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Addierer

Die Aufgabe eines Addierers ist es zwei binäre ganze Zahlen zusammen zu zählen. Um diese Aufgabe zu erfüllen können Volladdierer hintereinander geschalten werden, wie in einem Paralleladdierer oder die einzelnen Stellen nacheinander addiert werden, wie in einem seriellen Addierer. Diese Schaltungen werden hier näher erklärt.
In Abbildung 1 ist eine Beispieladdition der Zahlen 6(=0110b) und 3(=0011b) dargestellt.
Addition zweier Binärzahlen
Abb.1: Addition zweier Binärzahlen
(Übertrag in rot)

Paralleladdierer

Ein Paralleladdierer besteht aus einem Halb- und mehreren Volladdierern. Das Ergebnis einer Stelle besteht aus der Summe der beiden Eingangsbits (an,bn) und dem Übertrag der vorherigen Stelle.
Paralleladdierer
Abb.1: Paralleladdierer
Somit arbeitet ein Paralleladdierer genau wie bei der schriftlichen Addition. Er addiert die letzte Stelle zusammen, gibt die Summe aus und den Übertrag an die vorletzte Stelle weiter. Hier addiert der Volladierer die zwei Bits und den erhaltenen Übertrag zusammen. Seinen Übertrag gibt er an die Stelle vor sich weiter. Dies wird bis zur vordersten (höchstwertigsten) Stelle fortgesetzt.

Nachteile des Paralleladdierers

Angenommen ein Volladdierer hat eine Druchlaufzeit von 20ns, dann ist die Summe und das Carry 20ns nach anlegen des Eingangssignals stabil. Bevor diese 20ns abgelaufen sind haben Summe und Carry den Wert 0.
Die Durchlaufzeit für einen Paralleladdierer ist 20ns mal die Anzahl der Bits (=Anzahl der Addierer). Abbildung 2 verdeutlicht dies.
Paralleladdierer
Abb.2: Paralleladdierer
Mit Hilfe eines 4-Bit-Paralleladdieres werden die Zahlen 3(=0011b) und 5(=0101b) zusammengezählt. Zu Beginn (nach 0ns) sind alle Ausgänge auf 0. Nach 20ns hat jeder Addierer das Ergebnis seiner Eingänge berechnet. Allerdings hat er zu diesem Zeitpunkt noch nicht die richtigen Werte der Überträge der vorhergehenden Stelle. Somit ist nur die hinterste Stelle korrekt berechnet (stabil).
Nach 40ns ist auch die vorletzte Stelle stabil, da das Carry der letzten Stelle bereits 20ns lang anliegt.
Die folgende Rechnung zeigt diese Addition, wobei stabile Werte schwarz und nicht stabile Werte rot dargestellt sind. Die Einganssignale sind blau und grün.
zeitlicher Ablauf der Addition
Zeitlicher Ablauf bei der Addtionon von 0011 und 0101
Abb.3: Zeitlicher Ablauf bei der Addtionon von 0011 und 0101
(schwarz=stabile Werte, rot=nicht stabile Werte)
Bei einem 4-Bit-Paralleladdierer werden somit 80ns benötigt, bis das Ergebnis vorliegt. Bei einem 64-Bit Addierer wären es schon 64*20ns = 1280ns =1,28ms und bei einem 128-Bit Addierer wären es 2560ns = 2,56ms. In der Digitaltechnik ist das lange.

Serielladdierer

Bei einer seriellen Addition werden die einzelnen Stellen der zwei Binärzahlen nacheinander addiert. Begonnen wird mit der hintersten Stelle. Hierzu befinden sich die zu addierenden Werte, sowie das Ergebnis in Schieberegistern. Der Volladdierer ist mit den letzten Stellen der Schieberegister verbunden und addiert diese. Zusätzlich ist der Übertrag der vorhergehenden Addition in einem D-Flip-Flop gespeichert, welches ebenfalls mit dem Volladdierer verbunden ist.
Bei jeder Taktflanke übernimmt das Flip-Flop den neuen Übertrag und es werden die drei Schieberegister eins nach rechts geschoben.
