184,260 research outputs found
Colored operads, series on colored operads, and combinatorial generating systems
We introduce bud generating systems, which are used for combinatorial
generation. They specify sets of various kinds of combinatorial objects, called
languages. They can emulate context-free grammars, regular tree grammars, and
synchronous grammars, allowing us to work with all these generating systems in
a unified way. The theory of bud generating systems uses colored operads.
Indeed, an object is generated by a bud generating system if it satisfies a
certain equation in a colored operad. To compute the generating series of the
languages of bud generating systems, we introduce formal power series on
colored operads and several operations on these. Series on colored operads are
crucial to express the languages specified by bud generating systems and allow
us to enumerate combinatorial objects with respect to some statistics. Some
examples of bud generating systems are constructed; in particular to specify
some sorts of balanced trees and to obtain recursive formulas enumerating
these.Comment: 48 page
Homology and closure properties of autostackable groups
Autostackability for finitely presented groups is a topological property of
the Cayley graph combined with formal language theoretic restrictions, that
implies solvability of the word problem. The class of autostackable groups is
known to include all asynchronously automatic groups with respect to a
prefix-closed normal form set, and all groups admitting finite complete
rewriting systems. Although groups in the latter two classes all satisfy the
homological finiteness condition , we show that the class of
autostackable groups includes a group that is not of type . We also show
that the class of autostackable groups is closed under graph products and
extensions.Comment: 20 page
Replica methods for loopy sparse random graphs
I report on the development of a novel statistical mechanical formalism for
the analysis of random graphs with many short loops, and processes on such
graphs. The graphs are defined via maximum entropy ensembles, in which both the
degrees (via hard constraints) and the adjacency matrix spectrum (via a soft
constraint) are prescribed. The sum over graphs can be done analytically, using
a replica formalism with complex replica dimensions. All known results for
tree-like graphs are recovered in a suitable limit. For loopy graphs, the
emerging theory has an appealing and intuitive structure, suggests how message
passing algorithms should be adapted, and what is the structure of theories
describing spin systems on loopy architectures. However, the formalism is still
largely untested, and may require further adjustment and refinement.Comment: 11 pages, no figures. To be published in Proceedings of The
International Meeting on High-Dimensional Data-Driven Science (HD3-2015),
Kyoto, Japan, on 14-17 December, 201
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