On the Order Type of Scattered Context-Free Orderings
We show that if a context-free grammar generates a language whose lexicographic ordering is well-ordered of type less than $ω^2$, then its order type is effectively computable.
cs.FL↗
arXiv subjects
Kitti Gelle
Publications and source records attributed to Kitti Gelle.
The order type of scattered context-free orderings of rank one is computable
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.
cs.FL↗
The ordinal generated by an ordinal grammar is 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).
cs.FL↗
Maintaning maximal matching with lookahead
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.
cs.DS↗
Reversible Languages Having Finitely Many Reduced Automata
Reversible forms of computations are often interesting from an energy efficiency point of view. When the computation device in question is an automaton, it is known that the minimal reversible automaton recognizing a given language is not necessarily unique, moreover, there are languages having arbitrarily large reversible recognizers possessing no nontrivial reversible congruence. However, the exact characterization of this class of languages was open. In this paper we give a forbidden pattern capturing the reversible regular languages having only finitely many reduced reversible automata, allowing an efficient (NL) decision procedure.
cs.FL↗
Recognizing Union-Find trees built up using union-by-rank strategy 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) 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.
cs.DS↗
Algebraic Characterization of Forest Logics
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).
cs.FL↗
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 $\NP$-complete.
cs.CC↗