Ternary logic: Difference between revisions
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#define forall(a) for(a = t_F; a <= t_T; a++) |
#define forall(a) for(a = t_F; a <= t_T; a++) |
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void show_op(trit (*f)(trit, trit), char *name) { |
void show_op(trit (*f)(trit, trit), const char *name) { |
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trit a, b; |
trit a, b; |
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printf("\n[%s]\n F ? T\n -------", name); |
printf("\n[%s]\n F ? T\n -------", name); |
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Revision as of 08:54, 26 December 2012
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at Ternary logic. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
In logic, a three-valued logic (also trivalent, ternary, or trinary logic, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Łukasiewicz, Lewis and Sulski. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.
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Task:
- Define a new type that emulates ternary logic by storing data trits.
- Given all the binary logic operators of the original programming language, reimplement these operators for the new Ternary logic type trit.
- Generate a sampling of results using trit variables.
- Kudos for actually thinking up a test case algorithm where ternary logic is intrinsically useful, optimises the test case algorithm and is preferable to binary logic.
Note: Setun (Сетунь) was a balanced ternary computer developed in 1958 at Moscow State University. The device was built under the lead of Sergei Sobolev and Nikolay Brusentsov. It was the only modern ternary computer, using three-valued ternary logic
Ada
We first specify a package "Logic" for three-valued logic. Observe that predefined Boolean functions, "and" "or" and "not" are overloaded: <lang Ada>package Logic is
type Ternary is (True, Unknown, False);
-- logic functions function "and"(Left, Right: Ternary) return Ternary; function "or"(Left, Right: Ternary) return Ternary; function "not"(T: Ternary) return Ternary; function Equivalent(Left, Right: Ternary) return Ternary; function Implies(Condition, Conclusion: Ternary) return Ternary;
-- conversion functions function To_Bool(X: Ternary) return Boolean; function To_Ternary(B: Boolean) return Ternary; function Image(Value: Ternary) return Character;
end Logic;</lang>
Next, the implementation of the package:
<lang Ada>package body Logic is
-- type Ternary is (True, Unknown, False);
function Image(Value: Ternary) return Character is
begin
case Value is
when True => return 'T';
when False => return 'F';
when Unknown => return '?';
end case;
end Image;
function "and"(Left, Right: Ternary) return Ternary is
begin
return Ternary'max(Left, Right);
end "and";
function "or"(Left, Right: Ternary) return Ternary is
begin
return Ternary'min(Left, Right);
end "or";
function "not"(T: Ternary) return Ternary is
begin
case T is
when False => return True;
when Unknown => return Unknown;
when True => return False;
end case;
end "not";
function To_Bool(X: Ternary) return Boolean is
begin
case X is
when True => return True;
when False => return False;
when Unknown => raise Constraint_Error;
end case;
end To_Bool;
function To_Ternary(B: Boolean) return Ternary is
begin
if B then
return True;
else
return False;
end if;
end To_Ternary;
function Equivalent(Left, Right: Ternary) return Ternary is
begin
return To_Ternary(To_Bool(Left) = To_Bool(Right));
exception
when Constraint_Error => return Unknown;
end Equivalent;
function Implies(Condition, Conclusion: Ternary) return Ternary is
begin
return (not Condition) or Conclusion;
end Implies;
end Logic;</lang>
Finally, a sample program: <lang Ada>with Ada.Text_IO, Logic;
procedure Test_Tri_Logic is
use Logic;
type F2 is access function(Left, Right: Ternary) return Ternary; type F1 is access function(Trit: Ternary) return Ternary;
procedure Truth_Table(F: F1; Name: String) is
begin
Ada.Text_IO.Put_Line("X | " & Name & "(X)");
for T in Ternary loop
Ada.Text_IO.Put_Line(Image(T) & " | " & Image(F(T)));
end loop;
end Truth_Table;
procedure Truth_Table(F: F2; Name: String) is
begin
Ada.Text_IO.New_Line;
Ada.Text_IO.Put_Line("X | Y | " & Name & "(X,Y)");
for X in Ternary loop
for Y in Ternary loop
Ada.Text_IO.Put_Line(Image(X) & " | " & Image(Y) & " | " & Image(F(X,Y)));
end loop;
end loop;
end Truth_Table;
begin
Truth_Table(F => "not"'Access, Name => "Not"); Truth_Table(F => "and"'Access, Name => "And"); Truth_Table(F => "or"'Access, Name => "Or"); Truth_Table(F => Equivalent'Access, Name => "Eq"); Truth_Table(F => Implies'Access, Name => "Implies");
end Test_Tri_Logic;</lang>
The output:
X | Not(X) T | F ? | ? F | T X | Y | And(X,Y) T | T | T T | ? | ? T | F | F ? | T | ? ? | ? | ? ? | F | F F | T | F F | ? | F F | F | F ... (and so on)
ALGOL 68
File: Ternary_logic.a68 <lang algol68># -*- coding: utf-8 -*- #
INT trit width = 1, trit base = 3; MODE TRIT = STRUCT(BITS trit); CO FORMAT trit fmt = $c("?","⌈","⌊",#|"~"#)$; CO
- These values treated are as per "Balanced ternary" #
- eg true=1, maybe=0, false=-1 #
TRIT true =INITTRIT 4r1, maybe=INITTRIT 4r0,
false=INITTRIT 4r2;
- Warning: redefines standard builtins flip & flop #
LONGCHAR flap="?", flip="⌈", flop="⌊";
OP REPR = (TRIT t)LONGCHAR:
[]LONGCHAR(flap, flip, flop)[1+ABS trit OF t];
- Define some OPerators for coercing MODES #
OP INITTRIT = (BOOL in)TRIT:
(in|true|false);
OP INITBOOL = (TRIT in)BOOL:
(trit OF in=trit OF true|TRUE|:trit OF in=trit OF false|FALSE|
raise value error(("vague TRIT to BOOL coercion: """, REPR in,""""));~
);
OP B = (TRIT in)BOOL: INITBOOL in;
- These values treated are as per "Balanced ternary" #
- n.b true=1, maybe=0, false=-1 #
- Warning: BOOL ABS FALSE (0) is not the same as TRIT ABS false (-1) #
OP INITINT = (TRIT t)INT:
CASE 1+ABS trit OF t
IN #maybe# 0, #true # 1, #false#-1
OUT raise value error(("invalid TRIT value",REPR t)); ~
ESAC;
OP INITTRIT = (INT in)TRIT: (
TRIT out;
trit OF out:= trit OF
CASE 2+in
IN false, maybe, true
OUT raise value error(("invalid TRIT value",in)); ~
ESAC;
out
);
OP INITTRIT = (BITS b)TRIT:
(TRIT out; trit OF out:=b; out);
- Define the LOGICAL OPerators for the TRIT MODE #
MODE LOGICAL = TRIT; PR READ "Template_operators_logical_mixin.a68" PR
COMMENT
Kleene logic truth tables:
END COMMENT
OP AND = (TRIT a,b)TRIT: (
[,]TRIT( # ∧ ## false, maybe, true # #false# (false, false, false), #maybe# (false, maybe, maybe), #true # (false, maybe, true ) )[@-1,@-1][INITINT a, INITINT b]
);
OP OR = (TRIT a,b)TRIT: (
[,]TRIT( # ∨ ## false, maybe, true # #false# (false, maybe, true), #maybe# (maybe, maybe, true), #true # (true, true, true) )[@-1,@-1][INITINT a, INITINT b]
);
PRIO IMPLIES = 1; # PRIO = 1.9 # OP IMPLIES = (TRIT a,b)TRIT: (
[,]TRIT( # ⊃ ## false, maybe, true # #false# (true, true, true), #maybe# (maybe, maybe, true), #true # (false, maybe, true) )[@-1,@-1][INITINT a, INITINT b]
);
PRIO EQV = 1; # PRIO = 1.8 # OP EQV = (TRIT a,b)TRIT: (
[,]TRIT( # ≡ ## false, maybe, true # #false# (true, maybe, false), #maybe# (maybe, maybe, maybe), #true # (false, maybe, true ) )[@-1,@-1][INITINT a, INITINT b]
);</lang>File: Template_operators_logical_mixin.a68 <lang algol68># -*- coding: utf-8 -*- #
OP & = (LOGICAL a,b)LOGICAL: a AND b; CO # not included as they are treated as SCALAR # OP EQ = (LOGICAL a,b)LOGICAL: a = b,
NE = (LOGICAL a,b)LOGICAL: a /= b, ≠ = (TRIT a,b)TRIT: a /= b, ¬= = (TRIT a,b)TRIT: a /= b;
END CO
- IF html entities possible THEN
¢ "parked" operators for completeness ¢ OP ¬ = (LOGICAL a)LOGICAL: NOT a,
∧ = (LOGICAL a,b)LOGICAL: a AND b, /\ = (LOGICAL a,b)LOGICAL: a AND b, ∨ = (LOGICAL a,b)LOGICAL: a OR b, \/ = (LOGICAL a,b)LOGICAL: a OR b, ⊃ = (TRIT a,b)TRIT: a IMPLIES b, ≡ = (TRIT a,b)TRIT: a EQV b;
FI#
- IF algol68c THEN
OP ~ = (LOGICAL a)LOGICAL: NOT a,
~= = (LOGICAL a,b)LOGICAL: a /= b; SCALAR!
