Theoretical and Experimental DNA Computation (Natural ...

42 2 Theoretical Computer Science: A Primer
The function F computed by this circuit is (x OR y) AND (NOT x OR
NOT y). In formal notation, this is expressed as F =(x ∨ y) ∧ (x ∨ y). OR is
denoted by ∧, AND by ∧, and NOT by overlining the variable.
F is made up of two bracketed clauses, and the problem is to find values
for the variables so that both clauses have the value 1, giving a value of 1 for
F (observe that the values of the clauses are ANDed). This problem is known
as the “satisfiability” problem because making all the clauses true is viewed
as “satisfying” the clauses.
The most obvious method of solving this problem is to generate all possible
choices for the variable values. That is, construct the truth table for the circuit.
Table 2.5 shows a breakdown of the truth table for the circuit depicted in
Fig. 2.15.
Table 2.5. Truth table for satisfiability circuit
xy(x ∨ y) (x ∨ y) z
00 0 1 0
01 1 1 1
10 1 1 1
11 1 0 0
Clearly, by inspection, only the combinations “x = 0, y = 1” and “x = 1, y =
0” are valid solutions to the problem. The current best algorithm essentially
performs just this process of trying all possible 2 n choices for n variables.
Therein lies the problem.
Consider a situation where the number of variables doubles to four. There
are therefore 2 4 = 16 combinations to check. Doubling the circuit size again
means checking 2 8 = 256 combinations. However, what if the function has
100 variables? The number of combinations to be checked is now a staggering
1,267,650,600,228,229,401,496,703,205,376!
This is what mathematicians refer to as a combinatorial explosion. Thisis
a situation where work (or space required) increases by a factor of two, three,
or more for each successive value of n.
Table 2.6 shows the time required by different algorithms with different
time complexities for different values of n. Weassumethealgorithmsarerun
by a machine capable of executing a million steps a second.
Clearly, algorithms requiring n (linear) orn 2 (quadratic) time may be
feasible for relatively large problem instances, while algorithms requiring 2 n
(exponential) time are clearly impractical for even small values of n.