Lower Bounds for RAMs and Quantifier Elimination

We are considering RAMs $N_{n}$, with wordlength $n=2^{d}$, whose arithmetic instructions are the arithmetic operations multiplication and addition modulo $2^{n}$, the unary function $\min\lbrace 2^{x}, 2^{n}-1\rbrace$, the binary functions $\lfloor x/y\rfloor$ (with $\lfloor x/0 \rfloor =0$), $\max(x,y)$, $\min(x,y)$, and the boolean vector operations $\wedge,\vee,\neg$ defined on $0,1$ sequences of length $n$. It also has the other RAM instructions. The size of the memory is restricted only by the address space, that is, it is $2^{n}$ words. The RAMs has a finite instruction set, each instruction is encoded by a fixed natural number independently of $n$. Therefore a program $P$ can run on each machine $N_{n}$, if $n=2^{d}$ is sufficiently large. We show that there exists an $\epsilon>0$ and a program $P$, such that it satisfies the following two conditions. (i) For all sufficiently large $n=2^{d}$, if $P$ running on $N_{n}$ gets an input consisting of two words $a$ and $b$, then, in constant time, it gives a $0,1$ output $P_{n}(a,b)$. (ii) Suppose that $Q$ is a program such that for each sufficiently large $n=2^{d}$, if $Q$, running on $N_{n}$, gets a word $a$ of length $n$ as an input, then it decides whether there exists a word $b$ of length $n$ such that $P_{n}(a,b)=0$. Then, for infinitely many positive integers $d$, there exists a word $a$ of length $n=2^{d}$, such that the running time of $Q$ on $N_{n}$ at input $a$ is at least $\epsilon (\log d)^{\frac{1}{2}} (\log \log d)^{-1}$

Datalog vs first-order logic

Our main result is that every datalog query expressible in first-order logic is bounded; in terms of classical model theory it is a kind of compactness theorem for finite structures. In addition, we give some counter-examples delimiting the main result

The White-Box Adversarial Data Stream Model

We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the $L_1$-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our $L_1$-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for {\it randomized} algorithms robust to a white-box adversary. In particular, our results show that for all $p\ge 0$, there exists a constant $C_p>1$ such that any $C_p$-approximation algorithm for $F_p$ moment estimation in insertion-only streams with a white-box adversary requires $\Omega(n)$ space for a universe of size $n$. Similarly, there is a constant $C>1$ such that any $C$-approximation algorithm in an insertion-only stream for matrix rank requires $\Omega(n)$ space with a white-box adversary. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. (Abstract shortened to meet arXiv limits.)Comment: PODS 202

Reachability is harder for directed than for undirected finite graphs

Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “built-in ” relations, such as the successor relation). The proof makes use of Ehrenfeucht-Frai’sse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence. $1. Introduction. If s and t denote distinguished points in a directed (resp. undirected) graph, then we say that a graph is (s, t)-connected if there is a directed (undirected) path from s to t. We sometimes refer to the problem of deciding whether a given directed (undirected) graph with two given points sand t is (s, t)-connected as the directed (undirected) reachability problem

A Public-Key Cryptosystem with Worst-Case/Average-Case Equivalence (Extended Abstract)

We present a probabilistic public key cryptosystem which is secure unless the worst case of the following lattice problem can be solved in polynomial time: "Find the shortest nonzero vector in an n dimensional lattice L where the shortest vector v is unique in the sense that any other vector whose length is at most n c kvk is parallel to v.&quot