FI#</lang>File: test_Ternary_logic.a68 <lang algol68>#!/usr/local/bin/a68g --script #
- -*- coding: utf-8 -*- #
PR READ "prelude/general.a68" PR PR READ "Ternary_logic.a68" PR
[]TRIT trits = (false, maybe, true);
FORMAT
col fmt = $" "g" "$,
row fmt = $l3(f(col fmt)"|")f(col fmt)$,
row sep fmt = $l3("---+")"---"l$;
PROC row sep = VOID:
printf(row sep fmt);
PROC title = (STRING op name, LONGCHAR op char)VOID:(
print(("Operator: ",op name));
printf((row fmt,op char,REPR false, REPR maybe, REPR true))
);
PROC print trit op table = (LONGCHAR op char, STRING op name, PROC(TRIT,TRIT)TRIT op)VOID: (
printf($l$);
title(op name, op char);
FOR i FROM LWB trits TO UPB trits DO
row sep;
TRIT ti = trits[i];
printf((col fmt, REPR ti));
FOR j FROM LWB trits TO UPB trits DO
TRIT tj = trits[j];
printf(($"|"$, col fmt, REPR op(ti,tj)))
OD
OD;
printf($l$)
);
printf((
$"Comparitive table of coercions:"l$, $" TRIT BOOL INT"l$
));
FOR it FROM LWB trits TO UPB trits DO
TRIT t = trits[it]; printf(( $" "g" "$, REPR t, IF trit OF t = trit OF maybe THEN " " ELSE B t FI, INITINT t, $l$))
OD;
printf((
$l"Specific test of the IMPLIES operator:"l$,
$" "g" implies "g" is "b("not ","")"a contradiction!"l$,
B false, B false, B(false IMPLIES false),
B false, B true, B(false IMPLIES true),
B false, REPR maybe, B(false IMPLIES maybe),
B true, B false, B(true IMPLIES false),
B true, B true, B(true IMPLIES true),
REPR maybe, Btrue, B(maybe IMPLIES true),
$" "g" implies "g" is "g" a contradiction!"l$,
B true, REPR maybe, REPR (true IMPLIES maybe),
REPR maybe, B false, REPR (maybe IMPLIES false),
REPR maybe, REPR maybe, REPR (maybe IMPLIES maybe),
$l$
));
printf($"Kleene logic truth table samples:"l$);
print trit op table("≡","EQV", (TRIT a,b)TRIT: a EQV b); print trit op table("⊃","IMPLIES", (TRIT a,b)TRIT: a IMPLIES b); print trit op table("∧","AND", (TRIT a,b)TRIT: a AND b); print trit op table("∨","OR", (TRIT a,b)TRIT: a OR b)</lang> Output:
Comparitive table of coercions: TRIT BOOL INT ⌊ F -1 ? +0 ⌈ T +1 Specific test of the IMPLIES operator: F implies F is not a contradiction! F implies T is not a contradiction! F implies ? is not a contradiction! T implies F is a contradiction! T implies T is not a contradiction! ? implies T is not a contradiction! T implies ? is ? a contradiction! ? implies F is ? a contradiction! ? implies ? is ? a contradiction! Kleene logic truth table samples: Operator: EQV ≡ | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌈ | ? | ⌊ ---+---+---+--- ? | ? | ? | ? ---+---+---+--- ⌈ | ⌊ | ? | ⌈ Operator: IMPLIES ⊃ | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌈ | ⌈ | ⌈ ---+---+---+--- ? | ? | ? | ⌈ ---+---+---+--- ⌈ | ⌊ | ? | ⌈ Operator: AND ∧ | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌊ | ⌊ | ⌊ ---+---+---+--- ? | ⌊ | ? | ? ---+---+---+--- ⌈ | ⌊ | ? | ⌈ Operator: OR ∨ | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌊ | ? | ⌈ ---+---+---+--- ? | ? | ? | ⌈ ---+---+---+--- ⌈ | ⌈ | ⌈ | ⌈
BBC BASIC
<lang bbcbasic> INSTALL @lib$ + "CLASSLIB"
REM Create a ternary class:
DIM trit{tor, tand, teqv, tnot, tnor, s, v}
DEF PRIVATE trit.s (t&) LOCAL t$():DIM t$(2):t$()="FALSE","MAYBE","TRUE":=t$(t&)
DEF PRIVATE trit.v (t$) = INSTR("FALSE MAYBE TRUE", t$) DIV 6
DEF trit.tnot (t$) = FN(trit.s)(2 - FN(trit.v)(t$))
DEF trit.tor (a$,b$) LOCAL t:t=FN(trit.v)(a$)ORFN(trit.v)(b$):=FN(trit.s)(t+(t>2))
DEF trit.tnor (a$,b$) = FN(trit.tnot)(FN(trit.tor)(a$,b$))
DEF trit.tand (a$,b$) = FN(trit.tnor)(FN(trit.tnot)(a$),FN(trit.tnot)(b$))
DEF trit.teqv (a$,b$) = FN(trit.tor)(FN(trit.tand)(a$,b$),FN(trit.tnor)(a$,b$))
PROC_class(trit{})
PROC_new(mytrit{}, trit{})
REM Test it:
PRINT "Testing NOT:"
PRINT "NOT FALSE = " FN(mytrit.tnot)("FALSE")
PRINT "NOT MAYBE = " FN(mytrit.tnot)("MAYBE")
PRINT "NOT TRUE = " FN(mytrit.tnot)("TRUE")
PRINT '"Testing OR:"
PRINT "FALSE OR FALSE = " FN(mytrit.tor)("FALSE","FALSE")
PRINT "FALSE OR MAYBE = " FN(mytrit.tor)("FALSE","MAYBE")
PRINT "FALSE OR TRUE = " FN(mytrit.tor)("FALSE","TRUE")
PRINT "MAYBE OR MAYBE = " FN(mytrit.tor)("MAYBE","MAYBE")
PRINT "MAYBE OR TRUE = " FN(mytrit.tor)("MAYBE","TRUE")
PRINT "TRUE OR TRUE = " FN(mytrit.tor)("TRUE","TRUE")
PRINT '"Testing AND:"
PRINT "FALSE AND FALSE = " FN(mytrit.tand)("FALSE","FALSE")
PRINT "FALSE AND MAYBE = " FN(mytrit.tand)("FALSE","MAYBE")
PRINT "FALSE AND TRUE = " FN(mytrit.tand)("FALSE","TRUE")
PRINT "MAYBE AND MAYBE = " FN(mytrit.tand)("MAYBE","MAYBE")
PRINT "MAYBE AND TRUE = " FN(mytrit.tand)("MAYBE","TRUE")
PRINT "TRUE AND TRUE = " FN(mytrit.tand)("TRUE","TRUE")
PRINT '"Testing EQV (similar to EOR):"
PRINT "FALSE EQV FALSE = " FN(mytrit.teqv)("FALSE","FALSE")
PRINT "FALSE EQV MAYBE = " FN(mytrit.teqv)("FALSE","MAYBE")
PRINT "FALSE EQV TRUE = " FN(mytrit.teqv)("FALSE","TRUE")
PRINT "MAYBE EQV MAYBE = " FN(mytrit.teqv)("MAYBE","MAYBE")
PRINT "MAYBE EQV TRUE = " FN(mytrit.teqv)("MAYBE","TRUE")
PRINT "TRUE EQV TRUE = " FN(mytrit.teqv)("TRUE","TRUE")
PROC_discard(mytrit{})</lang>
Output:
Testing NOT: NOT FALSE = TRUE NOT MAYBE = MAYBE NOT TRUE = FALSE Testing OR: FALSE OR FALSE = FALSE FALSE OR MAYBE = MAYBE FALSE OR TRUE = TRUE MAYBE OR MAYBE = MAYBE MAYBE OR TRUE = TRUE TRUE OR TRUE = TRUE Testing AND: FALSE AND FALSE = FALSE FALSE AND MAYBE = FALSE FALSE AND TRUE = FALSE MAYBE AND MAYBE = MAYBE MAYBE AND TRUE = MAYBE TRUE AND TRUE = TRUE Testing EQV (similar to EOR): FALSE EQV FALSE = TRUE FALSE EQV MAYBE = MAYBE FALSE EQV TRUE = FALSE MAYBE EQV MAYBE = MAYBE MAYBE EQV TRUE = MAYBE TRUE EQV TRUE = TRUE
C
Implementing logic using lookup tables
<lang c>#include <stdio.h>
typedef enum {
TRITTRUE, /* In this enum, equivalent to integer value 0 */ TRITMAYBE, /* In this enum, equivalent to integer value 1 */ TRITFALSE /* In this enum, equivalent to integer value 2 */
} trit;
/* We can trivially find the result of the operation by passing
the trinary values as indeces into the lookup tables' arrays. */
trit tritNot[3] = {TRITFALSE , TRITMAYBE, TRITTRUE}; trit tritAnd[3][3] = { {TRITTRUE, TRITMAYBE, TRITFALSE},
{TRITMAYBE, TRITMAYBE, TRITFALSE},
{TRITFALSE, TRITFALSE, TRITFALSE} };
trit tritOr[3][3] = { {TRITTRUE, TRITTRUE, TRITTRUE},
{TRITTRUE, TRITMAYBE, TRITMAYBE},
{TRITTRUE, TRITMAYBE, TRITFALSE} };
trit tritThen[3][3] = { { TRITTRUE, TRITMAYBE, TRITFALSE},
{ TRITTRUE, TRITMAYBE, TRITMAYBE},
{ TRITTRUE, TRITTRUE, TRITTRUE } };
trit tritEquiv[3][3] = { { TRITTRUE, TRITMAYBE, TRITFALSE},
{ TRITMAYBE, TRITMAYBE, TRITMAYBE},
{ TRITFALSE, TRITMAYBE, TRITTRUE } };
/* Everything beyond here is just demonstration */
const char* tritString[3] = {"T", "?", "F"};
void demo_binary_op(trit operator[3][3], const char* name) {
trit operand1 = TRITTRUE; /* Declare. Initialize for CYA */ trit operand2 = TRITTRUE; /* Declare. Initialize for CYA */
/* Blank line */
printf("\n");
/* Demo this operator */
for( operand1 = TRITTRUE; operand1 <= TRITFALSE; ++operand1 )
{
for( operand2 = TRITTRUE; operand2 <= TRITFALSE; ++operand2 )
{
printf("%s %s %s: %s\n", tritString[operand1],
name,
tritString[operand2],
tritString[operator[operand1][operand2]]);
}
}
}
int main() {
trit op1 = TRITTRUE; /* Declare. Initialize for CYA */
trit op2 = TRITTRUE; /* Declare. Initialize for CYA */
/* Demo 'not' */
for( op1 = TRITTRUE; op1 <= TRITFALSE; ++op1 )
{
printf("Not %s: %s\n", tritString[op1], tritString[tritNot[op1]]);
}
demo_binary_op(tritAnd, "And");
demo_binary_op(tritOr, "Or");
demo_binary_op(tritThen, "Then");
demo_binary_op(tritEquiv, "Equiv");
return 0;
}</lang>
Output: <lang text>Not T: F Not ?: ? Not F: T
T And T: T T And ?: ? T And F: F ? And T: ? ? And ?: ? ? And F: F F And T: F F And ?: F F And F: F
T Or T: T T Or ?: T T Or F: T ? Or T: T ? Or ?: ? ? Or F: ? F Or T: T F Or ?: ? F Or F: F
T Then T: T T Then ?: ? T Then F: F ? Then T: T ? Then ?: ? ? Then F: ? F Then T: T F Then ?: T F Then F: T
T Equiv T: T T Equiv ?: ? T Equiv F: F ? Equiv T: ? ? Equiv ?: ? ? Equiv F: ? F Equiv T: F F Equiv ?: ? F Equiv F: T</lang>
Using functions
<lang c>#include <stdio.h>
typedef enum { t_F = -1, t_M, t_T } trit;
trit t_not (trit a) { return -a; } trit t_and (trit a, trit b) { return a < b ? a : b; } trit t_or (trit a, trit b) { return a > b ? a : b; } trit t_eq (trit a, trit b) { return a * b; } trit t_imply(trit a, trit b) { return -a > b ? -a : b; } char t_s(trit a) { return "F?T"[a + 1]; }
- define forall(a) for(a = t_F; a <= t_T; a++)
void show_op(trit (*f)(trit, trit), const char *name) { trit a, b; printf("\n[%s]\n F ? T\n -------", name); forall(a) { printf("\n%c |", t_s(a)); forall(b) printf(" %c", t_s(f(a, b))); } puts(""); }
int main(void) { trit a;
puts("[Not]"); forall(a) printf("%c | %c\n", t_s(a), t_s(t_not(a)));
show_op(t_and, "And"); show_op(t_or, "Or"); show_op(t_eq, "Equiv"); show_op(t_imply, "Imply");
return 0; }</lang>output<lang>[Not] F | T ? | ? T | F
[And]
F ? T -------
F | F F F ? | F ? ? T | F ? T
[Or]
F ? T -------
F | F ? T ? | ? ? T T | T T T
[Equiv]
F ? T -------
F | T ? F ? | ? ? ? T | F ? T
[Imply]
F ? T -------
F | T T T ? | ? ? T T | F ? T</lang>
Variable truthfulness
Represent each possible truth value as a floating point value x, where the var has x chance of being true and 1 - x chance of being false. When using if3 conditional on a potential truth varible, the result is randomly sampled to true or false according to the chance. (This description is definitely very confusing perhaps).
