Bruno Loff

Derandomizing from Random Strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K has high Kolmogorov complexity. As a partial converse to our claim we show that strings of very high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.

Computing with a full memory: catalytic space

We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace. In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1,..., xn and work registers r1,..., rm. The instructions available are of the form ri ← ri±u×v, with u, v registers (distinct from ri) or constants. We wish to compute a function f(x1,..., xn) through a sequence of such instructions. The working registers have some arbitrary initial value ri = τi, and they may be altered throughout the computation, but by the end all registers must be returned to their initial value τi, except for, say, r1 which must hold τ1 + f(x1,..., xn). We show that all of Valiants class VP, and more, can be computed in this model. This significantly extends the framework and techniques of Ben-Or and Cleve [6]. Upper bounding the power of catalytic computation we show that catalytic logspace is contained in ZPP. We further construct an oracle world where catalytic logpace is equal to PSPACE, and show that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.

Reductions to the set of random strings: the resource-bounded case

This paper is motivated by a conjecture [1,5] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [5] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.

Reductions to the set of random strings : The resource-bounded case

This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.

Catalytic space: non-determinism and hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Lower Bounds for Elimination via Weak Regularity

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.

Catalytic Space: Non-determinism and Hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Hardness of Approximation for Knapsack Problems

We show various hardness results for knapsack and related problems; in particular we will show that unless the Exponential-Time Hypothesis is false, subset-sum cannot be approximated any better than with an FPTAS. We also provide new unconditional lower bounds for approximating knapsack in Ketan Mulmuley’s parallel PRAM model. Furthermore, we give a simple new algorithm for approximating knapsack and subset-sum, that can be adapted to work for small space, or in small parallel time.

Learning Reductions to Sparse Sets

We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (Sat \(\leq_m^p \mathrm{LT}_1\)). They claim that P= NP follows as a consequence, but unfortunately their proof was incorrect.

Simulation beats richness: new data-structure lower bounds

We develop a new technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC’94) and Miltersen, Nisan, Safra and Wigderson (STOC’95). At the core of our technique is the first simulation theorem in the asymmetric setting, where Alice gets a p × n matrix x over F2 and Bob gets a vector y ∈ F2n. Alice and Bob need to evaluate f(x· y) for a Boolean function f: {0,1}p → {0,1}. Our simulation theorems show that a deterministic/randomized communication protocol exists for this problem, with cost C· n for Alice and C for Bob, if and only if there exists a deterministic/randomized *parity decision tree* of cost Θ(C) for evaluating f. As applications of this technique, we obtain the following results: 1. The first strong lower-bounds against randomized data-structure schemes for the Vector-Matrix-Vector product problem over F2. Moreover, our method yields strong lower bounds even when the data-structure scheme has tiny advantage over random guessing. 2. The first lower bounds against randomized data-structures schemes for two natural Boolean variants of Orthogonal Vector Counting. 3. We construct an asymmetric communication problem and obtain a deterministic lower-bound for it which is provably better than any lower-bound that may be obtained by the classical Richness Method of Miltersen et al. (STOC ’95). This seems to be the first known limitation of the Richness Method in the context of proving deterministic lower bounds.