Balanced ternary
Balanced ternary is a way of representing numbers. Unlike the prevailing binary representation, a balanced ternary integer is in base 3, and each digit can have the values 1, 0, or −1. For example, decimal 11 = 32 + 31 − 30, thus can be written as "++−", while 6 = 32 − 31 + 0 × 30, i.e., "+−0".
You are encouraged to solve this task according to the task description, using any language you may know.
For this task, implement balanced ternary representation of integers with the following
Requirements
- Support arbitrarily large integers, both positive and negative;
- Provide ways to convert to and from text strings, using digits '+', '-' and '0' (unless you are already using strings to represent balanced ternary; but see requirement 5).
- Provide ways to convert to and from native integer type (unless, improbably, your platform's native integer type is balanced ternary). If your native integers can't support arbitrary length, overflows during conversion must be indicated.
- Provide ways to perform addition, negation and multiplication directly on balanced ternary integers; do not convert to native integers first.
- Make your implementation efficient, with a reasonable definition of "efficient" (and with a reasonable definition of "reasonable").
Test case With balanced ternaries a from string "+-0++0+", b from native integer -436, c "+-++-":
- write out a, b and c in decimal notation;
- calculate a × (b − c), write out the result in both ternary and decimal notations.
Note: The pages generalised floating point addition and generalised floating point multiplication have code implementing arbitrary precision floating point balanced ternary.
ALGOL 68
See also:
Ada
Specifications (bt.ads): <lang Ada>with Ada.Finalization;
package BT is
type Balanced_Ternary is private;
-- conversions
function To_Balanced_Ternary (Num : Integer) return Balanced_Ternary;
function To_Balanced_Ternary (Str : String) return Balanced_Ternary;
function To_Integer (Num : Balanced_Ternary) return Integer;
function To_string (Num : Balanced_Ternary) return String;
-- Arithmetics
-- unary minus
function "-" (Left : in Balanced_Ternary)
return Balanced_Ternary;
-- subtraction function "-" (Left, Right : in Balanced_Ternary)
return Balanced_Ternary;
-- addition function "+" (Left, Right : in Balanced_Ternary)
return Balanced_Ternary;
-- multiplication function "*" (Left, Right : in Balanced_Ternary)
return Balanced_Ternary;
private
-- a balanced ternary number is a unconstrained array of (1,0,-1)
-- dinamically allocated, least significant trit leftmost
type Trit is range -1..1;
type Trit_Array is array (Positive range <>) of Trit;
pragma Pack(Trit_Array);
type Trit_Access is access Trit_Array;
type Balanced_Ternary is new Ada.Finalization.Controlled
with record
Ref : Trit_access;
end record; procedure Initialize (Object : in out Balanced_Ternary); procedure Adjust (Object : in out Balanced_Ternary); procedure Finalize (Object : in out Balanced_Ternary);
end BT;</lang>
Implementation (bt.adb): <lang Ada>with Ada.Unchecked_Deallocation;
package body BT is
procedure Free is new Ada.Unchecked_Deallocation (Trit_Array, Trit_Access);
-- Conversions
-- String to BT
function To_Balanced_Ternary (Str: String) return Balanced_Ternary is
J : Positive := 1;
Tmp : Trit_Access;
begin
Tmp := new Trit_Array (1..Str'Last);
for I in reverse Str'Range loop
case Str(I) is when '+' => Tmp (J) := 1; when '-' => Tmp (J) := -1; when '0' => Tmp (J) := 0; when others => raise Constraint_Error; end case; J := J + 1;
end loop;
return (Ada.Finalization.Controlled with Ref => Tmp);
end To_Balanced_Ternary;
-- Integer to BT
function To_Balanced_Ternary (Num: Integer) return Balanced_Ternary is
K : Integer := 0;
D : Integer;
Value : Integer := Num;
Tmp : Trit_Array(1..19); -- 19 trits is enough to contain
-- a 32 bits signed integer
begin
loop
D := (Value mod 3**(K+1))/3**K; if D = 2 then D := -1; end if; Value := Value - D*3**K; K := K + 1; Tmp(K) := Trit(D); exit when Value = 0;
end loop;
return (Ada.Finalization.Controlled
with Ref => new Trit_Array'(Tmp(1..K)));
end To_Balanced_Ternary;
-- BT to Integer --
-- If the BT number is too large Ada will raise CONSTRAINT ERROR
function To_Integer (Num : Balanced_Ternary) return Integer is
Value : Integer := 0;
Pos : Integer := 1;
begin
for I in Num.Ref.all'Range loop
Value := Value + Integer(Num.Ref(I)) * Pos; Pos := Pos * 3;
end loop;
return Value;
end To_Integer;
-- BT to String --
function To_String (Num : Balanced_Ternary) return String is
I : constant Integer := Num.Ref.all'Last;
Result : String (1..I);
begin
for J in Result'Range loop
case Num.Ref(I-J+1) is when 0 => Result(J) := '0'; when -1 => Result(J) := '-'; when 1 => Result(J) := '+'; end case;
end loop;
return Result;
end To_String;
-- unary minus -- function "-" (Left : in Balanced_Ternary)
return Balanced_Ternary is
Result : constant Balanced_Ternary := Left;
begin
for I in Result.Ref.all'Range loop
Result.Ref(I) := - Result.Ref(I);
end loop;
return Result;
end "-";
-- addition -- Carry : Trit; function Add (Left, Right : in Trit)
return Trit is
begin
if Left /= Right then
Carry := 0; return Left + Right;
else
Carry := Left; return -Right;
end if; end Add; pragma Inline (Add); function "+" (Left, Right : in Trit_Array)
return Balanced_Ternary is
Max_Size : constant Integer :=
Integer'Max(Left'Last, Right'Last);
Tmp_Left, Tmp_Right : Trit_Array(1..Max_Size) := (others => 0);
Result : Trit_Array(1..Max_Size+1) := (others => 0);
begin
Tmp_Left (1..Left'Last) := Left;
Tmp_Right(1..Right'Last) := Right;
for I in Tmp_Left'Range loop
Result(I) := Add (Result(I), Tmp_Left(I)); Result(I+1) := Carry; Result(I) := Add(Result(I), Tmp_Right(I)); Result(I+1) := Add(Result(I+1), Carry);
end loop;
-- remove trailing zeros
for I in reverse Result'Range loop
if Result(I) /= 0 then return (Ada.Finalization.Controlled with Ref => new Trit_Array'(Result(1..I))); end if;
end loop;
return (Ada.Finalization.Controlled
with Ref => new Trit_Array'(1 => 0));
end "+"; function "+" (Left, Right : in Balanced_Ternary)
return Balanced_Ternary is
begin
return Left.Ref.all + Right.Ref.all;
end "+";
-- Subtraction
function "-" (Left, Right : in Balanced_Ternary)
return Balanced_Ternary is
begin
return Left + (-Right);
end "-";
-- multiplication function "*" (Left, Right : in Balanced_Ternary)
return Balanced_Ternary is
A, B : Trit_Access;
Result : Balanced_Ternary;
begin
if Left.Ref.all'Length > Right.Ref.all'Length then
A := Right.Ref; B := Left.Ref;
else
B := Right.Ref; A := Left.Ref;
end if;
for I in A.all'Range loop
if A(I) /= 0 then declare Tmp_Result : Trit_Array (1..I+B.all'Length-1) := (others => 0); begin for J in B.all'Range loop Tmp_Result(I+J-1) := B(J) * A(I); end loop; Result := Result.Ref.all + Tmp_Result; end; end if;
end loop;
return Result;
end "*";
procedure Adjust (Object : in out Balanced_Ternary) is
begin
Object.Ref := new Trit_Array'(Object.Ref.all);
end Adjust;
procedure Finalize (Object : in out Balanced_Ternary) is
begin
Free (Object.Ref);
end Finalize;
procedure Initialize (Object : in out Balanced_Ternary) is
begin
Object.Ref := new Trit_Array'(1 => 0);
end Initialize;
end BT;</lang>
Test task requirements (testbt.adb): <lang Ada>with Ada.Text_Io; use Ada.Text_Io; with Ada.Integer_Text_Io; use Ada.Integer_Text_Io; with BT; use BT;
procedure TestBT is
Result, A, B, C : Balanced_Ternary;
begin
A := To_Balanced_Ternary("+-0++0+");
B := To_Balanced_Ternary(-436);
C := To_Balanced_Ternary("+-++-");
Result := A * (B - C);
Put("a = "); Put(To_integer(A), 4); New_Line;
Put("b = "); Put(To_integer(B), 4); New_Line;
Put("c = "); Put(To_integer(C), 4); New_Line;
Put("a * (b - c) = "); Put(To_integer(Result), 4);
Put_Line (" " & To_String(Result));
end TestBT;</lang> Output:
a = 523 b = -436 c = 65 a * (b - c) = -262023 ----0+--0++0
ATS
<lang ATS> (*
- This one is
- translated into ATS from the Ocaml entry
- )
(* ****** ****** *) // // How to compile: // patscc -DATS_MEMALLOC_LIBC -o bternary bternary.dats // (* ****** ****** *)
- include
"share/atspre_staload.hats"
(* ****** ****** *)
datatype btd = P | Z | N; typedef btern = List0(btd)
(* ****** ****** *)
fun btd2int (d: btd): int =
(case+ d of P() => 1 | Z() => 0 | N() => ~1)
(* ****** ****** *)
fun btd2string (d:btd): string = ( case+ d of P() => "+" | Z() => "0" | N() => "-" )
(* ****** ****** *)
fun btern2string (
ds: btern
) : string =
strptr2string(res) where
{
val xs = list_map_cloref (ds, lam d => btd2string(d))
val xs = list_vt_reverse (xs)
val res = stringlst_concat($UNSAFE.castvwtp1{List(string)}(xs))
val () = list_vt_free<string> (xs)
}
(* ****** ****** *)
fun from_string
(inp: string): btern = let
// fun loop{n:nat} (
inp: string(n), ds: btern
) : btern = ( //
if isneqz(inp)
then let
val c = inp.head()
val d =
(case- c of '+' => P | '0' => Z | '-' => N): btd
// end of [val]
in
loop (inp.tail(), list_cons(d, ds))
end // end of [then]
else ds // end of [else]
// ) (* end of [loop] *) // in
loop (g1ofg0(inp), list_nil)
end // end of [from_string]
(* ****** ****** *)
fun to_int (ds: btern): int = ( case+ ds of | list_nil () => 0 | list_cons (d, ds) => 3*to_int(ds) + btd2int(d) ) (* end of [to_int] *)
fun from_int (n: int): btern = ( if n = 0 then list_nil else let
val r = n mod 3
in
if r = 0
then list_cons (Z, from_int (n/3))
else if (r = 1 || r = ~2)
then list_cons (P, from_int ((n-1)/3))
else list_cons (N, from_int ((n+1)/3))
end // end of [else] ) (* end of [from_int] *)
(* ****** ****** *)
fun neg_btern
(ds: btern): btern =
list_vt2t ( list_map_cloref<btd><btd>
(ds, lam d => case+ d of P() => N() | Z() => Z() | N() => P())
) (* end of [neg_btern] *)
overload ~ with neg_btern
(* ****** ****** *) // extern fun add_btern_btern: (btern, btern) -> btern and sub_btern_btern: (btern, btern) -> btern overload + with add_btern_btern of 100 overload - with sub_btern_btern of 100 // extern fun mul_btern_btern: (btern, btern) -> btern overload * with mul_btern_btern of 110 // (* ****** ****** *)
- define :: list_cons
(* ****** ****** *)
local
fun aux0 (ds: btern): btern = (
case+ ds of nil() => ds | _ => Z()::ds
)
fun succ(ds:btern) = ds+list_sing(P()) fun pred(ds:btern) = ds+list_sing(N())
in (* in-of-local *)
implement add_btern_btern
(ds1, ds2) =
( case+ (ds1, ds2) of | (nil(), _) => ds2 | (_, nil()) => ds1 | (P()::ds1, N()::ds2) => aux0 (ds1+ds2) | (Z()::ds1, Z()::ds2) => aux0 (ds1+ds2) | (N()::ds1, P()::ds2) => aux0 (ds1+ds2) | (P()::ds1, P()::ds2) => N() :: succ(ds1 + ds2) | (N()::ds1, N()::ds2) => P() :: pred(ds1 + ds2) | (Z()::ds1, btd::ds2) => btd :: (ds1 + ds2) | (btd::ds1, Z()::ds2) => btd :: (ds1 + ds2) )
implement sub_btern_btern (ds1, ds2) = ds1 + (~ds2)
implement mul_btern_btern (ds1, ds2) = ( case+ ds2 of | nil() => nil() | Z()::ds2 => aux0 (ds1 * ds2) | P()::ds2 => aux0 (ds1 * ds2) + ds1 | N()::ds2 => aux0 (ds1 * ds2) - ds1 )
end // end of [local]
(* ****** ****** *)
typedef charptr = $extype"char*"
(* ****** ****** *)
implement main0 () = { // val a = from_string "+-0++0+" // val b = from_int (~436) val c = from_string "+-++-" // val d = a * (b - c) // val () = $extfcall (
void
, "printf" , "a = %d\nb = %d\nc = %d\na * (b - c) = %s = %d\n" , to_int(a) , to_int(b) , to_int(c) , $UNSAFE.cast{charptr}(btern2string(d)) , to_int(d) ) (* end of [val] *) // } (* end of [main0] *) </lang> Output:
a = 523 b = -436 c = 65 a * (b - c) = -262023 ----0+--0++0
AutoHotkey
<lang AutoHotkey>BalancedTernary(n){ k = 0 if abs(n)<2 return n=1?"+":n=0?"0":"-" if n<1 negative := true, n:= -1*n while !break { d := Mod(n, 3**(k+1)) / 3**k d := d=2?-1:d n := n - (d * 3**k) r := (d=-1?"-":d=1?"+":0) . r k++ if (n = 3**k) r := "+" . r , break := true } if negative { StringReplace, r, r, -,n, all StringReplace, r, r, `+,-, all StringReplace, r, r, n,+, all } return r }</lang> Examples:<lang AutoHotkey>data = ( 523 -436 65 -262023 ) loop, Parse, data, `n result .= A_LoopField " : " BalancedTernary(A_LoopField) "`n" MsgBox % result return</lang>
Outputs:
523 : +-0++0+ -436 : -++-0-- 65 : +-++- -262023 : ----0+--0++0
C++
<lang cpp>
- include <iostream>
- include <string>
- include <climits>
using namespace std;
class BalancedTernary { protected: // Store the value as a reversed string of +, 0 and - characters string value;
// Helper function to change a balanced ternary character to an integer int charToInt(char c) const { if (c == '0') return 0; return 44 - c; }
