰) :test (lst (+1)) (t⁺) :test (lst (+42)) (t⁰) # extracts most significant trit from balanced ternary numbers # TODO: Find a more elegant way to do this (and resolve list import loop?) mst [fix (last (list! %0))] last z [[<>?0 [false] [<>?(~1) ^1 (2 ~1)] i]] <>?‣ [0 [[[false]]] true] fix [((t⁻? 0) ⋁? ((t⁺? 0) ⋁? (t⁰? 0))) 0 t⁰] :test (mst (-1)) (t⁻) :test (mst (+0)) (t⁰) :test (mst (+1)) (t⁺) :test (mst (+42)) (t⁺) # returns true if balanced ternary number is negative negative? [t⁻? (mst 0)] ?‣ positive? :test (>?(+0)) (false) :test (>?(-1)) (false) :test (>?(+1)) (true) :test (>?(+42)) (true) # checks true if balanced ternary number is zero zero? [0 true [false] [false] i] =?‣ zero? :test (=?(+0)) (true) :test (=?(-1)) (false) :test (=?(+1)) (false) :test (=?(+42)) (false) # converts the normal balanced ternary representation into abstract # → the abstract representation is used in eq?/add/sub/mul abstract! [0 z a⁻ a⁺ a⁰] z (+0) a⁻ [[[[[2 4]]]]] a⁺ [[[[[1 4]]]]] a⁰ [[[[[0 4]]]]] →^‣ abstract! :test (→^(-3)) ([[[[0 [[[[2 [[[[3]]]]]]]]]]]]) :test (→^(+0)) ([[[[3]]]]) :test (→^(+3)) ([[[[0 [[[[1 [[[[3]]]]]]]]]]]]) # converts the abstracted balanced ternary representation back to normal # using ω to solve recursion normal! ω rec rec [[0 (+0) [↑⁻([3 3 0] 0)] [↑⁺([3 3 0] 0)] [↑⁰([3 3 0] 0)]]] →_‣ normal! :test (→_[[[[3]]]]) ((+0)) :test (→_(→^(+42))) ((+42)) :test (→_(→^(-42))) ((-42)) # checks whether two balanced ternary numbers are equal # larger numbers should be second argument (performance) # → ignores leading 0s! eq? [[abs 1 →^0]] abs [0 z a⁻ a⁺ a⁰] z [=?(→_0)] a⁻ [[0 false [2 0] [false] [false]]] a⁺ [[0 false [false] [2 0] [false]]] a⁰ [[0 (1 0) [false] [false] [2 0]]] …=?… eq? …/=?… not! .. eq? :test ((-42) =? (-42)) (true) :test ((-1) =? (-1)) (true) :test ((-1) =? (+0)) (false) :test ((+0) =? (+0)) (true) :test ((+1) =? (+0)) (false) :test ((+1) =? (+1)) (true) :test ((+42) =? (+42)) (true) :test ([[[[(1 (0 (0 (0 (0 3)))))]]]] =? (+1)) (true) :test ((+1) /=? (+0)) (true) :test ((-42) /=? (+42)) (true) # I believe Mogensen's Paper has an error in its inc/dec/add/mul/eq definitions. # They use 3 instead of 2 abstractions in the functions, also we use switched # +/0 in comparison to their implementation, yet the order of neg/pos/zero is # the same. Something's weird. # adds (+1) to a balanced ternary number (can introduce leading 0s) inc [~(0 z a⁻ a⁺ a⁰)] z (+0) : (+1) a⁻ [0 [[↑⁻1 : ↑⁰1]]] a⁺ [0 [[↑⁺1 : ↑⁻0]]] a⁰ [0 [[↑⁰1 : ↑⁺1]]] ++‣ inc # adds (+1) to a balanced ternary number and strips leading 0s ssinc strip . inc :test ((++(-42)) =? (-41)) (true) :test ((++(-1)) =? (+0)) (true) :test ((++(+0)) =? (+1)) (true) :test ((++(++(++(++(++(+0)))))) =? (+5)) (true) :test ((++(+42)) =? (+43)) (true) # subs (+1) from a balanced ternary number (can introduce leading 0s) dec [~(0 z a⁻ a⁺ a⁰)] z (+0) : (-1) a⁻ [0 [[↑⁻1 : ↑⁺0]]] a⁺ [0 [[↑⁺1 : ↑⁰1]]] a⁰ [0 [[↑⁰1 : ↑⁻1]]] --‣ dec # subs (+1) from a balanced ternary number and strips leading 0s sdec strip . dec :test ((--(-42)) =? (-43)) (true) :test ((--(+0)) =? (-1)) (true) :test ((--(--(--(--(--(+5)))))) =? (+0)) (true) :test ((--(+1)) =? (+0)) (true) :test ((--(+42)) =? (+41)) (true) # adds two balanced ternary numbers (can introduce leading 0s) # larger numbers should be second argument (performance) add [[abs 1 →^0]] abs [c (0 z a⁻ a⁺ a⁰)] b⁻ [1 ↑⁺(3 0 t⁻) ↑⁰(3 0 t⁰) ↑⁻(3 0 t⁰)] b⁰ [up 1 (3 0 t⁰)] b⁺ [1 ↑⁰(3 0 t⁰) ↑⁻(3 0 t⁺) ↑⁺(3 0 t⁰)] a⁻ [[[1 (b⁻ 1) b⁻' b⁰ b⁻]]] b⁻' [1 ↑⁰(3 0 t⁻) ↑⁻(3 0 t⁰) ↑⁺(3 0 t⁻)] a⁺ [[[1 (b⁺ 1) b⁰ b⁺' b⁺]]] b⁺' [1 ↑⁺(3 0 t⁰) ↑⁰(3 0 t⁺) ↑⁻(3 0 t⁺)] a⁰ [[[1 (b⁰ 1) b⁻ b⁺ b⁰]]] z [[0 --(→_1) ++(→_1) →_1]] c [[1 0 t⁰]] …+… add # adds two balanced ternary numbers and strips leading 0s sadd strip .. add :test (((-42) + (-1)) =? (-43)) (true) :test (((-5) + (+6)) =? (+1)) (true) :test (((-1) + (+0)) =? (-1)) (true) :test (((+0) + (+0)) =? (+0)) (true) :test (((+1) + (+2)) =? (+3)) (true) :test (((+42) + (+1)) =? (+43)) (true) # subs two balanced ternary numbers (can introduce leading 0s) # larger numbers should be second argument (performance) sub [[1 + -0]] …-… sub # subs two balanced ternary numbers and strips leading 0s ssub strip .. sub :test (((-42) - (-1)) =? (-41)) (true) :test (((-5) - (+6)) =? (-11)) (true) :test (((-1) - (+0)) =? (-1)) (true) :test (((+0) - (+0)) =? (+0)) (true) :test (((+1) - (+2)) =? (-1)) (true) :test (((+42) - (+1)) =? (+41)) (true) # returns true if number is greater than other number # larger numbers should be second argument (performance) gre? [[>?(1 - 0)]] …>?… gre? :test ((+1) >? (+2)) (false) :test ((+2) >? (+2)) (false) :test ((+3) >? (+2)) (true) # returns true if number is less than other number # smaller numbers should be second argument (performance) les? \gre? …? 0)]] …≤?… leq? :test ((+1) ≤? (+2)) (true) :test ((+2) ≤? (+2)) (true) :test ((+3) ≤? (+2)) (false) # returns true if number is greater than or equal to other number # smaller numbers should be second argument (performance) geq? \leq? …≥?… geq? :test ((+1) ≥? (+2)) (false) :test ((+2) ≥? (+2)) (true) :test ((+3) ≥? (+2)) (true) # returns max number of two max [[(1 ≤? 0) 0 1]] # returns min number of two min [[(1 ≤? 0) 1 0]] # muls two balanced ternary numbers (can introduce leading 0s) mul [[1 (+0) a⁻ a⁺ a⁰]] a⁻ [↑⁰0 - 1] a⁺ [↑⁰0 + 1] a⁰ [↑⁰0] …*… mul smul strip .. mul :test (((+42) * (+0)) =? (+0)) (true) :test (((-1) * (+42)) =? (-42)) (true) :test (((+3) * (+11)) =? (+33)) (true) :test (((+42) * (-4)) =? (-168)) (true)