Expected polynomial-time

I also expect that computational restrictions only make sense in combination with allowing error probabilities, at least in models where the complexity of an interactive entity is regarded as a function of its initial state alone or where honest users are modeled as computationally restricted. Then the correct part of the system is polynomial-time in its initial state and therefore only reacts on parts of polynomial length of the input from an unrestricted attacker. Hence with mere guessing, a computationally restricted attacker has a very small chance of doing exactly what a certain unrestricted attacker would do, as far as it is seen by the correct entities. Hence if a requirement is not fulfilled information-theoretically without error probability, such a restricted attacker has a small probability of success, too. 

Let P be the set of all decision problems that can be solved in deterministic polynomial time. Let NP be the set of decision problems solvable in polynomial time by nondeterministic algorithms. Clearly, P cNP. It is not known whether P = NP or P NP. The P = NP problem is important because it is related to the complexity of many interesting problems. There exist many problems that cannot be solved in polynomial time unless P = NP. Since, intuitively, one expects that P c NP, these problems are in all probability not solvable in polynomial time. The first problem that was shown to be related to the P = NP problem, in this way, was the problem of determining whether a propositional formula is sat-isfiable. This problem is referred to as the satisfiability problem. 

The importance of showing that a problem A is NP-hard lies in the P=NP problem. Since we do not expect that P=NP, we do not expect. NP-hard problems to be solvable by algorithms with a worst-case complexity that is polynomial in the size of the problem instance. From Table n, it is apparent that, if a problem cannot be solved in polynomial time (in particular, low-order polynomial time), it is intractable, for all practical purposes. If A is NP-complete and if it does turn out that P =NP, then A will be polynomially solvable. However, if A is only NP-hard, it is possible for P to equal NP and for A not to be inP. 

We now put one and one together. The key is that we can "read" the poles—telling what the form of the time-domain function is. We should have a pretty good idea from our exercises in partial fractions. Here, we provide the results one more time in general notation. Suppose we have taken a characteristic polynomial, found its roots and completed the partial fraction expansion, this is what we expect in the time-domain for each of the terms ... 

In real life, we expect probable simultaneous reference and disturbance inputs. As far as analysis goes, the mathematics is much simpler if we consider one case at a time. In addition, either case shares the same closed-loop characteristic polynomial. Hence they should also share the same stability and dynamic response characteristics. Later when we talk about integral error criteria in controller design, there are minor differences, but not sufficient to justify analyzing a problem with simultaneous reference and load inputs. 

Inspection of the curves shows that the two solutions cross each other. This is expected since the polynomial approximate solution satisfies the mass balance equation only in some average sense. Now, the exact solution behaves roughly as t/r for small times (see Eq. 12.4), so Fig. 12.1 also shows plots of the average concentrations versus /r. The polynomial solution fails to exhibit the linear Vr dependence at short times, but rather it takes a sigmoidal shape. 

We may have expected poor performance at short times, since we may have recognized from our earlier work that many terms are necessary for the short time period. To see this more clearly, we plot the exact solution y(x, t) (Eq. 12.2) and the polynomial approximate solution yjix, t) (Eq. 12.17) at two... 

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