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Weighted automata is a basic tool for specification in quantitative verification, which allows to express quantitative features of analysed systems such as resource consumption. Quantitative specification can be assisted by automata learning as there are classic results on Angluin-style learning of weighted automata. The existing work assumes perfect information about the values returned by the target weighted automaton. In assisted synthesis of a quantitative specification, knowledge of the exact values is a strong assumption and may be infeasible. In our work, we address this issue by introducing a new framework of partially-observable deterministic weighted automata, in which weighted automata return intervals containing the computed values of words instead of the exact values. We study the basic properties of this framework with the particular focus on the challenges of

Linear Temporal Logic (LTL) interpreted on finite traces is a robust specification framework popular in formal verification. However, despite the high interest in the logic in recent years, the topic of their quantitative extensions is not yet fully explored. The main goal of this work is to study the effect of adding weak forms of percentage constraints (e.g. that most of the positions in the past satisfy a given condition, or that sigma is the most-frequent letter occurring in the past) to fragments of LTL. Such extensions could potentially be used for the verification of influence networks or statistical reasoning. Unfortunately, as we prove in the paper, it turns out that percentage extensions of even tiny fragments of LTL have undecidable satisfiability and model-checking problems. Our undecidability proofs not only sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple. We also show that the undecidability can be avoided by restricting the allowed usage of the negation, and briefly discuss how the undecidability results transfer to first-order logic on words., Comment: Full version of FOSSACS 2021 paper

We propose a new approach to querying graph databases. Our approach balances competing goals of expressive power, language clarity and computational complexity. A distinctive feature of our approach is the ability to express properties of minimal (e.g. shortest) and maximal (e.g. most valuable) paths satisfying given criteria. To express complex properties in a modular way, we introduce labelling-generating ontologies. The resulting formalism is computationally attractive - queries can be answered in non-deterministic logarithmic space in the size of the database., Comment: arXiv admin note: text overlap with arXiv:1710.04419

We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables. The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose approximation algorithms for the probabilistic questions, which work in exponential time in the size of the automaton and polynomial time in the size of the Markov chain and the given precision. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric.

Limit-average automata are weighted automata on infinite words that use average to aggregate the weights seen in infinite runs. We study approximate learning problems for limit-average automata in two settings: passive and active. In the passive learning case, we show that limit-average automata are not PAC-learnable as samples must be of exponential-size to provide (with good probability) enough details to learn an automaton. We also show that the problem of finding an automaton that fits a given sample is NP-complete. In the active learning case, we show that limit-average automata can be learned almost-exactly, i.e., we can learn in polynomial time an automaton that is consistent with the target automaton on almost all words. On the other hand, we show that the problem of learning an automaton that approximates the target automaton (with perhaps fewer states) is NP-complete. The abovementioned results are shown for the uniform distribution on words. We briefly discuss learning over different distributions., Comment: This is a full version of a paper accepted to CONCUR 2019

We study multimodal logics over universally first-order definable classes of frames. We show that even for bimodal logics, there are universal Horn formulas that define set of frames such that the satisfiability problem is undecidable, even if one or two of the binary relations are transitive.

We study the average stack cost of Buechi pushdown automata (Buechi PDA). We associate a non-negative price with each stack symbol and define the cost of a stack as the sum of costs of all its elements. We introduce and study the average stack cost problem (ASC), which asks whether there exists an accepting run of a given Buechi PDA such that the long-run average of stack costs is below some given threshold. The ASC problem generalizes mean-payoff objective and can be used to express quantitative properties of pushdown systems. In particular, we can compute the average response time using the ASC problem. We show that the ASC problem can be solved in polynomial time., Comment: A conference version of this paper has been accepted to FSTTCS 2017

Querying graph databases has recently received much attention. We propose a new approach to this problem, which balances competing goals of expressive power, language clarity and computational complexity. A distinctive feature of our approach is the ability to express properties of minimal (e.g. shortest) and maximal (e.g. most valuable) paths satisfying given criteria. To express complex properties in a modular way, we introduce labelling-generating ontologies. The resulting formalism is computationally attractive -- queries can be answered in non-deterministic logarithmic space in the size of the database., Comment: A conference version fo this paper has been accepted to FSTTCS 2017

The Epistemic Halpern-Shoham logic (EHS) is a temporal-epistemic logic that combines the interval operators of the Halpern-Shoham logic with epistemic modalities. The semantics of EHS is based on interpreted systems whose labelling function is defined on the endpoints of intervals. We show that this definition can be generalised by allowing the labelling function to be based on the whole interval by means of regular expressions. We prove that all the positive results known for EHS, notably the attractive complexity of its model checking problem for some of its fragments, still hold for its generalisation. We also propose the new logic EHSre which operates on standard Kripke structures and has expressive power equivalent to that of EHS with regular expressions. We compare the expressive power of EHSre with standard temporal logics., Comment: 19 pages

We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our attention to finite structures. Many decidability and undecidability results for the elementary modal logics were proved separately for general satisfiability and for finite satisfiability [11, 12, 16, 17]. In this paper, we show that there is a reason why we must deal with both kinds of satisfiability separately -- we prove that there is a universal first-order formula that defines an elementary modal logic with decidable (global) satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal formula that defines an elementary modal logic with decidable finite satisfiability problem, but undecidable general satisfiability problem., Comment: In Proceedings GandALF 2012, arXiv:1210.2028

We study three different kinds of embeddings of tree patterns: weakly-injective, ancestor-preserving, and lca-preserving. While each of them is often referred to as injective embedding, they form a proper hierarchy and their computational properties vary (from P to NP-complete). We present a thorough study of the complexity of the model checking problem i.e., is there an embedding of a given tree pattern in a given tree, and we investigate the impact of various restrictions imposed on the tree pattern: bound on the degree of a node, bound on the height, and type of allowed labels and edges., Comment: Under conference submission

The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We contribute to this effort by showing - what we believe is quite an unexpected result - that the logic of subintervals, the fragment of the Halpern-Shoham where only the operator "during", or D, is allowed, is undecidable over discrete structures. This is surprising as this logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures.

It is well known, and easy to see, that not each nondeterministic Buchi automaton on infinite words can be simulated by a nondeterministic co-Buchi automaton. We show that in the cases when such a simulation is possible, the number of states needed for it can grow nonlinearly. More precisely, we show a sequence of - as we believe, simple and elegant - languages which witness the existence of a nondeterministic Buchi automaton with n states, which can be simulated by a nondeterministic co-Buchi automaton, but cannot be simulated by any nondeterministic co-Buchi automaton with less than c*n^{7/6} states for some constant c. This improves on the best previously known lower bound of 3(n-1)/2., Comment: 12 pages, 4 figures; added information about grand