<lang c>#include <stdio.h>
- include <stdlib.h>
typedef double half_truth, maybe;
inline maybe not3(maybe a) { return 1 - a; }
inline maybe and3(maybe a, maybe b) { return a * b; }
inline maybe or3(maybe a, maybe b) { return a + b - a * b; }
inline maybe eq3(maybe a, maybe b) { return 1 - a - b + 2 * a * b; }
inline maybe imply3(maybe a, maybe b) { return or3(not3(a), b); }
- define true3(x) ((x) * RAND_MAX > rand())
- define if3(x) if (true3(x))
int main() { maybe roses_are_red = 0.25; /* they can be white or black, too */ maybe violets_are_blue = 1; /* aren't they just */ int i;
puts("Verifying flowery truth for 40 times:\n");
puts("Rose is NOT red:"); /* chance: .75 */ for (i = 0; i < 40 || !puts("\n"); i++) printf( true3( not3(roses_are_red) ) ? "T" : "_");
/* pick a rose and a violet; */ puts("Rose is red AND violet is blue:"); /* chance of rose being red AND violet being blue is .25 */ for (i = 0; i < 40 || !puts("\n"); i++) printf( true3( and3(roses_are_red, violets_are_blue) ) ? "T" : "_");
/* chance of rose being red OR violet being blue is 1 */ puts("Rose is red OR violet is blue:"); for (i = 0; i < 40 || !puts("\n"); i++) printf( true3( or3(roses_are_red, violets_are_blue) ) ? "T" : "_");
/* pick two roses; chance of em being both red or both not red is .625 */ puts("This rose is as red as that rose:"); for (i = 0; i < 40 || !puts("\n"); i++) if3(eq3(roses_are_red, roses_are_red)) putchar('T'); else putchar('_');
return 0; }</lang>
C#
<lang csharp>using System;
/// <summary> /// Extension methods on nullable bool. /// </summary> /// <remarks> /// The operators !, & and | are predefined. /// </remarks> public static class NullableBoolExtension {
public static bool? Implies(this bool? left, bool? right)
{
return !left | right;
}
public static bool? IsEquivalentTo(this bool? left, bool? right)
{
return left.HasValue && right.HasValue ? left == right : default(bool?);
}
public static string Format(this bool? value)
{
return value.HasValue ? value.Value.ToString() : "Maybe";
}
}
public class Program {
private static void Main()
{
var values = new[] { true, default(bool?), false };
foreach (var left in values)
{
Console.WriteLine("¬{0} = {1}", left.Format(), (!left).Format());
foreach (var right in values)
{
Console.WriteLine("{0} & {1} = {2}", left.Format(), right.Format(), (left & right).Format());
Console.WriteLine("{0} | {1} = {2}", left.Format(), right.Format(), (left | right).Format());
Console.WriteLine("{0} → {1} = {2}", left.Format(), right.Format(), left.Implies(right).Format());
Console.WriteLine("{0} ≡ {1} = {2}", left.Format(), right.Format(), left.IsEquivalentTo(right).Format());
}
}
}
}</lang> Output:
¬True = False True & True = True True | True = True True → True = True True ≡ True = True True & Maybe = Maybe True | Maybe = True True → Maybe = Maybe True ≡ Maybe = Maybe True & False = False True | False = True True → False = False True ≡ False = False ¬Maybe = Maybe Maybe & True = Maybe Maybe | True = True Maybe → True = True Maybe ≡ True = Maybe Maybe & Maybe = Maybe Maybe | Maybe = Maybe Maybe → Maybe = Maybe Maybe ≡ Maybe = Maybe Maybe & False = False Maybe | False = Maybe Maybe → False = Maybe Maybe ≡ False = Maybe ¬False = True False & True = False False | True = True False → True = True False ≡ True = False False & Maybe = False False | Maybe = Maybe False → Maybe = True False ≡ Maybe = Maybe False & False = False False | False = False False → False = True False ≡ False = True
Common Lisp
<lang lisp>(defun tri-not (x) (- 1 x)) (defun tri-and (&rest x) (apply #'* x)) (defun tri-or (&rest x) (tri-not (apply #'* (mapcar #'tri-not x)))) (defun tri-eq (x y) (+ (tri-and x y) (tri-and (- 1 x) (- 1 y)))) (defun tri-imply (x y) (tri-or (tri-not x) y))
(defun tri-test (x) (< (random 1e0) x)) (defun tri-string (x) (if (= x 1) "T" (if (= x 0) "F" "?")))
- to say (tri-if (condition) (yes) (no))
(defmacro tri-if (tri ifcase &optional elsecase)
`(if (tri-test ,tri) ,ifcase ,elsecase))
(defun print-table (func header)
(let ((vals '(1 .5 0)))
(format t "~%~a:~%" header)
(format t " ~{~a ~^~}~%---------~%" (mapcar #'tri-string vals))
(loop for row in vals do
(format t "~a | " (tri-string row)) (loop for col in vals do (format t "~a " (tri-string (funcall func row col)))) (write-line ""))))
(write-line "NOT:") (loop for row in '(1 .5 0) do
(format t "~a | ~a~%" (tri-string row) (tri-string (tri-not row))))
(print-table #'tri-and "AND") (print-table #'tri-or "OR") (print-table #'tri-imply "IMPLY") (print-table #'tri-eq "EQUAL")</lang>output<lang>NOT: T | F ? | ? F | T
AND:
T ? F
T | T ? F ? | ? ? F F | F F F
OR:
T ? F
T | T T T ? | T ? ? F | T ? F
IMPLY:
T ? F
T | T ? F ? | T ? ? F | T T T
EQUAL:
T ? F
T | T ? F ? | ? ? ? F | F ? T</lang>
D
Partial translation of a C entry: <lang d>import std.stdio;
struct Trit {
private enum Val : byte { F = -1, M, T }
private Val t;
alias t this;
static immutable Trit[3] vals = [{Val.F}, {Val.M}, {Val.T}];
static immutable F = Trit(Val.F); // Not necessary but handy.
static immutable M = Trit(Val.M);
static immutable T = Trit(Val.T);
string toString() const pure nothrow {
return "F?T"[t + 1 .. t + 2];
}
Trit opUnary(string op)() const pure nothrow
if (op == "~") {
return Trit(-t);
}
Trit opBinary(string op)(in Trit b) const pure nothrow
if (op == "&") {
return t < b ? this : b;
}
Trit opBinary(string op)(in Trit b) const pure nothrow
if (op == "|") {
return t > b ? this : b;
}
Trit opBinary(string op)(in Trit b) const pure nothrow
if (op == "^") {
return ~(this == b);
}
Trit opEquals(in Trit b) const pure nothrow {
return Trit(cast(Val)(t * b));
}
Trit imply(in Trit b) const pure nothrow {
return -t > b ? ~this : b;
}
}
void showOperation(string op)(in string opName) {
writef("\n[%s]\n F ? T\n -------", opName);
foreach (immutable a; Trit.vals) {
writef("\n%s |", a);
foreach (immutable b; Trit.vals)
static if (op == "==>")
writef(" %s", a.imply(b));
else
writef(" %s", mixin("a " ~ op ~ " b"));
}
writeln();
}
void main() {
writeln("[Not]");
foreach (const a; Trit.vals)
writefln("%s | %s", a, ~a);
showOperation!"&"("And");
showOperation!"|"("Or");
showOperation!"^"("Xor");
showOperation!"=="("Equiv");
showOperation!"==>"("Imply");
}</lang>
- Output:
[Not]
F | T
? | ?
T | F
[And]
F ? T
-------
F | F F F
? | F ? ?
T | F ? T
[Or]
F ? T
-------
F | F ? T
? | ? ? T
T | T T T
[Xor]
F ? T
-------
F | F ? T
? | ? ? ?
T | T ? F
[Equiv]
F ? T
-------
F | T ? F
? | ? ? ?
T | F ? T
[Imply]
F ? T
-------
F | T T T
? | ? ? T
T | F ? T
Delphi
<lang delphi>unit TrinaryLogic;
interface
//Define our own type for ternary logic. //This is actually still a Boolean, but the compiler will use distinct RTTI information. type
TriBool = type Boolean;
const
TTrue:TriBool = True; TFalse:TriBool = False; TMaybe:TriBool = TriBool(2);
function TVL_not(Value: TriBool): TriBool; function TVL_and(A, B: TriBool): TriBool; function TVL_or(A, B: TriBool): TriBool; function TVL_xor(A, B: TriBool): TriBool; function TVL_eq(A, B: TriBool): TriBool;
implementation
Uses
SysUtils;
function TVL_not(Value: TriBool): TriBool; begin
if Value = True Then
Result := TFalse
else If Value = False Then
Result := TTrue
else
Result := Value;
end;
function TVL_and(A, B: TriBool): TriBool; begin
Result := TriBool(Iff(Integer(A * B) > 1, Integer(TMaybe), A * B));
end;
function TVL_or(A, B: TriBool): TriBool; begin
Result := TVL_not(TVL_and(TVL_not(A), TVL_not(B)));
end;
function TVL_xor(A, B: TriBool): TriBool; begin
Result := TVL_and(TVL_or(A, B), TVL_not(TVL_or(A, B)));
end;
function TVL_eq(A, B: TriBool): TriBool; begin
Result := TVL_not(TVL_xor(A, B));
end;
end.</lang>
And that's the reason why you never on no account ever should compare against the values of True or False unless you intent ternary logic!
An alternative version would be using an enum type <lang delphi>type TriBool = (tbFalse, tbMaybe, tbTrue);</lang> and defining a set of constants implementing the above tables: <lang delphi>const
tvl_not: array[TriBool] = (tbTrue, tbMaybe, tbFalse);
tvl_and: array[TriBool, TriBool] = (
(tbFalse, tbFalse, tbFalse),
(tbFalse, tbMaybe, tbMaybe),
(tbFalse, tbMaybe, tbTrue),
);
tvl_or: array[TriBool, TriBool] = (
(tbFalse, tbMaybe, tbTrue),
(tbMaybe, tbMaybe, tbTrue),
(tbTrue, tbTrue, tbTrue),
);
tvl_xor: array[TriBool, TriBool] = (
(tbFalse, tbMaybe, tbTrue),
(tbMaybe, tbMaybe, tbMaybe),
(tbTrue, tbMaybe, tbFalse),
);
tvl_eq: array[TriBool, TriBool] = (
(tbTrue, tbMaybe, tbFalse),
(tbMaybe, tbMaybe, tbMaybe),
(tbFalse, tbMaybe, tbTrue),
);
</lang>
That's no real fun, but lookup can then be done with <lang delphi>Result := tvl_and[A, B];</lang>
Factor
For boolean logic, Factor uses t and f with the words >boolean, not, and, or, xor. For ternary logic, we add m and define the words >trit, tnot, tand, tor, txor and t=. Our new class, trit, is the union class of t, m and f.
