Alternating register automata on finite words and trees

We study alternating register automata on data words and data trees in relation to logics. A data word (resp. data tree) is a word (resp. tree) whose every position carries a label from a finite alphabet and a data value from an infinite domain. We investigate one-way automata with alternating control over data words or trees, with one register for storing data and comparing them for equality. This is a continuation of the study started by Demri, Lazic and Jurdzinski. From the standpoint of register automata models, this work aims at two objectives: (1) simplifying the existent decidability proofs for the emptiness problem for alternating register automata; and (2) exhibiting decidable extensions for these models. From the logical perspective, we show that (a) in the case of data words, satisfiability of LTL with one register and quantification over data values is decidable; and (b) the satisfiability problem for the so-called forward fragment of XPath on XML documents is decidable, even in the presence of DTDs and even of key constraints. The decidability is obtained through a reduction to the automata model introduced. This fragment contains the child, descendant, next-sibling and following-sibling axes, as well as data equality and inequality tests

Focus-style proofs for the two-way alternation-free μ\mu-calculus

We introduce a cyclic proof system for the two-way alternation-free modal $\mu$-calculus. The system manipulates one-sided Gentzen sequents and locally deals with the backwards modalities by allowing analytic applications of the cut rule. The global effect of backwards modalities on traces is handled by making the semantics relative to a specific strategy of the opponent in the evaluation game. This allows us to augment sequents by so-called trace atoms, describing traces that the proponent can construct against the opponent's strategy. The idea for trace atoms comes from Vardi's reduction of alternating two-way automata to deterministic one-way automata. Using the multi-focus annotations introduced earlier by Marti and Venema, we turn this trace-based system into a path-based system. We prove that our system is sound for all sequents and complete for sequents not containing trace atoms.Comment: To appear in proceedings of WoLLIC 202

The Complexity of Flat Freeze LTL

We consider the model-checking problem for freeze LTL on one-counter automata (OCAs). Freeze LTL extends LTL with the freeze quantifier, which allows one to store different counter values of a run in registers so that they can be compared with one another. As the model-checking problem is undecidable in general, we focus on the flat fragment of freeze LTL, in which the usage of the freeze quantifier is restricted. Recently, Lechner et al. showed that model checking for flat freeze LTL on OCAs with binary encoding of counter updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in unary or binary. Like Lechner et al., we rely on a reduction to the reachability problem in OCAs with parameterized tests (OCAPs). The new aspect is that we simulate OCAPs by alternating two-way automata over words. This implies an exponential upper bound on the parameter values that we exploit towards an NP algorithm for reachability in OCAPs with unary updates. We obtain our main result as a corollary

Automaten und Logiken zur Beschreibung zeitabhängiger Systeme

When speaking of a 'real-time system' we are interested in a system's evolution in time where time is viewed as linear and measured in terms of non-negative real numbers. The thesis deals with automata-theoretic models of real-time systems and their description in monadic second-order and temporal logics. A parametrized automaton model is introduced and for this model a logical description in terms of a family of existential monadic second-order logics is obtained. This characterization is used to give a logical description of the behaviour of well-known models of real-time systems: timed automata (Alur & Dill), timed automata with halting feature, and linear hybrid automata. The corresponding logics incorporate distance, duration, and integration formulas, respectively. For instance, timed automata are captured by the {\em monadic logic of relative distance.} Its signature contains for every relation symbol ~ such as =, , $=$, or and every natural number k a binary predicate d(.,.)~k taking a set of natural numbers and a single natural number as arguments. The atomic formula d(X,y)~k is true in a timed state sequence if X contains a position smaller than y and the distance (in time) between position y and the last position before y that belongs to X satisfies the condition ~k. The monadic logic of relative distance turns out to have two important properties. First, its satisfiability problem is decidable, for its equivalence to timed automata allows a reduction of the satisfiability problem to the emptiness problem for such automata and this, in turn, is decidable due to Alur and Dill. Second, the monadic logic of relative distance is a powerful logic. One evidence for this is given by showing that the logic is strictly more expressive than the most powerful logic (for the specification of real-time systems) previously known to be decidable, namely the logic MITL^P introduced by Alur and Henzinger. By effectively embedding the latter logic in the former an alternative proof of Alur's and Henzinger's decidability result concerning MITL^P is obtained. Using embedding techniques also the decidability of Manna's and Pnueli's logic TL_Gamma is proved. Timed automata and the languages recognised by them, the so-called timed regular languages, are analysed in more detail. Several aspects are considered. A pumping lemma for timed automata is given, resulting in a formal proof that timed regular languages are not closed under complementation. It is shown that the number of clocks used in timed automata gives rise to an infinite hierarchy of timed regular languages, that the minimal number of clocks required for the recognition of a timed regular language is not computable, and that the property of a two-way timed automaton (Alur & Henzinger) to be reversal bounded is undecidable. Furthermore, unambiguous timed automata are considered, and an inherently ambiguous language is presented. Finally, variations of the emptiness problem for the three types of automata aforementioned and different restrictions concerning the event duration (bounded variation, minimal duration, and unit duration) are discussed. In particular, it is shown that bounded variation leads to a decidable emptiness problem in the case of timed automata, which implies that the full monadic logic of distance is decidable when restricted to timed state sequences of bounded variation. The obtained undecidability results give evidence that the monadic logic of relative distance is a good choice with respect to expressiveness and the requirement of a decidable satisfiability problem

The Complexity of Flat Freeze LTL

We consider the model-checking problem for freeze LTL on one-counter automata (OCAs). Freeze LTL extends LTL with the freeze quantifier, which allows one to store different counter values of a run in registers so that they can be compared with one another. As the model-checking problem is undecidable in general, we focus on the flat fragment of freeze LTL, in which the usage of the freeze quantifier is restricted. Recently, Lechner et al. showed that model checking for flat freeze LTL on OCAs with binary encoding of counter updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in unary or binary. Like Lechner et al., we rely on a reduction to the reachability problem in OCAs with parameterized tests (OCAPs). The new aspect is that we simulate OCAPs by alternating two-way automata over words. This implies an exponential upper bound on the parameter values that we exploit towards an NP algorithm for reachability in OCAPs with unary updates. We obtain our main result as a corollary

Synchronization Problems in Automata without Non-trivial Cycles

We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We investigate the complexity of finding a synchronizing set of states of maximum size. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889. Conference version was published at CIAA 2017, LNCS vol. 10329, pages 188-200, 201

Reachability in Higher-Order-Counters

Higher-order counter automata (\HOCS) can be either seen as a restriction of higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of \HOCS: those that can test whether the topmost counter value is zero and those which cannot. We show that control-state reachability for level $k$ \HOCS with $0$-test is complete for \mbox{$(k-2)$}-fold exponential space; leaving out the $0$-test leads to completeness for \mbox{$(k-2)$}-fold exponential time. Restricting \HOCS (without $0$-test) to level $2$, we prove that global (forward or backward) reachability analysis is \PTIME-complete. This enhances the known result for pushdown systems which are subsumed by level $2$ \HOCS without $0$-test. We transfer our results to the formal language setting. Assuming that \PTIME \subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet's constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201