arXiv subjects

Luisa Herrmann

Publications and source records attributed to Luisa Herrmann.

6 recordsLinked to original sources

arXiv2026

On smallest synchronizing terms over constant alphabets

We show a subexponential lower bound on the reset threshold of synchronizing deterministic finite tree automata (DTA) over alphabets of just two symbols. This significantly improves the previous one, which was quadratic in the number of states. Our result also narrows the gap towards the lower bound for DTA over alphabets that grow linearly with the number of states, as well as the best known upper bound, both of which are currently exponential.

cs.FL↗

arXiv2024

Non-Global Parikh Tree Automata

Parikh (tree) automata are an expressive and yet computationally well-behaved extension of finite automata -- they allow to increment a number of counters during their computations, which are finally tested by a semilinear constraint. In this work, we introduce and investigate a new perspective on Parikh tree automata (PTA): instead of testing one counter configuration that results from the whole input tree, we implement a non-global automaton model. Here, we copy and distribute the current configuration at each node to all its children, incrementing the counters pathwise, and check the arithmetic constraint at each leaf. We obtain that the classes of tree languages recognizable by global PTA and non-global PTA are incomparable. In contrast to global PTA, the non-emptiness problem is undecidable for non-global PTA if we allow the automata to work with at least three counters, whereas the membership problem stays decidable. However, for a restriction of the model, where counter configurations are passed in a linear fashion to at most one child node, we can prove decidability of the non-emptiness problem.

cs.FL↗

arXiv2024

Global One-Counter Tree Automata

We introduce global one-counter tree automata (GOCTA) which deviate from usual counter tree automata by working on only one counter which is passed through the tree in lexicographical order, rather than duplicating the counter at every branching position. We compare the capabilities of GOCTA to those of counter tree automata and obtain that their classes of recognizable tree languages are incomparable. Moreover, we show that the emptiness problem of GOCTA is undecidable while, in stark contrast, their membership problem is in P.

cs.FL↗

arXiv2023

Decidable (Ac)counting with Parikh and Muller: Adding Presburger Arithmetic to Monadic Second-Order Logic over Tree-Interpretable Structures

We propose $\omega$MSO$\Join$BAPA, an expressive logic for describing countable structures, which subsumes and transcends both Counting Monadic Second-Order Logic (CMSO) and Boolean Algebra with Presburger Arithmetic (BAPA). We show that satisfiability of $\omega$MSO$\Join$BAPA is decidable over the class of labeled infinite binary trees, whereas it becomes undecidable even for a rather mild relaxations. The decidability result is established by an elaborate multi-step transformation into a particular normal form, followed by the deployment of Parikh-Muller Tree Automata, a novel kind of automaton for infinite labeled binary trees, integrating and generalizing both Muller and Parikh automata while still exhibiting a decidable (in fact PSpace-complete) emptiness problem. By means of MSO-interpretations, we lift the decidability result to all tree-interpretable classes of structures, including the classes of finite/countable structures of bounded treewidth/cliquewidth/partitionwidth. We generalize the result further by showing that decidability is even preserved when coupling width-restricted $\omega$MSO$\Join$BAPA with width-unrestricted two-variable logic with advanced counting. A final showcase demonstrates how our results can be leveraged to harvest decidability results for expressive $\mu$-calculi extended by global Presburger constraints.

cs.LO↗

arXiv2017

Weighted Regular Tree Grammars with Storage

We introduce weighted regular tree grammars with storage as combination of (a) regular tree grammars with storage and (b) weighted tree automata over multioperator monoids. Each weighted regular tree grammar with storage generates a weighted tree language, which is a mapping from the set of trees to the multioperator monoid. We prove that, for multioperator monoids canonically associated to particular strong bi-monoids, the support of the generated weighted tree languages can be generated by (unweighted) regular tree grammars with storage. We characterize the class of all generated weighted tree languages by the composition of three basic concepts. Moreover, we prove results on the elimination of chain rules and of finite storage types, and we characterize weighted regular tree grammars with storage by a new weighted MSO-logic.

cs.FL↗

arXiv2015

Linear Context-Free Tree Languages and Inverse Homomorphisms

We prove that the class of linear context-free tree languages is not closed under inverse linear tree homomorphisms. The proof is by contradiction: we encode Dyck words into a context-free tree language and prove that its preimage under a certain linear tree homomorphism cannot be generated by any context-free tree grammar. A positive result can still be obtained: the linear monadic context-free tree languages are closed under inverse linear tree homomorphisms.

cs.FL↗