arXiv subjects
Publications and source records attributed to Szabolcs Ivan.
A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that scattered context-free (regular, resp.) orderings have rank less than $ω^ω$ ($ω$, resp). In this paper we confirm the conjecture that one-counter languages have rank less than $ω^2$.
A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that the isomorphism problem of scattered context-free orderings is undecidable, if one of them has a rank at least two. In this paper we show that it is decidable whether a context-free ordering has rank at most one, and if so, its order type is effectively computable.
A prefix grammar is a context-free grammar whose nonterminals generate prefix-free languages. A prefix grammar $G$ is an ordinal grammar if the language $L(G)$ is well-ordered with respect to the lexicographic ordering. It is known that from a finite system of parametric fixed point equations one can construct an ordinal grammar $G$ such that the lexicographic order of $G$ is isomorphic with the least solution of the system, if this solution is well-ordered. In this paper we show that given an ordinal grammar, one can compute (the Cantor normal form of) the order type of the lexicographic order of its language, yielding that least solutions of fixed point equation systems defining algebraic ordinals are effectively computable (and thus, their isomorphism problem is also decidable).
In this paper we study the problem of fully dynamic maximal matching with lookahead. In a fully dynamic $n$-vertex graph setting, we have to handle updates (insertions and removals of edges), and answer queries regarding the current graph, preferably with a better time bound than that when running the trivial deterministic algorithm with worst-case time of $O(m)$ (where $m$ is the all-time maximum number of the edges) and recompute the matching from scratch each time a query arrives. We show that a maximal matching can be maintained in an (undirected) general graph with a deterministic amortized update cost of $O(\log m)$, provided that a lookahead of length $m$ is available, i.e. we can ``take a peek'' at the next $m$ update operations in advance.
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) for the case assuming union-by-rank strategy for merging. In this paper we provide such a characterization by means of a simple push operation and show that the decision problem whether a given tree (along with the rank info of its nodes) is a Union-Find tree is NP-complete, complementing our earlier similar result for the union-by-size strategy.
In this paper we define future-time branching temporal logics evaluated over forests, that is, ordered tuples of ordered, but unranked, finite trees. We associate a rich class FL[$\mathcal{L}$] of temporal logics to each set L of (regular) modalities. Then, we define an algebraic product operation which we call the Moore product, which operates on forest automata, algebraic devices recognizing forest languages. We show a lattice isomorphism between the pseudovarieties of finite forest automata, closed under the Moore product, and the classes of languages of the form FL[$\mathcal{L}$]. We demonstrate the usefulness of the algebraic approach by showing the decidability of the membership problem of a specific pseudovariety of finite forest automata, implying the decidability of the definability problem of the FL[EF] fragment of the logic CTL. Then, using the same approach, we also formulate a conjecture regarding a decidable characterization of the FL[AF] fragment which has currently an unknown decidability status (also in the setting of ranked trees).
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 $\NP$-complete.
Atoms of a (regular) language $L$ were introduced by Brzozowski and Tamm in 2011 as intersections of complemented and uncomplemented quotients of $L$. They derived tight upper bounds on the complexity of atoms in 2013. In 2014, Brzozowski and Davies characterized the regular languages meeting these bounds. To achieve these results, they used the so-called "atomaton" of a language, introduced by Brzozowski and Tamm in 2011. In this note we give an alternative proof of their characterization, via a purely combinatorial approach.
It has been conjectured in 2011 by Brzozowski et al. that if $K$ and $L$ are factor-free regular languages over a binary alphabet having state complexity $m$ and $n$, resp, then the state complexity of $K\cup L$ is at most $mn-(m+n)+3-\min\{m,n\}$. We disprove this conjecture by giving a lower bound of $mn-(m+n)-2-\lfloor\frac{\min\{m,n\}-2}{2}\rfloor$, which exceeds the conjectured bound whenever $\min\{m,n\}\geq 10$.
We give an upper bound of $n((n-1)!-(n-3)!)$ for the possible largest size of a subsemigroup of the full transformational semigroup over $n$ elements consisting only of nonpermutational transformations. As an application we gain the same upper bound for the syntactic complexity of (generalized) definite languages as well.
We give a Kleene-type operational characterization of Muller context-free languages (MCFLs) of well-ordered and scattered words.
We give a forbidden pattern characterization for the class of generalized definite languages, show that the corresponding problem is NL-complete and can be solved in quadratic time. We also show that their syntactic complexity coincides with that of the definite languages and give an upper bound of n! for this measure.
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 $ω^ω$. As a corollary of this result we obtain that an ordinal is the order type of a well-ordered context-free language iff it is less than $ω^{ω^ω}$.