85 research outputs found
Varieties of Data Languages
We establish an Eilenberg-type correspondence for data languages, i.e.
languages over an infinite alphabet. More precisely, we prove that there is a
bijective correspondence between varieties of languages recognized by
orbit-finite nominal monoids and pseudovarieties of such monoids. This is the
first result of this kind for data languages. Our approach makes use of nominal
Stone duality and a recent category theoretic generalization of Birkhoff-type
HSP theorems that we instantiate here for the category of nominal sets. In
addition, we prove an axiomatic characterization of weak pseudovarieties as
those classes of orbit-finite monoids that can be specified by sequences of
nominal equations, which provides a nominal version of a classical theorem of
Eilenberg and Sch\"utzenberger
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
Varieties of Data Languages
We establish an Eilenberg-type correspondence for data languages, i.e.
languages over an infinite alphabet. More precisely, we prove that there is a
bijective correspondence between varieties of languages recognized by
orbit-finite nominal monoids and pseudovarieties of such monoids. This is the
first result of this kind for data languages. Our approach makes use of nominal
Stone duality and a recent category theoretic generalization of Birkhoff-type
HSP theorems that we instantiate here for the category of nominal sets. In
addition, we prove an axiomatic characterization of weak pseudovarieties as
those classes of orbit-finite monoids that can be specified by sequences of
nominal equations, which provides a nominal version of a classical theorem of
Eilenberg and Sch\"utzenberger
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