In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x 1 ; : : : ; xn which are related to y by a known relation y = f(x 1 ; : : : ; xn ). Measurements are never 100% accurate; hence, the measured values e x i are different from x i , and the resulting estimate e y = f(ex 1 ; : : : ; e xn ) is different from the desired value y = f(x 1 ; : : : ; xn ). How different? Traditional engineering to error estimation in data processing assumes that we know the probabilities of different measurement error \Deltax i = e x i \Gamma x i . In many practical situations, we only know the upper bound \Delta i for this error; hence, after the measurement, the only information that we have about x i is that it belongs to the interval x i = [ex i \Gamma \Delta i ; e x i + \Delta i ]. In this case, it is important to find the range y of all possible values of y = f(x 1 ; : : : ; xn ) when x i 2 x i . We start the paper with a brief overview of the computational complexity of the corresponding interval computation problems.

) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomial-time hierarchy, ffl has polynomial-size circuits and ffl has publishable proofs (EXPTIME=MA). We also show that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we show BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA n P . The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not ...

We consider the problem of approximating a Boolean function f : f0; 1g n ! f0; 1g by the sign of an integer polynomial p of degree k. For us, a polynomial p(x) predicts the value of f(x) if, whenever p(x) 0, f(x) = 1, and whenever p(x) ! 0, f(x) = 0. A low-degree polynomial p is a good approximator for f if it predicts f at almost all points. Given a positive integer k, and a Boolean function f , we ask, "how good is the best degree k approximation to f?" We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the case f is parity. Minsky and Papert [10] proved that a perceptron can not compute parity; our bounds indicate exactly how well Yale University, Dept. of Computer Science, P.O. Box 208285, New Haven CT 06520-8285. y Email: aspnes-james@cs.yale.edu. z Email: beigel-richard@cs.yale.edu. Supported in part by NSF grants CCR-8808949 and CCR-8958528. x Carnegie-Mellon University, Schoo...

The representation of functions as low-degree polynomials over various rings has provided many insights in the theory of small-depth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of AND-OR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...

We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

Measure-theoretic aspects of the P m -reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m - hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m -hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m -complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m -degree (e.g., the degree of all P m -complete languages for NP) has measure 0 in E and in E 2 .

Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NP-complete sets---the k-creative sets---and defined a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman--Hartmanis and Joseph--Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...

We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes.

We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept \Sigma . ffl Every total multivalued nondeterministic function has a polynomial-time computable refinement. ffl In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. ffl In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. 1 Introduction Understanding the power of nondeterminism has been one of the pri...

. Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision trees of a certain type or for machines with polylogarithmic restrictions on time. The following question arises: Are these methods of proving "relativized" results universal? In the first part of the present paper we propose a general framework in which assertions of universality of this kind may be formulated and proved as convenient criteria. Using these criteria we obtain, as easy consequences of the known results on boolean decision trees, some new "relativized" results and new proofs of some known results. In the second part of the present paper we apply these general criteria to many particular cases. For example, for many of the complexity classes studied in the literature all relativiza...