// Helper function to negate a string of ternary characters string negate(string s) const { for (int i = 0; i < s.length(); ++i) { if (s[i] == '+') s[i] = '-'; else if (s[i] == '-') s[i] = '+'; } return s; }
public: // Default constructor BalancedTernary() { value = "0"; }
// Construct from a string BalancedTernary(string s) { value = string(s.rbegin(), s.rend()); }
// Construct from an integer BalancedTernary(long long n) { if (n == 0) { value = "0"; return; }
bool neg = n < 0; if (neg) n = -n;
value = ""; while (n != 0) { int r = n % 3; if (r == 0) value += "0"; else if (r == 1) value += "+"; else { value += "-"; ++n; }
n /= 3; }
if (neg) value = negate(value); }
// Copy constructor BalancedTernary(const BalancedTernary &n) { value = n.value; }
// Addition operators BalancedTernary operator+(BalancedTernary n) const { n += *this; return n; }
BalancedTernary& operator+=(const BalancedTernary &n) { static char *add = "0+-0+-0"; static char *carry = "--000++";
int lastNonZero = 0; char c = '0'; for (int i = 0; i < value.length() || i < n.value.length(); ++i) { char a = i < value.length() ? value[i] : '0'; char b = i < n.value.length() ? n.value[i] : '0';
int sum = charToInt(a) + charToInt(b) + charToInt(c) + 3; c = carry[sum];
if (i < value.length()) value[i] = add[sum]; else value += add[sum];
if (add[sum] != '0') lastNonZero = i; }
if (c != '0') value += c; else value = value.substr(0, lastNonZero + 1); // Chop off leading zeroes
return *this; }
// Negation operator BalancedTernary operator-() const { BalancedTernary result; result.value = negate(value); return result; }
// Subtraction operators BalancedTernary operator-(const BalancedTernary &n) const { return operator+(-n); }
BalancedTernary& operator-=(const BalancedTernary &n) { return operator+=(-n); }
// Multiplication operators BalancedTernary operator*(BalancedTernary n) const { n *= *this; return n; }
BalancedTernary& operator*=(const BalancedTernary &n) { BalancedTernary pos = *this; BalancedTernary neg = -pos; // Storing an extra copy to avoid negating repeatedly value = "0";
for (int i = 0; i < n.value.length(); ++i) { if (n.value[i] == '+') operator+=(pos); else if (n.value[i] == '-') operator+=(neg); pos.value = '0' + pos.value; neg.value = '0' + neg.value; }
return *this; }
// Stream output operator friend ostream& operator<<(ostream &out, const BalancedTernary &n) { out << n.toString(); return out; }
// Convert to string string toString() const { return string(value.rbegin(), value.rend()); }
// Convert to integer long long toInt() const { long long result = 0; for (long long i = 0, pow = 1; i < value.length(); ++i, pow *= 3) result += pow * charToInt(value[i]); return result; }
// Convert to integer if possible bool tryInt(long long &out) const { long long result = 0; bool ok = true;
for (long long i = 0, pow = 1; i < value.length() && ok; ++i, pow *= 3) { if (value[i] == '+') { ok &= LLONG_MAX - pow >= result; // Clear ok if the result overflows result += pow; } else if (value[i] == '-') { ok &= LLONG_MIN + pow <= result; // Clear ok if the result overflows result -= pow; } }
if (ok) out = result; return ok; } };
int main() { BalancedTernary a("+-0++0+"); BalancedTernary b(-436); BalancedTernary c("+-++-");
cout << "a = " << a << " = " << a.toInt() << endl; cout << "b = " << b << " = " << b.toInt() << endl; cout << "c = " << c << " = " << c.toInt() << endl;
BalancedTernary d = a * (b - c);
cout << "a * (b - c) = " << d << " = " << d.toInt() << endl;
BalancedTernary e("+++++++++++++++++++++++++++++++++++++++++");
long long n; if (e.tryInt(n)) cout << "e = " << e << " = " << n << endl; else cout << "e = " << e << " is too big to fit in a long long" << endl;
return 0; } </lang>
Output
a = +-0++0+ = 523 b = -++-0-- = -436 c = +-++- = 65 a * (b - c) = ----0+--0++0 = -262023 e = +++++++++++++++++++++++++++++++++++++++++ is too big to fit in a long long
C#
<lang csharp>using System; using System.Text; using System.Collections.Generic;
public class BalancedTernary { public static void Main() { BalancedTernary a = new BalancedTernary("+-0++0+"); System.Console.WriteLine("a: " + a + " = " + a.ToLong()); BalancedTernary b = new BalancedTernary(-436); System.Console.WriteLine("b: " + b + " = " + b.ToLong()); BalancedTernary c = new BalancedTernary("+-++-"); System.Console.WriteLine("c: " + c + " = " + c.ToLong()); BalancedTernary d = a * (b - c); System.Console.WriteLine("a * (b - c): " + d + " = " + d.ToLong()); }
private enum BalancedTernaryDigit { MINUS = -1, ZERO = 0, PLUS = 1 }
private BalancedTernaryDigit[] value;
// empty = 0 public BalancedTernary() { this.value = new BalancedTernaryDigit[0]; }
// create from String public BalancedTernary(String str) { this.value = new BalancedTernaryDigit[str.Length]; for (int i = 0; i < str.Length; ++i) { switch (str[i]) { case '-': this.value[i] = BalancedTernaryDigit.MINUS; break; case '0': this.value[i] = BalancedTernaryDigit.ZERO; break; case '+': this.value[i] = BalancedTernaryDigit.PLUS; break; default: throw new ArgumentException("Unknown Digit: " + str[i]); } } Array.Reverse(this.value); }
// convert long integer public BalancedTernary(long l) { List<BalancedTernaryDigit> value = new List<BalancedTernaryDigit>(); int sign = Math.Sign(l); l = Math.Abs(l);
while (l != 0) { byte rem = (byte)(l % 3); switch (rem) { case 0: case 1: value.Add((BalancedTernaryDigit)rem); l /= 3; break; case 2: value.Add(BalancedTernaryDigit.MINUS); l = (l + 1) / 3; break; } }
this.value = value.ToArray(); if (sign < 0) { this.Invert(); } }
// copy constructor public BalancedTernary(BalancedTernary origin) { this.value = new BalancedTernaryDigit[origin.value.Length]; Array.Copy(origin.value, this.value, origin.value.Length); }
// only for internal use private BalancedTernary(BalancedTernaryDigit[] value) { int end = value.Length - 1; while (value[end] == BalancedTernaryDigit.ZERO) --end; this.value = new BalancedTernaryDigit[end + 1]; Array.Copy(value, this.value, end + 1); }
// invert the values private void Invert() { for (int i=0; i < this.value.Length; ++i) { this.value[i] = (BalancedTernaryDigit)(-(int)this.value[i]); } }
// convert to string override public String ToString() { StringBuilder result = new StringBuilder(); for (int i = this.value.Length - 1; i >= 0; --i) { switch (this.value[i]) { case BalancedTernaryDigit.MINUS: result.Append('-'); break; case BalancedTernaryDigit.ZERO: result.Append('0'); break; case BalancedTernaryDigit.PLUS: result.Append('+'); break; } } return result.ToString(); }
// convert to long public long ToLong() { long result = 0; int digit; for (int i = 0; i < this.value.Length; ++i) { result += (long)this.value[i] * (long)Math.Pow(3.0, (double)i); } return result; }
// unary minus public static BalancedTernary operator -(BalancedTernary origin) { BalancedTernary result = new BalancedTernary(origin); result.Invert(); return result; }
// addition of digits private static BalancedTernaryDigit carry = BalancedTernaryDigit.ZERO; private static BalancedTernaryDigit Add(BalancedTernaryDigit a, BalancedTernaryDigit b) { if (a != b) { carry = BalancedTernaryDigit.ZERO; return (BalancedTernaryDigit)((int)a + (int)b); } else { carry = a; return (BalancedTernaryDigit)(-(int)b); } }
// addition of balanced ternary numbers public static BalancedTernary operator +(BalancedTernary a, BalancedTernary b) { int maxLength = Math.Max(a.value.Length, b.value.Length); BalancedTernaryDigit[] resultValue = new BalancedTernaryDigit[maxLength + 1]; for (int i=0; i < maxLength; ++i) { if (i < a.value.Length) { resultValue[i] = Add(resultValue[i], a.value[i]); resultValue[i+1] = carry; } else { carry = BalancedTernaryDigit.ZERO; }
if (i < b.value.Length) { resultValue[i] = Add(resultValue[i], b.value[i]); resultValue[i+1] = Add(resultValue[i+1], carry); } } return new BalancedTernary(resultValue); }
// subtraction of balanced ternary numbers public static BalancedTernary operator -(BalancedTernary a, BalancedTernary b) { return a + (-b); }
// multiplication of balanced ternary numbers public static BalancedTernary operator *(BalancedTernary a, BalancedTernary b) { BalancedTernaryDigit[] longValue = a.value; BalancedTernaryDigit[] shortValue = b.value; BalancedTernary result = new BalancedTernary(); if (a.value.Length < b.value.Length) { longValue = b.value; shortValue = a.value; }
for (int i = 0; i < shortValue.Length; ++i) { if (shortValue[i] != BalancedTernaryDigit.ZERO) { BalancedTernaryDigit[] temp = new BalancedTernaryDigit[i + longValue.Length]; for (int j = 0; j < longValue.Length; ++j) { temp[i+j] = (BalancedTernaryDigit)((int)shortValue[i] * (int)longValue[j]); } result = result + new BalancedTernary(temp); } } return result; } }</lang> output:
a: +-0++0+ = 523 b: -++-0-- = -436 c: +-++- = 65 a * (b - c): ----0+--0++0 = -262023
Common Lisp
<lang lisp>;;; balanced ternary
- represented as a list of 0, 1 or -1s, with least significant digit first
- convert ternary to integer
(defun bt-integer (b)
(reduce (lambda (x y) (+ x (* 3 y))) b :from-end t :initial-value 0))
- convert integer to ternary
(defun integer-bt (n)
(if (zerop n) nil
(case (mod n 3)
(0 (cons 0 (integer-bt (/ n 3))))
(1 (cons 1 (integer-bt (floor n 3))))
(2 (cons -1 (integer-bt (floor (1+ n) 3)))))))
- convert string to ternary
(defun string-bt (s)
(loop with o = nil for c across s do
(setf o (cons (case c (#\+ 1) (#\- -1) (#\0 0)) o)) finally (return o)))
- convert ternary to string
(defun bt-string (bt)
(if (not bt) "0" (let* ((l (length bt))
(s (make-array l :element-type 'character)))
(mapc (lambda (b)
(setf (aref s (decf l)) (case b (-1 #\-) (0 #\0) (1 #\+)))) bt)
s)))
- arithmetics
(defun bt-neg (a) (map 'list #'- a)) (defun bt-sub (a b) (bt-add a (bt-neg b)))
(let ((tbl #((0 -1) (1 -1) (-1 0) (0 0) (1 0) (-1 1) (0 1))))
(defun bt-add-digits (a b c) (values-list (aref tbl (+ 3 a b c)))))
(defun bt-add (a b &optional (c 0))
(if (not (and a b))
(if (zerop c) (or a b)
(bt-add (list c) (or a b)))
(multiple-value-bind (d c)
(bt-add-digits (if a (car a) 0) (if b (car b) 0) c)
(let ((res (bt-add (cdr a) (cdr b) c)))
;; trim leading zeros (if (or res (not (zerop d))) (cons d res))))))
(defun bt-mul (a b)
(if (not (and a b)) nil (bt-add (case (car a)
(-1 (bt-neg b)) ( 0 nil) ( 1 b)) (cons 0 (bt-mul (cdr a) b)))))
- division with quotient/remainder, for completeness
(defun bt-truncate (a b)
(let ((n (- (length a) (length b)))
(d (car (last b))))
(if (minusp n)
(values nil a)
(labels ((recur (a b x)
(multiple-value-bind (quo rem) (if (plusp x) (recur a (cons 0 b) (1- x)) (values nil a))
(loop with g = (car (last rem)) with quo = (cons 0 quo) while (= (length rem) (length b)) do (cond ((= g d) (setf rem (bt-sub rem b) quo (bt-add '(1) quo))) ((= g (- d)) (setf rem (bt-add rem b) quo (bt-add '(-1) quo)))) (setf x (car (last rem))) finally (return (values quo rem))))))
(recur a b n)))))
- test case
(let* ((a (string-bt "+-0++0+"))
(b (integer-bt -436))
(c (string-bt "+-++-"))
(d (bt-mul a (bt-sub b c))))
(format t "a~5d~8t~a~%b~5d~8t~a~%c~5d~8t~a~%a × (b − c) = ~d ~a~%"
(bt-integer a) (bt-string a) (bt-integer b) (bt-string b) (bt-integer c) (bt-string c) (bt-integer d) (bt-string d)))</lang>output<lang>a 523 +-0++0+ b -436 -++-0-- c 65 +-++- a × (b − c) = -262023 ----0+--0++0</lang>
D
<lang d>import std.stdio, std.bigint, std.range, std.algorithm;
struct BalancedTernary {
// Represented as a list of 0, 1 or -1s,
// with least significant digit first.
enum Dig : byte { N=-1, Z=0, P=+1 } // Digit.
const Dig[] digits;
// This could also be a BalancedTernary template argument. static immutable string dig2str = "-0+";
immutable static Dig[dchar] str2dig; // = ['+': Dig.P, ...];
nothrow static this() {
str2dig = ['+': Dig.P, '-': Dig.N, '0': Dig.Z];
}
immutable pure nothrow static Dig[2][] table =
[[Dig.Z, Dig.N], [Dig.P, Dig.N], [Dig.N, Dig.Z],
[Dig.Z, Dig.Z], [Dig.P, Dig.Z], [Dig.N, Dig.P],
[Dig.Z, Dig.P]];
this(in string inp) const pure {
this.digits = inp.retro.map!(c => str2dig[c]).array;
}
this(in long inp) const pure nothrow {
this.digits = _bint2ternary(inp.BigInt);
}
this(in BigInt inp) const pure nothrow {
this.digits = _bint2ternary(inp);
}
this(in BalancedTernary inp) const pure nothrow {
// No need to dup, they are virtually immutable.
this.digits = inp.digits;
}
private this(in Dig[] inp) pure nothrow {
this.digits = inp;
}
static Dig[] _bint2ternary(in BigInt n) pure nothrow {
static py_div(T1, T2)(in T1 a, in T2 b) pure nothrow {
if (a < 0) {
return (b < 0) ?
-a / -b :
-(-a / b) - (-a % b != 0 ? 1 : 0);
} else {
return (b < 0) ?
