9,222 research outputs found
On the Learnability of Shuffle Ideals
PAC learning of unrestricted regular languages is long known to be a difficult problem. The class of shuffle ideals is a very restricted subclass of regular languages, where the shuffle ideal generated by a string u is the collection of all strings containing u as a subsequence. This fundamental language family is of theoretical interest in its own right and provides the building blocks for other important language families. Despite its apparent simplicity, the class of shuffle ideals appears quite difficult to learn. In particular, just as for unrestricted regular languages, the class is not properly PAC learnable in polynomial time if RP 6= NP, and PAC learning the class improperly in polynomial time would imply polynomial time algorithms for certain fundamental problems in cryptography. In the positive direction, we give an efficient algorithm for properly learning shuffle ideals in the statistical query (and therefore also PAC) model under the uniform distribution.T-Party Projec
On shuffle ideals of general algebras
We extend a word language concept called shuffle ideal to general algebras. For this purpose, we introduce the relation SH and show that there exists a natural connection between this relation and the homeomorphic embedding order on trees. We establish connections between shuffle ideals, monotonically ordered algebras and automata, and piecewise testable tree languages
On shuffle products, acyclic automata and piecewise-testable languages
We show that the shuffle L \unicode{x29E2} F of a piecewise-testable
language and a finite language is piecewise-testable. The proof relies
on a classic but little-used automata-theoretic characterization of
piecewise-testable languages. We also discuss some mild generalizations of the
main result, and provide bounds on the piecewise complexity of L
\unicode{x29E2} F
Generalized Matsumoto-Tits sections and quantum quasi-shuffle algebras
In this paper generalized Matsumoto-Tits sections lifting permutations to the
algebra associated to a generalized virtual braid monoid are defined. They are
then applied to study the defining relations of the quantum quasi-shuffle
algebras via the total symmetrization operator.Comment: 18 page
Anick-type resolutions and consecutive pattern avoidance
For permutations avoiding consecutive patterns from a given set, we present a
combinatorial formula for the multiplicative inverse of the corresponding
exponential generating function. The formula comes from homological algebra
considerations in the same sense as the corresponding inversion formula for
avoiding word patterns comes from the well known Anick's resolution.Comment: 16 pages. Preliminary version, comments are welcom
On defining ideals and differential algebras of Nichols algebras
This paper is devoted to understanding the defining ideal of a Nichols
algebra from the decomposition of specific elements in the group algebra of
braid groups. A family of primitive elements are found and algorithms are
proposed. To prove the main result, the differential algebra of a Nichols
algebra is constructed. Moreover, another point of view on Serre relations is
provided.Comment: 34 pages, including a list of notation
- …