Inflations of geometric grid classes of permutations

All three authors were partially supported by EPSRC via the grant EP/J006440/1.Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than κ ≈ 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than κ has a rational generating function. This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.PostprintPeer reviewe

Controlled Fuzzy Parallel Rewriting

We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of cells) in which developmental errors may occur. In essence this model is the fuzzy analogue of the derivation-controlled iteration grammar. Under minor assumptions on the family of control languages and on the family of fuzzy languages in the underlying iteration grammar, we show (i) regular control does not provide additional generating power to the model, (ii) the number of fuzzy substitutions in the underlying iteration grammar can be reduced to two, and (iii) the resulting family of fuzzy languages possesses strong closure properties, viz. it is a full hyper-AFFL, i.e., a hyper-algebraically closed full Abstract Family of Fuzzy Languages

A translational theorem for the class of EOL languages

If K is not a context-free language, then sh(K, a*) is not an EOL language (where sh(K1, K2) denotes the shuffle of the languages K1 and K2, and a is a symbol not in the alphabet of K). Hence the class of context-free languages is the largest full AFL inside the class of EOL languages

Partially-commutative context-free languages

The paper is about a class of languages that extends context-free languages (CFL) and is stable under shuffle. Specifically, we investigate the class of partially-commutative context-free languages (PCCFL), where non-terminal symbols are commutative according to a binary independence relation, very much like in trace theory. The class has been recently proposed as a robust class subsuming CFL and commutative CFL. This paper surveys properties of PCCFL. We identify a natural corresponding automaton model: stateless multi-pushdown automata. We show stability of the class under natural operations, including homomorphic images and shuffle. Finally, we relate expressiveness of PCCFL to two other relevant classes: CFL extended with shuffle and trace-closures of CFL. Among technical contributions of the paper are pumping lemmas, as an elegant completion of known pumping properties of regular languages, CFL and commutative CFL.Comment: In Proceedings EXPRESS/SOS 2012, arXiv:1208.244