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.

This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps read-once branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem.

We prove time-space tradeoffs for traversing undi-rected graphs. One of these is a quadratic lower bound on a deterministic model that closely matches the recent probabilistic upper bound of Broder, Karlin, Raghavan, and Upfal. The models used are variants of Cook and Rackoff’s “Jumping Automata for Graphs".

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 derive bounds on the product of the communication C and space S for communicating circuits. The first bound applies to quantum circuits and follows from a "bipartite product" result for the discrepancy of communication problems. If for any problem f : XY the multicolor discrepancy of the communication matrix of f is 1/2 then the problem in which Alice receives some l inputs, Bob r inputs, and their task is to compute f(x i , y j ) for the l r pairs of inputs (x i , y j ), has a quantum communication-space tradeo# CS (lrd log |Z|).

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

Directed and undirected st-connectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (log n) space. The first result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graphs), these upper bounds are not simultaneously achievable. This uses new entropy techniques to prove tight bounds on a game involving a helper and a player that models a computation having precomputed information about the input stored in its bounded space. The second result proves that a JAG requires a time-space tradeoff of T \Theta S 1 2 2\Omega i mn 1 2 j to compute directed st-connectivity. The third result proves a time-space tradeoff of T \Theta S 1 3 2\Omega i m 2 3 n 2 3 j on a version of the...

Abstract. In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs (QBPs), which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between QBPs and nonuniform quantum Turing machines are presented, which allow to transfer lower and upper bound results between the two models. Using additional insights about the connection between running time and the precision of amplitudes, it is shown that nonuniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Quantum ordered binary decision diagrams (QOBDDs) are a restricted variant of QBPs, which can be considered as nonuniform analog of one-way quantum finite automata. In the second part of the paper, lower and upper bound results for QOBDDs are presented in order to compare variants of QOBDDs with their deterministic and randomized counterparts. In the third part QBPs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.

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