Logická funkce

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Logická funkce je funkce, která pro konečný počet vstupních parametrů vrací logické hodnoty.

Používá se v matematické logice, v oboru teorie řízení a číslicové techniky, v praxi pak například v mikroprocesorové technice. Parametry logické funkce jsou logické proměnné.

Přiřazuje-li logická funkce výstupní hodnoty všem kombinacím vstupních logických proměnných, pak se nazývá úplně zadaná logická funkce; v opačném případě se nazývá neúplně zadaná logická funkce. Kombinace vstupních logických proměnných, k níž není určena hodnota výstupní logické funkce, se nazývá neurčitý stav.

Pro n logických proměnných lze definovat 22n logických funkcí. Pro n logických proměnných obsahuje tabulka 2n řádků.

Počet proměnných Počet funkcí
1 4
2 16
3 256
atd.

Funkce jedné proměnné

A f0 f1 f2 f3
0 0 0 1 1
1 0 1 0 1

f0 = 0 konstanta
f1 = x přímá proměnná
f2 = ¬x negovaná proměnná
f3 = 1 konstanta
¬ = negace
Funkce dvou proměnných

B A f0 f1 f2 f3 f4 f5 f6 f7 osa f8 f9 f10 f11 f12 f13 f14 f15
0 0 0 0 0 0 0 0 0 0 " 1 1 1 1 1 1 1 1
0 1 0 0 0 0 1 1 1 1 * 0 0 0 0 1 1 1 1
1 0 0 0 1 1 0 0 1 1 " 0 0 1 1 0 0 1 1
1 1 0 1 0 1 0 1 0 1 * 0 1 0 1 0 1 0 1

osa = osa negace Za touto osou se nacházejí tytéž funkce, ale v znegovaném tvaru.

f0 = 0 konstanta
f1 = A*B (logický součin, AND)
f2 = ¬(A implikuje B)
f3 = A přímá proměnná
f4 = ¬(B implikuje A)
f5 = B přímá proměnná
f6 = ¬A*B+A*¬B nonekvivalence
f7 = A+B (logický součet, OR)

f13 = implikace
¬ = negace

Úplný systém logických funkcí

pomocí daných operátorů lze realizovat tři logické funkce AND, NOT (NON)a OR

Úplný systém logických funkcí

AND a NOT
chybí OR
použijeme De Morganův zákon

OR a NOT
chybí AND
použijeme De Morganův zákon

NAND
chybí NOT, OR, AND
NOT :
AND :
OR :

NOR
chybí NOT, OR, AND
NOT :
AND :
OR :

Zápis logických funkcí

  1. pravdivostní tabulkou
  2. logickou rovnicí
  3. mapou (Karnaughova)

Pravdivostní tabulka

Je-li n počet vstupních logických proměnných a m počet výstupních funkcí, bude mít pravdivostní tabulka 2n řádků a n + m sloupců. V prvních n sloupcích budou zobrazeny všechny možné kombinace vstupních logických hodnot. Pravdivostní tabulka pro funkci disjunkce (A or B) vypadá takto:

A B Y = A or B
0 0 0
0 1 1
1 0 1
1 1 1

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