Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

Conservative Extensions in the Lightweight Description Logic $\mathcal{EL}$

Abstract: We bring together two recent trends in description logic (DL): lightweight DLsin which the subsumption problem is tractable and conservative extensionsas a central tool for formalizing notions of ontology design such as refinement and modularity. Our aim is to investigate conservative extensions as an automated reasoning problem for the basic tractable DL $\mathcal{EL}$. The main result is that deciding (deductive) conservative extensions is ExpTime -complete, thus more difficult than subsumption in $\mathcal{EL}$, but not more difficult than subsumption in expressive DLs. We also show that if conservative extensions are defined model-theoretically, the associated decision problem for $\mathcal{EL}$ is undecidable.

Automatic Verification of Recursive Procedures with One Integer Parameter

Abstract: Context-free processes (BPA) have been used for dataflow analysis in recursive procedures with applications in optimizing compilers (Proceedings of FOSSaCS'99, Lecture Notes in Computer Science, Vol. 1578, Springer, Berlin, 1999, pp. 14-30). We introduce a more refined model called BPA(Z) that can model not only recursive dependencies, but also the passing of an integer parameter to a subroutine. Moreover, this parameter can be tested against conditions expressible in Presburger arithmetic. This new and more expressive model can still be analyzed automatically. We define Z-input 1-CM, a new class of 1-counter machines (cm) that take integer numbers as input, to describe sets of configurations of BPA(Z). We show that the Post* (the set of successors) of a set of BPA(Z)-configurations described by a Z-input 1-CM can be effectively constructed. The Pre* (set of predecessors) of a regular set can be effectively constructed as well. However, the Pre* of a set described by a Z-input 1-CM cannot be represented by a Z-input 1-CM, in general, and has an undecidable membership problem. Then we develop a new temporal logic based on reversal-bounded counter machines (i.e. machines which use counters such that the change between increasing and decreasing mode of each counter is bounded (J. Assoc. Comput. Mach. 25 (1978) 116) that can be used to describe properties of BPA(Z) and show that the model-checking problem is decidable.

Real-Time Model-Checking: Parameters Everywhere

Abstract: In this paper, we study the model-checking and parameter synthesis problems of the logic TCTL over discrete-timed automata where parameters are allowed both in the model (timed automaton) and in the property (temporal formula). Our results are as follows. On the negative side, we show that the model-checking problem of TCTL extended with parameters is undecidable over discrete-timed automata with only one parametric clock. The undecidability result needs equality in the logic. On the positive side, we show that the model-checking and the parameter synthesis problems become decidable for a fragment of the logic where equality is not allowed. Our method is based on automata theoretic principles and an extension of our method to express durations of runs in timed automata using Presburger arithmetic.

Active Diagnosis for Probabilistic Systems

Abstract: The diagnosis problem amounts to deciding whether some specific “fault” event occurred or not in a system, given the observations collected on a run of this system. This system is then diagnosable if the fault can always be detected, and the active diagnosis problem consists in controlling the system in order to ensure its diagnosability. We consider here a stochastic framework for this problem: once a control is selected, the system becomes a stochastic process. In this setting, the active diagnosis problem consists in deciding whether there exists some observation-based strategy that makes the system diagnosable with probability one. We prove that this problem is EXPTIME-complete, and that the active diagnosis strategies are belief-based. The safe active diagnosis problem is similar, but aims at enforcing diagnosability while preserving a positive probability to non faulty runs, i.e. without enforcing the occurrence of a fault. We prove that this problem requires non belief-based strategies, and that it is undecidable. However, it belongs to NEXPTIME when restricted to belief-based strategies. Our work also refines the decidability/undecidability frontier for verification problems on partially observed Markov decision processes.

Quantitative Languages Defined by Functional Automata

Abstract: A weighted automaton is functional if any two accepting runs on the same finite word have the same value. In this paper, we investigate functional weighted automata for four different measures: the sum, the mean, the discounted sum of weights along edges and the ratio between rewards and costs. On the positive side, we show that functionality is decidable for the four measures. Furthermore, the existential and universal threshold problems, the language inclusion problem and the equivalence problem are all decidable when the weighted automata are functional. On the negative side, we also study the quantitative extension of the realizability problem and show that it is undecidable for sum, mean and ratio. We finally show how to decide whether the language associated with a given functional automaton can be defined with a deterministic one, for sum, mean and discounted sum. The results on functionality and determinizability are expressed for the more general class of functional group automata. This allows one to formulate within the same framework new results related to discounted sum automata and known results on sum and mean automata. Ratio automata do not fit within this general scheme and different techniques are required to decide functionality.