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...

We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose random-access Turing machines in time n 1.618 and space n o(1) . This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less than the golden ratio, we prove that satisfiability cannot be solved in time n a and space n b for some positive constant b. Our techniques allow us to establish this result for b < 1 2 ( a+2 a 2 - a). We can do better for a close to the golden ratio, for example, satisfiability cannot be solved by a random-access Turing machine using n 1.46 time and n .11 space. We also show tradeoffs for nondeterministic linear time computations using sublinear space. For example, there exists a language computable in nondeterministic linear time and n .619 space that cannot be computed in deterministic n 1.618 time and n o(1) space. Higher up the polynomial-time hierarchy we can get be...

We establish the first polynomial time-space lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic random-access Turing machine can solve satisfiability in time n c and space n d, where d approaches 1 when c does. On conondeterministic instead of deterministic machines, we prove the same for any constant c less than √ 2. Our lower bounds apply to nondeterministic linear time and almost all natural NP-complete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n 1/c. Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove time-space lower bounds for languages higher up in the polynomial-time hierarchy.

We extend the lower bound techniques of [14], to the unbounded-error probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspace-uniform TC 0 circuits for iterated multiplication [9]. Here is an

this paper are strengthenings of the results in [4] and [9] for Turing machines. The results in [4] and [9] hold for SAT but our results hold for 2-SAT also, since the formulae we reduce the language L to belong to 2-SAT. Therefore our techniques are less promising if the ultimate goal is to prove that SAT does not belong to P, since it is known that 2-SAT belongs to P. Moreover we obtain the same lower bounds for NTMs as for DTMs, which indicates that our techniques may not be useful in separating nondeterministic time and deterministic time

wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constant-factor overhead ; if g = O(t log t) it has a factor-of-O(log t) overhead , and so on. The simulation is on-line if each step of M 1 i

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.

Abstract. A realistic model of computation called the Block Move (BM) model is developed. The BM regards computation as a sequence of finite transductions in memory, and operations are timed according to a memory cost parameter µ. Unlike previous memory-cost models, the BM provides a rich theory of linear time, and in contrast to what is known for Turing machines, the BM is proved to be highly robust for linear time. Under a wide range of µ parameters, many forms of the BM model, ranging from a fixed-wordsize RAM down to a single finite automaton iterating itself on a single tape, are shown to simulate each other up to constant factors in running time. The BM is proved to enjoy efficient universal simulation, and to have a tight deterministic time hierarchy. Relationships among BM and TM time complexity classes are studied. Key words. Computational complexity, theory of computation, machine models, Turing machines, random-access machines, simulation, memory hierarchies, finite automata, linear time, caching. AMS/MOS classification: 68Q05,68Q10,68Q15,68Q68.

We survey the recent lower bounds on the running time of general-purpose random-access machines that solve satisfiability in a small amount of work space, and related lower bounds for satisfiability in nonuniform models.

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.