Ternary logic
You are encouraged to solve this task according to the task description, using any language you may know.
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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.
Task:
- Define a new type that emulates Ternary logic by storing data trits.
- Given all the binary 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.
ALGOL 68
File: Ternary_logic.a68 <lang algol68># -*- coding: utf-8 -*- #
MODE TRIT = STRUCT(BITS trit); INT trit width = 1, trit base = 3; FORMAT trit fmt = $c("⌊","⌈","?" #|"~"#)$;
- 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 #⌊#;
TRIT flip=true, flop=false, flap=maybe;
OP REPR = (TRIT t)STRING:
IF t = false THEN "⌊"
ELIF t = maybe THEN "?"
ELIF t = true THEN "⌈"
ELSE raise value error(("invalid TRIT value",INITINT t));~
FI;
- Define some OPerators for coercing MODES #
OP INITTRIT = (BOOL in)TRIT:
(in|true|false);
OP B = (TRIT in)BOOL:
(in=true|TRUE|:in=false|FALSE|
raise value error(("invalid TRIT to BOOL coercion: """, REPR 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:
IF t=true THEN 1
ELIF t=maybe THEN 0
ELIF t=false THEN -1
ELSE raise value error(("invalid TRIT value",REPR t));~
FI;
OP INITTRIT = (INT in)TRIT: (
TRIT out;
trit OF out:= trit OF
IF in= 1 THEN true
ELIF in= 0 THEN maybe
ELIF in=-1 THEN false
ELSE raise value error(("invalid TRIT value",in));~
FI;
out
);
OP INITTRIT = (BITS b)TRIT:
(TRIT out; trit OF out:=b; out);
- Define the OPerators for the TRIT MODE #
- These can be optimised by peekng at the binary value #
- These operators are as per "Balanced ternary" #
- Warning: "both" is ignored as it isn't Ternary #
OP LT = (TRIT a,b)BOOL: a EQ false AND b NE false OR a EQ maybe AND b EQ true,
LE = (TRIT a,b)BOOL: a EQ b OR a LT b, EQ = (TRIT a,b)BOOL: trit OF a = trit OF b, NE = (TRIT a,b)BOOL: NOT (a EQ b), GE = (TRIT a,b)BOOL: NOT (a LT b), GT = (TRIT a,b)BOOL: NOT (a LE b);
- A solo, unique and rather confusing CMP OPerator #
PRIO CMP = 5; OP CMP = (TRIT a,b)TRIT:
IF a < b THEN false ELIF a = b THEN maybe ELIF a > b THEN true FI;
- ASCII OPerators #
OP < = (TRIT a,b)BOOL: a LT b,
<= = (TRIT a,b)BOOL: a LE b, = = (TRIT a,b)BOOL: a EQ b, /= = (TRIT a,b)BOOL: a NE b, >= = (TRIT a,b)BOOL: a GE b, > = (TRIT a,b)BOOL: a GT b;
- Non ASCII OPerators
OP ≤ = (TRIT a,b)BOOL: a LE b,
≠ = (TRIT a,b)BOOL: a NE b, ≥ = (TRIT a,b)BOOL: a GE b;
OP - = (TRIT t)TRIT:
IF t=maybe THEN maybe
ELIF t=true THEN false
ELIF t=false THEN true
ELSE raise value error(("invalid TRIT value",REPR t)); ~
FI;
- Warning: This routine ASSIGNS "out" AND returns "carry" #
OP +:= = (REF TRIT out, TRIT arg)TRIT:
IF out = maybe THEN out := arg; maybe ELIF arg = maybe THEN # out:= out# arg ELIF out = arg THEN out := -out; arg ELIF out = -arg THEN out:=maybe; maybe ELSE raise value error((REPR out," + ",REPR arg)); ~ FI;
OP + = (TRIT a, b)TRIT:
(TRIT out:=a; VOID(out+:=b); out);
OP - = (TRIT a, b)TRIT:
a + -b;
OP * = (TRIT a, b)TRIT:
IF a = maybe OR b = maybe THEN maybe ELIF a = b THEN true ELSE false FI;
OP ODD = (TRIT t)BOOL:
t /= maybe;
COMMENT
Kleene logic truth tables:
END COMMENT
OP AND = (TRIT a,b)TRIT: (