<lang factor>! rosettacode/ternary/ternary.factor ! http://rosettacode.org/wiki/Ternary_logic USING: combinators kernel ; IN: rosettacode.ternary
SINGLETON: m UNION: trit t m POSTPONE: f ;
GENERIC: >trit ( object -- trit ) M: trit >trit ;
- tnot ( trit1 -- trit )
>trit { { t [ f ] } { m [ m ] } { f [ t ] } } case ;
- tand ( trit1 trit2 -- trit )
>trit {
{ t [ >trit ] }
{ m [ >trit { { t [ m ] } { m [ m ] } { f [ f ] } } case ] }
{ f [ >trit drop f ] }
} case ;
- tor ( trit1 trit2 -- trit )
>trit {
{ t [ >trit drop t ] }
{ m [ >trit { { t [ t ] } { m [ m ] } { f [ m ] } } case ] }
{ f [ >trit ] }
} case ;
- txor ( trit1 trit2 -- trit )
>trit {
{ t [ tnot ] }
{ m [ >trit drop m ] }
{ f [ >trit ] }
} case ;
- t= ( trit1 trit2 -- trit )
{
{ t [ >trit ] }
{ m [ >trit drop m ] }
{ f [ tnot ] }
} case ;</lang>
Example use: <lang factor>( scratchpad ) CONSTANT: trits { t m f } ( scratchpad ) trits [ tnot ] map . { f m t } ( scratchpad ) trits [ trits swap [ tand ] curry map ] map . { { t m f } { m m f } { f f f } } ( scratchpad ) trits [ trits swap [ tor ] curry map ] map . { { t t t } { t m m } { t m f } } ( scratchpad ) trits [ trits swap [ txor ] curry map ] map . { { f m t } { m m m } { t m f } } ( scratchpad ) trits [ trits swap [ t= ] curry map ] map . { { t m f } { m m m } { f m t } }</lang>
Go
Go has four operators for the bool type: ==, &&, ||, and !. <lang go>package main
import "fmt"
type trit int8
const (
trFalse = iota - 1 trMaybe trTrue
)
func (t trit) String() string {
switch t {
case trFalse:
return "False"
case trMaybe:
return "Maybe"
case trTrue:
return "True "
}
panic("Invalid trit")
}
func trNot(t trit) trit {
return -t
}
func trAnd(s, t trit) trit {
if s < t {
return s
}
return t
}
func trOr(s, t trit) trit {
if s > t {
return s
}
return t
}
func trEq(s, t trit) trit {
return s * t
}
func main() {
trSet := []trit{trFalse, trMaybe, trTrue}
fmt.Println("t not t")
for _, t := range trSet {
fmt.Println(t, trNot(t))
}
fmt.Println("\ns t s and t")
for _, s := range trSet {
for _, t := range trSet {
fmt.Println(s, t, trAnd(s, t))
}
}
fmt.Println("\ns t s or t")
for _, s := range trSet {
for _, t := range trSet {
fmt.Println(s, t, trOr(s, t))
}
}
fmt.Println("\ns t s eq t")
for _, s := range trSet {
for _, t := range trSet {
fmt.Println(s, t, trEq(s, t))
}
}
}</lang> Output:
t not t False True Maybe Maybe True False s t s and t False False False False Maybe False False True False Maybe False False Maybe Maybe Maybe Maybe True Maybe True False False True Maybe Maybe True True True s t s or t False False False False Maybe Maybe False True True Maybe False Maybe Maybe Maybe Maybe Maybe True True True False True True Maybe True True True True s t s eq t False False True False Maybe Maybe False True False Maybe False Maybe Maybe Maybe Maybe Maybe True Maybe True False False True Maybe Maybe True True True
Groovy
Solution: <lang groovy>enum Trit {
TRUE, MAYBE, FALSE
private Trit nand(Trit that) {
switch ([this,that]) {
case { FALSE in it }: return TRUE
case { MAYBE in it }: return MAYBE
default : return FALSE
}
}
private Trit nor(Trit that) { this.or(that).not() }
Trit and(Trit that) { this.nand(that).not() }
Trit or(Trit that) { this.not().nand(that.not()) }
Trit not() { this.nand(this) }
Trit imply(Trit that) { this.nand(that.not()) }
Trit equiv(Trit that) { this.and(that).or(this.nor(that)) }
}</lang>
Test: <lang groovy>printf 'AND\n ' Trit.values().each { b -> printf ('%6s', b) } println '\n ----- ----- -----' Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.and(b)) }
println()
}
printf '\nOR\n ' Trit.values().each { b -> printf ('%6s', b) } println '\n ----- ----- -----' Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.or(b)) }
println()
}
println '\nNOT' Trit.values().each {
printf ('%6s | %6s\n', it, it.not())
}
printf '\nIMPLY\n ' Trit.values().each { b -> printf ('%6s', b) } println '\n ----- ----- -----' Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.imply(b)) }
println()
}
printf '\nEQUIV\n ' Trit.values().each { b -> printf ('%6s', b) } println '\n ----- ----- -----' Trit.values().each { a ->
printf ('%6s | ', a)
Trit.values().each { b -> printf ('%6s', a.equiv(b)) }
println()
}</lang>
Output:
AND
TRUE MAYBE FALSE
----- ----- -----
TRUE | TRUE MAYBE FALSE
MAYBE | MAYBE MAYBE FALSE
FALSE | FALSE FALSE FALSE
OR
TRUE MAYBE FALSE
----- ----- -----
TRUE | TRUE TRUE TRUE
MAYBE | TRUE MAYBE MAYBE
FALSE | TRUE MAYBE FALSE
NOT
TRUE | FALSE
MAYBE | MAYBE
FALSE | TRUE
IMPLY
TRUE MAYBE FALSE
----- ----- -----
TRUE | TRUE MAYBE FALSE
MAYBE | TRUE MAYBE MAYBE
FALSE | TRUE TRUE TRUE
EQUIV
TRUE MAYBE FALSE
----- ----- -----
TRUE | TRUE MAYBE FALSE
MAYBE | MAYBE MAYBE MAYBE
FALSE | FALSE MAYBE TRUE
Haskell
All operations given in terms of NAND, the functionally-complete operation.
<lang Haskell>import Prelude hiding (Bool(..), not, (&&), (||), (==))
main = mapM_ (putStrLn . unlines . map unwords)
[ table "not" $ unary not , table "and" $ binary (&&) , table "or" $ binary (||) , table "implies" $ binary (=->) , table "equals" $ binary (==) ]
data Trit = False | Maybe | True deriving (Show)
False `nand` _ = True _ `nand` False = True True `nand` True = False _ `nand` _ = Maybe
not a = nand a a
a && b = not $ a `nand` b
a || b = not a `nand` not b
a =-> b = a `nand` not b
a == b = (a && b) || (not a && not b)
inputs1 = [True, Maybe, False] inputs2 = [(a,b) | a <- inputs1, b <- inputs1]
unary f = map (\a -> [a, f a]) inputs1 binary f = map (\(a,b) -> [a, b, f a b]) inputs2
table name xs = map (map pad) . (header :) $ map (map show) xs
where header = map (:[]) (take ((length $ head xs) - 1) ['A'..]) ++ [name]
pad s = s ++ replicate (5 - length s) ' '</lang>
Output:
A not True False Maybe Maybe False True A B and True True True True Maybe Maybe True False False Maybe True Maybe Maybe Maybe Maybe Maybe False False False True False False Maybe False False False False A B or True True True True Maybe True True False True Maybe True True Maybe Maybe Maybe Maybe False Maybe False True True False Maybe Maybe False False False A B implies True True True True Maybe Maybe True False False Maybe True True Maybe Maybe Maybe Maybe False Maybe False True True False Maybe True False False True A B equals True True True True Maybe Maybe True False False Maybe True Maybe Maybe Maybe Maybe Maybe False Maybe False True False False Maybe Maybe False False True
Icon and Unicon
The following example works in both Icon and Unicon. There are a couple of comments on the code that pertain to the task requirements:
- Strictly speaking there are no binary values in Icon and Unicon. There are a number of flow control operations that result in expression success (and a result) or failure which affects flow. As a result there really isn't a set of binary operators to map into ternary. The example provides the minimum required by the task plus xor.
- The code below does not define a data type as it doesn't really make sense in this case. Icon and Unicon can create records which would be overkill and clumsy in this case. Unicon can create objects which would also be overkill. The only remaining option is to reinterpret one of the existing types as ternary values. The code below implements balanced ternary values as integers in order to simplify several of the functions.
- The use of integers doesn't really support strings of trits well. While there is a function showtrit to ease display a converse function to decode character trits in a string is not included.
<lang Icon>$define TRUE 1
$define FALSE -1
$define UNKNOWN 0
invocable all link printf
procedure main() # demonstrate ternary logic
ufunc := ["not3"] bfunc := ["and3", "or3", "xor3", "eq3", "ifthen3"]
every f := !ufunc do { # display unary functions
printf("\nunary function=%s:\n",f)
every t1 := (TRUE | FALSE | UNKNOWN) do
printf(" %s : %s\n",showtrit(t1),showtrit(not3(t1)))
}
every f := !bfunc do { # display binary functions
printf("\nbinary function=%s:\n ",f)
every t1 := (&null | TRUE | FALSE | UNKNOWN) do {
printf(" %s : ",showtrit(\t1))
every t2 := (TRUE | FALSE | UNKNOWN | &null) do {
if /t1 then printf(" %s",showtrit(\t2)|"\n")
else printf(" %s",showtrit(f(t1,\t2))|"\n")
}
}
}
end
procedure showtrit(a) #: return printable trit of error if invalid return case a of {TRUE:"T";FALSE:"F";UNKNOWN:"?";default:runerr(205,a)} end
procedure istrit(a) #: return value of trit or error if invalid return (TRUE|FALSE|UNKNOWN|runerr(205,a)) = a end
procedure not3(a) #: not of trit or error if invalid return FALSE * istrit(a) end
procedure and3(a,b) #: and of two trits or error if invalid return min(istrit(a),istrit(b)) end
procedure or3(a,b) #: or of two trits or error if invalid return max(istrit(a),istrit(b)) end
procedure eq3(a,b) #: equals of two trits or error if invalid return istrit(a) * istrit(b) end
procedure ifthen3(a,b) #: if trit then trit or error if invalid return case istrit(a) of { TRUE: istrit(b) ; UNKNOWN: or3(a,b); FALSE: TRUE } end
procedure xor3(a,b) #: xor of two trits or error if invalid return not3(eq3(a,b)) end</lang>
printf.icn provides support for the printf family of functions
Output:
unary function=not3:
T : F
F : T
? : ?
binary function=and3:
T F ?
T : T F ?
F : F F F
? : ? F ?
binary function=or3:
T F ?
T : T T T
F : T F ?
? : T ? ?
binary function=xor3:
T F ?
T : F T ?
F : T F ?
? : ? ? ?
binary function=eq3:
T F ?
T : T F ?
F : F T ?
? : ? ? ?
binary function=ifthen3:
T F ?
T : T F ?
F : T T T
? : T ? ?
J
The designers of J felt that user defined types were harmful, so that part of the task will not be supported here.
Instead:
true: 1 false: 0 maybe: 0.5
<lang j>not=: -. and=: <. or =: >. if =: (>. -.)"0~ eq =: (<.&-. >. <.)"0</lang>
Example use:
<lang j> not 0 0.5 1 1 0.5 0
0 0.5 1 and/ 0 0.5 1
0 0 0 0 0.5 0.5 0 0.5 1
0 0.5 1 or/ 0 0.5 1 0 0.5 1
0.5 0.5 1
1 1 1
0 0.5 1 if/ 0 0.5 1 1 1 1
0.5 0.5 1
0 0.5 1
0 0.5 1 eq/ 0 0.5 1 1 0.5 0
0.5 0.5 0.5
0 0.5 1</lang>
Note that this implementation is a special case of "fuzzy logic" (using a limited set of values).