-(a / -b) - (a % -b != 0 ? 1 : 0) :
a / b;
}
}
if (n == 0) return [];
// This final switch in D v.2.064 is fake, not enforced.
final switch (((n % 3) + 3) % 3) { // (n % 3) is the remainder.
case 0: return Dig.Z ~ _bint2ternary(py_div(n, 3));
case 1: return Dig.P ~ _bint2ternary(py_div(n, 3));
case 2: return Dig.N ~ _bint2ternary(py_div(n + 1, 3));
}
}
@property BigInt toBint() const pure nothrow {
return reduce!((y, x) => x + 3 * y)(0.BigInt, digits.retro);
}
string toString() const pure nothrow {
if (digits.empty) return "0";
return digits.retro.map!(d => dig2str[d + 1]).array;
}
static const(Dig)[] neg_(in Dig[] digs) pure nothrow {
return digs.map!(a => -a).array;
}
BalancedTernary opUnary(string op:"-")() const pure nothrow {
return BalancedTernary(neg_(this.digits));
}
static const(Dig)[] add_(in Dig[] a, in Dig[] b, in Dig c=Dig.Z)
pure nothrow {
const a_or_b = a.length ? a : b;
if (a.empty || b.empty) {
if (c == Dig.Z)
return a_or_b;
else
return BalancedTernary.add_([c], a_or_b);
} else {
// (const d, c) = table[...];
const dc = table[3 + (a.length ? a[0] : 0) +
(b.length ? b[0] : 0) + c];
const res = add_(a[1 .. $], b[1 .. $], dc[1]);
// Trim leading zeros.
if (res.length || dc[0] != Dig.Z)
return [dc[0]] ~ res;
else
return res;
}
}
BalancedTernary opBinary(string op:"+")(in BalancedTernary b)
const pure nothrow {
return BalancedTernary(add_(this.digits, b.digits));
}
BalancedTernary opBinary(string op:"-")(in BalancedTernary b)
const pure nothrow {
return this + (-b);
}
static const(Dig)[] mul_(in Dig[] a, in Dig[] b) pure nothrow {
if (a.empty || b.empty) {
return [];
} else {
const y = Dig.Z ~ mul_(a[1 .. $], b);
final switch (a[0]) {
case Dig.N: return add_(neg_(b), y);
case Dig.Z: return add_([], y);
case Dig.P: return add_(b, y);
}
}
}
BalancedTernary opBinary(string op:"*")(in BalancedTernary b)
const pure nothrow {
return BalancedTernary(mul_(this.digits, b.digits));
}
}
void main() {
immutable a = BalancedTernary("+-0++0+");
writeln("a: ", a.toBint, ' ', a);
immutable b = BalancedTernary(-436);
writeln("b: ", b.toBint, ' ', b);
immutable c = BalancedTernary("+-++-");
writeln("c: ", c.toBint, ' ', c);
const /*immutable*/ r = a * (b - c);
writeln("a * (b - c): ", r.toBint, ' ', r);
}</lang>
- Output:
a: 523 +-0++0+ b: -436 -++-0-- c: 65 +-++- a * (b - c): -262023 ----0+--0++0
Elixir
<lang elixir>defmodule Ternary do
def to_string(t), do: ( for x <- t, do: to_char(x) ) |> List.to_string
def from_string(s), do: ( for x <- to_char_list(s), do: from_char(x) )
defp to_char(-1), do: ?-
defp to_char(0), do: ?0
defp to_char(1), do: ?+
defp from_char(?-), do: -1
defp from_char(?0), do: 0
defp from_char(?+), do: 1
def to_ternary(n) when n > 0, do: to_ternary(n,[])
def to_ternary(n), do: neg(to_ternary(-n))
defp to_ternary(0,acc), do: acc
defp to_ternary(n,acc) when rem(n, 3) == 0, do: to_ternary(div(n, 3), [0|acc])
defp to_ternary(n,acc) when rem(n, 3) == 1, do: to_ternary(div(n, 3), [1|acc])
defp to_ternary(n,acc), do: to_ternary(div((n+1), 3), [-1|acc])
def from_ternary(t), do: from_ternary(t,0)
defp from_ternary([],acc), do: acc
defp from_ternary([h|t],acc), do: from_ternary(t, acc*3 + h)
def mul(a,b), do: mul(b,a,[])
defp mul(_,[],acc), do: acc
defp mul(b,[a|as],acc) do
bp = case a do
-1 -> neg(b)
0 -> [0]
1 -> b
end
a = add(bp, acc ++ [0])
mul(b,as,a)
end
defp neg(t), do: ( for h <- t, do: -h )
def sub(a,b), do: add(a,neg(b))
def add(a,b) when length(a) < length(b),
do: add(List.duplicate(0, length(b)-length(a)) ++ a, b)
def add(a,b) when length(a) > length(b), do: add(b,a)
def add(a,b), do: add(Enum.reverse(a), Enum.reverse(b), 0, [])
defp add([],[],0,acc), do: acc
defp add([],[],c,acc), do: [c|acc]
defp add([a|as],[b|bs],c,acc) do
[c1,d] = add_util(a+b+c)
add(as,bs,c1,[d|acc])
end
defp add_util(-3), do: [-1,0]
defp add_util(-2), do: [-1,1]
defp add_util(-1), do: [0,-1]
defp add_util(3), do: [1,0]
defp add_util(2), do: [1,-1]
defp add_util(1), do: [0,1]
defp add_util(0), do: [0,0]
end
as = "+-0++0+"; at = Ternary.from_string(as); a = Ternary.from_ternary(at) b = -436; bt = Ternary.to_ternary(b); bs = Ternary.to_string(bt) cs = "+-++-"; ct = Ternary.from_string(cs); c = Ternary.from_ternary(ct) rt = Ternary.mul(at,Ternary.sub(bt,ct)) r = Ternary.from_ternary(rt) rs = Ternary.to_string(rt) IO.puts "a = #{as} -> #{a}" IO.puts "b = #{bs} -> #{b}" IO.puts "c = #{cs} -> #{c}" IO.puts "a x (b - c) = #{rs} -> #{r}"</lang>
- Output:
a = +-0++0+ -> 523 b = -++-0-- -> -436 c = +-++- -> 65 a x (b - c) = 0----0+--0++0 -> -262023
Erlang
<lang erlang> -module(ternary). -compile(export_all).
test() ->
AS = "+-0++0+", AT = from_string(AS), A = from_ternary(AT),
B = -436, BT = to_ternary(B), BS = to_string(BT),
CS = "+-++-", CT = from_string(CS), C = from_ternary(CT),
RT = mul(AT,sub(BT,CT)),
R = from_ternary(RT),
RS = to_string(RT),
io:fwrite("A = ~s -> ~b~n",[AS, A]),
io:fwrite("B = ~s -> ~b~n",[BS, B]),
io:fwrite("C = ~s -> ~b~n",[CS, C]),
io:fwrite("A x (B - C) = ~s -> ~b~n", [RS, R]).
to_string(T) -> [to_char(X) || X <- T].
from_string(S) -> [from_char(X) || X <- S].
to_char(-1) -> $-; to_char(0) -> $0; to_char(1) -> $+.
from_char($-) -> -1; from_char($0) -> 0; from_char($+) -> 1.
to_ternary(N) when N > 0 ->
to_ternary(N,[]);
to_ternary(N) ->
neg(to_ternary(-N)).
to_ternary(0,Acc) ->
Acc;
to_ternary(N,Acc) when N rem 3 == 0 ->
to_ternary(N div 3, [0|Acc]);
to_ternary(N,Acc) when N rem 3 == 1 ->
to_ternary(N div 3, [1|Acc]);
to_ternary(N,Acc) ->
to_ternary((N+1) div 3, [-1|Acc]).
from_ternary(T) -> from_ternary(T,0).
from_ternary([],Acc) ->
Acc;
from_ternary([H|T],Acc) ->
from_ternary(T,Acc*3 + H).
mul(A,B) -> mul(B,A,[]).
mul(_,[],Acc) ->
Acc;
mul(B,[A|As],Acc) ->
BP = case A of
-1 -> neg(B);
0 -> [0];
1 -> B
end,
A1 = Acc++[0],
A2=add(BP,A1),
mul(B,As,A2).
neg(T) -> [ -H || H <- T].
sub(A,B) -> add(A,neg(B)).
add(A,B) when length(A) < length(B) ->
add(lists:duplicate(length(B)-length(A),0)++A,B);
add(A,B) when length(A) > length(B) ->
add(B,A);
add(A,B) ->
add(lists:reverse(A),lists:reverse(B),0,[]).
add([],[],0,Acc) ->
Acc;
add([],[],C,Acc) ->
[C|Acc];
add([A|As],[B|Bs],C,Acc) ->
[C1,D] = add_util(A+B+C), add(As,Bs,C1,[D|Acc]).
add_util(-3) -> [-1,0]; add_util(-2) -> [-1,1]; add_util(-1) -> [0,-1]; add_util(3) -> [1,0]; add_util(2) -> [1,-1]; add_util(1) -> [0,1]; add_util(0) -> [0,0]. </lang> Output <lang erlang> 234> ternary:test(). A = +-0++0+ -> 523 B = -++-0-- -> -436 C = +-++- -> 65 A x (B - C) = 0----0+--0++0 -> -262023 ok </lang>
Glagol
ОТДЕЛ Сетунь+;
ИСПОЛЬЗУЕТ
Параметр ИЗ "...\Отделы\Обмен\",
Текст ИЗ "...\Отделы\Числа\",
Вывод ИЗ "...\Отделы\Обмен\";
ПЕР
зч: РЯД 10 ИЗ ЗНАК;
счпоз: ЦЕЛ;
число: ЦЕЛ;
память: ДОСТУП К НАБОР
ячейки: РЯД 20 ИЗ ЦЕЛ;
размер: УЗКЦЕЛ;
отрицательное: КЛЮЧ
КОН;
ЗАДАЧА СоздатьПамять;
УКАЗ
СОЗДАТЬ(память);
память.размер := 0;
память.отрицательное := ОТКЛ
КОН СоздатьПамять;
ЗАДАЧА ДобавитьВПамять(что: ЦЕЛ);
УКАЗ
память.ячейки[память.размер] := что;
УВЕЛИЧИТЬ(память.размер)
КОН ДобавитьВПамять;
ЗАДАЧА ОбратитьПамять;
ПЕР
зчсл: ЦЕЛ;
сч: ЦЕЛ;
УКАЗ
ОТ сч := 0 ДО память.размер ДЕЛИТЬ 2 - 1 ВЫП
зчсл := память.ячейки[сч];
память.ячейки[сч] := память.ячейки[память.размер-сч-1];
память.ячейки[память.размер-сч-1] := зчсл
КОН
КОН ОбратитьПамять;
ЗАДАЧА ВывестиПамять;
ПЕР
сч: ЦЕЛ;
УКАЗ
ОТ сч := 0 ДО память.размер-1 ВЫП
ЕСЛИ память.ячейки[сч] < 0 ТО
Вывод.Цепь("-")
АЕСЛИ память.ячейки[сч] > 0 ТО
Вывод.Цепь("+")
ИНАЧЕ Вывод.Цепь("0") КОН
КОН
КОН ВывестиПамять;
ЗАДАЧА УдалитьПамять;
УКАЗ
память := ПУСТО
КОН УдалитьПамять;
ЗАДАЧА Перевести(число: ЦЕЛ);
ПЕР
о: ЦЕЛ;
з: КЛЮЧ;
ЗАДАЧА ВПамять(что: ЦЕЛ);
УКАЗ
ЕСЛИ память.отрицательное ТО
ЕСЛИ что < 0 ТО ДобавитьВПамять(1)
АЕСЛИ что > 0 ТО ДобавитьВПамять(-1)
ИНАЧЕ ДобавитьВПамять(0) КОН
ИНАЧЕ
ДобавитьВПамять(что)
КОН
КОН ВПамять;
УКАЗ
ЕСЛИ число < 0 ТО память.отрицательное := ВКЛ КОН;
число := МОДУЛЬ(число);
з := ОТКЛ;
ПОКА число > 0 ВЫП
о := число ОСТАТОК 3;
число := число ДЕЛИТЬ 3;
ЕСЛИ з ТО
ЕСЛИ о = 2 ТО ВПамять(0) АЕСЛИ о = 1 ТО ВПамять(-1) ИНАЧЕ ВПамять(1); з := ОТКЛ КОН
ИНАЧЕ
ЕСЛИ о = 2 ТО ВПамять(-1); з := ВКЛ ИНАЧЕ ВПамять(о) КОН
КОН
КОН;
ЕСЛИ з ТО ВПамять(1) КОН;
ОбратитьПамять;
ВывестиПамять(ВКЛ);
КОН Перевести;
ЗАДАЧА ВЧисло(): ЦЕЛ;
ПЕР
сч, мн: ЦЕЛ;
результат: ЦЕЛ;
УКАЗ
результат := 0;
мн := 1;
ОТ сч := 0 ДО память.размер-1 ВЫП
УВЕЛИЧИТЬ(результат, память.ячейки[память.размер-сч-1]*мн);
мн := мн * 3
КОН;
ВОЗВРАТ результат
КОН ВЧисло;
УКАЗ
Параметр.Текст(1, зч); счпоз := 0;
число := Текст.ВЦел(зч, счпоз);
СоздатьПамять;
Перевести(число);
Вывод.ЧЦел(" = %d.", ВЧисло(), 0, 0, 0);
УдалитьПамять
КОН Сетунь.
