Szabolcs Iván

On Müller context-free grammars

We define context-free grammars with Muller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Muller context-free languages including closure properties and others. We show that every Muller context-free grammar can be transformed into a normal form grammar in polynomial space, and then we show that many decision problems can be decided in polynomial time for Muller context-free grammars in normal form. These decision problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result, we establish a limitedness property of Muller context-free grammars: if the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.

Büchi context-free languages

We define context-free grammars with Buchi acceptance condition generating languages of countable words. We establish several closure properties and decidability results for the class of Buchi context-free languages generated by these grammars. We also define context-free grammars with Muller acceptance condition and show that there is a language generated by a grammar with Muller acceptance condition which is not a Buchi context-free language.

On Nonpermutational Transformation Semigroups with an Application to Syntactic Complexity

We give an upper bound of nn-1!-n-3! for the possible largestsize of a subsemigroup of the full transformational semigroup overn elements consisting only of nonpermutational transformations.As an application we gain the same upper bound for the syntacticcomplexity of generalized definite languages as well.

Hausdorff rank of scattered context-free linear orders

We consider context-free languages equipped with the lexicographic ordering. We show that when the lexicographic ordering of a context-free language is scattered, then its Hausdorff rank is less than ωω. As an application of this result, we obtain that an ordinal is the order type of the lexicographic ordering of a context-free language if and only if it is less than ωωω.

Context-Free Languages of Countable Words

We define context-free grammars with Buchi acceptance condition generating languages of countable words. We establish several closure properties and decidability results for the class of Buchi context-free languages generated by these grammars. We also define context-free grammars with Muller acceptance condition and show that there is a language generated by a grammar with Muller acceptance condition which is not a Buchi context-free language.

Aperiodicity in tree automata

We define and compare several different notions of aperiodicity in tree automata. We also relate these notions to the cascade product and logical definability of tree languages.

Synchronizing weighted automata.

We introduce two generalizations of synchronizability to automata with transitions weighted in an arbitrary semiring K=(K,+,*,0,1). (or equivalently, to finite sets of matrices in K^nxn.) Let us call a matrix A location-synchronizing if there exists a column in A consisting of nonzero entries such that all the other columns of A are filled by zeros. If additionally all the entries of this designated column are the same, we call A synchronizing. Note that these notions coincide for stochastic matrices and also in the Boolean semiring. A set M of matrices in K^nxn is called (location-)synchronizing if M generates a matrix subsemigroup containing a (location-)synchronizing matrix. The K-(location-)synchronizability problem is the following: given a finite set M of nxn matrices with entries in K, is it (location-)synchronizing? Both problems are PSPACE-hard for any nontrivial semiring. We give sufficient conditions for the semiring K when the problems are PSPACE-complete and show several undecidability results as well, e.g. synchronizability is undecidable if 1 has infinite order in (K,+,0) or when the free semigroup on two generators can be embedded into (K,*,1).

Recognizing Union-Find trees is NP-complete

Disjoint-Set forests, consisting of Union-Find trees are data structures having a widespread practical application due to their efficiency. Despite them being well-known, no exact structural characterization of these trees is known (such a characterization exists for Union trees which are constructed without using path compression). In this paper we provide such a characterization and show that the decision problem whether a given tree is a Union-Find tree is

Recognizing Union-Find Trees Built Up Using Union-By-Rank Strategy is NP-Complete

\NP

Operational Characterization of Scattered MCFLs

-complete.