Seiichi Tani

VC-dimensions of finite automata and commutative finite automata with k letters and n states

Abstract An investigation is conducted of the Vapnik-Chervonenkis dimensions (VC-dimensions) of finite automata having k letters and n states. It is shown for a fixed positive integer k (⩾ 2), that (1) the VC-dimension of DFA k (n):= {L ⊂ {1, 2, …, k}∗ : some deterministic finite automaton with at most n states accepts L} is n + log2 n − O(log logn) for k = 1 and (k − 1 + 0(1))n log2 n for k ⩾ 2, and (2) the VC-dimension of CDFAk(n):= {L ϵ DFAk(n) : L is commutative} is n + o(n).

DEGREES OF THE JONES POLYNOMIALS OF CERTAIN PRETZEL LINKS

We calculate the highest and the lowest degrees of the Kauffman bracket polynomials of certain inadequate pretzel links and show that there is a knot K such that c(K) – r_deg VK=k for any nonnegative integer k, where c(K) is the crossing number of K and r_deg VK is the reduced degree of the Jones polynomial VK of K.

Fast algorithms for computing Jones polynomials of certain links

We give a fast algorithm for computing Jones polynomials of 2-bridge links. Given the Tait graph with n edges of a 2-bridge diagram, this algorithm runs with O(n) additions and multiplications in polynomials of degree O(n), namely in O(n^2logn) time. We also propose an algorithm that, given the Tait graph with n edges of a closed 3-braid diagram, computes the Jones polynomial of the closed 3-braid link in O(n^2logn) time.

A CHARACTERIZATION OF k-TH POWERS Pn,k OF PATHS IN TERMS OF k-TREES

Let G be a k-tree such that |{v ∈ V(G): degG(v) = k}| = 2, n = |V(G)| ≥ 2k + 2, and the maximum degree of G is at most 2k. In this paper, we will show that such a k-tree G is isomorphic to Pn,k. In this way, we give a new characterization of k-th power (i.e. Pn,k) of paths with n vertices in terms of k-trees.

Learning of Restricted RNLC Graph Languages

We consider learning of graph languages (sets of graphs) which are generated by restricted regular node label controlled (for short RNLC) graph grammars. We show that for any restricted RNLC graph languages L, given the Parikh image of L, one can construct a restricted RNLC graph grammar G such that L(G)=L in polynomial time. We present an algorithm to construct a restricted RNLC graph grammar which generates an unknown restricted RNLC graph language using restricted superset queries and restricted subset queries. We also showed that for a fixed nonnegative integer t, this algorithm halts in polynomial time when the Parikh image of the unknown restricted RNLC graph languages has at most t-periods.

Computing HOMFLY polynomials of 2-bridge links from 4-plat representation

We propose an algorithm for computing HOMFLY polynomials of 2-bridge links from 4-plat representation in O ( n 3 ) time, where n is the number of the crossings of a given input diagram.

Spacio-Temporal Analysis Using the Web Archive System Based on Ajax

Ajax is a new technology based on asynchronous communication between a client Web browser and a back-end server, allowing Web applications to request data without Web application to request and receive data ever reloading the page. We designed and implemented the Ajax Web archive application which supports to discover and analyze spacio-temporal structure of a sequence of sentences.