58 research outputs found
On the additive theory of prime numbers II
The undecidability of the additive theory of primes (with identity) as well
as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are
open questions. As a possible approach, we extend the latter theory by adding
some extra function. In this direction we show the undecidability of the
existential part of the theory Th(N, +, n -> p\_n, n -> r\_n), where r\_n is
the remainder of p\_n divided by n in the euclidian division
Tree inclusions in windows and slices
is an embedded subtree of if can be obtained by deleting some nodes from : if a node is deleted, all edges adjacent to are also deleted, and outgoing edges are replaced by edges going from the parent of (if it exists) to the children of . Deciding whether is an embedded subtree of is known to be NP-complete. Given two trees (a target and a pattern ) and a natural number , we address two problems: 1. counting the number of windows of having height exactly and containing pattern as an embedded subtree, and 2. counting the number of slices of having height exactly and containing pattern as an embedded subtree
Multiple serial episode matching
12In a previous paper we generalized the Knuth-Morris-Pratt (KMP) pattern matching algorithm and defined a non-conventional kind of RAM, the MP--RAMs (RAMS equipped with extra operations), and designed an on-line algorithm for solving the serial episode matching problem on MP--RAMs when there is only one single episode. We here give two extensions of this algorithm to the case when we search for several patterns simultaneously and compare them. More preciseley, given strings (a text of length and patterns ) and a natural number , the {\em multiple serial episode matching problem} consists in finding the number of size windows of text which contain patterns as subsequences, i.e. for each , if , the letters occur in the window, in the same order as in , but not necessarily consecutively (they may be interleaved with other letters).} The main contribution is an algorithm solving this problem on-line in time
The algebra of binary trees is affine complete
A function on an algebra is congruence preserving if, for any congruence, it
maps pairs of congruent elements onto pairs of congruent elements. We show that
on the algebra of binary trees whose leaves are labeled by letters of an
alphabet containing at least three letters, a function is congruence preserving
if and only if it is polynomial.Comment: 9 pages, 1 figur
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