Tatsuie Tsukiji

A Limit Law for Outputs in Random Recursive Circuits

Abstract. We study the structure of uniform random binary recursive circuits. We show that a suitably normalized version of the number of outputs converges in distribution to a normal random variate. We also discuss the connection of the number of outputs to a non-classical urn model, and our investigation provides a first solved instance of this new class of urns.

Finding Relevant Variables in PAC Model with Membership Queries

A new research frontier in AI and data mining seeks to develop methods to automatically discover relevant variables among many irrelevant ones. In this paper, we present four algorithms that output such crucial variables in PAC model with membership queries. The first algorithm executes the task under any unknown distribution by measuring the distance between virtual and real targets. The second algorithm exhausts virtual version space under an arbitrary distribution. The third algorithm exhausts universal set under the uniform distribution. The fourth algorithm measures influence of variables under the uniform distribution. Knowing the number r of relevant variables, the first algorithm runs in almost linear time for r. The second and the third ones use less membership queries than the first one, but runs in time exponential for r. The fourth one enumerates highly influential variables in quadratic time for r.

Computing bounded-degree phylogenetic roots of disconnected graphs

The Phylogenetic kth Root Problem (PRk) is the problem of finding a (phylogenetic) tree T from a given graph G = (V, E) such that (1)T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are exactly the elements of V, and (3) (u, v) ∈ E if and only if the distance between u and v in tree T is at most k, where k is some fixed threshold k. Such a tree T, if exists, is called a phylogenetic kth root of graph G. The computational complexity of PRk is open, except for k ≤ 4. Recently, Chen et al. investigated PRk under a natural restriction that the maximum degree of the phylogenetic root is bounded from above by a constant. They presented a linear-time algorithm that determines if a given connected G has such a phylogenetic kth root, and if so, demonstrates one. In this paper, we supplement their work by presenting a linear-time algorithm for disconnected graphs.

Computing Phylogenetic Roots with Bounded Degrees and Errors Is Hard

The Degree-ΔClosest Phylogenetic k th Root Problem (ΔCPR k ) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) the degree of each internal node of T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, |E ⊕ {{u,v} : u,v are leaves of T and d T (u,v) ≤ k}| does not exceed a given number, where d T (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.

The Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems

In this paper we separate many-one reducibility from truth-table reducibility for distributional problems in DistNP under the hypothesis that P≠NP . As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT with a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT with a less redundant distribution unless P = NP .We extend this separation result and define a distributional complexity class C with the following properties:(1) C is a subclass of DistNP, this relation is proper unless P = NP.(2) C contains DistP, but it is not contained in AveP unless DistNP\subseteq AveZPP.(3) C has a ≤pm -complete set.(4) C has a ≤ptt -complete set that is not ≤pm -complete unless P = NP.This shows that under the assumption that P ≠ NP, the two completeness notions differ on some nontrivial subclass of DistNP.

Limit laws for terminal nodes in random circuits with restricted fan-out: a family of graphs generalizing binary search trees

Abstract.We introduce a family of graphs C(n,i,s,a) that generalizes the binary search tree. The graphs represent logic circuits with fan-in i, restricted fan-out s, and arising by n progressive additions of random gates to a starting circuit of a isolated nodes. We show via martingales that a suitably normalized version of the number of terminal nodes in binary circuits converges in distribution to a normal random variate.

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

In 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.

Learning DNF by approximating inclusion-exclusion formulae

We analyze upper and lower bounds on size of Boolean conjunctions necessary and sufficient to approximate a given DNF formula by accuracy slightly better than 1/2 (here we define the size of a Boolean conjunction as the number of distinct variables on which it depends). Such an analysis determines the performance of a naive search algorithm that exhausts Boolean conjunctions in the order of their sizes. In fact, our analysis does not depend on kinds of symmetric functions to be exhausted: instead of conjunctions, counting either disjunctions, parity functions, majority functions, or even general symmetric functions, derives the same learning results from similar analyses.

Recognizing the Repeatable Configurations of Time-Reversible Generalized Langton’s Ant Is PSPACE-Hard

Chris Langton proposed a model of an artificial life that he named “ant”: an agent- called ant- that is over a square of a grid moves by turning to the left (or right) accordingly to black (or white) color of the square where it is heading, and the square then reverses its color. Bunimovich and Troubetzkoy proved that an ant’s trajectory is always unbounded, or equivalently, there exists no repeatable configuration of the ant’s system. On the other hand, by introducing a new type of color where the ant goes straight ahead and the color never changes, repeatable configurations are known to exist. In this paper, we prove that determining whether a given finite configuration of generalized Langton’s ant is repeatable or not is PSPACE-hard. We also prove the PSPACE-hardness of the ant’s problem on a hexagonal grid.

On the internal structure of random recursive circuits

We study the joint probability distribution of the number of nodes of fan-out k in random recursive circuits. For suitable norming we obtain a limiting multivariate normal distribution for the numbers of node of fan-out at most k, where we compute explicitly the limiting covariance matrix by solving a recurrence satisfied among its entries.