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CHAPTER THREE using vs .net tocompose code39 with web,windows application Data Capacity of QR Code Variations on P and NP Cast a cold eye On life, on death. Horseman, pass by! W. B.

Yeats, Under Ben Bulben In this chapter we consider variations on the complexity classes P and NP. We refer speci cally to the non-uniform version of P, and to the Polynomial-time Hierarchy (which extends NP). These variations are motivated by relatively technical considerations; still, the resulting classes are referred to quite frequently in the literature.

Summary: Non-uniform polynomial-time (P/poly) captures ef cient computations that are carried out by devices that can each handle only inputs of a speci c length. The basic formalism ignores the complexity of constructing such devices (i.e.

, a uniformity condition). A ner formalism that allows for quantifying the amount of non-uniformity refers to so-called machines that take advice. The Polynomial-time Hierarchy (PH) generalizes NP by considering statements expressed by quanti ed Boolean formulae with a xed number of alternations of existential and universal quanti ers.

It is widely believed that each quanti er alternation adds expressive power to the class of such formulae. An interesting result that refers to both classes asserts that if NP is contained in P/poly then the Polynomial-time Hierarchy collapses to its second level. This result is commonly interpreted as supporting the common belief that non-uniformity is irrelevant to the P-vs-NP Question; that is, although P/poly extends beyond the class P, it is believed that P/poly does not contain NP.

Except for the latter result, which is presented in Section 3.2.3, the treatments of P/poly (in Section 3.

1) and of the Polynomial-time Hierarchy (in Section 3.2) are independent of one another..

3.1. Non-uniform Po .

NET 39 barcode lynomial Time (P/poly). In this section we consider two formulations of the notion of non-uniform polynomial time, based on the two models of non-uniform computing devices that were. 3.1 NON-UNIFORM POLYNOMIAL TIME presented in Sectio n 1.2.4.

That is, we specialize the treatment of non-uniform computing devices, provided in Section 1.2.4, to the case of polynomially bounded complexities.

It turns out that both (polynomially bounded) formulations allow for solving the same class of computational problems, which is a strict superset of the class of problems solvable by polynomial-time algorithms. The two models of non-uniform computing devices are Boolean circuits and machines that take advice (cf. 1.

2.4.1 and 1.

2.4.2, respectively).

We will focus on the restriction of both models to the case of polynomial complexities, considering (non-uniform) polynomial-size circuits and polynomial-time algorithms that take (non-uniform) advice of polynomially bounded length. The main motivation for considering non-uniform polynomial-size circuits is that their computational limitations imply analogous limitations on polynomial-time algorithms. The hope is that, as is often the case in mathematics and science, disposing of an auxiliary condition (i.

e., uniformity) that seems secondary1 and is not well understood may turn out to be fruitful. In particular, the (non-uniform) circuit model facilitates a low-level analysis of the evolution of a computation, and allows for the application of combinatorial techniques.

The bene t of this approach has been demonstrated in the study of restricted classes of circuits (see Appendix B.2.2 and B.

2.3). The main motivation for considering polynomial-time algorithms that take polynomially bounded advice is that such devices are useful in modeling auxiliary information that is available to possible ef cient strategies that are of interest to us.

We mention two such settings. In cryptography (see Appendix C), the advice is used for accounting for auxiliary information that is available to an adversary. In the context of derandomization (see Section 8.

3), the advice is used for accounting for the main input to the randomized algorithm. In addition, the model of polynomial-time algorithms that take advice allows for a quantitative study of the amount of non-uniformity, ranging from zero to polynomial..

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