Note that while >. and <. could be used for boolean operations instead of J's +. and *., the identity elements for >. and <. are not boolean values, but are negative and positive infinity. See also: Boolean ring
Note that we might instead define values between 0 and 1 to represent independent probabilities:
<lang J>not=: -. and=: * or=: *&.-. if =: (or -.)"0~ eq =: (*&-. or *)"0</lang>
However, while this might be a more intellectually satisfying approach, this gives us some different results from the task requirement, for the combination of two "maybe" values:
<lang J> not 0 0.5 1 1 0.5 0
0 0.5 1 and/ 0 0.5 1
0 0 0 0 0.25 0.5 0 0.5 1
0 0.5 1 or/ 0 0.5 1 0 0.5 1
0.5 0.75 1
1 1 1
0 0.5 1 if/ 0 0.5 1 1 1 1
0.5 0.75 1
0 0.5 1
0 0.5 1 eq/ 0 0.5 1 1 0.5 0
0.5 0.4375 0.5
0 0.5 1</lang>
Another interesting possibility would involve using George Boole's original operations. This leaves us without any "not", (if we include the definition of logical negation which was later added to the definition of Boolean algebra, then the only numbers which can be used with Boolean algebra are 1 and 0). So, it's not clear how we would implement "if" or "eq". However, "and" and "or" would look like this:
<lang J>and=: *. or=: +.</lang>
And, the boolean result tables would look like this:
<lang J> 0 0.5 1 and/ 0 0.5 1 0 0 0 0 0.5 1 0 1 1
0 0.5 1 or/ 0 0.5 1 0 0.5 1
0.5 0.5 0.5
1 0.5 1</lang>
Java
<lang java5>public class Logic{ public static enum Trit{ TRUE, MAYBE, FALSE;
public Trit and(Trit other){ if(this == TRUE){ return other; }else if(this == MAYBE){ return (other == FALSE) ? FALSE : MAYBE; }else{ return FALSE; } }
public Trit or(Trit other){ if(this == TRUE){ return TRUE; }else if(this == MAYBE){ return (other == TRUE) ? TRUE : MAYBE; }else{ return other; } }
public Trit tIf(Trit other){ if(this == TRUE){ return other; }else if(this == MAYBE){ return (other == TRUE) ? TRUE : MAYBE; }else{ return TRUE; } }
public Trit not(){ if(this == TRUE){ return FALSE; }else if(this == MAYBE){ return MAYBE; }else{ return TRUE; } }
public Trit equals(Trit other){ if(this == TRUE){ return other; }else if(this == MAYBE){ return MAYBE; }else{ return other.not(); } } } public static void main(String[] args){ for(Trit a:Trit.values()){ System.out.println("not " + a + ": " + a.not()); } for(Trit a:Trit.values()){ for(Trit b:Trit.values()){ System.out.println(a+" and "+b+": "+a.and(b)+ "\t "+a+" or "+b+": "+a.or(b)+ "\t "+a+" implies "+b+": "+a.tIf(b)+ "\t "+a+" = "+b+": "+a.equals(b)); } } } }</lang> Output:
not TRUE: FALSE not MAYBE: MAYBE not FALSE: TRUE TRUE and TRUE: TRUE TRUE or TRUE: TRUE TRUE implies TRUE: TRUE TRUE = TRUE: TRUE TRUE and MAYBE: MAYBE TRUE or MAYBE: TRUE TRUE implies MAYBE: MAYBE TRUE = MAYBE: MAYBE TRUE and FALSE: FALSE TRUE or FALSE: TRUE TRUE implies FALSE: FALSE TRUE = FALSE: FALSE MAYBE and TRUE: MAYBE MAYBE or TRUE: TRUE MAYBE implies TRUE: TRUE MAYBE = TRUE: MAYBE MAYBE and MAYBE: MAYBE MAYBE or MAYBE: MAYBE MAYBE implies MAYBE: MAYBE MAYBE = MAYBE: MAYBE MAYBE and FALSE: FALSE MAYBE or FALSE: MAYBE MAYBE implies FALSE: MAYBE MAYBE = FALSE: MAYBE FALSE and TRUE: FALSE FALSE or TRUE: TRUE FALSE implies TRUE: TRUE FALSE = TRUE: FALSE FALSE and MAYBE: FALSE FALSE or MAYBE: MAYBE FALSE implies MAYBE: TRUE FALSE = MAYBE: MAYBE FALSE and FALSE: FALSE FALSE or FALSE: FALSE FALSE implies FALSE: TRUE FALSE = FALSE: TRUE
Liberty BASIC
<lang lb> 'ternary logic '0 1 2 'F ? T 'False Don't know True 'LB has NOT AND OR XOR, so we implement them. 'LB has no EQ, but XOR could be expressed via EQ. In 'normal' boolean at least.
global tFalse, tDontKnow, tTrue tFalse = 0 tDontKnow = 1 tTrue = 2
print "Short and long names for ternary logic values" for i = tFalse to tTrue
print shortName3$(i);" ";longName3$(i)
next print
print "Single parameter functions" print "x";" ";"=x";" ";"not(x)" for i = tFalse to tTrue
print shortName3$(i);" ";shortName3$(i);" ";shortName3$(not3(i))
next print
print "Double parameter fuctions" print "x";" ";"y";" ";"x AND y";" ";"x OR y";" ";"x EQ y";" ";"x XOR y" for a = tFalse to tTrue
for b = tFalse to tTrue
print shortName3$(a);" ";shortName3$(b);" "; _
shortName3$(and3(a,b));" "; shortName3$(or3(a,b));" "; _
shortName3$(eq3(a,b));" "; shortName3$(xor3(a,b))
next
next
function and3(a,b)
and3 = min(a,b)
end function
function or3(a,b)
or3 = max(a,b)
end function
function eq3(a,b)
select case
case a=tDontKnow or b=tDontKnow
eq3 = tDontKnow
case a=b
eq3 = tTrue
case else
eq3 = tFalse
end select
end function
function xor3(a,b)
xor3 = not3(eq3(a,b))
end function
function not3(b)
not3 = 2-b
end function
'------------------------------------------------ function shortName3$(i)
shortName3$ = word$("F ? T", i+1)
end function
function longName3$(i)
longName3$ = word$("False,Don't know,True", i+1, ",")
end function
</lang>
Output:
Short and long names for ternary logic values F False ? Don't know T True Single parameter functions x =x not(x) F F T ? ? ? T T F Double parameter fuctions x y x AND y x OR y x EQ y x XOR y F F F F T F F ? F ? ? ? F T F T F T ? F F ? ? ? ? ? ? ? ? ? ? T ? T ? ? T F F T F T T ? ? T ? ? T T T T T F
Mathematica
Type definition is not allowed in Mathematica. We can just use the build-in symbols "True" and "False", and add a new symbol "Maybe". <lang mathematica>Maybe /: ! Maybe = Maybe; Maybe /: (And | Or | Nand | Nor | Xor | Xnor | Implies | Equivalent)[Maybe, Maybe] = Maybe;</lang> Example: <lang mathematica>trits = {True, Maybe, False}; Print@Grid[
ArrayFlatten[{{Template:Not, Template:Null}, {List /@ trits,
List /@ Not /@ trits}}]];
Do[Print@Grid[
ArrayFlatten[{{Template:Operator, {{Null, Null,
Null}}}, {Template:Null, {trits}}, {List /@ trits,
Outer[operator, trits, trits]}}]], {operator, {And, Or, Nand,
Nor, Xor, Xnor, Implies, Equivalent}}]</lang>
Output:
Not True False Maybe Maybe False True And True Maybe False True True Maybe False Maybe Maybe Maybe False False False False False Or True Maybe False True True True True Maybe True Maybe Maybe False True Maybe False Nand True Maybe False True False Maybe True Maybe Maybe Maybe True False True True True Nor True Maybe False True False False False Maybe False Maybe Maybe False False Maybe True Xor True Maybe False True False Maybe True Maybe Maybe Maybe Maybe False True Maybe False Xnor True Maybe False True True Maybe False Maybe Maybe Maybe Maybe False False Maybe True Implies True Maybe False True True Maybe False Maybe True Maybe Maybe False True True True Equivalent True Maybe False True True Maybe False Maybe Maybe Maybe Maybe False False Maybe True
OCaml
<lang ocaml>type trit = True | False | Maybe
let t_not = function
| True -> False | False -> True | Maybe -> Maybe
let t_and a b = match (a,b) with
| (True,True) -> True | (False,_) | (_,False) -> False | _ -> Maybe
let t_or a b = t_not (t_and (t_not a) (t_not b))
let t_eq a b = match (a,b) with
| (True,True) | (False,False) -> True | (False,True) | (True,False) -> False | _ -> Maybe
let t_imply a b = t_or (t_not a) b
let string_of_trit = function
| True -> "True" | False -> "False" | Maybe -> "Maybe"
let () =
let values = [| True; Maybe; False |] in
let f = string_of_trit in
Array.iter (fun v -> Printf.printf "Not %s: %s\n" (f v) (f (t_not v))) values;
print_newline ();
let print op str =
Array.iter (fun a ->
Array.iter (fun b ->
Printf.printf "%s %s %s: %s\n" (f a) str (f b) (f (op a b))
) values
) values;
print_newline ()
in
print t_and "And";
print t_or "Or";
print t_imply "Then";
print t_eq "Equiv";
- </lang>
Output:
Not True: False Not Maybe: Maybe Not False: True True And True: True True And Maybe: Maybe True And False: False Maybe And True: Maybe Maybe And Maybe: Maybe Maybe And False: False False And True: False False And Maybe: False False And False: False True Or True: True True Or Maybe: True True Or False: True Maybe Or True: True Maybe Or Maybe: Maybe Maybe Or False: Maybe False Or True: True False Or Maybe: Maybe False Or False: False True Then True: True True Then Maybe: Maybe True Then False: False Maybe Then True: True Maybe Then Maybe: Maybe Maybe Then False: Maybe False Then True: True False Then Maybe: True False Then False: True True Equiv True: True True Equiv Maybe: Maybe True Equiv False: False Maybe Equiv True: Maybe Maybe Equiv Maybe: Maybe Maybe Equiv False: Maybe False Equiv True: False False Equiv Maybe: Maybe False Equiv False: True
Using a general binary -> ternary transform
Instead of writing all of the truth-tables by hand, we can construct a general binary -> ternary transform and apply it to any logical function we want: <lang OCaml>type trit = True | False | Maybe
let to_bin = function True -> [true] | False -> [false] | Maybe -> [true;false]
let eval f x =
List.fold_left (fun l c -> List.fold_left (fun m d -> ((d c) :: m)) l f) [] x
let rec from_bin =
function [true] -> True | [false] -> False
| h :: t -> (match (h, from_bin t) with
(true,True) -> True | (false,False) -> False | _ -> Maybe)
| _ -> Maybe
let to_ternary1 uop = fun x -> from_bin (eval [uop] (to_bin x)) let to_ternary2 bop = fun x y -> from_bin (eval (eval [bop] (to_bin x)) (to_bin y))
let t_not = to_ternary1 (not) let t_and = to_ternary2 (&&) let t_or = to_ternary2 (||) let t_equiv = to_ternary2 (=) let t_imply = to_ternary2 (fun p q -> (not p) || q)
let str = function True -> "True " | False -> "False" | Maybe -> "Maybe" let iterv f = List.iter f [True; False; Maybe]
let table1 s u =
print_endline ("\n"^s^":");
iterv (fun v -> print_endline (" "^(str v)^" -> "^(str (u v))));;
let table2 s b =
print_endline ("\n"^s^":");
iterv (fun u ->
iterv (fun v ->
print_endline (" "^(str u)^" "^(str v)^" -> "^(str (b u v)))));;
table1 "not" t_not;; table2 "and" t_and;; table2 "or" t_or;; table2 "equiv" t_equiv;; table2 "implies" t_imply;;</lang> Output:
