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.

Restricting the search space f0; 1g to the set of truth tables of "easy" Boolean functions on log n variables, as well as using some known hardness-randomness tradeoffs, we establish a number of results relating the complexity of exponential-time and probabilistic polynomialtime complexity classes. In particular, we show that NEXP ae P=poly , NEXP = MA; this can be interpreted as saying that no derandomization of MA (and, hence, of promise-BPP) is possible unless NEXP contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show EXP = BPP , EE = BPE, where EE is the double-exponential time class and BPE is the exponential-time analogue of BPP.

Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1

. Circuit size lower bounds were previously established for functions in p 2 , ZPP NP , exp 2 , ZPEXP NP and MA exp . We ask the general question: Given a time bound t(n). What is the best circuit size lower bound that can be currently shown for the classes MA-TIME[t(n)], ZP-TIME NP [t(n)]; . . .? For the classes MA exp , ZPEXP NP and exp 2 , the answer turns out to be \half-exponential". Informally, a function f is said to be half-exponential when f f is exponential. Such functions were constructed by Szekeres. 1 Introduction One main issue of complexity theory is how powerful non-uniform (e.g. circuit based) computation is, compared to uniform (machine based) computation. In particular, a 64K dollar question is whether exponential time has polynomial size circuits. This being a challenging open question, a series of papers have looked at circuit size of functions further up the exponential hierarchy. In the early eighties, Kannan [11] established that there is ...

We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type-2 classes are distinct.

1 Introduction The symmetric alternation class (Sp2) was introduced independently by Russelland Sundaram [16] and by Canetti [5]. The class S p 2 contains languages havingan interactive proof system of the following type. The proof system consists of

I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!

We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the average-case setting, i.e., we show that MA/1 cannot be solved on more than 1/2 + 1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudo-random generators computable using O(1) bits of advice. 1

We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k> 0, we give an explicit Σ p 2 language, acceptable by a Σp2-machine with running time O(nk2 +k), that requires circuit size> nk. This provides a constructive version of an existence theorem of Kannan [Kan82]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomialtime hierarchy requires super polynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and Nisan-Wigderson pseudorandom generator.

Theoretical computer scientists have been debating the role of oracles since the 1970’s. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low-degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of “traditional ” complexity classes, as opposed to interactive proof classes such as MIP and MAEXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP NP | | and even BPP NP | | have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P = NP. Thus, classes such as ZPP NP inhabit a “twilight zone, ” where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem. 1