We establish the first polynomial-strength time-space lower bounds for problems in the lineartime hierarchy on randomized machines with two-sided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machines in time n c and space n d, where QSAT ℓ denotes the problem of deciding the validity of a quantified Boolean formula with at most ℓ − 1 quantifier alternations. Moreover, d approaches 1/2 from below as c approaches 1 from above for ℓ = 2, and d approaches 1 from below as c approaches 1 from above for ℓ ≥ 3. In fact, we establish the stronger result that for any constants a ≤ 1 and c < 1+(ℓ −1)a, there exists a positive constant d such that linear-time alternating machines using space n a and ℓ − 1 alternations cannot be simulated by randomized machines with two-sided error running in time n c and space n d, where d approaches a/2 from below as c approaches 1 from above for ℓ = 2 and d approaches a from below as c approaches 1 from above for ℓ ≥ 3. Corresponding to ℓ = 1, we prove that there exists a positive constant d such that the set of Boolean tautologies cannot be decided by a randomized machine with one-sided error in time n 1.759 and space n d. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result. 1

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 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

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.

Complexity theory is the part of theoretical computer science that attempts to prove that certain transformations from input to output are impossible to compute using a reasonable amount of resources. Theorem 1 below illustrates the type of ‘‘impossibility’ ’ proof that can sometimes be obtained (1); it talks about the problem of determining whether a logic formula in a certain formalism (abbreviated WS1S) is true.

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

Nepomnjaˇsčiǐ’s Theorem states that for all 0 ≤ ǫ < 1 and k> 0 the class of languages recognized in nondeterministic time n k and space n ǫ, NTISP[n k, n ǫ], is contained in the linear time hierarchy. By considering restrictions on the size of the universal quantifiers in the linear time hierarchy, this paper refines Nepomnjaˇsčiǐ’s result to give a subhierarchy, Eu-LinH, of the linear time hierarchy that is contained in NP and which contains NTISP[n k, n ǫ]. Hence, Eu-LinH contains NL and SC. This paper investigates basic structural properties of Eu-LinH. Then the relationships between Eu-LinH and the classes NL, SC, and NP are considered to see if they can shed light on the NL = NP or SC = NP questions. Finally, a new hierarchy, ξ-LinH, is defined to reduce the space requirements needed for the upper bound on Eu-LinH. Mathematics Subject Classification: 03F30, 68Q15 Keywords: structural complexity, linear time hierarchy, Nepomnjaˇsčiǐ’s Theorem