A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership. Similarly we present languages where checking is harder than computing membership. Each of the following properties --- checkability, random-self-reducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself --- implies coherence, one of the weakest forms of self-reducibility. Under assumptions about triple-exponential time, we construct incoherent sets in NP....

We show how to construct proof systems for NP languages where a deterministic polynomialtime verifier can check membership, given any N (2=3)+ffl bits of an N -bit witness of membership.

Goldwasser and Sipser [GS89] proved that every interactive proof system can be transformed into a public-coin one (a.k.a., an Arthur--Merlin game). Their transformation has the drawback that the computational complexity of the prover's strategy is not preserved. We show that this is inherent, by proving that the same must be true of any transformation which only uses the original prover and verifier strategies as "black boxes." Our negative result holds even if the original proof system is restricted to be honest-verifier perfect zero knowledge and the transformation can also use the simulator as a black box. We also examine a similar deficiency in a transformation of Furer et al. [FGM + 89] from interactive proofs to ones with perfect completeness. We argue that the increase in prover complexity incurred by their transformation is necessary, given that their construction is a black-box transformation which works regardless of the verifier's computational complexity. Keywor...

Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems -- interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs -- stressing the essential role of randomness in each of them.

for continuous guidance and wise advice throughout the course of the dissertation, as well as for teaching me to write. I wish to thank Scott Page for his numerous insightful comments and assistance during the course of the dissertation. I thank Mike Alvarez and Rod Kiewiet for their comments and assistance during the course of the dissertation. I also thank Gary King, Daniel Lowenstein and Ken McCue for their comments on individual chapters. Redistricting is always political, increasingly controversial, and often ugly. Politicians have always fought tooth-and-nail over district lines, while the courts, for most of their history, considered the subject a thicket too political even to enter. Three decades ago the courts finally entered the political thicket, ruling in Baker v. Carr (1962) that redistricting was justiciable. A decade ago, the court showed signs that it wanted to chop the thicket down, ruling in Davis v. Bandemer (1986) that partisan gerrymanders were actionable. But, in fact, few suits followed this potentially

The existence of cryptographically secure one-way functions is related to the measure of a subclass of NP. This subclass, called fiNP ("balanced NP"), contains 3SAT and other standard NP problems. The hypothesis that fiNP is not a subset of P is equivalent to the P 6= NP conjecture. A stronger hypothesis, that fiNP is not a measure 0 subset of E 2 = DTIME(2 polynomial ) is shown to have the following two consequences. 1. For every k, there is a polynomial time computable, honest function f that is (2 n k =n k )-one-way with exponential security. (That is, no 2 n k -time-bounded algorithm with n k bits of nonuniform advice inverts f on more than an exponentially small set of inputs. ) 2. If DTIME(2 n ) "separates all BPP pairs," then there is a (polynomial time computable) pseudorandom generator that passes all probabilistic polynomial-time statistical tests. (This result is a partial converse of Yao, Boppana, and Hirschfeld's theorem, that the existence of pseudorandom ge...

Assuming that k 2 and \Delta P k does not have p-measure 0, it is shown that BP \Delta \Delta P k = \Delta P k . This implies that the following conditions hold if \Delta P 2 does not have p-measure 0. (i) AM " co-AM is low for \Delta P 2 . (Thus BPP and the graph isomorphism problem are low for \Delta P 2 .) (ii) If \Delta P 2 6= PH, then NP does not have polynomial-size circuits. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation. 1 Introduction Many widely believed conjectures in computational complexity are "strong" in the sense that they are known to imply that P 6= NP, but are not known to follow from the P 6= NP hypothesis. Recent investigations have shown that a number of these conjectures do follow from the (apparently) stronger hypothesis that NP does not have p-measure 0. (This hypothesis, written ¯ p (NP) 6= 0, is defined in...

r output 0 if none exists. The CLIQUE problem we have looked at, and shown to be NP-complete, is the decision problem. In the search problem, you actually have to find the clique, meaning identify a set C which forms a clique. Notice that if you can solve the search problem you can certainly solve the decision problem. Why? Take SAT. Given a formula ', if I can find a satisfying assignment or tell that one does not exist, I can certainly say whether or not there exists a satisfying assignment. So sarch is harder; if you can solve it you can certainly solve decision. So then the question is why do we focus on decision problems? After all the real object of interest is the search problem. What use is it to know a solution exists if you can't find it? CSE 200 Wi99, Decision versus Search 2 2 Self-reducibility of SAT One of the reasons we consider decision problems is that if we can solve them, we can often s

Introduction Investigation of the measure-theoretic structure of complexity classes began with the development of resource-bounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resource-bounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], Ambos-Spies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resource-bounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resource-bounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their

In previous works, Juba and Sudan [6] and Goldreich, Juba and Sudan [4] considered the idea of “semantic communication”, wherein two players, a user and a server, attempt to communicate with each other without any prior common language (or communication) protocol. They showed that if communication was goal-oriented and the user could sense progress towards the goal (or verify when it has been achieved), then meaningful communication is possible, in that the user’s goal can be achieved whenever the server is helpful. A principal criticism of their result has been that it is inefficient: in order to determine the “right ” protocol to communicate with the server, the user enumerates protocols and tries them out with the server until it finds one that allows it to achieve its goal. They also show settings in which such enumeration is essentially the best possible solution. In this work we introduce definitions which allow for efficient behavior in practice. Roughly, we measure the performance of users and servers against their own “beliefs ” about natural protocols. We show that if user and server are efficient with respect to their own beliefs and their beliefs are (even just slightly) compatible with each other, then they can achieve their goals very efficiently. We show that this model allows sufficiently “broad-minded ” servers to talk with “exponentially” many different users in polynomial time, while dismissing the “counterexamples” in the previous work as being “narrow-minded, ” or based on “incompatible beliefs.” Portions of this work are presented in modified form in the first author’s Ph.D. thesis. [5, Chapter 4]. Research