Die Abbildungen 4 bis 8 zeigen die serielle Addition der Werte 5(=0101b) und 3(=0011b).

Zu Beginn sind die zwei Werte 5 und 3 in den linken Schieberegistern, das Ergebnis auf der rechten Seite ist noch leer und im Flip-Flop ist 0 gespeichert.
Serielladdierer
Abb.4: Serielladdierer zu Beginn
Da am Volladdierer die Werte 1, 1 und 0 anliegen ergibt sich die Summe 0 und der Übertrag 1, welcher mit der nächsten Taktflanke in das rechte Schieberegister und das Flip-Flop übernommen werden. In Abbildung 5 ist der Zustand des Addierers nach der ersten Flanke abgebildet.
Serielladdierer
Abb.5: Serielladdierer nach erster Taktflanke
Nun liegt am Volladdierer 0, 1 und 1 an, womit eine 0 in das Ergebnis geschoben wird und eine 1 in das Flip-Flop.
Serielladdierer
Abb.6: Serielladdierer nach zweiter Taktflanke
Nun liegt am Volladdierer 1, 0 und 1 an, womit eine 0 in das Ergebnis geschoben wird und eine 1 in das Flip-Flop.
Serielladdierer
Abb.7: Serielladdierer nach dritter Taktflanke
Vor der letzten Taktflanke liegt 0, 0 und 1 am Volladdierer an und somit ergibt sich als Summe der Wert 1 und als Übertrag eine 0. Da nach 4 Taktflanken alle Bits addiert wurden ist das Ergebnis stabil mit dem Wert 8 (=1000b).
Serielladdierer
Abb.8: Serielladdierer nach vierter Taktflanke
Unter der Annahme, dass der Volladdierer 20ns zur Berechnung benötigt, ist auch beim seriellen Addierer das Ergebnis nach 80ns stabil. Er kommt somit bei gleicher Berechnungszeit mit deutlich weniger Schaltelementen aus (nur ein Volladdierer und ein Flip-Flop).

Carry-Lookahead

Das Problem beim Parallel- und Serielladdierer ist, dass die vorderste Stelle erst stabil ist nachdem das Carry stabil ist. Dies ist jedoch erst der Fall, wenn alle vorhergehenden Stellen berechnet wurden. Um die Laufzeit eines Addieres zu verbessern, werden sogenannte Carry-Lookahead-Schaltungen verwendet. Hierbei handelt es sich um Schaltnetze, welche den Carry einer Stelle direkt aus den eingehenden Zahlen berechnen und nicht aus dem vorherigen Volladdierer übernehmen.
Dieses Schaltnetz wird vor den Addierer geschaltet, wie in Abbildung 9 dargestellt.
Paralleladdierer mit Carry-Lookahead-Logik
Abb.9: Paralleladdierer mit Carry-Lookahead-Logik
Eine weitere Alternative ist, den kompletten Addierer als Schaltnetz zu entwerfen und nicht aus einzelnen Komponenten aufzubauen. Da sich jede DNF mit drei Schichten aufbauen lässt (Negationsschicht, Und-Schicht und Oder-Schicht) ist ein solcher Addierer in 30ns von einem Signal durchlaufen.
Ein solches Vorgehen erzeugt jedoch weit aus längere Terme für jedes Ergebnis-Bit als es bei einer Carry-Lookahead-Logik der Fall ist. Für einen 4-Bit-Addierer sind hier die minimierten Formeln für die Ergebnisbits (e3..e0) für eine Addition von a (a3..a0) und b (b3..b0) angegeben. Bei größeren Zahlen (64 Bit, 128 Bit, ...) wachsen die Formeln noch stärker an.
