Decision Problems for Deterministic Pushdown Automata on Infinite Words

The article surveys some decidability results for DPDAs on infinite words (omega-DPDA). We summarize some recent results on the decidability of the regularity and the equivalence problem for the class of weak omega-DPDAs. Furthermore, we present some new results on the parity index problem for omega-DPDAs. For the specification of a parity condition, the states of the omega-DPDA are assigned priorities (natural numbers), and a run is accepting if the highest priority that appears infinitely often during a run is even. The basic simplification question asks whether one can determine the minimal number of priorities that are needed to accept the language of a given omega-DPDA. We provide some decidability results on variations of this question for some classes of omega-DPDAs.Comment: In Proceedings AFL 2014, arXiv:1405.527

Synthesis of Deterministic Top-down Tree Transducers from Automatic Tree Relations

We consider the synthesis of deterministic tree transducers from automaton definable specifications, given as binary relations, over finite trees. We consider the case of specifications that are deterministic top-down tree automatic, meaning the specification is recognizable by a deterministic top-down tree automaton that reads the two given trees synchronously in parallel. In this setting we study tree transducers that are allowed to have either bounded delay or arbitrary delay. Delay is caused whenever the transducer reads a symbol from the input tree but does not produce output. We provide decision procedures for both bounded and arbitrary delay that yield deterministic top-down tree transducers which realize the specification for valid input trees. Similar to the case of relations over words, we use two-player games to obtain our results.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Ambiguity, Weakness, and Regularity in Probabilistic B\"uchi Automata

Probabilistic B\"uchi automata are a natural generalization of PFA to infinite words, but have been studied in-depth only rather recently and many interesting questions are still open. PBA are known to accept, in general, a class of languages that goes beyond the regular languages. In this work we extend the known classes of restricted PBA which are still regular, strongly relying on notions concerning ambiguity in classical omega-automata. Furthermore, we investigate the expressivity of the not yet considered but natural class of weak PBA, and we also show that the regularity problem for weak PBA is undecidable

Transforming structures by set interpretations

We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.Comment: 36 page

On Nondeterministic Unranked Tree Automata with Sibling Constraints

We continue the study of bottom-up unranked tree automata with equality and disequality constraints between direct subtrees. In particular, we show that the emptiness problem for the nondeterministic automata is decidable. In addition, we show that the universality problem, in contrast, is undecidable

Resynchronized Uniformization and Definability Problems for Rational Relations

Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of uniformization and definability problems for subclasses of rational relations defined in terms of such synchronization languages. We rephrase known results in this setting and complete the picture by adding several new decidability and undecidability results

Determinization of B\"uchi Automata: Unifying the Approaches of Safra and Muller-Schupp

Determinization of B\"uchi automata is a long-known difficult problem and after the seminal result of Safra, who developed the first asymptotically optimal construction from B\"uchi into Rabin automata, much work went into improving, simplifying or avoiding Safra's construction. A different, less known determinization construction was derived by Muller and Schupp and appears to be unrelated to Safra's construction on the first sight. In this paper we propose a new meta-construction from nondeterministic B\"uchi to deterministic parity automata which strictly subsumes both the construction of Safra and the construction of Muller and Schupp. It is based on a correspondence between structures that are encoded in the macrostates of the determinization procedures - Safra trees on one hand, and levels of the split-tree, which underlies the Muller and Schupp construction, on the other. Our construction allows for combining the mentioned constructions and opens up new directions for the development of heuristics.Comment: Full version of ICALP 2019 pape

Constructing Deterministic Parity Automata from Positive and Negative Examples

We present a polynomial time algorithm that constructs a deterministic parity automaton (DPA) from a given set of positive and negative ultimately periodic example words. We show that this algorithm is complete for the class of $\omega$-regular languages, that is, it can learn a DPA for each regular $\omega$-language. For use in the algorithm, we give a definition of a DPA, that we call the precise DPA of a language, and show that it can be constructed from the syntactic family of right congruences for that language (introduced by Maler and Staiger in 1997). Depending on the structure of the language, the precise DPA can be of exponential size compared to a minimal DPA, but it can also be a minimal DPA. The upper bound that we obtain on the number of examples required for our algorithm to find a DPA for $L$ is therefore exponential in the size of a minimal DPA, in general. However we identify two parameters of regular $\omega$-languages such that fixing these parameters makes the bound polynomial.Comment: Changes from v1: - integrate appendix into paper - extend introduction to cover related work in more detail - add a second (more involved) example - minor change

Decision Problems for Subclasses of Rational Relations over Finite and Infinite Words

We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.Comment: v1: 31 pages, submitted to DMTCS, extended version of the paper with the same title published in the conference proceedings of FCT 2017; v2: 32 pages, minor revision of v1 (DMTCS review process), results unchanged; v3: 32 pages, enabled hyperref for Figure 1; v4: 32 pages, add reference for known complexity results for the slenderness problem; v5: 32 pages, added DMTCS metadat

On Minimization and Learning of Deterministic ω\omega-Automata in the Presence of Don't Care Words

We study minimization problems for deterministic $\omega$-automata in the presence of don't care words. We prove that the number of priorities in deterministic parity automata can be efficiently minimized under an arbitrary set of don't care words. We derive that from a more general result from which one also obtains an efficient minimization algorithm for deterministic parity automata with informative right-congruence (without don't care words). We then analyze languages of don't care words with a trivial right-congruence. For such sets of don't care words it is known that weak deterministic B\"uchi automata (WDBA) have a unique minimal automaton that can be efficiently computed from a given WDBA (Eisinger, Klaedtke 2006). We give a congruence-based characterization of the corresponding minimal WDBA, and show that the don't care minimization results for WDBA do not extend to deterministic $\omega$-automata with informative right-congruence: for this class there is no unique minimal automaton for a given don't care set with trivial right congruence, and the minimization problem is NP-hard. Finally, we extend an active learning algorithm for WDBA (Maler, Pnueli 1995) to the setting with an additional set of don't care words with trivial right-congruence.Comment: Version 2 is a minor revision with a few references added, some additional explanations, and a few typos corrected Version 3: Added "On" to title, and added a reference for Corollary 4.