Theoretical and Experimental DNA Computation (Natural ...

Boolean algebra
2.2 Algorithms and Automata 25
A Boolean function has one or more input variables and yields a result that
depends only on the values of those variables. One function may be defined
by saying that f(X) is1ifX is 0, and F (X) is0ifX is 1. This is obviously
the NOT function described above.
Because each Boolean function of n variables has only 2 n possible combinations
of input values, we can describe the function completely by listing a
table of 2 n rows, each specifying a unique set of input values with the corresponding
value for the function. Such tables are known as truth tables, and
the tables of Fig. 2.1 are examples of these.
We may combine gates to form circuits to evaluate more complicated functions.
For example, we may want to say “if the car ignition is on and the
seatbelt is not fastened then sound the buzzer.” The circuit to implement this
function is depicted in Fig. 2.2.
X
Y
X Y Z
0 0 0
0 1 0
1 0 1
1 1 0
Fig. 2.2. Example circuit
Here, X represents the state of the ignition, and Y the state of the seatbelt.
The buzzer sounds (Z =1)onlyifX is 1 (i.e., the ignition is on) and Y is 0
(i.e., not 1, or the seatbelt is not on).
Boolean logic circuits implement computations by taking in inputs, applying
some function, and producing an output. We now examine the nature of
computation in more detail.
2.2 Algorithms and Automata
An algorithm is a mathematical procedure for performing a computation. To
phrase it another way, an algorithm is “a computable set of steps to achieve a
Z