e0 = (!b0&a0)#(b0&!a0)
e1 = (!b1&!a1&b0&a0)#(!b1&a1&!b0)#(!b1&a1&!a0)#(b1&!a1&!b0)#(b1&!a1&!a0)#(b1&a1&b0&a0)
e2 = (!b2&!a2&b1&a1)#(!b2&!a2&b1&b0&a0)#(!b2&!a2&a1&b0&a0)#(!b2&a2&!b1&!a1)#(!b2&a2&!b1&!b0)#(!b2&a2&!b1&!a0)#(!b2&a2&!a1&!b0)#(!b2&a2&!a1&!a0)#(b2&!a2&!b1&!a1)#(b2&!a2&!b1&!b0)#(b2&!a2&!b1&!a0)#(b2&!a2&!a1&!b0)#(b2&!a2&!a1&!a0)#(b2&a2&b1&a1)#(b2&a2&b1&b0&a0)#(b2&a2&a1&b0&a0)
e3 = (!b3&!a3&b2&a2)#(!b3&!a3&b2&b1&a1)#(!b3&!a3&b2&b1&b0&a0)#(!b3&!a3&b2&a1&b0&a0)#(!b3&!a3&a2&b1&a1)#(!b3&!a3&a2&b1&b0&a0)#(!b3&!a3&a2&a1&b0&a0)#(!b3&a3&!b2&!a2)#(!b3&a3&!b2&!b1&!a1)#(!b3&a3&!b2&!b1&!b0)#(!b3&a3&!b2&!b1&!a0)#(!b3&a3&!b2&!a1&!b0)#(!b3&a3&!b2&!a1&!a0)#(!b3&a3&!a2&!b1&!a1)#(!b3&a3&!a2&!b1&!b0)#(!b3&a3&!a2&!b1&!a0)#(!b3&a3&!a2&!a1&!b0)#(!b3&a3&!a2&!a1&!a0)#(b3&!a3&!b2&!a2)#(b3&!a3&!b2&!b1&!a1)#(b3&!a3&!b2&!b1&!b0)#(b3&!a3&!b2&!b1&!a0)#(b3&!a3&!b2&!a1&!b0)#(b3&!a3&!b2&!a1&!a0)#(b3&!a3&!a2&!b1&!a1)#(b3&!a3&!a2&!b1&!b0)#(b3&!a3&!a2&!b1&!a0)#(b3&!a3&!a2&!a1&!b0)#(b3&!a3&!a2&!a1&!a0)#(b3&a3&b2&a2)#(b3&a3&b2&b1&a1)#(b3&a3&b2&b1&b0&a0)#(b3&a3&b2&a1&b0&a0)#(b3&a3&a2&b1&a1)#(b3&a3&a2&b1&b0&a0)#(b3&a3&a2&a1&b0&a0)
Hier noch die minimierten DNF für einen 8-Bit Addierer:
e0 = (!b0&a0)#(b0&!a0)
e1 = (!b1&!a1&b0&a0)#(!b1&a1&!b0)#(!b1&a1&!a0)#(b1&!a1&!b0)#(b1&!a1&!a0)#(b1&a1&b0&a0)
e2 = (!b2&!a2&b1&a1)#(!b2&!a2&b1&b0&a0)#(!b2&!a2&a1&b0&a0)#(!b2&a2&!b1&!a1)#(!b2&a2&!b1&!b0)#(!b2&a2&!b1&!a0)#(!b2&a2&!a1&!b0)#(!b2&a2&!a1&!a0)#(b2&!a2&!b1&!a1)#(b2&!a2&!b1&!b0)#(b2&!a2&!b1&!a0)#(b2&!a2&!a1&!b0)#(b2&!a2&!a1&!a0)#(b2&a2&b1&a1)#(b2&a2&b1&b0&a0)#(b2&a2&a1&b0&a0)