not: True -> False False -> True Maybe -> Maybe and: True True -> True True False -> False True Maybe -> Maybe False True -> False False False -> False False Maybe -> False Maybe True -> Maybe Maybe False -> False Maybe Maybe -> Maybe or: True True -> True True False -> True True Maybe -> True False True -> True False False -> False False Maybe -> Maybe Maybe True -> True Maybe False -> Maybe Maybe Maybe -> Maybe equiv: True True -> True True False -> False True Maybe -> Maybe False True -> False False False -> True False Maybe -> Maybe Maybe True -> Maybe Maybe False -> Maybe Maybe Maybe -> Maybe implies: True True -> True True False -> False True Maybe -> Maybe False True -> True False False -> True False Maybe -> True Maybe True -> True Maybe False -> Maybe Maybe Maybe -> Maybe
ooRexx
<lang ooRexx> tritValues = .array~of(.trit~true, .trit~false, .trit~maybe) tab = '09'x
say "not operation (\)" loop a over tritValues
say "\"a":" (\a)
end
say say "and operation (&)" loop aa over tritValues
loop bb over tritValues
say (aa" & "bb":" (aa&bb))
end
end
say say "or operation (|)" loop aa over tritValues
loop bb over tritValues
say (aa" | "bb":" (aa|bb))
end
end
say say "implies operation (&&)" loop aa over tritValues
loop bb over tritValues
say (aa" && "bb":" (aa&&bb))
end
end
say say "equals operation (=)" loop aa over tritValues
loop bb over tritValues
say (aa" = "bb":" (aa=bb))
end
end
- class trit
-- making this a private method so we can control the creation -- of these. We only allow 3 instances to exist
- method new class private
forward class(super)
- method init class
expose true false maybe -- delayed creation true = .nil false = .nil maybe = .nil
-- read only attribute access to the instances. -- these methods create the appropriate singleton on the first call
- attribute true class get
expose true
if true == .nil then true = self~new("True")
return true
- attribute false class get
expose false
if false == .nil then false = self~new("False")
return false
- attribute maybe class get
expose maybe
if maybe == .nil then maybe = self~new("Maybe")
return maybe
-- create an instance
- method init
expose value use arg value
-- string method to return the value of the instance
- method string
expose value return value
-- "and" method using the operator overload
- method "&"
use strict arg other
if self == .trit~true then return other
else if self == .trit~maybe then do
if other == .trit~false then return .trit~false
else return .trit~maybe
end
else return .trit~false
-- "or" method using the operator overload
- method "|"
use strict arg other
if self == .trit~true then return .trit~true
else if self == .trit~maybe then do
if other == .trit~true then return .trit~true
else return .trit~maybe
end
else return other
-- implies method...using the XOR operator for this
- method "&&"
use strict arg other
if self == .trit~true then return other
else if self == .trit~maybe then do
if other == .trit~true then return .trit~true
else return .trit~maybe
end
else return .trit~true
-- "not" method using the operator overload
- method "\"
if self == .trit~true then return .trit~false else if self == .trit~maybe then return .trit~maybe else return .trit~true
-- "equals" using the "=" override. This makes a distinction between -- the "==" operator, which is real equality and the "=" operator, which -- is trinary equality.
- method "="
use strict arg other if self == .trit~true then return other else if self == .trit~maybe then return .trit~maybe else return \other
</lang>
not operation (\) \True: False \False: True \Maybe: Maybe and operation (&) True & True: True True & False: False True & Maybe: Maybe False & True: False False & False: False False & Maybe: False Maybe & True: Maybe Maybe & False: False Maybe & Maybe: Maybe or operation (|) True | True: True True | False: True True | Maybe: True False | True: True False | False: False False | Maybe: Maybe Maybe | True: True Maybe | False: Maybe Maybe | Maybe: Maybe implies operation (&&) True && True: True True && False: False True && Maybe: Maybe False && True: True False && False: True False && Maybe: True Maybe && True: True Maybe && False: Maybe Maybe && Maybe: Maybe equals operation (=) True = True: True True = False: False True = Maybe: Maybe False = True: False False = False: True False = Maybe: Maybe Maybe = True: Maybe Maybe = False: Maybe Maybe = Maybe: Maybe
Pascal
<lang pascal>Program TernaryLogic (output);
type
trit = (terTrue, terMayBe, terFalse);
function terNot (a: trit): trit;
begin
case a of
terTrue: terNot := terFalse;
terMayBe: terNot := terMayBe;
terFalse: terNot := terTrue;
end;
end;
function terAnd (a, b: trit): trit;
begin
terAnd := terMayBe;
if (a = terFalse) or (b = terFalse) then
terAnd := terFalse
else
if (a = terTrue) and (b = terTrue) then
terAnd := terTrue;
end;
function terOr (a, b: trit): trit;
begin
terOr := terMayBe;
if (a = terTrue) or (b = terTrue) then
terOr := terTrue
else
if (a = terFalse) and (b = terFalse) then
terOr := terFalse;
end;
function terEquals (a, b: trit): trit;
begin
if a = b then
terEquals := terTrue
else
if a <> b then
terEquals := terFalse;
if (a = terMayBe) or (b = terMayBe) then
terEquals := terMayBe;
end;
function terIfThen (a, b: trit): trit;
begin
terIfThen := terMayBe;
if (a = terTrue) or (b = terFalse) then
terIfThen := terTrue
else
if (a = terFalse) and (b = terTrue) then
terIfThen := terFalse;
end;
function terToStr(a: trit): string;
begin
case a of
terTrue: terToStr := 'True ';
terMayBe: terToStr := 'Maybe';
terFalse: terToStr := 'False';
end;
end;
begin
writeln('Ternary logic test:');
writeln;
writeln('NOT ', ' True ', ' Maybe', ' False');
writeln(' ', terToStr(terNot(terTrue)), ' ', terToStr(terNot(terMayBe)), ' ', terToStr(terNot(terFalse)));
writeln;
writeln('AND ', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terAnd(terTrue,terTrue)), ' ', terToStr(terAnd(terMayBe,terTrue)), ' ', terToStr(terAnd(terFalse,terTrue)));
writeln('Maybe ', terToStr(terAnd(terTrue,terMayBe)), ' ', terToStr(terAnd(terMayBe,terMayBe)), ' ', terToStr(terAnd(terFalse,terMayBe)));
writeln('False ', terToStr(terAnd(terTrue,terFalse)), ' ', terToStr(terAnd(terMayBe,terFalse)), ' ', terToStr(terAnd(terFalse,terFalse)));
writeln;
writeln('OR ', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terOR(terTrue,terTrue)), ' ', terToStr(terOR(terMayBe,terTrue)), ' ', terToStr(terOR(terFalse,terTrue)));
writeln('Maybe ', terToStr(terOR(terTrue,terMayBe)), ' ', terToStr(terOR(terMayBe,terMayBe)), ' ', terToStr(terOR(terFalse,terMayBe)));
writeln('False ', terToStr(terOR(terTrue,terFalse)), ' ', terToStr(terOR(terMayBe,terFalse)), ' ', terToStr(terOR(terFalse,terFalse)));
writeln;
writeln('IFTHEN', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terIfThen(terTrue,terTrue)), ' ', terToStr(terIfThen(terMayBe,terTrue)), ' ', terToStr(terIfThen(terFalse,terTrue)));
writeln('Maybe ', terToStr(terIfThen(terTrue,terMayBe)), ' ', terToStr(terIfThen(terMayBe,terMayBe)), ' ', terToStr(terIfThen(terFalse,terMayBe)));
writeln('False ', terToStr(terIfThen(terTrue,terFalse)), ' ', terToStr(terIfThen(terMayBe,terFalse)), ' ', terToStr(terIfThen(terFalse,terFalse)));
writeln;
writeln('EQUAL ', ' True ', ' Maybe', ' False');
writeln('True ', terToStr(terEquals(terTrue,terTrue)), ' ', terToStr(terEquals(terMayBe,terTrue)), ' ', terToStr(terEquals(terFalse,terTrue)));
writeln('Maybe ', terToStr(terEquals(terTrue,terMayBe)), ' ', terToStr(terEquals(terMayBe,terMayBe)), ' ', terToStr(terEquals(terFalse,terMayBe)));
writeln('False ', terToStr(terEquals(terTrue,terFalse)), ' ', terToStr(terEquals(terMayBe,terFalse)), ' ', terToStr(terEquals(terFalse,terFalse)));
writeln;
end.</lang> Output:
:> ./TernaryLogic
Ternary logic test:
NOT True Maybe False
False Maybe True
AND True Maybe False
True True Maybe False
Maybe Maybe Maybe False
False False False False
OR True Maybe False
True True True True
Maybe True Maybe Maybe
False True Maybe False
IFTHEN True Maybe False
True True Maybe False
Maybe True Maybe Maybe
False True True True
EQUAL True Maybe False
True True Maybe False
Maybe Maybe Maybe Maybe
False False Maybe True
Perl
File Trit.pm: <lang perl>package Trit;
- -1 = false ; 0 = maybe ; 1 = true
use Exporter 'import';
our @EXPORT_OK = qw(TRUE FALSE MAYBE is_true is_false is_maybe); our %EXPORT_TAGS = ( all => \@EXPORT_OK, const => [qw(TRUE FALSE MAYBE)], bool => [qw(is_true is_false is_maybe)], );
use List::Util qw(min max);
use overload '=' => sub { $_[0]->clone() }, '<=>'=> sub { $_[0]->cmp($_[1]) }, 'cmp'=> sub { $_[0]->cmp($_[1]) }, '==' => sub { ${$_[0]} == ${$_[1]} }, 'eq' => sub { $_[0]->equiv($_[1]) }, '>' => sub { ${$_[0]} > ${$_[1]} }, '<' => sub { ${$_[0]} < ${$_[1]} }, '>=' => sub { ${$_[0]} >= ${$_[1]} }, '<=' => sub { ${$_[0]} <= ${$_[1]} }, '|' => sub { $_[0]->or($_[1]) }, '&' => sub { $_[0]->and($_[1]) }, '!' => sub { $_[0]->not() }, '~' => sub { $_[0]->not() }, '""' => sub { $_[0]->tostr() }, '0+' => sub { $_[0]->tonum() },
sub new { my ($class, $v) = @_; my $ret = !defined($v) ? 0 : $v eq 'true' ? 1 : $v eq 'false'? -1 : $v eq 'maybe'? 0 : $v > 0 ? 1 : $v < 0 ? -1 : 0; return bless \$ret, $class; }
sub TRUE() { new Trit( 1) } sub FALSE() { new Trit(-1) } sub MAYBE() { new Trit( 0) }
sub clone { my $ret = ${$_[0]}; return bless \$ret, ref($_[0]); }
sub tostr { ${$_[0]} > 0 ? "true" : ${$_[0]} < 0 ? "false" : "maybe" } sub tonum { ${$_[0]} }
sub is_true { ${$_[0]} > 0 } sub is_false { ${$_[0]} < 0 } sub is_maybe { ${$_[0]} == 0 }
sub cmp { ${$_[0]} <=> ${$_[1]} } sub not { new Trit(-${$_[0]}) } sub and { new Trit(min(${$_[0]}, ${$_[1]}) ) } sub or { new Trit(max(${$_[0]}, ${$_[1]}) ) }
sub equiv { new Trit( ${$_[0]} * ${$_[1]} ) }</lang> File test.pl: <lang perl>use Trit ':all';
my @a = (TRUE(), MAYBE(), FALSE());
print "\na\tNOT a\n"; print "$_\t".(!$_)."\n" for @a; # Example of use of prefix operator NOT. Tilde ~ also can be used.