A crude English/Pidgin Algol translation of the above Category:Glagol code. <lang algol68>PROGRAM Setun+; USES
Parameter IS "...\Departments\Exchange\" Text IS "...\Departments\Numbers\" Output IS "...\Departments\Exchange\";
VAR
AF: RANGE 10 IS SIGN; mfpos: INT; number: INT; memory ACCESS TO STRUCT cell: RANGE 20 IS INT; size: UZKEL; negative: BOOL END;
PROC Create.Memory; BEGIN
CREATE(memory); memory.size := 0; memory.negative := FALSE
END Create.Memory;
PROC Add.Memory(that: INT) BEGIN
memory.cells[memory.size] := that; ZOOM(memory.size)
END Add.Memory;
PROC Invert.Memory; VAR
zchsl: INT; account: INT;
BEGIN
FOR cq := 0 TO memory.size DIVIDE 2 - 1 DO zchsl := memory.cells[cq]; memory.cells[cq] := memory.cells[memory.size-size-1]; memory.cells[memory.size-MF-1] := zchsl END
END Invert.Memory;
PROC Withdraw.Memory; VAR
account: INT;
BEGIN
FOR cq := 0 TO memory.size-1 DO
IF memory.cells[cq] < 0 THEN
Output.Append("-")
ANDIF memory.cells[cq] > 0 THEN
Output.Append("+")
ELSE Output.Append("0") END
END
END Withdraw.Memory;
PROC Remove.Memory; BEGIN
memory := Empty
END Remove.Memory;
PROC Translate(number: INT) VAR
about: INT;
s: BOOL;
PROC B.Memory(that: INT)
BEGIN
IF memory.negative THEN
IF that < 0 THEN Add.Memory(1)
ANDIF that > 0 THEN Add.Memory(1)
ELSE Add.Memory(0) END
ELSE
Add.Memory(that)
END
END B.Memory;
BEGIN
IF number < 0 THEN memory.negative := TRUE END;
number := UNIT(number)
s := FALSE;
WHILE number > 0 DO
about := number BALANCE 3;
number := number DIVIDE 3;
IF s THEN
IF about = 2 THEN B.Memory(0) ANDIF about = 1 THEN B.Memory(1) ELSE B.Memory(1) s := FALSE END
ELSE
IF about = 2 THEN B.Memory(-1) s := TRUE ELSE B.Memory(a) END
END
END;
IF s THEN B.Memory(1) END;
Invert.Memory;
Withdraw.Memory(TRUE)
END Translate;
PROC InNumber(): INT; VAR
MF, MN: INT; result: INT;
BEGIN
result := 0 pl := 1; FOR cq := 0 TO memory.size-1 DO ZOOM(result, memory.Cells[memory.size-cq-1] * mn); pl := pl * 3 END; RETURN result;
END InNumber;
BEGIN
Parameter.Text(1, AF); mfpos := 0;
number := Text.Whole(AF, mfpos);
Create.Memory;
Translate(number);
Output.ChTarget(" = %d.", InNumber(), 0, 0, 0);
Remove.Memory
END Setun.</lang>
Go
<lang go>package main
import (
"fmt" "strings"
)
// R1: representation is a slice of int8 digits of -1, 0, or 1. // digit at index 0 is least significant. zero value of type is // representation of the number 0. type bt []int8
// R2: string conversion:
// btString is a constructor. valid input is a string of any length // consisting of only '+', '-', and '0' characters. // leading zeros are allowed but are trimmed and not represented. // false return means input was invalid. func btString(s string) (*bt, bool) {
s = strings.TrimLeft(s, "0")
b := make(bt, len(s))
for i, last := 0, len(s)-1; i < len(s); i++ {
switch s[i] {
case '-':
b[last-i] = -1
case '0':
b[last-i] = 0
case '+':
b[last-i] = 1
default:
return nil, false
}
}
return &b, true
}
// String method converts the other direction, returning a string of // '+', '-', and '0' characters representing the number. func (b bt) String() string {
if len(b) == 0 {
return "0"
}
last := len(b) - 1
r := make([]byte, len(b))
for i, d := range b {
r[last-i] = "-0+"[d+1]
}
return string(r)
}
// R3: integer conversion // int chosen as "native integer"
// btInt is a constructor like btString. func btInt(i int) *bt {
if i == 0 {
return new(bt)
}
var b bt
var btDigit func(int)
btDigit = func(digit int) {
m := int8(i % 3)
i /= 3
switch m {
case 2:
m = -1
i++
case -2:
m = 1
i--
}
if i == 0 {
b = make(bt, digit+1)
} else {
btDigit(digit + 1)
}
b[digit] = m
}
btDigit(0)
return &b
}
// Int method converts the other way, returning the value as an int type. // !ok means overflow occurred during conversion, not necessarily that the // value is not representable as an int. (Of course there are other ways // of doing it but this was chosen as "reasonable.") func (b bt) Int() (r int, ok bool) {
pt := 1
for _, d := range b {
dp := int(d) * pt
neg := r < 0
r += dp
if neg {
if r > dp {
return 0, false
}
} else {
if r < dp {
return 0, false
}
}
pt *= 3
}
return r, true
}
// R4: negation, addition, and multiplication
func (z *bt) Neg(b *bt) *bt {
if z != b {
if cap(*z) < len(*b) {
*z = make(bt, len(*b))
} else {
*z = (*z)[:len(*b)]
}
}
for i, d := range *b {
(*z)[i] = -d
}
return z
}
func (z *bt) Add(a, b *bt) *bt {
if len(*a) < len(*b) {
a, b = b, a
}
r := *z
r = r[:cap(r)]
var carry int8
for i, da := range *a {
if i == len(r) {
n := make(bt, len(*a)+4)
copy(n, r)
r = n
}
sum := da + carry
if i < len(*b) {
sum += (*b)[i]
}
carry = sum / 3
sum %= 3
switch {
case sum > 1:
sum -= 3
carry++
case sum < -1:
sum += 3
carry--
}
r[i] = sum
}
last := len(*a)
if carry != 0 {
if len(r) == last {
n := make(bt, last+4)
copy(n, r)
r = n
}
r[last] = carry
*z = r[:last+1]
return z
}
for {
if last == 0 {
*z = nil
break
}
last--
if r[last] != 0 {
*z = r[:last+1]
break
}
}
return z
}
func (z *bt) Mul(a, b *bt) *bt {
if len(*a) < len(*b) {
a, b = b, a
}
var na bt
for _, d := range *b {
if d == -1 {
na.Neg(a)
break
}
}
r := make(bt, len(*a)+len(*b))
for i := len(*b) - 1; i >= 0; i-- {
switch (*b)[i] {
case 1:
p := r[i:]
p.Add(&p, a)
case -1:
p := r[i:]
p.Add(&p, &na)
}
}
i := len(r)
for i > 0 && r[i-1] == 0 {
i--
}
*z = r[:i]
return z
}
func main() {
a, _ := btString("+-0++0+")
b := btInt(-436)
c, _ := btString("+-++-")
show("a:", a)
show("b:", b)
show("c:", c)
show("a(b-c):", a.Mul(a, b.Add(b, c.Neg(c))))
}
func show(label string, b *bt) {
fmt.Printf("%7s %12v ", label, b)
if i, ok := b.Int(); ok {
fmt.Printf("%7d\n", i)
} else {
fmt.Println("int overflow")
}
}</lang>
- Output:
a: +-0++0+ 523
b: -++-0-- -436
c: +-++- 65
a(b-c): ----0+--0++0 -262023
Groovy
Solution: <lang groovy>enum T {
m('-', -1), z('0', 0), p('+', 1)
final String symbol final int value
private T(String symbol, int value) {
this.symbol = symbol
this.value = value
}
static T get(Object key) {
switch (key) {
case [m.value, m.symbol] : return m
case [z.value, z.symbol] : return z
case [p.value, p.symbol] : return p
default: return null
}
}
T negative() {
T.get(-this.value)
}
String toString() { this.symbol }
}
class BalancedTernaryInteger {
static final MINUS = new BalancedTernaryInteger(T.m) static final ZERO = new BalancedTernaryInteger(T.z) static final PLUS = new BalancedTernaryInteger(T.p) private static final LEADING_ZEROES = /^0+/ final String value
BalancedTernaryInteger(String bt) {
assert bt && bt.toSet().every { T.get(it) }
value = bt ==~ LEADING_ZEROES ? T.z : bt.replaceAll(LEADING_ZEROES, );
}
BalancedTernaryInteger(BigInteger i) {
this(i == 0 ? T.z.symbol : valueFromInt(i));
}
BalancedTernaryInteger(T...tArray) {
this(tArray.sum{ it.symbol });
}
BalancedTernaryInteger(List<T> tList) {
this(tList.sum{ it.symbol });
}
private static String valueFromInt(BigInteger i) {
assert i != null
if (i < 0) return negate(valueFromInt(-i))
if (i == 0) return
int bRem = (((i % 3) - 2) ?: -3) + 2
valueFromInt((i - bRem).intdiv(3)) + T.get(bRem)
}
private static String negate(String bt) {
bt.collect{ T.get(it) }.inject() { str, t ->
str + (-t)
}
}
private static final Map INITIAL_SUM_PARTS = [carry:T.z, sum:[]]
private static final prepValueLen = { int len, String s ->
s.padLeft(len + 1, T.z.symbol).collect{ T.get(it) }
}
private static final partCarrySum = { partialSum, carry, trit ->
[carry: carry, sum: [trit] + partialSum]
}
private static final partSum = { parts, trits ->
def carrySum = partCarrySum.curry(parts.sum)
switch ((trits + parts.carry).sort()) {
case T.m, T.m, T.m: return carrySum(T.m, T.z) //-3
case T.m, T.m, T.z: return carrySum(T.m, T.p) //-2
case [[T.m, T.z, T.z], [T.m, T.m, T.p]]: return carrySum(T.z, T.m) //-1
case [[T.z, T.z, T.z], [T.m, T.z, T.p]]: return carrySum(T.z, T.z) //+0
case [[T.z, T.z, T.p], [T.m, T.p, T.p]]: return carrySum(T.z, T.p) //+1
case T.z, T.p, T.p: return carrySum(T.p, T.m) //+2
case T.p, T.p, T.p: default: return carrySum(T.p, T.z) //+3
}
}
BalancedTernaryInteger plus(BalancedTernaryInteger that) {
assert that != null
if (this == ZERO) return that
if (that == ZERO) return this
def prep = prepValueLen.curry([value.size(), that.value.size()].max())
List values = [prep(value), prep(that.value)].transpose()
new BalancedTernaryInteger(values[-1..(-values.size())].inject(INITIAL_SUM_PARTS, partSum).sum)
}
BalancedTernaryInteger negative() {
!this ? this : new BalancedTernaryInteger(negate(value))
}
BalancedTernaryInteger minus(BalancedTernaryInteger that) {
assert that != null
this + -that
}
private static final INITIAL_PRODUCT_PARTS = [sum:ZERO, pad:]
private static final sigTritCount = { it.value.replaceAll(T.z.symbol,).size() }
private BalancedTernaryInteger paddedValue(String pad) {
new BalancedTernaryInteger(value + pad)
}
private BalancedTernaryInteger partialProduct(T multiplier, String pad){
switch (multiplier) {
case T.z: return ZERO
case T.m: return -paddedValue(pad)
case T.p: default: return paddedValue(pad)
}
}
BalancedTernaryInteger multiply(BalancedTernaryInteger that) {
assert that != null
if (that == ZERO) return ZERO
if (that == PLUS) return this
if (that == MINUS) return -this
if (this.value.size() == 1 || sigTritCount(this) < sigTritCount(that)) {
return that.multiply(this)
}
that.value.collect{ T.get(it) }[-1..(-value.size())].inject(INITIAL_PRODUCT_PARTS) { parts, multiplier ->
[sum: parts.sum + partialProduct(multiplier, parts.pad), pad: parts.pad + T.z]
}.sum
}
BigInteger asBigInteger() {
value.collect{ T.get(it) }.inject(0) { i, trit -> i * 3 + trit.value }
}
def asType(Class c) {
switch (c) {
case Integer: return asBigInteger() as Integer
case Long: return asBigInteger() as Long
case [BigInteger, Number]: return asBigInteger()
case Boolean: return this != ZERO
case String: return toString()
default: return super.asType(c)
}
}
boolean equals(Object that) {
switch (that) {
case BalancedTernaryInteger: return this.value == that?.value
default: return super.equals(that)
}
}
int hashCode() { this.value.hashCode() }
String toString() { value }
}</lang>
Test: <lang groovy>BalancedTernaryInteger a = new BalancedTernaryInteger('+-0++0+') BalancedTernaryInteger b = new BalancedTernaryInteger(-436) BalancedTernaryInteger c = new BalancedTernaryInteger(T.p, T.m, T.p, T.p, T.m) BalancedTernaryInteger bmc = new BalancedTernaryInteger(-436 - (c as Integer)) BalancedTernaryInteger atbmc = new BalancedTernaryInteger((a as Integer) * (-436 - (c as Integer)))
printf ("%9s = %12s %8d\n", 'a', "${a}", a as Number) printf ("%9s = %12s %8d\n", 'b', "${b}", b as Number) printf ("%9s = %12s %8d\n", 'c', "${c}", c as Number) assert (b-c) == bmc printf ("%9s = %12s %8d\n", 'b-c', "${b-c}", (b-c) as Number) assert (a * (b-c)) == atbmc printf ("%9s = %12s %8d\n", 'a * (b-c)', "${a * (b-c)}", (a * (b-c)) as Number)
println "\nDemonstrate failure:" assert (a * (b-c)) == a</lang>
Output:
a = +-0++0+ 523
b = -++-0-- -436
c = +-++- 65
b-c = -+0-++0 -501
a * (b-c) = ----0+--0++0 -262023
Demonstrate failure:
Caught: Assertion failed:
assert (a * (b-c)) == a
| | ||| | |
| | ||| | +-0++0+
| | ||| false
| | ||+-++-
| | |-+0-++0
| | -++-0--
| ----0+--0++0
+-0++0+
...
Haskell
BTs are represented internally as lists of digits in integers from -1 to 1, but displayed as "+-0" strings. <lang haskell>data BalancedTernary = Bt [Int]
zeroTrim a = if null s then [0] else s where s = fst $ foldl f ([],[]) a f (x,y) 0 = (x, y++[0]) f (x,y) z = (x++y++[z], [])
btList (Bt a) = a
instance Eq BalancedTernary where (==) a b = btList a == btList b
btNormalize = listBt . _carry 0 where _carry c [] = if c == 0 then [] else [c] _carry c (a:as) = r:_carry cc as where (cc, r) = f $ (a+c) `quotRem` 3 where f (x, 2) = (x + 1, -1) f (x, -2) = (x - 1, 1) f x = x
listBt = Bt . zeroTrim
instance Show BalancedTernary where show = reverse . map (\d->case d of -1->'-'; 0->'0'; 1->'+') . btList
strBt = Bt . zeroTrim.reverse.map (\c -> case c of '-' -> -1; '0' -> 0; '+' -> 1)
intBt :: Integral a => a -> BalancedTernary intBt = fromIntegral . toInteger
btInt = foldr (\a z -> a + 3 * z) 0 . btList
listAdd a b = take (max (length a) (length b)) $ zipWith (+) (a++[0,0..]) (b++[0,0..])
-- mostly for operators, also small stuff to make GHC happy instance Num BalancedTernary where negate = Bt . map negate . btList (+) x y = btNormalize $ listAdd (btList x) (btList y) (*) x y = btNormalize $ mul_ (btList x) (btList y) where mul_ _ [] = []
mul_ as b = foldr (\a z -> listAdd (map (a*) b) (0:z)) [] as
-- we don't need to define binary "-" by hand
signum (Bt a) = if a == [0] then 0 else Bt [last a] abs x = if signum x == Bt [-1] then negate x else x
fromInteger = btNormalize . f where f 0 = [] f x = fromInteger (rem x 3) : f (quot x 3)
main = let (a,b,c) = (strBt "+-0++0+", intBt (-436), strBt "+-++-")
r = a * (b - c)
in do
print $ map btInt [a,b,c]
print $ r
print $ btInt r</lang>
Icon and Unicon
Works in both languages: <lang unicon>procedure main()
a := "+-0++0+"
write("a = +-0++0+"," = ",cvtFromBT("+-0++0+"))
write("b = -436 = ",b := cvtToBT(-436))
c := "+-++-"
write("c = +-++- = ",cvtFromBT("+-++-"))
d := mul(a,sub(b,c))
write("a(b-c) = ",d," = ",cvtFromBT(d))
end
procedure bTrim(s)
return s[upto('+-',s):0] | "0"
end
procedure cvtToBT(n)
if n=0 then return "0"
if n<0 then return map(cvtToBT(-n),"+-","-+")
return bTrim(case n%3 of {
0: cvtToBT(n/3)||"0"
1: cvtToBT(n/3)||"+"
2: cvtToBT((n+1)/3)||"-"
})
end
procedure cvtFromBT(n)
sum := 0
i := -1
every c := !reverse(n) do {
sum +:= case c of {
"+" : 1
"-" : -1
"0" : 0
}*(3^(i+:=1))
}
return sum
end
procedure neg(n)
return map(n,"+-","-+")
end
procedure add(a,b)
if *b > *a then a :=: b
b := repl("0",*a-*b)||b
c := "0"
sum := ""
every place := 1 to *a do {
ds := addDigits(a[-place],b[-place],c)
c := if *ds > 1 then c := ds[1] else "0"
sum := ds[-1]||sum
}
return bTrim(c||sum)
end
procedure addDigits(a,b,c)
sum1 := addDigit(a,b) sum2 := addDigit(sum1[-1],c) if *sum1 = 1 then return sum2 if *sum2 = 1 then return sum1[1]||sum2 return sum1[1]
end
procedure addDigit(a,b)
return case(a||b) of {
"00"|"0+"|"0-": b
"+0"|"-0" : a
"++" : "+-"
"+-"|"-+" : "0"
"--" : "-+"
}
end
procedure sub(a,b)
return add(a,neg(b))
end
procedure mul(a,b)
if b[1] == "-" then {
b := neg(b)
negate := "yes"
}
b := cvtFromBT(b)
i := "+"
mul := "0"
while cvtFromBT(i) <= b do {
mul := add(mul,a)
i := add(i,"+")
}
return (\negate,map(mul,"+-","-+")) | mul
end</lang>
Output:
->bt a = +-0++0+ = 523 b = -436 = -++-0-- c = +-++- = 65 a(b-c) = ----0+--0++0 = -262023 ->
J
Implementation:
<lang j>trigits=: 1+3 <.@^. 2 * 1&>.@| trinOfN=: |.@((_1 + ] #: #.&1@] + [) #&3@trigits) :. nOfTrin nOfTrin=: p.&3 :. trinOfN trinOfStr=: 0 1 _1 {~ '0+-'&i.@|. :. strOfTrin strOfTrin=: {&'0+-'@|. :. trinOfStr
carry=: +//.@:(trinOfN"0)^:_ trimLead0=: (}.~ i.&1@:~:&0)&.|.
add=: carry@(+/@,:) neg=: - mul=: trimLead0@carry@(+//.@(*/))</lang>
trinary numbers are represented as a sequence of polynomial coefficients. The coefficient values are limited to 1, 0, and -1. The polynomial's "variable" will always be 3 (which happens to illustrate an interesting absurdity in the terminology we use to describe polynomials -- one which might be an obstacle for learning, for some people).
trigits computes the number of trinary "digits" (that is, the number of polynomial coefficients) needed to represent an integer. pseudocode: 1+floor(log3(2*max(1,abs(n))). Note that floating point inaccuracies combined with comparison tolerance may lead to a [harmless] leading zero when converting incredibly large numbers.
fooOfBar converts a bar into a foo. These functions are all invertable (so we can map from one domain to another, perform an operation, and map back using J's under). This aspect is not needed for this task and the definitions could be made simpler by removing it (removing the :. obverse clauses), but it made testing and debugging easier.
carry performs carry propagation. (Intermediate results will have overflowed trinary representation and become regular integers, so we convert them back into trinary and then perform a polynomial sum, repeating until the result is the same as the argument.)
trimLead0 removes leading zeros from a sequence of polynomial coefficients.
add adds these polynomials.
neg negates these polynomials. Note that it's just a name for J's -
mul multiplies these polynomials.