e3 = (!b3&!a3&b2&a2)#(!b3&!a3&b2&b1&a1)#(!b3&!a3&b2&b1&b0&a0)#(!b3&!a3&b2&a1&b0&a0)#(!b3&!a3&a2&b1&a1)#(!b3&!a3&a2&b1&b0&a0)#(!b3&!a3&a2&a1&b0&a0)#(!b3&a3&!b2&!a2)#(!b3&a3&!b2&!b1&!a1)#(!b3&a3&!b2&!b1&!b0)#(!b3&a3&!b2&!b1&!a0)#(!b3&a3&!b2&!a1&!b0)#(!b3&a3&!b2&!a1&!a0)#(!b3&a3&!a2&!b1&!a1)#(!b3&a3&!a2&!b1&!b0)#(!b3&a3&!a2&!b1&!a0)#(!b3&a3&!a2&!a1&!b0)#(!b3&a3&!a2&!a1&!a0)#(b3&!a3&!b2&!a2)#(b3&!a3&!b2&!b1&!a1)#(b3&!a3&!b2&!b1&!b0)#(b3&!a3&!b2&!b1&!a0)#(b3&!a3&!b2&!a1&!b0)#(b3&!a3&!b2&!a1&!a0)#(b3&!a3&!a2&!b1&!a1)#(b3&!a3&!a2&!b1&!b0)#(b3&!a3&!a2&!b1&!a0)#(b3&!a3&!a2&!a1&!b0)#(b3&!a3&!a2&!a1&!a0)#(b3&a3&b2&a2)#(b3&a3&b2&b1&a1)#(b3&a3&b2&b1&b0&a0)#(b3&a3&b2&a1&b0&a0)#(b3&a3&a2&b1&a1)#(b3&a3&a2&b1&b0&a0)#(b3&a3&a2&a1&b0&a0)
e4 = (!b4&!a4&b3&a3)#(!b4&!a4&b3&b2&a2)#(!b4&!a4&b3&b2&b1&a1)#(!b4&!a4&b3&b2&b1&b0&a0)#(!b4&!a4&b3&b2&a1&b0&a0)#(!b4&!a4&b3&a2&b1&a1)#(!b4&!a4&b3&a2&b1&b0&a0)#(!b4&!a4&b3&a2&a1&b0&a0)#(!b4&!a4&a3&b2&a2)#(!b4&!a4&a3&b2&b1&a1)#(!b4&!a4&a3&b2&b1&b0&a0)#(!b4&!a4&a3&b2&a1&b0&a0)#(!b4&!a4&a3&a2&b1&a1)#(!b4&!a4&a3&a2&b1&b0&a0)#(!b4&!a4&a3&a2&a1&b0&a0)#(!b4&a4&!b3&!a3)#(!b4&a4&!b3&!b2&!a2)#(!b4&a4&!b3&!b2&!b1&!a1)#(!b4&a4&!b3&!b2&!b1&!b0)#(!b4&a4&!b3&!b2&!b1&!a0)#(!b4&a4&!b3&!b2&!a1&!b0)#(!b4&a4&!b3&!b2&!a1&!a0)#(!b4&a4&!b3&!a2&!b1&!a1)#(!b4&a4&!b3&!a2&!b1&!b0)#(!b4&a4&!b3&!a2&!b1&!a0)#(!b4&a4&!b3&!a2&!a1&!b0)#(!b4&a4&!b3&!a2&!a1&!a0)#(!b4&a4&!a3&!b2&!a2)#(!b4&a4&!a3&!b2&!b1&!a1)#(!b4&a4&!a3&!b2&!b1&!b0)#(!b4&a4&!a3&!b2&!b1&!a0)#(!b4&a4&!a3&!b2&!a1&!b0)#(!b4&a4&!a3&!b2&!a1&!a0)#(!b4&a4&!a3&!a2&!b1&!a1)#(!b4&a4&!a3&!a2&!b1&!b0)#(!b4&a4&!a3&!a2&!b1&!a0)#(!b4&a4&!a3&!a2&!a1&!b0)#(!b4&a4&!a3&!a2&!a1&!a0)#(b4&!a4&!b3&!a3)#(b4&!a4&!b3&!b2&!a2)#(b4&!a4&!b3&!b2&!b1&!a1)#(b4&!a4&!b3&!b2&!b1&!b0)#(b4&!a4&!b3&!b2&!b1&!a0)#(b4&!a4&!b3&!b2&!a1&!b0)#(b4&!