print "\nAND\t".join("\t",@a)."\n";
for my $a (@a) {
print $a;
for my $b (@a) {
print "\t".($a & $b); # Example of use of infix & (and)
}
print "\n";
}
print "\nOR\t".join("\t",@a)."\n"; for my $a (@a) { print $a; for my $b (@a) { print "\t".($a | $b); # Example of use of infix | (or) } print "\n"; }
print "\nEQV\t".join("\t",@a)."\n"; for my $a (@a) { print $a; for my $b (@a) { print "\t".($a eq $b); # Example of use of infix eq (equivalence) } print "\n"; }
print "\n==\t".join("\t",@a)."\n"; for my $a (@a) { print $a; for my $b (@a) { print "\t".($a == $b); # Example of use of infix == (equality) } print "\n"; }</lang> Output:
a NOT a true false maybe maybe false true AND true maybe false true true maybe false maybe maybe maybe false false false false false OR true maybe false true true true true maybe true maybe maybe false true maybe false EQV true maybe false true true maybe false maybe maybe maybe maybe false false maybe true == true maybe false true 1 maybe 1 false 1
Perl 6
Implementation: <lang perl6>enum Trit <Foo Moo Too>;
sub prefix:<¬> (Trit $a) { Trit(1-($a-1)) }
sub infix:<∧> is equiv(&infix:<*>) (Trit $a, Trit $b) { $a min $b } sub infix:<∨> is equiv(&infix:<+>) (Trit $a, Trit $b) { $a max $b }
sub infix:<→> is equiv(&infix:<..>) (Trit $a, Trit $b) { ¬$a max $b } sub infix:<≡> is equiv(&infix:<eq>) (Trit $a, Trit $b) { Trit(1 + ($a-1) * ($b-1)) }</lang> The precedence of each operator is specified as equivalent to an existing operator. We've taken the liberty of using an arrow for implication, to avoid confusing it with ⊃, (U+2283 SUPERSET OF).
To test, we use this code: <lang perl6>say '¬'; say "Too {¬Too}"; say "Moo {¬Moo}"; say "Foo {¬Foo}";
sub tbl (&op,$name) {
say ;
say "$name Too Moo Foo";
say " ╔═══════════";
say "Too║{op Too,Too} {op Too,Moo} {op Too,Foo}";
say "Moo║{op Moo,Too} {op Moo,Moo} {op Moo,Foo}";
say "Foo║{op Foo,Too} {op Foo,Moo} {op Foo,Foo}";
}
tbl(&infix:<∧>, '∧'); tbl(&infix:<∨>, '∨'); tbl(&infix:<→>, '→'); tbl(&infix:<≡>, '≡');
say ; say 'Precedence tests should all print "Too":'; say ~(
Foo ∧ Too ∨ Too ≡ Too, Foo ∧ (Too ∨ Too) ≡ Foo, Too ∨ Too ∧ Foo ≡ Too, (Too ∨ Too) ∧ Foo ≡ Foo,
¬Too ∧ Too ∨ Too ≡ Too, ¬Too ∧ (Too ∨ Too) ≡ ¬Too, Too ∨ Too ∧ ¬Too ≡ Too, (Too ∨ Too) ∧ ¬Too ≡ ¬Too, Foo ∧ Too ∨ Foo → Foo ≡ Too, Foo ∧ Too ∨ Too → Foo ≡ Foo,
);</lang> Output:
¬ Too Foo Moo Moo Foo Too ∧ Too Moo Foo ╔═══════════ Too║Too Moo Foo Moo║Moo Moo Foo Foo║Foo Foo Foo ∨ Too Moo Foo ╔═══════════ Too║Too Too Too Moo║Too Moo Moo Foo║Too Moo Foo → Too Moo Foo ╔═══════════ Too║Too Moo Foo Moo║Too Moo Moo Foo║Too Too Too ≡ Too Moo Foo ╔═══════════ Too║Too Moo Foo Moo║Moo Moo Moo Foo║Foo Moo Too Precedence tests should all print "Too": Too Too Too Too Too Too Too Too Too Too
PicoLisp
In addition for the standard T (for "true") and NIL (for "false") we define 0 (zero, for "maybe"). <lang PicoLisp>(de 3not (A)
(or (=0 A) (not A)) )
(de 3and (A B)
(cond
((=T A) B)
((=0 A) (and B 0)) ) )
(de 3or (A B)
(cond
((=T A) T)
((=0 A) (or (=T B) 0))
(T B) ) )
(de 3impl (A B)
(cond
((=T A) B)
((=0 A) (or (=T B) 0))
(T T) ) )
(de 3equiv (A B)
(cond
((=T A) B)
((=0 A) 0)
(T (3not B)) ) )</lang>
Test: <lang PicoLisp>(for X '(T 0 NIL)
(println 'not X '-> (3not X)) )
(for Fun '((and . 3and) (or . 3or) (implies . 3impl) (equivalent . 3equiv))
(for X '(T 0 NIL)
(for Y '(T 0 NIL)
(println X (car Fun) Y '-> ((cdr Fun) X Y)) ) ) )</lang>
Output:
not T -> NIL not 0 -> 0 not NIL -> T T and T -> T T and 0 -> 0 T and NIL -> NIL 0 and T -> 0 0 and 0 -> 0 0 and NIL -> NIL NIL and T -> NIL NIL and 0 -> NIL NIL and NIL -> NIL T or T -> T T or 0 -> T T or NIL -> T 0 or T -> T 0 or 0 -> 0 0 or NIL -> 0 NIL or T -> T NIL or 0 -> 0 NIL or NIL -> NIL T implies T -> T T implies 0 -> 0 T implies NIL -> NIL 0 implies T -> T 0 implies 0 -> 0 0 implies NIL -> 0 NIL implies T -> T NIL implies 0 -> T NIL implies NIL -> T T equivalent T -> T T equivalent 0 -> 0 T equivalent NIL -> NIL 0 equivalent T -> 0 0 equivalent 0 -> 0 0 equivalent NIL -> 0 NIL equivalent T -> NIL NIL equivalent 0 -> 0 NIL equivalent NIL -> T
Python
In Python, the keywords 'and', 'not', and 'or' are coerced to always work as boolean operators. I have therefore overloaded the boolean bitwise operators &, |, ^ to provide the required functionality. <lang python>class Trit(int):
def __new__(cls, value):
if value == 'TRUE':
value = 1
elif value == 'FALSE':
value = 0
elif value == 'MAYBE':
value = -1
return super(Trit, cls).__new__(cls, value // (abs(value) or 1))
def __repr__(self):
if self > 0:
return 'TRUE'
elif self == 0:
return 'FALSE'
return 'MAYBE'
def __str__(self):
return repr(self)
def __bool__(self):
if self > 0:
return True
elif self == 0:
return False
else:
raise ValueError("invalid literal for bool(): '%s'" % self)
def __or__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][1]
else:
try:
return _ttable[(self, Trit(bool(other)))][1]
except:
return NotImplemented
def __ror__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][1]
else:
try:
return _ttable[(self, Trit(bool(other)))][1]
except:
return NotImplemented
def __and__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][0]
else:
try:
return _ttable[(self, Trit(bool(other)))][0]
except:
return NotImplemented
def __rand__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][0]
else:
try:
return _ttable[(self, Trit(bool(other)))][0]
except:
return NotImplemented
def __xor__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][2]
else:
try:
return _ttable[(self, Trit(bool(other)))][2]
except:
return NotImplemented
def __rxor__(self, other):
if isinstance(other, Trit):
return _ttable[(self, other)][2]
else:
try:
return _ttable[(self, Trit(bool(other)))][2]
except:
return NotImplemented
def __invert__(self):
return _ttable[self]
def __getattr__(self, name):
if name in ('_n', 'flip'):
# So you can do x._n == x.flip; the inverse of x
# In Python 'not' is strictly boolean so we can't write `not x`
# Same applies to keywords 'and' and 'or'.
return _ttable[self]
else:
raise AttributeError
TRUE, FALSE, MAYBE = Trit(1), Trit(0), Trit(-1)
_ttable = {
# A: -> flip_A
TRUE: FALSE,
FALSE: TRUE,
MAYBE: MAYBE,
# (A, B): -> (A_and_B, A_or_B, A_xor_B)
(MAYBE, MAYBE): (MAYBE, MAYBE, MAYBE),
(MAYBE, FALSE): (FALSE, MAYBE, MAYBE),
(MAYBE, TRUE): (MAYBE, TRUE, MAYBE),
(FALSE, MAYBE): (FALSE, MAYBE, MAYBE),
(FALSE, FALSE): (FALSE, FALSE, FALSE),
(FALSE, TRUE): (FALSE, TRUE, TRUE),
( TRUE, MAYBE): (MAYBE, TRUE, MAYBE),
( TRUE, FALSE): (FALSE, TRUE, TRUE),
( TRUE, TRUE): ( TRUE, TRUE, FALSE),
}
values = ('FALSE', 'TRUE ', 'MAYBE')
print("\nTrit logical inverse, '~'") for a in values:
expr = '~%s' % a
print(' %s = %s' % (expr, eval(expr)))
for op, ophelp in (('&', 'and'), ('|', 'or'), ('^', 'exclusive-or')):
print("\nTrit logical %s, '%s'" % (ophelp, op))
for a in values:
for b in values:
expr = '%s %s %s' % (a, op, b)
print(' %s = %s' % (expr, eval(expr)))</lang>
- Output
Trit logical inverse, '~' ~FALSE = TRUE ~TRUE = FALSE ~MAYBE = MAYBE Trit logical and, '&' FALSE & FALSE = FALSE FALSE & TRUE = FALSE FALSE & MAYBE = FALSE TRUE & FALSE = FALSE TRUE & TRUE = TRUE TRUE & MAYBE = MAYBE MAYBE & FALSE = FALSE MAYBE & TRUE = MAYBE MAYBE & MAYBE = MAYBE Trit logical or, '|' FALSE | FALSE = FALSE FALSE | TRUE = TRUE FALSE | MAYBE = MAYBE TRUE | FALSE = TRUE TRUE | TRUE = TRUE TRUE | MAYBE = TRUE MAYBE | FALSE = MAYBE MAYBE | TRUE = TRUE MAYBE | MAYBE = MAYBE Trit logical exclusive-or, '^' FALSE ^ FALSE = FALSE FALSE ^ TRUE = TRUE FALSE ^ MAYBE = MAYBE TRUE ^ FALSE = TRUE TRUE ^ TRUE = FALSE TRUE ^ MAYBE = MAYBE MAYBE ^ FALSE = MAYBE MAYBE ^ TRUE = MAYBE MAYBE ^ MAYBE = MAYBE
- Extra doodling in the Python shell
>>> values = (TRUE, FALSE, MAYBE) >>> for a in values: for b in values: assert (a & ~b) | (b & ~a) == a ^ b >>>
Ruby
Ruby, like Smalltalk, has two boolean classes: TrueClass for true and FalseClass for false. We add a third class, MaybeClass for MAYBE, and define ternary logic for all three classes.