[,]TRIT( # ∧ maybe, true, false, # #maybe# (maybe, maybe, false), #true# (maybe, true, false), #false# (false, false, false) )[@0,@0][ABS trit OF a, ABS trit OF b]
);
OP OR = (TRIT a,b)TRIT: (
[,]TRIT( # ∨ maybe, true, false, # #maybe# (maybe, true, maybe), #true# (true, true, true), #false# (maybe, true, false) )[@0,@0][ABS trit OF a, ABS trit OF b]
);
PRIO IMPLIES = 1; # 1.9 # OP IMPLIES = (TRIT a,b)TRIT: (
[,]TRIT( # ⊃ maybe, true, false, # #maybe# (maybe, true, maybe), #true# (maybe, true, false), #false# (true, true, true) )[@0,@0][ABS trit OF a, ABS trit OF b]
);
PRIO EQV = 1; # 1.8 # OP EQV = (TRIT a,b)TRIT: (
[,]TRIT( # ≡ maybe, true, false, # #maybe# (maybe, maybe, maybe), #true# (maybe, true, false), #false# (maybe, false, true) )[@0,@0][ABS trit OF a, ABS trit OF b]
);
- Non ASCII OPerators
OP ¬ = (TRIT a)TRIT: NOT b,
∨ = (TRIT a,b)TRIT: a OR b, ∧ = (TRIT a,b)TRIT: a AND b, & = (TRIT a,b)TRIT: a AND b, ⊃ = (TRIT a,b)TRIT: a IMPLIES b, ≡ = (TRIT a,b)TRIT: a EQV b;
- </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 = $" "g" "$; FORMAT row fmt = $l3(f(col fmt)"|")f(col fmt)$; FORMAT row sep fmt = $l3("---+")"---"l$;
PROC row sep = VOID:
printf(row sep fmt);
PROC title = (UTF op)VOID:(
print(("Operator: ",op));
printf((row fmt," ",REPR false, REPR maybe, REPR true));
row sep
);
PROC print bool op table = (STRING op name, PROC(TRIT,TRIT)BOOL op)VOID: (
title(op name);
FOR i FROM LWB trits TO UPB trits DO
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, op(ti,tj)))
OD;
row sep
OD;
print(new line)
);
PROC print trit op table = (STRING op name, PROC(TRIT,TRIT)TRIT op)VOID: (
title(op name);
FOR i FROM LWB trits TO UPB trits DO
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;
row sep
OD;
print(new line)
);
printf((
$"Comparitive table of coercions:"l$, $" TRIT BOOL INT"l$
));
FOR it FROM LWB trits TO UPB trits DO
TRIT t = trits[it]; IF t = maybe THEN printf(($" "g" "$, REPR t, " ", INITINT t, $l$)) ELSE printf(($" "g" "$, REPR t, B t, INITINT t, $l$)) FI
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($l"Kleene logic truth table samples:"l$);
print trit op table("CMP", (TRIT a,b)TRIT: a CMP b); 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) CO; print trit op table("+", (TRIT a,b)TRIT: a + b); print trit op table("-", (TRIT a,b)TRIT: a - b); print trit op table("*", (TRIT a,b)TRIT: a * b); print bool op table("EQ", (TRIT a,b)BOOL: a EQ b); print bool op table("<=", (TRIT a,b)BOOL: a <= b) END CO</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: CMP | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ? | ⌊ | ⌊ ---+---+---+--- ? | ⌈ | ? | ⌊ ---+---+---+--- ⌈ | ⌈ | ⌈ | ? ---+---+---+--- Operator: EQV | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌈ | ? | ⌊ ---+---+---+--- ? | ? | ? | ? ---+---+---+--- ⌈ | ⌊ | ? | ⌈ ---+---+---+--- Operator: IMPLIES | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌈ | ⌈ | ⌈ ---+---+---+--- ? | ? | ? | ⌈ ---+---+---+--- ⌈ | ⌊ | ? | ⌈ ---+---+---+--- Operator: AND | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌊ | ⌊ | ⌊ ---+---+---+--- ? | ⌊ | ? | ? ---+---+---+--- ⌈ | ⌊ | ? | ⌈ ---+---+---+--- Operator: OR | ⌊ | ? | ⌈ ---+---+---+--- ⌊ | ⌊ | ? | ⌈ ---+---+---+--- ? | ? | ? | ⌈ ---+---+---+--- ⌈ | ⌈ | ⌈ | ⌈ ---+---+---+---