We obtain the first non-trivial time-space tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0. We also give the first separation result between the syntactic and semantic read-k models [BRS93] for k > 1 by showing that polynomial-size semantic read-twice branching programs can compute functions that require exponential size on any syntactic read-k branching program. We also show...

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his time-space tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems.

We show that non-deterministic time NT IME(n) is not contained in deterministic time n # 2-# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2-# ) and poly-logarithmic space. A similar result is presented for uniform circuits.

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...

Automata-based representations and related algorithms have been applied to address several problems in information security, and often the automata had to be augmented with additional information. For example, extended finite-state automata (EFSA) augment finitestate automata (FSA) with variables to track dependencies between arguments of system calls. In this paper, we introduce extended finite automata (XFAs) which augment FSAs with finite scratch memory and instructions to manipulate this memory. Our primary motivation for introducing XFAs is signature matching in Network Intrusion Detection Systems (NIDS). Representing NIDS signatures as deterministic finite-state automata (DFAs) results in very fast signature matching but for several classes of signatures DFAs can blowup in space. Using nondeterministic finite-state automata (NFA) to represent NIDS signatures results in a succinct representation but at the expense of higher time complexity for signature matching. In other words, DFAs are time-efficient but space-inefficient, and NFAs are spaceefficient but time-inefficient. In our experiments we have noticed that for a large class of NIDS signatures XFAs have time complexity similar to DFAs and space complexity similar to NFAs. For our test set, XFAs use 10 times less memory than a DFA-based solution, yet achieve 20 times higher matching speeds. 1.

Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalternating computation, on both subpolynomial (n o(1) ) space RAMs and sequential one-tape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NP-complete problems that have efficient reductions from SAT, and Σk-SAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using n √ 3 time and subpolynomial space. 2. We show how indirect diagonalization leads to time-space lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k ≥ 1, there is a constant ck> 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic n ck time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in n k·ck time and subpolynomial space.

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.

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