We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�-size depth-3 circuits for Approximate Majority on n bits have bottom fan-in Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomial-size circuits of depth 3. Prior to our work, the only known polynomial-size depth-3 circuits for Approximate Majority were non-uniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a random-access input tape and a sequential-access work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR-0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the

Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving non-trivial lower bounds on the computational complexity of satisfiability. In this article we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the state-of-the-art results and present the underlying arguments in a unified framework. 1

We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known time-space tradeoffs for Sat. Let m> 0 be an integer, and define MODm-Sat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODp-Sat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6-Sat, as well as MODm-Sat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.

no. DGE-0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.

We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, Σ O(1)Time(t). Our main results are the following: 1. We prove that BPTime(t) ⊆ Σ3Time(t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time � t 2 · log t � (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime(t) ⊆ ΣcTime(t) for a large constant c> 3 (Ajtai, Adv. in Comp. Complexity Theory ’93). 2. We prove that BPTime(t) � ⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements our result (1), and shows that the running time of the Sipser-Gács-Lautemann simulation is optimal, up to a log t factor, for relativizing techniques. (All the results in (1) relativize.) This result is obtained as a corollary from a new circuit lower bound for approximate majority: poly(n)-size depth-3 circuits for approximate majority have bottom fan-in Ω(log n). 3. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on probabilistic Turing machines using space n.9, with random access to input and work tapes, and two-way sequential access to the random-bit tape. This is the first lower bound of the form t = n 1+Ω(1) on a model with random access to the input and two-way access to the random bits. 4. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on Turing machines with an input tape and a sequential work tape that is initialized with random bits. This is the first lower bound on a probabilistic extension of the off-line Turing machine model with one work tape.

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real ǫ there exists a real c> 1 such that either: • MajMajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n c, or • MajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n d and space n 1−ǫ. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time n 1+o(1) and space n 1−ǫ for any ǫ> 0. The key technical novelty is a time- and space-efficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation. 1

We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every real d < 3 √ 4 there exists a positive real e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e. 1

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. These lower bound proofs all follow a certain diagonalization-based proof-by-contradiction strategy. A pressing open problem has been to determine how powerful such proofs can possibly be. We propose an automated theorem-proving methodology for studying these lower bound problems. In particular, we prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. We describe an implementation of a small-scale theorem prover and discover surprising experimental results. In some settings, our program provides strong evidence that the best known lower bound proofs are already optimal for the current framework, contradicting the consensus intuition; in others, the program guides us to improved lower bounds where none had been known for years.

We prove a lower bound for swapping the order of Arthur and Merlin in two-round Merlin-Arthur games using black-box techniques. Namely, we show that any AM-game requires time Ω(t 2) to black-box simulate MA-games running in time t. Thus, the known simulations of MA by AM with quadratic overhead, dating back to Babai’s original paper on Arthur-Merlin games, are tight within this setting. The black-box lower bound also yields an oracle relative to which MA-TIME[n] � AM-TIME[o(n 2)]. Complementing our lower bounds for swapping Merlin in MA-games, we prove a time-space lower bound for simulations that drop Merlin entirely. We show that for any c &lt; √ 2, there exists a positive d such that there is a language recognized by linear-time MA-games with one-sided error but not by probabilistic random-access machines with two-sided error that run in time n c and space n d. This improves recent results that give such lower bounds for problems in the second level of the polynomial-time hierarchy.

We give two time- and space-efficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a bounded-error quantum algorithm running in time t and space s is also solvable by an unbounded-error randomized algorithm running in time O(t · log t) and space O(s + log t), as well as by a bounded-error quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylog t) and space O(s + log t), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are the problems of determining the truth value of a given Boolean formula whose variables are fully quantified by one or two majority quantifiers, respectively. We prove that for every real d and every positive real δ there exists a real c> 1 such that either • MajMajSAT does not have a bounded-error quantum algorithm running in time O(n c), or • MajSAT does not have a bounded-error quantum algorithm running in time O(n d) and space O(n 1−δ). In particular, MajMajSAT does not have a bounded-error quantum algorithm running in time O(n 1+o(1) ) and space O(n 1−δ) for any δ> 0. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop. 1