a4&!b3&!b2&!a1&!a0)#(b4&!a4&!b3&!a2&!b1&!a1)#(b4&!a4&!b3&!a2&!b1&!b0)#(b4&!a4&!b3&!a2&!b1&!a0)#(b4&!a4&!b3&!a2&!a1&!b0)#(b4&!a4&!b3&!a2&!a1&!a0)#(b4&!a4&!a3&!b2&!a2)#(b4&!a4&!a3&!b2&!b1&!a1)#(b4&!a4&!a3&!b2&!b1&!b0)#(b4&!a4&!a3&!b2&!b1&!a0)#(b4&!a4&!a3&!b2&!a1&!b0)#(b4&!a4&!a3&!b2&!a1&!a0)#(b4&!a4&!a3&!a2&!b1&!a1)#(b4&!a4&!a3&!a2&!b1&!b0)#(b4&!a4&!a3&!a2&!b1&!a0)#(b4&!a4&!a3&!a2&!a1&!b0)#(b4&!a4&!a3&!a2&!a1&!a0)#(b4&a4&b3&a3)#(b4&a4&b3&b2&a2)#(b4&a4&b3&b2&b1&a1)#(b4&a4&b3&b2&b1&b0&a0)#(b4&a4&b3&b2&a1&b0&a0)#(b4&a4&b3&a2&b1&a1)#(b4&a4&b3&a2&b1&b0&a0)#(b4&a4&b3&a2&a1&b0&a0)#(b4&a4&a3&b2&a2)#(b4&a4&a3&b2&b1&a1)#(b4&a4&a3&b2&b1&b0&a0)#(b4&a4&a3&b2&a1&b0&a0)#(b4&a4&a3&a2&b1&a1)#(b4&a4&a3&a2&b1&b0&a0)#(b4&a4&a3&a2&a1&b0&a0)
e5 = (!b5&!a5&b4&a4)#(!b5&!a5&b4&b3&a3)#(!b5&!a5&b4&b3&b2&a2)#(!b5&!a5&b4&b3&b2&b1&a1)#(!b5&!a5&b4&b3&b2&b1&b0&a0)#(!b5&!a5&b4&b3&b2&a1&b0&a0)#(!b5&!a5&b4&b3&a2&b1&a1)#(!b5&!a5&b4&b3&a2&b1&b0&a0)#(!b5&!a5&b4&b3&a2&a1&b0&a0)#(!b5&!a5&b4&a3&b2&a2)#(!b5&!a5&b4&a3&b2&b1&a1)#(!b5&!a5&b4&a3&b2&b1&b0&a0)#(!b5&!a5&b4&a3&b2&a1&b0&a0)#(!b5&!a5&b4&a3&a2&b1&a1)#(!b5&!a5&b4&a3&a2&b1&b0&a0)#(!b5&!a5&b4&a3&a2&a1&b0&a0)#(!b5&!a5&a4&b3&a3)#(!b5&!a5&a4&b3&b2&a2)#(!b5&!a5&a4&b3&b2&b1&a1)#(!b5&!a5&a4&b3&b2&b1&b0&a0)#(!b5&!a5&a4&b3&b2&a1&b0&a0)#(!b5&!a5&a4&b3&a2&b1&a1)#(!b5&!a5&a4&b3&a2&b1&b0&a0)#(!b5&!a5&a4&b3&a2&a1&b0&a0)#(!b5&!a5&a4&a3&b2&a2)#(!b5&!a5&a4&a3&b2&b1&a1)#(!b5&!a5&a4&a3&b2&b1&b0&a0)#(!b5&!a5&a4&a3&b2&a1&b0&a0)#(!b5&!a5&a4&a3&a2&b1&a1)#(!b5&!a5&a4&a3&a2&b1&b0&a0)#(!b5&!a5&a4&a3&a2&a1&b0&a0)#(!b5&a5&!b4&!a4)#(!b5&a5&!b4&!b3&!a3)#(!b5&a5&!b4&!b3&!b2&!a2)#(!b5&a5&!b4&!b3&!b2&!b1&!a1)#(!b5&a5&!b4&!b3&!b2&!b1&!b0)#(!b5&a5&!b4&!b3&!b2&!b1&