Definitions for example:
<lang j>a=: trinOfStr '+-0++0+' b=: trinOfN -436 c=: trinOfStr '+-++-'</lang>
Required example:
<lang j> nOfTrin&> a;b;c 523 _436 65
strOfTrin a mul b (add -) c
0+--0++0
nOfTrin a mul b (add -) c
_262023</lang>
Java
<lang java> /*
* Test case * With balanced ternaries a from string "+-0++0+", b from native integer -436, c "+-++-": * Write out a, b and c in decimal notation; * Calculate a × (b − c), write out the result in both ternary and decimal notations. */
public class BalancedTernary { public static void main(String[] args) {
BTernary a=new BTernary("+-0++0+");
BTernary b=new BTernary(-436); BTernary c=new BTernary("+-++-");
System.out.println("a="+a.intValue()); System.out.println("b="+b.intValue()); System.out.println("c="+c.intValue()); System.out.println();
//result=a*(b-c) BTernary result=a.mul(b.sub(c));
System.out.println("result= "+result+" "+result.intValue()); }
public static class BTernary
{
String value;
public BTernary(String s)
{
int i=0;
while(s.charAt(i)=='0')
i++;
this.value=s.substring(i);
}
public BTernary(int v)
{
this.value="";
this.value=convertToBT(v);
}
private String convertToBT(int v) { if(v<0) return flip(convertToBT(-v)); if(v==0) return ""; int rem=mod3(v); if(rem==0) return convertToBT(v/3)+"0"; if(rem==1) return convertToBT(v/3)+"+"; if(rem==2) return convertToBT((v+1)/3)+"-"; return "You can't see me"; } private String flip(String s) { String flip=""; for(int i=0;i<s.length();i++) { if(s.charAt(i)=='+') flip+='-'; else if(s.charAt(i)=='-') flip+='+'; else flip+='0'; } return flip; } private int mod3(int v) { if(v>0) return v%3; v=v%3; return (v+3)%3; }
public int intValue() { int sum=0; String s=this.value; for(int i=0;i<s.length();i++) { char c=s.charAt(s.length()-i-1); int dig=0; if(c=='+') dig=1; else if(c=='-') dig=-1; sum+=dig*Math.pow(3, i); } return sum; }
public BTernary add(BTernary that)
{
String a=this.value;
String b=that.value;
String longer=a.length()>b.length()?a:b; String shorter=a.length()>b.length()?b:a;
while(shorter.length()<longer.length()) shorter=0+shorter;
a=longer; b=shorter;
char carry='0'; String sum=""; for(int i=0;i<a.length();i++) { int place=a.length()-i-1; String digisum=addDigits(a.charAt(place),b.charAt(place),carry); if(digisum.length()!=1) carry=digisum.charAt(0); else carry='0'; sum=digisum.charAt(digisum.length()-1)+sum; } sum=carry+sum;
return new BTernary(sum); } private String addDigits(char a,char b,char carry) { String sum1=addDigits(a,b); String sum2=addDigits(sum1.charAt(sum1.length()-1),carry); //System.out.println(carry+" "+sum1+" "+sum2); if(sum1.length()==1) return sum2; if(sum2.length()==1) return sum1.charAt(0)+sum2; return sum1.charAt(0)+""; } private String addDigits(char a,char b) { String sum=""; if(a=='0') sum=b+""; else if (b=='0') sum=a+""; else if(a=='+') { if(b=='+') sum="+-"; else sum="0"; } else { if(b=='+') sum="0"; else sum="-+"; } return sum; }
public BTernary neg() { return new BTernary(flip(this.value)); }
public BTernary sub(BTernary that) { return this.add(that.neg()); }
public BTernary mul(BTernary that) { BTernary one=new BTernary(1); BTernary zero=new BTernary(0); BTernary mul=new BTernary(0);
int flipflag=0; if(that.compareTo(zero)==-1) { that=that.neg(); flipflag=1; } for(BTernary i=new BTernary(1);i.compareTo(that)<1;i=i.add(one)) mul=mul.add(this);
if(flipflag==1) mul=mul.neg(); return mul; }
public boolean equals(BTernary that) { return this.value.equals(that.value); } public int compareTo(BTernary that) { if(this.intValue()>that.intValue()) return 1; else if(this.equals(that)) return 0; return -1; }
public String toString() { return value; } } } </lang>
Output:
a=523 b=-436 c=65 result= ----0+--0++0 -262023
Kotlin
This is based on the Java entry. However, I've added 'BigInteger' support as this is a current requirement of the task description even though it's not actually needed to process the test case: <lang scala>// version 1.1.3
import java.math.BigInteger
val bigZero = BigInteger.ZERO val bigOne = BigInteger.ONE val bigThree = BigInteger.valueOf(3L)
class BTernary(private var value: String) : Comparable<BTernary> {
init {
require(value.all { it in "0+-" } )
value = value.trimStart('0')
}
constructor(v: Int): this(BigInteger.valueOf(v.toLong()))
constructor(v: Long): this(BigInteger.valueOf(v))
constructor(v: BigInteger): this("") {
value = toBT(v)
}
private fun toBT(v: BigInteger): String {
if (v < bigZero) return flip(toBT(-v))
if (v == bigZero) return ""
val rem = mod3(v)
return when (rem) {
bigZero -> toBT(v / bigThree) + "0"
bigOne -> toBT(v / bigThree) + "+"
else -> toBT((v + bigOne) / bigThree) + "-"
}
}
private fun flip(s: String): String {
val sb = StringBuilder()
for (c in s) {
sb.append(when (c) {
'+' -> "-"
'-' -> "+"
else -> "0"
})
}
return sb.toString()
}
private fun mod3(v: BigInteger): BigInteger {
if (v > bigZero) return v % bigThree
return ((v % bigThree) + bigThree) % bigThree
}
fun toBigInteger(): BigInteger {
val len = value.length
var sum = bigZero
var pow = bigOne
for (i in 0 until len) {
val c = value[len - i - 1]
val dig = when (c) {
'+' -> bigOne
'-' -> -bigOne
else -> bigZero
}
if (dig != bigZero) sum += dig * pow
pow *= bigThree
}
return sum
}
private fun addDigits(a: Char, b: Char, carry: Char): String {
val sum1 = addDigits(a, b)
val sum2 = addDigits(sum1.last(), carry)
return when {
sum1.length == 1 -> sum2
sum2.length == 1 -> sum1.take(1) + sum2
else -> sum1.take(1)
}
}
private fun addDigits(a: Char, b: Char): String =
when {
a == '0' -> b.toString()
b == '0' -> a.toString()
a == '+' -> if (b == '+') "+-" else "0"
else -> if (b == '+') "0" else "-+"
}
operator fun plus(other: BTernary): BTernary {
var a = this.value
var b = other.value
val longer = if (a.length > b.length) a else b
var shorter = if (a.length > b.length) b else a
while (shorter.length < longer.length) shorter = "0" + shorter
a = longer
b = shorter
var carry = '0'
var sum = ""
for (i in 0 until a.length) {
val place = a.length - i - 1
val digisum = addDigits(a[place], b[place], carry)
carry = if (digisum.length != 1) digisum[0] else '0'
sum = digisum.takeLast(1) + sum
}
sum = carry.toString() + sum
return BTernary(sum)
}
operator fun unaryMinus() = BTernary(flip(this.value)) operator fun minus(other: BTernary) = this + (-other)
operator fun times(other: BTernary): BTernary {
var that = other
val one = BTernary(1)
val zero = BTernary(0)
var mul = zero
var flipFlag = false
if (that < zero) {
that = -that
flipFlag = true
}
var i = one
while (i <= that) {
mul += this
i += one
}
if (flipFlag) mul = -mul
return mul
}
fun equals(other: BTernary) = this.compareTo(other) == 0
override operator fun compareTo(other: BTernary) =
this.toBigInteger().compareTo(other.toBigInteger())
override fun toString() = value
}
fun main(args: Array<String>) {
val a = BTernary("+-0++0+")
val b = BTernary(-436)
val c = BTernary("+-++-")
println("a = ${a.toBigInteger()}")
println("b = ${b.toBigInteger()}")
println("c = ${c.toBigInteger()}")
val bResult = a * (b - c)
val iResult = bResult.toBigInteger()
println("a * (b - c) = $bResult = $iResult")
}</lang>
- Output:
a = 523 b = -436 c = 65 a * (b - c) = ----0+--0++0 = -262023
Liberty BASIC
<lang lb> global tt$ tt$="-0+" '-1 0 1; +2 -> 1 2 3, instr
'Test case: 'With balanced ternaries a from string "+-0++0+", b from native integer -436, c "+-++-": '* write out a, b and c in decimal notation; '* calculate a * (b - c), write out the result in both ternary and decimal notations.
a$="+-0++0+" a=deci(a$) print "a",a, a$
b=-436 b$=ternary$(b) print "b",b, b$
c$="+-++-" c=deci(c$) print "c",c, c$
'calculate in ternary
res$=multTernary$(a$, subTernary$(b$, c$)) print "a * (b - c)", res$ print "In decimal:",deci(res$)
print "Check:" print "a * (b - c)", a * (b - c) end
function deci(s$)
pow = 1
for i = len(s$) to 1 step -1
c$ = mid$(s$,i,1)
'select case c$
' case "+":sign= 1
' case "-":sign=-1
' case "0":sign= 0
'end select
sign = instr(tt$,c$)-2
deci = deci+pow*sign
pow = pow*3
next
end function
function ternary$(n)
while abs(n)>3^k/2
k=k+1
wend
k=k-1
pow = 3^k
for i = k to 0 step -1
sign = (n>0) - (n<0)
sign = sign * (abs(n)>pow/2)
ternary$ = ternary$+mid$(tt$,sign+2,1)
n = n - sign*pow
pow = pow/3
next
if ternary$ = "" then ternary$ ="0"
end function
function multTernary$(a$, b$)
c$ = "" t$ = "" shift$ = "" for i = len(a$) to 1 step -1
select case mid$(a$,i,1)
case "+": t$ = b$
case "0": t$ = "0"
case "-": t$ = negate$(b$)
end select
c$ = addTernary$(c$, t$+shift$)
shift$ = shift$ +"0" 'print d, t$, c$ next multTernary$ = c$
end function
function subTernary$(a$, b$)
subTernary$ = addTernary$(a$, negate$(b$))
end function
function negate$(s$)
negate$=""
for i = 1 to len(s$)
'print mid$(s$,i,1), instr(tt$, mid$(s$,i,1)), 4-instr(tt$, mid$(s$,i,1))
negate$=negate$+mid$(tt$, 4-instr(tt$, mid$(s$,i,1)), 1)
next
end function
function addTernary$(a$, b$) 'add a$ + b$, for now only positive
l = max(len(a$), len(b$))
a$=pad$(a$,l)
b$=pad$(b$,l)
c$ = "" 'result
carry = 0
for i = l to 1 step -1
a = instr(tt$,mid$(a$,i,1))-2
b = instr(tt$,mid$(b$,i,1))-2 '-1 0 1
c = a+b+carry
select case
case abs(c)<2
carry = 0
case c>0
carry =1: c=c-3
case c<0
carry =-1: c=c+3
end select
'print a, b, c
c$ = mid$(tt$,c+2,1)+c$
next
if carry<>0 then c$ = mid$(tt$,carry+2,1) +c$
'print c$
'have to trim leading 0's
i=0
while mid$(c$,i+1,1)="0"
i=i+1
wend
c$=mid$(c$,i+1)
if c$="" then c$="0"
addTernary$ = c$
end function
function pad$(a$,n) 'pad from right with 0 to length n
pad$ = a$
while len(pad$)<n
pad$ = "0"+pad$
wend
end function </lang>
- Output:
a 523 +-0++0+ b -436 -++-0-- c 65 +-++- a * (b - c) ----0+--0++0 In decimal: -262023 Check: a * (b - c) -262023
Mathematica / Wolfram Language
<lang mathematica>frombt = FromDigits[StringCases[#, {"+" -> 1, "-" -> -1, "0" -> 0}],
3] &;
tobt = If[Quotient[#, 3, -1] == 0,
"", #0@Quotient[#, 3, -1]] <> (Mod[#,
3, -1] /. {1 -> "+", -1 -> "-", 0 -> "0"}) &;
btnegate = StringReplace[#, {"+" -> "-", "-" -> "+"}] &; btadd = StringReplace[
StringJoin[
Fold[Sort@{#11,
Sequence @@ #2} /. {{x_, x_, x_} :> {x,
"0" <> #12}, {"-", "+", x_} | {x_, "-", "+"} | {x_,
"0", "0"} :> {"0", x <> #12}, {"+", "+", "0"} -> {"+",
"-" <> #12}, {"-", "-", "0"} -> {"-",
"+" <> #12}} &, {"0", ""},
Reverse@Transpose@PadLeft[Characters /@ {#1, #2}] /. {0 ->
"0"}]], StartOfString ~~ "0" .. ~~ x__ :> x] &;
btsubtract = btadd[#1, btnegate@#2] &; btmultiply =
btadd[Switch[StringTake[#2, -1], "0", "0", "+", #1, "-",
btnegate@#1],
If[StringLength@#2 == 1,
"0", #0[#1, StringDrop[#2, -1]] <> "0"]] &;</lang>
Examples: <lang mathematica>frombt[a = "+-0++0+"] b = tobt@-436 frombt[c = "+-++-"] btmultiply[a, btsubtract[b, c]]</lang> Outputs:
523 "-++-0--" 65 "----0+--0++0"
МК-61/52
<lang>П0 ЗН П2 0 П3 П4 1 П5 ИП0 /-/ x<0 78 ИП0 ^ ^ 3 / [x] П0 3 * - П1 ИП3 x#0 52 ИП1 x=0 36 1 ПП 86 0 П3 БП 08 ИП1 1 - x=0 47 1 /-/ ПП 86 БП 08 0 ПП 86 БП 08 ИП1 x=0 60 0 ПП 86 БП 08 ИП1 1 - x=0 70 1 ПП 86 БП 08 1 /-/ ПП 86 1 П3 БП 08 ИП3 x#0 85 1 ПП 86 ИП4 С/П ИП2 x<0 91 <-> /-/ <-> 8 + ИП5 * ИП4 + П4 ИП5 1 0 * П5 В/О</lang>
Note: the "-", "0", "+" denotes by digits, respectively, the "7", "8", "9".
