Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

A Polynomial Number of Random Points does not Determine the Volume of a Convex Body

We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in R^n, can approximate the volume of the body up to a constant factor with high probability

Computational analogues of entropy

Abstract. Min-entropy is a statistical measure of the amount of randomness that a particular distribution contains. In this paper we investigate the notion of computational min-entropy which is the computational analog of statistical min-entropy. We consider three possible definitions for this notion, and show equivalence and separation results for these definitions in various computational models. We also study whether or not certain properties of statistical min-entropy have a computational analog. In particular, we consider the following questions: 1. Let X be a distribution with high computational min-entropy. Does one get a pseudo-random distribution when applying a “randomness extractor ” on X? 2. Let X and Y be (possibly dependent) random variables. Is the computational min-entropy of (X, Y) at least as large as the computational min-entropy of X? 3. Let X be a distribution over {0, 1} n that is “weakly unpredictable” in the sense that it is hard to predict a constant fraction of the coordinates of X with a constant bias. Does X have computational min-entropy Ω(n)? We show that the answers to these questions depend on the computational model considered. In some natural models the answer is false and in others the answer is true. Our positive results for the third question exhibit models in which the “hybrid argument bottleneck ” in “moving from a distinguisher to a predictor ” can be avoided.

When Homomorphism Becomes a Liability

We show that an encryption scheme cannot have a simple decryption circuit and be homomorphic at the same time. Specifically, if a scheme can homomorphically evaluate the majority function, then its decryption circuit cannot be a linear function of the secret key (or even a succinct polynomial), even if decryption error is allowed. An immediate corollary is that known schemes that are based on the hardness of decoding in the presence of noise with low hamming weight cannot be fully homomorphic. This applies to known schemes such as LPN-based symmetric or public key encryption. An additional corollary is that the recent candidate fully homomorphic encryption, suggested by Bogdanov and Lee (ePrint ’11, henceforth BL), is insecure. In fact, we show two attacks on the BL scheme: One by applying the aforementioned general statement, and another by directly attacking one of the components of the scheme. An encryption scheme is called homomorphic if there is an efficient transformation that given Enc(m) for some message m, and a function f, produces Enc(f(m)) using only public information

Local contrast and maintained generalization.

Pigeons received variable-interval reinforcement for key pecking during presentations of horizontal and vertical line-orientation stimuli, while pecks during five intermediate orientations were extinguished. Lowest peck rates were observed during presentations of negative stimuli adjacent to the positive orientations while peck rate during 45 degrees (the intermediate negative orientation) was relatively high, i.e., there were negative contrast shoulders. When peck rates were manipulated in the positive orientations, peck rate in neithboring orientations changed in the opposite direction. Contrast shoulders faded after prolonged training. A second type of contrast, local contrast, was correlated with similarity of preceding stimulus and different average peck rates during different stages of the discrimination process. The data suggest that sequential local contrast accompanying the formation of a discrimination contributes to the form of generalization gradients. Blough's model of stimulus control predicts the changes in gradient form described here, but may not accurately depict the underlying process responsible for gradient form