We keep !a, a & b and so on for binary logic. We add !a.trit, a.trit & b and so on for ternary logic. The code for !a.trit uses def !, which works with Ruby 1.9, but fails as a syntax error with Ruby 1.8.
<lang ruby># trit.rb - ternary logic
require 'singleton'
- MAYBE, the only instance of MaybeClass, enables a system of ternary
- logic using TrueClass#trit, MaybeClass#trit and FalseClass#trit.
- !a.trit # ternary not
- a.trit & b # ternary and
- a.trit | b # ternary or
- a.trit ^ b # ternary exclusive or
- a.trit == b # ternary equal
- Though +true+ and +false+ are internal Ruby values, +MAYBE+ is not.
- Programs may want to assign +maybe = MAYBE+ in scopes that use
- ternary logic. Then programs can use +true+, +maybe+ and +false+.
class MaybeClass
include Singleton
# maybe.to_s # => "maybe" def to_s; "maybe"; end
end
MAYBE = MaybeClass.instance
class TrueClass
TritMagic = Object.new class << TritMagic def index; 0; end def !; false; end def & other; other; end def | other; true; end def ^ other; [false, MAYBE, true][other.trit.index]; end def == other; other; end end
# Performs ternary logic. See MaybeClass. # !true.trit # => false # true.trit & obj # => obj # true.trit | obj # => true # true.trit ^ obj # => false, maybe or true # true.trit == obj # => obj def trit; TritMagic; end
end
class MaybeClass
TritMagic = Object.new class << TritMagic def index; 1; end def !; MAYBE; end def & other; [MAYBE, MAYBE, false][other.trit.index]; end def | other; [true, MAYBE, MAYBE][other.trit.index]; end def ^ other; MAYBE; end def == other; MAYBE; end end
# Performs ternary logic. See MaybeClass. # !maybe.trit # => maybe # maybe.trit & obj # => maybe or false # maybe.trit | obj # => true or maybe # maybe.trit ^ obj # => maybe # maybe.trit == obj # => maybe def trit; TritMagic; end
end
class FalseClass
TritMagic = Object.new class << TritMagic def index; 2; end def !; true; end def & other; false; end def | other; other; end def ^ other; other; end def == other; [false, MAYBE, true][other.trit.index]; end end
# Performs ternary logic. See MaybeClass. # !false.trit # => true # false.trit & obj # => false # false.trit | obj # => obj # false.trit ^ obj # => obj # false.trit == obj # => false, maybe or true def trit; TritMagic; end
end</lang>
This IRB session shows ternary not, and, or, equal.
<lang ruby>$ irb irb(main):001:0> require './trit' => true irb(main):002:0> maybe = MAYBE => maybe irb(main):003:0> !true.trit => false irb(main):004:0> !maybe.trit => maybe irb(main):005:0> maybe.trit & false => false irb(main):006:0> maybe.trit | true => true irb(main):007:0> false.trit == true => false irb(main):008:0> false.trit == maybe => maybe</lang>
This program shows all 9 outcomes from a.trit ^ b.
<lang ruby>require 'trit' maybe = MAYBE
[true, maybe, false].each do |a|
[true, maybe, false].each do |b| printf "%5s ^ %5s => %5s\n", a, b, a.trit ^ b end
end</lang>
$ ruby -I. trit-xor.rb true ^ true => false true ^ maybe => maybe true ^ false => true maybe ^ true => maybe maybe ^ maybe => maybe maybe ^ false => maybe false ^ true => true false ^ maybe => maybe false ^ false => false
Seed7
The type boolean does not define separate xor, implies and equiv operators. But there are replacements for them:
| Instead of | Use |
|---|---|
| p xor q | p <> q |
| p implies q | p <= q |
| p equiv q | p = q |
Since ternary logic needs xor, implies and equiv with a trit result they are introduced as the operators xor, -> and ==. The trit operators and and or are defined as short circuit operators. A short circuit operator evaluates the second parameter only when necessary. This is analogous to the boolean operators and and or, which use also short circuit evaluation.
<lang seed7>$ include "seed7_05.s7i";
const type: trit is new enum
False, Maybe, True end enum;
- Enum types define comparisons (=, <, >, <=, >=, <>) and
- the conversions ord and conv.
const func string: str (in trit: aTrit) is
return [] ("False", "Maybe", "True")[succ(ord(aTrit))];
enable_output(trit); # Allow writing trit values
const array trit: tritNot is [] (True, Maybe, False); const array array trit: tritAnd is [] (
[] (False, False, False), [] (False, Maybe, Maybe), [] (False, Maybe, True ));
const array array trit: tritOr is [] (
[] (False, Maybe, True ), [] (Maybe, Maybe, True ), [] (True, True, True ));
const array array trit: tritXor is [] (
[] (False, Maybe, True ), [] (Maybe, Maybe, Maybe), [] (True, Maybe, False));
const array array trit: tritImplies is [] (
[] (True, True, True ), [] (Maybe, Maybe, True ), [] (False, Maybe, True ));
const array array trit: tritEquiv is [] (
[] (True, Maybe, False), [] (Maybe, Maybe, Maybe), [] (False, Maybe, True ));
const func trit: not (in trit: aTrit) is
return tritNot[succ(ord(aTrit))];
const func trit: (in trit: aTrit1) and (in trit: aTrit2) is
return tritAnd[succ(ord(aTrit1))][succ(ord(aTrit2))];
const func trit: (in trit: aTrit1) and (ref func trit: aTrit2) is func
result
var trit: res is False;
begin
if aTrit1 = True then
res := aTrit2;
elsif aTrit1 = Maybe and aTrit2 <> False then
res := Maybe;
end if;
end func;
const func trit: (in trit: aTrit1) or (in trit: aTrit2) is
return tritOr[succ(ord(aTrit1))][succ(ord(aTrit2))];
const func trit: (in trit: aTrit1) or (ref func trit: aTrit2) is func
result
var trit: res is True;
begin
if aTrit1 = False then
res := aTrit2;
elsif aTrit1 = Maybe and aTrit2 <> True then
res := Maybe;
end if;
end func;
$ syntax expr: .().xor.() is -> 15; const func trit: (in trit: aTrit1) xor (in trit: aTrit2) is
return tritImplies[succ(ord(aTrit1))][succ(ord(aTrit2))];
const func trit: (in trit: aTrit1) -> (in trit: aTrit2) is
return tritImplies[succ(ord(aTrit1))][succ(ord(aTrit2))];
const func trit: (in trit: aTrit1) == (in trit: aTrit2) is
return tritEquiv[succ(ord(aTrit1))][succ(ord(aTrit2))];
const func trit: rand (in trit: low, in trit: high) is
return trit conv (rand(ord(low), ord(high)));
const func integer: compare (in trit: aTrit1, in trit: aTrit2) is
return compare(ord(aTrit1), ord(aTrit2));
const func integer: hashCode (in trit: aTrit) is
return hashCode(ord(aTrit));
- Begin of test code
var trit: operand1 is False; var trit: operand2 is False;
const proc: writeTable (ref func trit: tritExpr, in string: name) is func
begin
writeln;
writeln(" " <& name rpad 7 <& " | False Maybe True");
writeln("---------+---------------------");
for operand1 range False to True do
write(" " <& operand1 rpad 7 <& " | ");
for operand2 range False to True do
write(tritExpr rpad 7);
end for;
writeln;
end for;
end func;
const proc: main is func
begin
writeln(" not" rpad 8 <& " | False Maybe True");
writeln("---------+---------------------");
write(" | ");
for operand1 range False to True do
write(not operand1 rpad 7);
end for;
writeln;
writeTable(operand1 and operand2, "and");
writeTable(operand1 or operand2, "or");
writeTable(operand1 xor operand2, "xor");
writeTable(operand1 -> operand2, "->");
writeTable(operand1 == operand2, "==");
end func;</lang>
Output:
not | False Maybe True
---------+---------------------
| True Maybe False
and | False Maybe True
---------+---------------------
False | False False False
Maybe | False Maybe Maybe
True | False Maybe True
or | False Maybe True
---------+---------------------
False | False Maybe True
Maybe | Maybe Maybe True
True | True True True
xor | False Maybe True
---------+---------------------
False | True True True
Maybe | Maybe Maybe True
True | False Maybe True
-> | False Maybe True
---------+---------------------
False | True True True
Maybe | Maybe Maybe True
True | False Maybe True
== | False Maybe True
---------+---------------------
False | True Maybe False
Maybe | Maybe Maybe Maybe
True | False Maybe True
Tcl
The simplest way of doing this is by constructing the operations as truth tables. The code below uses an abbreviated form of truth table. <lang tcl>package require Tcl 8.5 namespace eval ternary {
# Code generator
proc maketable {name count values} {
set sep "" for {set i 0; set c 97} {$i<$count} {incr i;incr c} { set v [format "%c" $c] lappend args $v; append key $sep "$" $v set sep "," } foreach row [split $values \n] { if {[llength $row]>1} { lassign $row from to lappend table $from [list return $to] } } proc $name $args \ [list ckargs $args]\;[concat [list switch -glob --] $key [list $table]] namespace export $name
}
# Helper command to check argument syntax
proc ckargs argList {
foreach var $argList { upvar 1 $var v switch -exact -- $v { true - maybe - false { continue } default { return -level 2 -code error "bad ternary value \"$v\"" } } }
}
# The "truth" tables; “*” means “anything”
maketable not 1 {
true false maybe maybe false true
}
maketable and 2 {
true,true true false,* false *,false false * maybe
}
maketable or 2 {
true,* true *,true true false,false false * maybe
}
maketable implies 2 {
false,* true *,true true true,false false * maybe
}
maketable equiv 2 {
*,maybe maybe maybe,* maybe true,true true false,false true * false
}
}</lang> Demonstrating: <lang tcl>namespace import ternary::* puts "x /\\ y == x \\/ y" puts " x | y || result" puts "-------+-------++--------" foreach x {true maybe false} {
foreach y {true maybe false} {
set z [equiv [and $x $y] [or $x $y]] puts [format " %-5s | %-5s || %-5s" $x $y $z]
}
}</lang> Output:
x /\ y == x \/ y x | y || result -------+-------++-------- true | true || true true | maybe || maybe true | false || false maybe | true || maybe maybe | maybe || maybe maybe | false || maybe false | true || false false | maybe || maybe false | false || true