46,601 research outputs found
Regular tree languages and quasi orders
Regular languages were characterized as sets closed with respect to monotone well-quasi orders. A similar result is proved here for tree languages. Moreover, families of quasi orders that correspond to positive varieties of tree languages and varieties of finite ordered algebras are characterized
On regularity of context-free languages
AbstractThis paper considers conditions under which a context-free language is regular and conditions which imposed on (productions of) a rewriting system generating a context-free language will guarantee that the generated language is regular. In particular: 1.(1) necessary and sufficient conditions on productions of a unitary grammar are given that guarantee the generated language to be regular (a unitary grammar is a semi-Thue system in which the left-hand of each production is the empty word), and2.(2) it is proved that commutativity of a linear language implies its regularity. To obtain the former result, we give a generalization of the Myhill–Nerode characterization of the regular languages in terms of well-quasi orders, along with a generalization of Higman's well-quasi order result concerning the subsequence embedding relation on Σ*. In obtaining the latter results, we introduce the class of periodic languages, and demonstrate how they can be used to characterize the commutative regular languages. Here we also utilize the theory of well-quasi orders
The FC-rank of a context-free language
We prove that the finite condensation rank (FC-rank) of the lexicographic
ordering of a context-free language is strictly less than
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
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