Partial Occam's Razor and its Applications

We introduce the notion of "partial Occam algorithm". A partial Occam algorithm produces a succinct hypothesis that is partially consistent with given examples, where the proportion of consistent examples is a bit more than half. By using this new notion, we propose one approach for obtaining a PAC learning algorithm. First, as shown in this paper, a partial Occam algorithm is equivalent to a weak PAC learning algorithm. Then by using boosting techniques of Schapire or Freund, we can obtain an ordinary PAC learning algorithm from this weak PAC learning algorithm. We demonstrate some examples that some improvement is possible by this approach. First we obtain a new (non-proper) PAC learning algorithm for k-DNF, which has similar sample complexity as Littlestone's Winnow, but produces hypothesis of size polynomial in d and log k for a k-DNF target with n variables and d terms (Cf. The hypothesis size of Winnow is O(n k )). Next we show that 1decision lists of length d with n variable..

On the Depth of Randomly Generated Circuits

This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean circuits with descriptions length d that consist of gates with a fixed fan-in f and a constant number of inputs. Assuming uniform distribution of descriptions, we show that such a circuit has expected depth O(log d). This improves on the best known result. More precisely, we prove for circuits of size n their depth is asymptotically ef ln n with extremely high probability. Our proof uses the coupling technique to bound circuit depth from above and below by those of two alternative discrete-time processes. We are able to establish the result by embedding the processes in suitable continuous-time branching processes. As a simple consequence of our result we obtain that monotone CVP is in the class average NC. Key Words: random circuits, depth, recursive trees, domination by coupling, continuous Poisson process. 1 The Problem and Motivation A circuit is a ..

Hardness of Approximation for Langton’s Ant on a Twisted Torus

Langton’s ant is a deterministic cellular automaton studied in many fields, artificial life, computational complexity, cryptography, emergent dynamics, Lorents lattice gas, and so forth, motivated by the hardness of predicting the ant’s macroscopic behavior from an initial microscopic configuration. Gajardo, Moreira, and Goles (2002) proved that Langton’s ant is PTIME -hard for reachability. On a twisted torus, we demonstrate that it is PSPACE hard to determine whether the ant will ever visit almost all vertices or nearly none of them

Computing phylogenetic roots with bounded degrees and errors is NP-complete

AbstractIn this paper we study the computational complexity of the following optimization problem: given a graph G=(V,E), we wish to find a tree T such that (1) the degree of each internal node of T is at least 3 and at most Δ, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E and {{u,v}:u,v are leaves of T and dT(u,v)≤k}, is as small as possible, where dT(u,v) denotes the distance between u and v in tree T. We show that this problem is NP-hard for all fixed constants Δ,k≥3.Let sΔ(k) be the size of the largest clique for which an error-free tree T exists. In the course of our proof, we will determine all trees (possibly with degree 2 nodes) that approximate the (sΔ(k)-1)-clique by errors at most 2