!a0)#(!b5&a5&!b4&!b3&!b2&!a1&!b0)#(!b5&a5&!b4&!b3&!b2&!a1&!a0)#(!b5&a5&!b4&!b3&!a2&!b1&!a1)#(!b5&a5&!b4&!b3&!a2&!b1&!b0)#(!b5&a5&!b4&!b3&!a2&!b1&!a0)#(!b5&a5&!b4&!b3&!a2&!a1&!b0)#(!b5&a5&!b4&!b3&!a2&!a1&!a0)#(!b5&a5&!b4&!a3&!b2&!a2)#(!b5&a5&!b4&!a3&!b2&!b1&!a1)#(!b5&a5&!b4&!a3&!b2&!b1&!b0)#(!b5&a5&!b4&!a3&!b2&!b1&!a0)#(!b5&a5&!b4&!a3&!b2&!a1&!b0)#(!b5&a5&!b4&!a3&!b2&!a1&!a0)#(!b5&a5&!b4&!a3&!a2&!b1&!a1)#(!b5&a5&!b4&!a3&!a2&!b1&!b0)#(!b5&a5&!b4&!a3&!a2&!b1&!a0)#(!b5&a5&!b4&!a3&!a2&!a1&!b0)#(!b5&a5&!b4&!a3&!a2&!a1&!a0)#(!b5&a5&!a4&!b3&!a3)#(!b5&a5&!a4&!b3&!b2&!a2)#(!b5&a5&!a4&!b3&!b2&!b1&!a1)#(!b5&a5&!a4&!b3&!b2&!b1&!b0)#(!b5&a5&!a4&!b3&!b2&!b1&!a0)#(!b5&a5&!a4&!b3&!b2&!a1&!b0)#(!b5&a5&!a4&!b3&!b2&!a1&!a0)#(!b5&a5&!a4&!b3&!a2&!b1&!a1)#(!b5&a5&!a4&!b3&!a2&!b1&!b0)#(!b5&a5&!a4&!b3&!a2&!b1&!a0)#(!b5&a5&!a4&!b3&!a2&!a1&!b0)#(!b5&a5&!a4&!b3&!a2&!a1&!a0)#(!b5&a5&!a4&!a3&!b2&!a2)#(!b5&a5&!a4&!a3&!b2&!b1&!a1)#(!b5&a5&!a4&!a3&!b2&!b1&!b0)#(!b5&a5&!a4&!a3&!b2&!b1&!a0)#(!b5&a5&!a4&!a3&!b2&!a1&!b0)#(!b5&a5&!a4&!a3&!b2&!a1&!a0)#(!b5&a5&!a4&!a3&!a2&!b1&!a1)#(!b5&a5&!a4&!a3&!a2&!b1&!b0)#(!b5&a5&!a4&!a3&!a2&!b1&!a0)#(!b5&a5&!a4&!a3&!a2&!a1&!b0)#(!b5&a5&!a4&!a3&!a2&!a1&!a0)#(b5&!a5&!b4&!a4)#(b5&!a5&!b4&!b3&!a3)#(b5&!a5&!b4&!b3&!b2&!a2)#(b5&!a5&!b4&!b3&!b2&!b1&!a1)#(b5&!a5&!b4&!b3&!b2&!b1&!b0)#(b5&!a5&!b4&!b3&!b2&!b1&!a0)#(b5&!a5&!b4&!b3&!b2&!a1&!b0)#(b5&!a5&!b4&!b3&!b2&!a1&!a0)#(b5&!a5&!b4&!b3&!a2&!b1&!a1)#(b5&!a5&!b4&!b3&!a2&!b1&!b0)#(b5&!a5&!b4&!b3&!a2&!b1&!a0)#(b5&!a5&!b4&!b3&!a2&!a1&!b0)#(b5&!a5&!b4&!b3&!a2&!a1&!a0)#(b5&!a5&!b4&!a3&!b2&!a2)#(b5&!a5&!b4&!a3&!b2&!b1&!a1)#(b5&!a5&!b4&!a3&!b2&!b1&!b0)#(b5&!a5&!b4&!a3&!b2&!b1&!a0)#(b5&!a5&!b4&!a3&!b2&!a1&!b0)#(