OCaml
<lang ocaml>type btdigit = Pos | Zero | Neg type btern = btdigit list
let to_string n =
String.concat ""
(List.rev_map (function Pos -> "+" | Zero -> "0" | Neg -> "-") n)
let from_string s =
let sl = ref [] in
let digit = function '+' -> Pos | '-' -> Neg | '0' -> Zero
| _ -> failwith "invalid digit" in
String.iter (fun c -> sl := (digit c) :: !sl) s; !sl
let rec to_int = function
| [Zero] | [] -> 0 | Pos :: t -> 1 + 3 * to_int t | Neg :: t -> -1 + 3 * to_int t | Zero :: t -> 3 * to_int t
let rec from_int n =
if n = 0 then [] else
match n mod 3 with
| 0 -> Zero :: from_int (n/3)
| 1 | -2 -> Pos :: from_int ((n-1)/3)
| 2 | -1 -> Neg :: from_int ((n+1)/3)
let rec (+~) n1 n2 = match (n1,n2) with
| ([], a) | (a,[]) -> a
| (Pos::t1, Neg::t2) | (Neg::t1, Pos::t2) | (Zero::t1, Zero::t2) ->
let sum = t1 +~ t2 in if sum = [] then [] else Zero :: sum
| (Pos::t1, Pos::t2) -> Neg :: t1 +~ t2 +~ [Pos]
| (Neg::t1, Neg::t2) -> Pos :: t1 +~ t2 +~ [Neg]
| (Zero::t1, h::t2) | (h::t1, Zero::t2) -> h :: t1 +~ t2
let neg = List.map (function Pos -> Neg | Neg -> Pos | Zero -> Zero) let (-~) a b = a +~ (neg b)
let rec ( *~) n1 = function
| [] -> [] | [Pos] -> n1 | [Neg] -> neg n1 | Pos::t -> (Zero :: t *~ n1) +~ n1 | Neg::t -> (Zero :: t *~ n1) -~ n1 | Zero::t -> Zero :: t *~ n1
let a = from_string "+-0++0+" let b = from_int (-436) let c = from_string "+-++-" let d = a *~ (b -~ c) let _ =
Printf.printf "a = %d\nb = %d\nc = %d\na * (b - c) = %s = %d\n" (to_int a) (to_int b) (to_int c) (to_string d) (to_int d);</lang>
Output:
a = 523 b = -436 c = 65 a * (b - c) = ----0+--0++0 = -262023
Perl
<lang perl>use strict; use warnings;
my @d = qw( 0 + - ); my @v = qw( 0 1 -1 );
sub to_bt {
my $n = shift;
my $b = ;
while( $n ) {
my $r = $n%3;
$b .= $d[$r];
$n -= $v[$r];
$n /= 3;
}
return scalar reverse $b;
}
sub from_bt {
my $n = 0;
for( split //, shift ) { # Horner
$n *= 3;
$n += "${_}1" if $_;
}
return $n;
}
my %addtable = (
'-0' => [ '-', ],
'+0' => [ '+', ],
'+-' => [ '0', ],
'00' => [ '0', ],
'--' => [ '+', '-' ],
'++' => [ '-', '+' ],
);
sub add {
my ($b1, $b2) = @_;
return ($b1 or $b2 ) unless ($b1 and $b2);
my $d = $addtable{ join , sort substr( $b1, -1, 1, ), substr( $b2, -1, 1, ) };
return add( add($b1, $d->[1]), $b2 ).$d->[0];
}
sub unary_minus {
my $b = shift; $b =~ tr/-+/+-/; return $b;
}
sub subtract {
my ($b1, $b2) = @_; return add( $b1, unary_minus $b2 );
}
sub mult {
my ($b1, $b2) = @_;
my $r = '0';
for( reverse split //, $b2 ){
$r = add $r, $b1 if $_ eq '+';
$r = subtract $r, $b1 if $_ eq '-';
$b1 .= '0';
}
$r =~ s/^0+//;
return $r;
}
my $a = "+-0++0+"; my $b = to_bt( -436 ); my $c = "+-++-"; my $d = mult( $a, subtract( $b, $c ) ); printf " a: %14s %10d\n", $a, from_bt( $a ); printf " b: %14s %10d\n", $b, from_bt( $b ); printf " c: %14s %10d\n", $c, from_bt( $c ); printf "a*(b-c): %14s %10d\n", $d, from_bt( $d ); </lang>
- Output:
a: +-0++0+ 523
b: -++-0-- -436
c: +-++- 65
a*(b-c): ----0+--0++0 -262023
Perl 6
<lang perl6>class BT {
has @.coeff;
my %co2bt = '-1' => '-', '0' => '0', '1' => '+'; my %bt2co = %co2bt.invert;
multi method new (Str $s) {
self.bless(coeff => %bt2co{$s.flip.comb});
}
multi method new (Int $i where $i >= 0) {
self.bless(coeff => carry $i.base(3).comb.reverse);
}
multi method new (Int $i where $i < 0) {
self.new(-$i).neg;
}
method Str () { %co2bt{@!coeff}.join.flip }
method Int () { [+] @!coeff Z* (1,3,9...*) }
multi method neg () {
self.new: coeff => carry self.coeff X* -1;
}
}
sub carry (*@digits is copy) {
loop (my $i = 0; $i < @digits; $i++) {
while @digits[$i] < -1 { @digits[$i] += 3; @digits[$i+1]--; } while @digits[$i] > 1 { @digits[$i] -= 3; @digits[$i+1]++; }
} pop @digits while @digits and not @digits[*-1]; @digits;
}
multi prefix:<-> (BT $x) { $x.neg }
multi infix:<+> (BT $x, BT $y) {
my ($b,$a) = sort +*.coeff, ($x, $y); BT.new: coeff => carry ($a.coeff Z+ |$b.coeff, |(0 xx $a.coeff - $b.coeff));
}
multi infix:<-> (BT $x, BT $y) { $x + $y.neg }
multi infix:<*> (BT $x, BT $y) {
my @x = $x.coeff;
my @y = $y.coeff;
my @z = 0 xx @x+@y-1;
my @safe;
for @x -> $xd {
@z = @z Z+ |(@y X* $xd), |(0 xx @z-@y); @safe.push: @z.shift;
} BT.new: coeff => carry @safe, @z;
}
my $a = BT.new: "+-0++0+"; my $b = BT.new: -436; my $c = BT.new: "+-++-"; my $x = $a * ( $b - $c );
say 'a == ', $a.Int; say 'b == ', $b.Int; say 'c == ', $c.Int; say "a × (b − c) == ", ~$x, ' == ', $x.Int;</lang>
- Output:
a == 523 b == -436 c == 65 a × (b − c) == ----0+--0++0 == -262023
Phix
Using strings to represent balanced ternary. Note that as implemented dec2bt and bt2dec are limited to Phix integers (~+/-1,000,000,000), but it would probably be pretty trivial (albeit quite a bit slower) to replace them with (say) ba2bt and bt2ba which use/yield bigatoms. <lang Phix>function bt2dec(string bt) integer res = 0
for i=1 to length(bt) do
res = 3*res+(bt[i]='+')-(bt[i]='-')
end for
return res
end function
function negate(string bt)
for i=1 to length(bt) do
if bt[i]!='0' then
bt[i] = '+'+'-'-bt[i]
end if
end for
return bt
end function
function dec2bt(integer n) string res = "0" integer neg, r
if n!=0 then
neg = n<0
if neg then n = -n end if
res = ""
while n!=0 do
r = mod(n,3)
res = "0+-"[r+1]&res
n = floor((n+(r=2))/3)
end while
if neg then res = negate(res) end if
end if
return res
end function
-- res,carry for a+b+carry lookup tables (not the fastest way to do it, I'm sure): constant {tadd,addres} = columnize({{"---","0-"},{"--0","+-"},{"--+","-0"},
{"-0-","+-"},{"-00","-0"},{"-0+","00"},
{"-+-","-0"},{"-+0","00"},{"-++","+0"},
{"0--","+-"},{"0-0","-0"},{"0-+","00"},
{"00-","-0"},{"000","00"},{"00+","+0"},
{"0+-","00"},{"0+0","+0"},{"0++","-+"},
{"+--","-0"},{"+-0","00"},{"+-+","+0"},
{"+0-","00"},{"+00","+0"},{"+0+","-+"},
{"++-","+0"},{"++0","-+"},{"+++","0+"}})
function bt_add(string a, string b)
integer pad = length(a)-length(b)
integer carry = '0'
if pad!=0 then
if pad<0 then
a = repeat('0',-pad)&a
else
b = repeat('0',pad)&b
end if
end if
for i=length(a) to 1 by -1 do
{a[i],carry} = addres[find(a[i]&b[i]&carry,tadd)]
end for
if carry!='0' then
a = carry&a
else
while length(a)>1 and a[1]='0' do
a = a[2..$]
end while
end if
return a
end function
function bt_mul(string a, string b) string pos = a, neg = negate(a), res = "0" integer ch
for i=length(b) to 1 by -1 do
ch = b[i]
if ch='+' then
res = bt_add(res,pos)
elsif ch='-' then
res = bt_add(res,neg)
end if
pos = pos&'0'
neg = neg&'0'
end for
return res
end function
string a = "+-0++0+", b = dec2bt(-436), c = "+-++-"
?{bt2dec(a),bt2dec(b),bt2dec(c)}
string res = bt_mul(a,bt_add(b,negate(c))) ?{res,bt2dec(res)}</lang>
- Output:
{523,-436,65}
{"----0+--0++0",-262023}
Proof of arbitrary large value support is provided by calculating 1000! and 999! and using a naive subtraction loop to effect division. The limit for factorials that can be held in native integers is a mere 12, and for atoms 170, mind you, inaccurate above 22. The timings show it manages a 5000+digit multiplication and subtraction in about 0.2s, which I say is "reasonable", given that I didn't try very hard, as evidenced by that daft addition lookup table! <lang Phix>atom t0 = time() string f999 = dec2bt(1) for i=2 to 999 do
f999 = bt_mul(f999,dec2bt(i))
end for string f1000 = bt_mul(f999,dec2bt(1000))
printf(1,"In balanced ternary, f999 has %d digits and f1000 has %d digits\n",{length(f999),length(f1000)})
integer count = 0 f999 = negate(f999) while f1000!="0" do
f1000 = bt_add(f1000,f999) count += 1
end while printf(1,"It took %d subtractions to reach 0. (%3.2fs)\n",{count,time()-t0})</lang>
- Output:
In balanced ternary, f999 has 5376 digits and f1000 has 5383 digits It took 1000 subtractions to reach 0. (9.30s)
PicoLisp
<lang PicoLisp>(seed (in "/dev/urandom" (rd 8)))
(setq *G '((0 -1) (1 -1) (-1 0) (0 0) (1 0) (-1 1) (0 1)))
- For humans
(de negh (L)
(mapcar
'((I)
(case I
(- '+)
(+ '-)
(T 0) ) )
L ) )
(de trih (X)
(if (num? X)
(let (S (lt0 X) X (abs X) R NIL)
(if (=0 X)
(push 'R 0)
(until (=0 X)
(push 'R
(case (% X 3)
(0 0)
(1 '+)
(2 (inc 'X) '-) ) )
(setq X (/ X 3)) ) )
(if S (pack (negh R)) (pack R)) )
(let M 1
(sum
'((C)
(prog1
(unless (= C "0") ((intern C) M))
(setq M (* 3 M)) ) )
(flip (chop X)) ) ) ) )
- For robots
(de neg (L)
(mapcar
'((I)
(case I (-1 1) (1 -1) (T 0)) )
L ) )
(de tri (X)
(if (num? X)
(let (S (lt0 X) X (abs X) R NIL)
(if (=0 X)
(push 'R 0)
(until (=0 X)
(push 'R
(case (% X 3)
(0 0)
(1 1)
(2 (inc 'X) (- 1)) ) )
(setq X (/ X 3)) ) )
(flip (if S (neg R) R)) )
(let M 1
(sum
'((C)
(prog1 (* C M) (setq M (* 3 M))) )
X ) ) ) )
(de add (D1 D2)
(let
(L (max (length D1) (length D2))
D1 (need (- L) D1 0)
D2 (need (- L) D2 0)
C 0 )
(mapcon
'((L1 L2)
(let R
(get
*G
(+ 4 (+ (car L1) (car L2) C)) )
(ifn (cdr L1)
R
(setq C (cadr R))
(cons (car R)) ) ) )
D1
D2 ) ) )
(de mul (D1 D2)
(ifn (and D1 D2)
0
(add
(case (car D1)
(0 0)
(1 D2)
(-1 (neg D2)) )
(cons 0 (mul (cdr D1) D2) ) ) ) )
(de sub (D1 D2)
(add D1 (neg D2)) )
- Random testing
(let (X 0 Y 0 C 2048)
(do C
(setq
X (rand (- C) C)
Y (rand (- C) C) )
(test X (trih (trih X)))
(test X (tri (tri X)))
(test
(+ X Y)
(tri (add (tri X) (tri Y))) )
(test
(- X Y)
(tri (sub (tri X) (tri Y))) )
(test
(* X Y)
(tri (mul (tri X) (tri Y))) ) ) )
(println 'A (trih 523) (trih "+-0++0+")) (println 'B (trih -436) (trih "-++-0--")) (println 'C (trih 65) (trih "+-++-")) (let R
(tri
(mul
(tri (trih "+-0++0+"))
(sub (tri -436) (tri (trih "+-++-"))) ) )
(println 'R (trih R) R) )
(bye)</lang>
Prolog
Works with SWI-Prolog and library clpfd written by Markus Triska.
Three modules, one for the conversion, one for the addition and one for the multiplication.
The conversion.
Library clpfd is used so that bt_convert works in both ways Decimal => Ternary and Ternary ==> Decimal.
<lang Prolog>:- module('bt_convert.pl', [bt_convert/2,
op(950, xfx, btconv),
btconv/2]).
- - use_module(library(clpfd)).
- - op(950, xfx, btconv).
X btconv Y :- bt_convert(X, Y).
% bt_convert(?X, ?L) bt_convert(X, L) :- ( (nonvar(L), \+is_list(L)) ->string_to_list(L, L1); L1 = L), convert(X, L1), ( var(L) -> string_to_list(L, L1); true).
% map numbers toward digits +, - 0 plus_moins( 1, 43). plus_moins(-1, 45). plus_moins( 0, 48).
convert(X, [48| L]) :-
var(X),
( L \= [] -> convert(X, L); X = 0, !).
convert(0, L) :- var(L), !, string_to_list(L, [48]).
convert(X, L) :- ( (nonvar(X), X > 0) ; (var(X), X #> 0, L = [43|_], maplist(plus_moins, L1, L))), !, convert(X, 0, [], L1), ( nonvar(X) -> maplist(plus_moins, L1, LL), string_to_list(L, LL) ; true).
convert(X, L) :- ( nonvar(X) -> Y is -X ; X #< 0, maplist(plus_moins, L2, L), maplist(mult(-1), L2, L1)), convert(Y, 0, [], L1), ( nonvar(X) -> maplist(mult(-1), L1, L2), maplist(plus_moins, L2, LL),
string_to_list(L, LL)
; X #= -Y).
mult(X, Y, Z) :- Z #= X * Y.
convert(0, 0, L, L) :- !.
convert(0, 1, L, [1 | L]) :- !.
convert(N, C, LC, LF) :-
R #= N mod 3 + C,
R #> 1 #<==> C1,
N1 #= N / 3,
R1 #= R - 3 * C1, % C1 #= 1,
convert(N1, C1, [R1 | LC], LF).
