We give the first nontrivial model-independent time-space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for log-space uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...

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

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have non-uniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasi-polynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have non-uniform ACC circuits of 2no(1) size. The lower bound gives an exponential size-depth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depth-d ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth-3 polynomial size circuits made out of only MOD6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.

This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilinear-time hierarchy.

This paper first surveys the near-total lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows non-local communication with memory at unit cost. We study a model that imposes a “fair cost ” for non-local communication, and obtain modest superlinear lower bounds for some problems via a Kolmogorov-complexity argument. Then we look to the larger picture of what it will take to prove really striking lower bounds, and pull from ours and others’ work a concept of information vicinity that may offer new tools and modes of analysis to a young field that rather lacks them.

Abstract. We give a reduction from arbitrary languages in alternating time t(n) to quantified Boolean formulas (QBF) describable in O(t(n)) bits. The reduction works for a reasonable succinct encoding of Boolean formulas and for several reasonable machine models, including multitape Turing machines and logarithmic-cost RAMs. By a simple diagonalization, it follows that our succinct QBF problem requires superlinear time on those models. To our knowledge this is the first known instance of a non-linear time lower bound (with no space restriction) for solving a natural linear space problem on a variety of computational models.

Succinct arguments for NP are proof systems that allow a weak verifier to retroactively check computation done by a more powerful prover. These protocols prove membership in languages (consisting of succinctlyrepresented very large constraint satisfaction problems) that, alas, are unnatural in the sense that the problems that arise in practice are not in such form. For general computation tasks, the most natural and efficient representation is typically as random-access machine (RAM) algorithms, because such a representation can be obtained very efficiently by applying a compiler to code written in a high-level programming language. We thus study efficient reductions from RAM to other problem representations for which succinct arguments are known. Specifically, we construct reductions from the correctness of computation of a T-step non-deterministic random-access machine to: 1. (succinct) circuit satisfiability with O(log T) overhead, and 2. (succinct) algebraic constraint satisfaction with O(log 2 T) overhead. On the latter problem representation, the best known Probabilistically Checkable Proofs can be directly invoked. Our constructions are explicit and do not hide large constants. To attain these, we develop a set of tools (both unconditional and leveraging computational assumptions) for generically and efficiently structuring and arithmetizing the computation of random-access machines.

Succinct non-interactive arguments of knowledge (SNARKs), and their generalization to distributed computations by proof-carrying data (PCD), are powerful tools for enforcing the correctness of dynamically evolving computations among multiple mutually-untrusting parties. We present recursive composition and bootstrapping techniques that: 1. Transform any SNARK with an expensive preprocessing phase into a SNARK without such a phase. 2. Transform any SNARK into a PCD system for constant-depth distributed computations. 3. Transform any PCD system for constant-depth distributed computations into a PCD system for distributed computation over paths of fixed polynomial length. Our transformations apply to both the public- and private-verification settings, and assume the existence of CRHs; for the private-verification setting, we additionally assume FHE. By applying our transformations to the NIZKs of [Groth, ASIACRYPT ’10], whose security is based on a Knowledge of Exponent assumption in bilinear groups, we obtain the first publicly-verifiable SNARKs and PCD without preprocessing in the plain model. (Previous constructions were either in the randomoracle model [Micali, FOCS ’94] or in a signature oracle model [Chiesa and Tromer, ICS ’10].) Interestingly,