b5&!a5&!b4&!a3&!b2&!a1&!a0)#(b5&!a5&!b4&!a3&!a2&!b1&!a1)#(b5&!a5&!b4&!a3&!a2&!b1&!b0)#(b5&!a5&!b4&!a3&!a2&!b1&!a0)#(b5&!a5&!b4&!a3&!a2&!a1&!b0)#(b5&!a5&!b4&!a3&!a2&!a1&!a0)#(b5&!a5&!a4&!b3&!a3)#(b5&!a5&!a4&!b3&!b2&!a2)#(b5&!a5&!a4&!b3&!b2&!b1&!a1)#(b5&!a5&!a4&!b3&!b2&!b1&!b0)#(b5&!a5&!a4&!b3&!b2&!b1&!a0)#(b5&!a5&!a4&!b3&!b2&!a1&!b0)#(b5&!a5&!a4&!b3&!b2&!a1&!a0)#(b5&!a5&!a4&!b3&!a2&!b1&!a1)#(b5&!a5&!a4&!b3&!a2&!b1&!b0)#(b5&!a5&!a4&!b3&!a2&!b1&!a0)#(b5&!a5&!a4&!b3&!a2&!a1&!b0)#(b5&!a5&!a4&!b3&!a2&!a1&!a0)#(b5&!a5&!a4&!a3&!b2&!a2)#(b5&!a5&!a4&!a3&!b2&!b1&!a1)#(b5&!a5&!a4&!a3&!b2&!b1&!b0)#(b5&!a5&!a4&!a3&!b2&!b1&!a0)#(b5&!a5&!a4&!a3&!b2&!a1&!b0)#(b5&!a5&!a4&!a3&!b2&!a1&!a0)#(b5&!a5&!a4&!a3&!a2&!b1&!a1)#(b5&!a5&!a4&!a3&!a2&!b1&!b0)#(b5&!a5&!a4&!a3&!a2&!b1&!a0)#(b5&!a5&!a4&!a3&!a2&!a1&!b0)#(b5&!a5&!a4&!a3&!a2&!a1&!a0)#(b5&a5&b4&a4)#(b5&a5&b4&b3&a3)#(b5&a5&b4&b3&b2&a2)#(b5&a5&b4&b3&b2&b1&a1)#(b5&a5&b4&b3&b2&b1&b0&a0)#(b5&a5&b4&b3&b2&a1&b0&a0)#(b5&a5&b4&b3&a2&b1&a1)#(b5&a5&b4&b3&a2&b1&b0&a0)#(b5&a5&b4&b3&a2&a1&b0&a0)#(b5&a5&b4&a3&b2&a2)#(b5&a5&b4&a3&b2&b1&a1)#(b5&a5&b4&a3&b2&b1&b0&a0)#(b5&a5&b4&a3&b2&a1&b0&a0)#(b5&a5&b4&a3&a2&b1&a1)#(b5&a5&b4&a3&a2&b1&b0&a0)#(b5&a5&b4&a3&a2&a1&b0&a0)#(b5&a5&a4&b3&a3)#(b5&a5&a4&b3&b2&a2)#(b5&a5&a4&b3&b2&b1&a1)#(b5&a5&a4&b3&b2&b1&b0&a0)#(b5&a5&a4&b3&b2&a1&b0&a0)#(b5&a5&a4&b3&a2&b1&a1)#(b5&a5&a4&b3&a2&b1&b0&a0)#(b5&a5&a4&b3&a2&a1&b0&a0)#(b5&a5&a4&a3&b2&a2)#(b5&a5&a4&a3&b2&b1&a1)#(b5&a5&a4&a3&b2&b1&b0&a0)#(b5&a5&a4&a3&b2&a1&b0&a0)#(b5&a5&a4&a3&a2&b1&a1)#(b5&a5&a4&a3&a2&b1&b0&a0)#(b5&a5&a4&a3&a2&a1&b0&a0)
e6 = 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e7 = 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