</lang>
The addition.
The same predicate is used for addition and substraction.
<lang Prolog>:- module('bt_add.pl', [bt_add/3,
bt_add1/3,
op(900, xfx, btplus),
op(900, xfx, btmoins),
btplus/2,
btmoins/2,
strip_nombre/3
]).
- - op(900, xfx, btplus).
- - op(900, xfx, btmoins).
% define operator btplus A is X btplus Y :- bt_add(X, Y, A).
% define operator btmoins % no need to define a predicate for the substraction A is X btmoins Y :-
X is Y btplus A.
% bt_add(?X, ?Y, ?R)
% R is X + Y
% X, Y, R are strings
% At least 2 args must be instantiated
bt_add(X, Y, R) :-
( nonvar(X) -> string_to_list(X, X1); true),
( nonvar(Y) -> string_to_list(Y, Y1); true),
( nonvar(R) -> string_to_list(R, R1); true),
bt_add1(X1, Y1, R1),
( var(X) -> string_to_list(X, X1); true),
( var(Y) -> string_to_list(Y, Y1); true),
( var(R) -> string_to_list(R, R1); true).
% bt_add1(?X, ?Y, ?R) % R is X + Y % X, Y, R are lists bt_add1(X, Y, R) :- % initialisation : X and Y must have the same length % we add zeros at the beginning of the shortest list ( nonvar(X) -> length(X, LX); length(R, LR)), ( nonvar(Y) -> length(Y, LY); length(R, LR)), ( var(X) -> LX is max(LY, LR) , length(X1, LX), Y1 = Y ; X1 = X), ( var(Y) -> LY is max(LX, LR) , length(Y1, LY), X1 = X ; Y1 = Y),
Delta is abs(LX - LY), ( LX < LY -> normalise(Delta, X1, X2), Y1 = Y2 ; LY < LX -> normalise(Delta, Y1, Y2), X1 = X2 ; X1 = X2, Y1 = Y2),
% if R is instancied, it must have, at least, the same length than X or Y
Max is max(LX, LY),
( (nonvar(R), length(R, LR), LR < Max) -> Delta1 is Max - LR, normalise(Delta1, R, R2)
; nonvar(R) -> R = R2
; true),
bt_add(X2, Y2, C, R2),
( C = 48 -> strip_nombre(R2, R, []), ( var(X) -> strip_nombre(X2, X, []) ; true), ( var(Y) -> strip_nombre(Y2, Y, []) ; true) ; var(R) -> strip_nombre([C|R2], R, []) ; ( select(C, [45,43], [Ca]), ( var(X) -> strip_nombre([Ca | X2], X, []) ; strip_nombre([Ca | Y2], Y, [])))).
% here we actually compute the sum
bt_add([], [], 48, []).
bt_add([H1|T1], [H2|T2], C3, [R2 | L]) :- bt_add(T1, T2, C, L), % add HH1 and H2 ternary_sum(H1, H2, R1, C1), % add first carry, ternary_sum(R1, C, R2, C2), % add second carry ternary_sum(C1, C2, C3, _).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ternary_sum % @arg1 : V1 % @arg2 : V2 % @arg3 : R is V1 + V2 % @arg4 : Carry ternary_sum(43, 43, 45, 43).
ternary_sum(43, 45, 48, 48).
ternary_sum(45, 43, 48, 48).
ternary_sum(45, 45, 43, 45).
ternary_sum(X, 48, X, 48).
ternary_sum(48, X, X, 48).
% if L has a length smaller than N, complete L with 0 (code 48)
normalise(0, L, L) :- !.
normalise(N, L1, L) :-
N1 is N - 1,
normalise(N1, [48 | L1], L).
% contrary of normalise
% remove leading zeros.
% special case of number 0 !
strip_nombre([48]) --> {!}, "0".
% enlève les zéros inutiles strip_nombre([48 | L]) --> strip_nombre(L).
strip_nombre(L) -->
L.
</lang>
The multiplication.
We give a predicate euclide(?A, +B, ?Q, ?R) which computes both the multiplication and the division, but it is very inefficient.
The predicates multiplication(+B, +Q, -A) and division(+A, +B, -Q, -R) are much more efficient.
<lang Prolog>:- module('bt_mult.pl', [op(850, xfx, btmult),
btmult/2,
multiplication/3
]).
- - use_module('bt_add.pl').
- - op(850, xfx, btmult).
A is B btmult C :- multiplication(B, C, A).
neg(A, B) :- maplist(opp, A, B).
opp(48, 48). opp(45, 43). opp(43, 45).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the multiplication (efficient) % multiplication(+BIn, +QIn, -AOut) % Aout is BIn * QIn % BIn, QIn, AOut are strings multiplication(BIn, QIn, AOut) :- string_to_list(BIn, B), string_to_list(QIn, Q),
% We work with positive numbers ( B = [45 | _] -> Pos0 = false, neg(B,BP) ; BP = B, Pos0 = true), ( Q = [45 | _] -> neg(Q, QP), select(Pos0, [true, false], [Pos1]); QP = Q, Pos1 = Pos0),
multiplication_(BP, QP, [48], A), ( Pos1 = false -> neg(A, A1); A1 = A), string_to_list(AOut, A1).
multiplication_(_B, [], A, A).
multiplication_(B, [H | T], A, AF) :- multiplication_1(B, H, B1), append(A, [48], A1), bt_add1(B1, A1, A2), multiplication_(B, T, A2, AF).
% by 1 (digit '+' code 43) multiplication_1(B, 43, B).
% by 0 (digit '0' code 48) multiplication_1(_, 48, [48]).
% by -1 (digit '-' code 45) multiplication_1(B, 45, B1) :- neg(B, B1).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the division (efficient)
% division(+AIn, +BIn, -QOut, -ROut)
%
division(AIn, BIn, QOut, ROut) :-
string_to_list(AIn, A),
string_to_list(BIn, B),
length(B, LB),
length(A, LA),
Len is LA - LB,
( Len < 0 -> Q = [48], R = A
; neg(B, NegB), division_(A, B, NegB, LB, Len, [], Q, R)),
string_to_list(QOut, Q),
string_to_list(ROut, R).
division_(A, B, NegB, LenB, LenA, QC, QF, R) :-
% if the remainder R is negative (last number A), we must decrease the quotient Q, annd add B to R
( LenA = -1 -> (A = [45 | _] -> positive(A, B, QC, QF, R) ; QF = QC, A = R)
; extract(LenA, _, A, AR, AF),
length(AR, LR),
( LR >= LenB -> ( AR = [43 | _] -> bt_add1(AR, NegB, S), Q0 = [43], % special case : R has the same length than B % and his first digit is + (1) % we must do another one substraction ( (length(S, LenB), S = [43|_]) -> bt_add1(S, NegB, S1), bt_add1(QC, [43], QC1), Q00 = [45] ; S1 = S, QC1 = QC, Q00 = Q0)
; bt_add1(AR, B, S1), Q00 = [45], QC1 = QC),
append(QC1, Q00, Q1),
append(S1, AF, A1),
strip_nombre(A1, A2, []),
LenA1 is LenA - 1,
division_(A2, B, NegB, LenB, LenA1, Q1, QF, R)
; append(QC, [48], Q1), LenA1 is LenA - 1, division_(A, B, NegB, LenB, LenA1, Q1, QF, R))).
% extract(+Len, ?N1, +L, -Head, -Tail) % remove last N digits from the list L % put them in Tail. extract(Len, Len, [], [], []).
extract(Len, N1, [H|T], AR1, AF1) :- extract(Len, N, T, AR, AF), N1 is N-1, ( N > 0 -> AR = AR1, AF1 = [H | AF]; AR1 = [H | AR], AF1 = AF).
positive(R, _, Q, Q, R) :- R = [43 | _].
positive(S, B, Q, QF, R ) :- bt_add1(S, B, S1), bt_add1(Q, [45], Q1), positive(S1, B, Q1, QF, R).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% "euclidian" division (inefficient)
% euclide(?A, +BIn, ?Q, ?R)
% A = B * Q + R
euclide(A, B, Q, R) :-
mult(A, B, Q, R).
mult(AIn, BIn, QIn, RIn) :-
( nonvar(AIn) -> string_to_list(AIn, A); A = AIn),
( nonvar(BIn) -> string_to_list(BIn, B); B = BIn),
( nonvar(QIn) -> string_to_list(QIn, Q); Q = QIn),
( nonvar(RIn) -> string_to_list(RIn, R); R = RIn),
% we use positive numbers ( B = [45 | _] -> Pos0 = false, neg(B,BP) ; BP = B, Pos0 = true), ( (nonvar(Q), Q = [45 | _]) -> neg(Q, QP), select(Pos0, [true, false], [Pos1]) ; nonvar(Q) -> Q = QP , Pos1 = Pos0 ; Pos1 = Pos0), ( (nonvar(A), A = [45 | _]) -> neg(A, AP) ; nonvar(A) -> AP = A ; true),
% is R instancied ? ( nonvar(R) -> R1 = R; true), % multiplication ? we add B to A and substract 1 (digit '-') to Q ( nonvar(Q) -> BC = BP, Ajout = [45], ( nonvar(R) -> bt_add1(BC, R, AP) ; AP = BC) % division ? we substract B to A and add 1 (digit '+') to Q ; neg(BP, BC), Ajout = [43], QP = [48]),
% do the real job mult_(BC, QP, AP, R1, Resultat, Ajout),
( var(QIn) -> (Pos1 = false -> neg(Resultat, QT); Resultat = QT), string_to_list(QIn, QT) ; true), ( var(AIn) -> (Pos1 = false -> neg(Resultat, AT); Resultat = AT), string_to_list(AIn, AT) ; true), ( var(RIn) -> string_to_list(RIn, R1); true).
% @arg1 : divisor % @arg2 : quotient % @arg3 : dividend % @arg4 : remainder % @arg5 : Result : receive either the dividend A % either the quotient Q mult_(B, Q, A, R, Resultat, Ajout) :- bt_add1(Q, Ajout, Q1), bt_add1(A, B, A1), ( Q1 = [48] -> Resultat = A % a multiplication ; ( A1 = [45 | _], Ajout = [43]) -> Resultat = Q, R = A % a division ; mult_(B, Q1, A1, R, Resultat, Ajout)) .
</lang> Example of output :
?- A btconv "+-0++0+". A = 523. ?- -436 btconv B. B = "-++-0--". ?- C btconv "+-++-". C = 65. ?- X is "-++-0--" btmoins "+-++-", Y is "+-0++0+" btmult X, Z btconv Y. X = "-+0-++0", Y = "----0+--0++0", Z = -262023 .
Python
<lang python>class BalancedTernary:
# Represented as a list of 0, 1 or -1s, with least significant digit first.
str2dig = {'+': 1, '-': -1, '0': 0} # immutable
dig2str = {1: '+', -1: '-', 0: '0'} # immutable
table = ((0, -1), (1, -1), (-1, 0), (0, 0), (1, 0), (-1, 1), (0, 1)) # immutable
def __init__(self, inp):
if isinstance(inp, str):
self.digits = [BalancedTernary.str2dig[c] for c in reversed(inp)]
elif isinstance(inp, int):
self.digits = self._int2ternary(inp)
elif isinstance(inp, BalancedTernary):
self.digits = list(inp.digits)
elif isinstance(inp, list):
if all(d in (0, 1, -1) for d in inp):
self.digits = list(inp)
else:
raise ValueError("BalancedTernary: Wrong input digits.")
else:
raise TypeError("BalancedTernary: Wrong constructor input.")
@staticmethod
def _int2ternary(n):
if n == 0: return []
if (n % 3) == 0: return [0] + BalancedTernary._int2ternary(n // 3)
if (n % 3) == 1: return [1] + BalancedTernary._int2ternary(n // 3)
if (n % 3) == 2: return [-1] + BalancedTernary._int2ternary((n + 1) // 3)
def to_int(self):
return reduce(lambda y,x: x + 3 * y, reversed(self.digits), 0)
def __repr__(self):
if not self.digits: return "0"
return "".join(BalancedTernary.dig2str[d] for d in reversed(self.digits))
@staticmethod
def _neg(digs):
return [-d for d in digs]
def __neg__(self):
return BalancedTernary(BalancedTernary._neg(self.digits))
@staticmethod
def _add(a, b, c=0):
if not (a and b):
if c == 0:
return a or b
else:
return BalancedTernary._add([c], a or b)
else:
(d, c) = BalancedTernary.table[3 + (a[0] if a else 0) + (b[0] if b else 0) + c]
res = BalancedTernary._add(a[1:], b[1:], c)
# trim leading zeros
if res or d != 0:
return [d] + res
else:
return res
def __add__(self, b):
return BalancedTernary(BalancedTernary._add(self.digits, b.digits))
def __sub__(self, b):
return self + (-b)
@staticmethod
def _mul(a, b):
if not (a and b):
return []
else:
if a[0] == -1: x = BalancedTernary._neg(b)
elif a[0] == 0: x = []
elif a[0] == 1: x = b
else: assert False
y = [0] + BalancedTernary._mul(a[1:], b)
return BalancedTernary._add(x, y)
def __mul__(self, b):
return BalancedTernary(BalancedTernary._mul(self.digits, b.digits))
def main():
a = BalancedTernary("+-0++0+")
print "a:", a.to_int(), a
b = BalancedTernary(-436) print "b:", b.to_int(), b
c = BalancedTernary("+-++-")
print "c:", c.to_int(), c
r = a * (b - c) print "a * (b - c):", r.to_int(), r
main()</lang>
- Output:
a: 523 +-0++0+ b: -436 -++-0-- c: 65 +-++- a * (b - c): -262023 ----0+--0++0
Racket
<lang racket>#lang racket
- Represent a balanced-ternary number as a list of 0's, 1's and -1's.
- e.g. 11 = 3^2 + 3^1 - 3^0 ~ "++-" ~ '(-1 1 1)
- 6 = 3^2 - 3^1 ~ "+-0" ~ '(0 -1 1)
- Note
- the list-rep starts with the least signifcant tert, while
- the string-rep starts with the most significsnt tert.
(define (bt->integer t)
(if (null? t)
0
(+ (first t) (* 3 (bt->integer (rest t))))))
(define (integer->bt n)
(letrec ([recur (λ (b r) (cons b (convert (floor (/ r 3)))))]
[convert (λ (n) (if (zero? n) null
(case (modulo n 3)
[(0) (recur 0 n)]
[(1) (recur 1 n)]
[(2) (recur -1 (add1 n))])))])
(convert n)))
(define (bt->string t)
(define (strip-leading-zeroes a)
(if (or (null? a) (not (= (first a) 0))) a (strip-leading-zeroes (rest a))))
(string-join (map (λ (u)
(case u
[(1) "+"]
[(-1) "-"]
[(0) "0"]))
(strip-leading-zeroes (reverse t))) ""))
(define (string->bt s)
(reverse
(map (λ (c)
(case c
[(#\+) 1]
[(#\-) -1]
[(#\0) 0]))
(string->list s))))
(define (bt-negate t)
(map (λ (u) (- u)) t))
(define (bt-add a b [c 0])
(cond [(and (null? a) (null? b)) (if (zero? c) null (list c))]
[(null? b) (if (zero? c) a (bt-add a (list c)))]
[(null? a) (bt-add b a c)]
[else (let* ([t (+ (first a) (first b) c)]
[carry (if (> (abs t) 1) (sgn t) 0)]
[v (case (abs t)
[(3) 0]
[(2) (- (sgn t))]
[else t])])
(cons v (bt-add (rest a) (rest b) carry)))]))
(define (bt-multiply a b)
(cond [(null? a) null]
[(null? b) null]
[else (bt-add (case (first a)
[(-1) (bt-negate b)]
[(0) null]
[(1) b])
(cons 0 (bt-multiply (rest a) b)))]))
- test case
(let* ([a (string->bt "+-0++0+")]
[b (integer->bt -436)]
[c (string->bt "+-++-")]
[d (bt-multiply a (bt-add b (bt-negate c)))])
(for ([bt (list a b c d)]
[description (list 'a 'b 'c "a×(b−c)")])
(printf "~a = ~a or ~a\n" description (bt->integer bt) (bt->string bt))))
</lang>
- Output:
a = 523 or +-0++0+ b = -436 or -++-0-- c = 65 or +-++- a×(b−c) = -262023 or ----0+--0++0
REXX
The REXX program could be optimized by using (procedure) with expose and having the $. and @. variables set only once. <lang rexx>/*REXX pgm converts decimal ◄───► balanced ternary; also performs arithmetic.*/ numeric digits 10000 /*be able to handle gihugic numbers. */ Ao = '+-0++0+' ; Abt = Ao /* [↓] 2 literals used by subroutine*/ Bo = '-436' ; Bbt = d2bt(Bo) ; @ = '(decimal)' Co = '+-++-' ; Cbt = Co ; @@ = 'balanced ternary ='
call btShow '[a]', Abt
call btShow '[b]', Bbt
call btShow '[c]', Cbt
say; $bt = btMul(Abt,btSub(Bbt,Cbt))
call btShow '[a*(b-c)]', $bt
exit /*stick a fork in it, we're all done. */ /*────────────────────────────────────────────────────────────────────────────*/ d2bt: procedure; parse arg x 1; p=0; $.='-'; $.1='+'; $.0=0; #=
x=x/1
do until x==0; _=(x//(3**(p+1)))%3**p
if _== 2 then _= -1
if _== -2 then _= 1
x=x-_*(3**p); p=p+1; #=$._ || #
end /*until ···*/
return #
/*────────────────────────────────────────────────────────────────────────────*/ bt2d: procedure; parse arg x; r=reverse(x); #=0; $.=-1; $.0=0; _='+'; $._=1
do j=1 for length(x); _=substr(r,j,1); #=#+$._*3**(j-1); end
return #
/*────────────────────────────────────────────────────────────────────────────*/ btAdd: procedure; parse arg x,y; rx=reverse(x); ry=reverse(y); carry=0
@.=0 ; _='-'; @._=-1; _="+"; @._=1
$.='-'; $.0=0; $.1='+'
#=; do j=1 for max(length(x),length(y))
x_=substr(rx,j,1); xn=@.x_
y_=substr(ry,j,1); yn=@.y_
s=xn+yn+carry ; carry= 0
if s== 2 then do; s=-1; carry= 1; end
if s== 3 then do; s= 0; carry= 1; end
if s==-2 then do; s= 1; carry=-1; end
#=$.s || #
end /*j*/
if carry\==0 then #=$.carry||#; return btNorm(#)
/*────────────────────────────────────────────────────────────────────────────*/ btMul: procedure; parse arg x 1 x1 2, y 1 y1 2; if x==0 | y==0 then return 0 S=1; x=btNorm(x); y=btNorm(y) /*handle: 0-xxx values.*/ if x1=='-' then do; x=btNeg(x); S=-S; end /*positate.*/ if y1=='-' then do; y=btNeg(y); S=-S; end /* " */ if length(y)>length(x) then parse value x y with y x /*optimize.*/ P=0
do until y==0 /*keep adding 'til done*/
P=btAdd(P,x) /*multiple the hard way*/
y=btSub(y,'+') /*subtract 1 from Y.*/
end /*until*/
if S==-1 then P=btNeg(P) /*adjust product sign. */ return P /*return the product P.*/ /*────────────────────────────────────────────────────────────────────────────*/ btNeg: return translate(arg(1), '-+', "+-") /*negate the bal_tern #*/ btNorm: _=strip(arg(1),'L',0); if _== then _=0; return _ /*normalize a #*/ btSub: return btAdd(arg(1), btNeg(arg(2))) /*subtract two BT args.*/ btShow: say center(arg(1),9) right(arg(2),20) @@ right(bt2d(arg(2)),9) @; return</lang> output when using the default inputs:
[a] +-0++0+ balanced ternary = 523 (decimal) [b] -++-0-- balanced ternary = -436 (decimal) [c] +-++- balanced ternary = 65 (decimal) [a*(b-c)] ----0+--0++0 balanced ternary = -262023 (decimal)
Ruby
<lang ruby>class BalancedTernary
include Comparable
def initialize(str = "")
if str =~ /[^-+0]+/
raise ArgumentError, "invalid BalancedTernary number: #{str}"
end
@digits = trim0(str)
end
I2BT = {0 => ["0",0], 1 => ["+",0], 2 => ["-",1]}
def self.from_int(value)
n = value.to_i
digits = ""
while n != 0
quo, rem = n.divmod(3)
bt, carry = I2BT[rem]
digits = bt + digits
n = quo + carry
end
new(digits)
end
BT2I = {"-" => -1, "0" => 0, "+" => 1}
def to_int
@digits.chars.inject(0) do |sum, char|
sum = 3 * sum + BT2I[char]
end
end
alias :to_i :to_int
def to_s
@digits.dup # String is mutable
end
alias :inspect :to_s
def <=>(other)
to_i <=> other.to_i
end
ADDITION_TABLE = {
"---" => ["-","0"], "--0" => ["-","+"], "--+" => ["0","-"],
"-0-" => ["-","+"], "-00" => ["0","-"], "-0+" => ["0","0"],
"-+-" => ["0","-"], "-+0" => ["0","0"], "-++" => ["0","+"],
"0--" => ["-","+"], "0-0" => ["0","-"], "0-+" => ["0","0"],
"00-" => ["0","-"], "000" => ["0","0"], "00+" => ["0","+"],
"0+-" => ["0","0"], "0+0" => ["0","+"], "0++" => ["+","-"],
"+--" => ["0","-"], "+-0" => ["0","0"], "+-+" => ["0","+"],
"+0-" => ["0","0"], "+00" => ["0","+"], "+0+" => ["+","-"],
"++-" => ["0","+"], "++0" => ["+","-"], "+++" => ["+","0"],
}
def +(other)
maxl = [to_s.length, other.to_s.length].max
a = pad0_reverse(to_s, maxl)
b = pad0_reverse(other.to_s, maxl)
carry = "0"
sum = a.zip( b ).inject("") do |sum, (c1, c2)|
carry, digit = ADDITION_TABLE[carry + c1 + c2]
sum = digit + sum
end
self.class.new(carry + sum)
end
MULTIPLICATION_TABLE = {
"-" => "+0-",
"0" => "000",
"+" => "-0+",
}
def *(other)
product = self.class.new
other.to_s.each_char do |bdigit|
row = to_s.tr("-0+", MULTIPLICATION_TABLE[bdigit])
product += self.class.new(row)
product << 1
end
product >> 1
end
# negation
def -@()
self.class.new(@digits.tr('-+','+-'))
end
# subtraction
def -(other)
self + (-other)
end
# shift left
def <<(count)
@digits = trim0(@digits + "0"*count)
self
end
# shift right
def >>(count)
@digits[-count..-1] = "" if count > 0
@digits = trim0(@digits)
self
end
private
def trim0(str)
str = str.sub(/^0+/, "")
str = "0" if str.empty?
str
end
def pad0_reverse(str, len)
str.rjust(len, "0").reverse.chars
end
end
a = BalancedTernary.new("+-0++0+") b = BalancedTernary.from_int(-436) c = BalancedTernary.new("+-++-")
%w[a b c a*(b-c)].each do |exp|
val = eval(exp) puts "%8s :%13s,%8d" % [exp, val, val.to_i]
end</lang>
- Output:
a : +-0++0+, 523
b : -++-0--, -436
c : +-++-, 65
a*(b-c) : ----0+--0++0, -262023
Scala
This implementation represents ternaries as a reversed list of bits. Also, there are plenty of implicit convertors <lang scala> object TernaryBit {
val P = TernaryBit(+1)
val M = TernaryBit(-1)
val Z = TernaryBit( 0)
implicit def asChar(t: TernaryBit): Char = t.charValue
implicit def valueOf(c: Char): TernaryBit = {
c match {
case '0' => 0
case '+' => 1
case '-' => -1
case nc => throw new IllegalArgumentException("Illegal ternary symbol " + nc)
}
}
implicit def asInt(t: TernaryBit): Int = t.intValue
implicit def valueOf(i: Int): TernaryBit = TernaryBit(i)
}
case class TernaryBit(val intValue: Int) {
def inverse: TernaryBit = TernaryBit(-intValue)
def charValue = intValue match {
case 0 => '0'
case 1 => '+'
case -1 => '-'
}
}
class Ternary(val bits: List[TernaryBit]) {
def + (b: Ternary) = {
val sumBits: List[Int] = bits.map(_.intValue).zipAll(b.bits.map(_.intValue), 0, 0).map(p => p._1 + p._2)
// normalize
val iv: Tuple2[List[Int], Int] = (List(), 0)
val (revBits, carry) = sumBits.foldLeft(iv)((accu: Tuple2[List[Int], Int], e: Int) => {
val s = e + accu._2
(((s + 1 + 3 * 100) % 3 - 1) :: accu._1 , (s + 1 + 3 * 100) / 3 - 100)
})
new Ternary(( TernaryBit(carry) :: revBits.map(TernaryBit(_))).reverse )
}
def - (b: Ternary) = {this + (-b)}
def <<<(a: Int): Ternary = { List.fill(a)(TernaryBit.Z) ++ bits}
def >>>(a: Int): Ternary = { bits.drop(a) }
def unary_- = { bits.map(_.inverse) }
def ** (b: TernaryBit): Ternary = {
b match {
case TernaryBit.P => this
case TernaryBit.M => - this
case TernaryBit.Z => 0
}
}
def * (mul: Ternary): Ternary = {
// might be done more efficiently - perform normalize only once
mul.bits.reverse.foldLeft(new Ternary(Nil))((a: Ternary, b: TernaryBit) => (a <<< 1) + (this ** b))
}
def intValue = bits.foldRight(0)((c, a) => a*3 + c.intValue)
override def toString = new String(bits.reverse.map(_.charValue).toArray)
}
object Ternary {
implicit def asString(t: Ternary): String = t.toString()
implicit def valueOf(s: String): Ternary = new Ternary(s.toList.reverse.map(TernaryBit.valueOf(_)))
implicit def asBits(t: Ternary): List[TernaryBit] = t.bits
implicit def valueOf(l: List[TernaryBit]): Ternary = new Ternary(l)
implicit def asInt(t: Ternary): BigInt = t.intValue
// XXX not tail recursive
implicit def valueOf(i: BigInt): Ternary = {
if (i < 0) -valueOf(-i)
else if (i == 0) new Ternary(List())
else if (i % 3 == 0) TernaryBit.Z :: valueOf(i / 3)
else if (i % 3 == 1) TernaryBit.P :: valueOf(i / 3)
else /*(i % 3 == 2)*/ TernaryBit.M :: valueOf((i + 1) / 3)
}
implicit def intToTernary(i: Int): Ternary = valueOf(i)
} </scala>
Then these classes can be used in the following way: <lang scala> object Main {
def main(args: Array[String]): Unit = {
val a: Ternary = "+-0++0+"
val b: Ternary = -436
val c: Ternary = "+-++-"
println(a.toString + " " + a.intValue)
println(b.toString + " " + b.intValue)
println(c.toString + " " + c.intValue)
val res = a * (b - c)
println(res.toString + " " + res.intValue)
}
} </lang>
- Output:
+-0++0+ 523 -++-0-- -436 +-++- 65 00000000----0+--0++0 -262023
Besides, we can easily check, that the code works for any input. This can be achieved with ScalaCheck: <lang scala> object TernarySpecification extends Properties("Ternary") {
property("sum") = forAll { (a: Int, b: Int) =>
val at: Ternary = a
val bt: Ternary = b
(at+bt).intValue == (at.intValue + bt.intValue)
}
property("multiply") = forAll { (a: Int, b: Int) =>
val at: Ternary = a
val bt: Ternary = b
(at*bt).intValue == (at.intValue * bt.intValue)
}
} </lang>
- Output:
+ Ternary.sum: OK, passed 100 tests. + Ternary.multiply: OK, passed 100 tests.
Tcl
This directly uses the printable representation of the balanced ternary numbers, as Tcl's string operations are reasonably efficient. <lang tcl>package require Tcl 8.5
proc bt-int b {
set n 0
foreach c [split $b ""] {
set n [expr {$n * 3}] switch -- $c { + { incr n 1 } - { incr n -1 } }
} return $n
} proc int-bt n {
if {$n == 0} {
return "0"
}
while {$n != 0} {
lappend result [lindex {0 + -} [expr {$n % 3}]] set n [expr {$n / 3 + ($n%3 == 2)}]
} return [join [lreverse $result] ""]
}
proc bt-neg b {
string map {+ - - +} $b
} proc bt-sub {a b} {
bt-add $a [bt-neg $b]
} proc bt-add-digits {a b c} {
if {$a eq ""} {set a 0}
if {$b eq ""} {set b 0}
if {$a ne 0} {append a 1}
if {$b ne 0} {append b 1}
lindex {{0 -1} {+ -1} {- 0} {0 0} {+ 0} {- 1} {0 1}} [expr {$a+$b+$c+3}]
} proc bt-add {a b} {
set c 0
set result {}
foreach ca [lreverse [split $a ""]] cb [lreverse [split $b ""]] {
lassign [bt-add-digits $ca $cb $c] d c lappend result $d
}
if {$c ne "0"} {lappend result [lindex {0 + -} $c]}
if {![llength $result]} {return "0"}
string trimleft [join [lreverse $result] ""] 0
} proc bt-mul {a b} {
if {$a eq "0" || $a eq "" || $b eq "0"} {return "0"}
set sub [bt-mul [string range $a 0 end-1] $b]0
switch -- [string index $a end] {
0 { return $sub } + { return [bt-add $sub $b] } - { return [bt-sub $sub $b] }
}
}</lang> Demonstration code: <lang tcl>for {set i 0} {$i<=10} {incr i} {puts "$i = [int-bt $i]"} puts "'+-+'+'+--' = [bt-add +-+ +--] = [bt-int [bt-add +-+ +--]]" puts "'++'*'++' = [bt-mul ++ ++] = [bt-int [bt-mul ++ ++]]"
set a "+-0++0+" set b [int-bt -436] set c "+-++-" puts "a = [bt-int $a], b = [bt-int $b], c = [bt-int $c]" set abc [bt-mul $a [bt-sub $b $c]] puts "a*(b-c) = $abc (== [bt-int $abc])"</lang> Output:
0 = 0 1 = + 2 = +- 3 = +0 4 = ++ 5 = +-- 6 = +-0 7 = +-+ 8 = +0- 9 = +00 10 = +0+ '+-+'+'+--' = ++0 = 12 '++'*'++' = +--+ = 16 a = 523, b = -436, c = 65 a*(b-c) = ----0+--0++0 (== -262023)