Undecidable problem

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

Translation-like Actions and Aperiodic Subshifts on Groups

Abstract: It is well known that if $G$ admits a f.g. subgroup $H$ with a weakly aperiodic SFT (resp. an undecidable domino problem), then $G$ itself has a weakly aperiodic SFT (resp. an undecidable domino problem). We prove that we can replace the property "$H$ is a subgroup of $G$" by "$H$ acts translation-like on $G$", provided $H$ is finitely presented. In particular: * If $G_1$ and $G_2$ are f.g. infinite groups, then $G_1 \times G_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-group admits a weakly aperiodic SFT.

Ordered Navigation on Multi-attributed Data Words

Abstract: We study temporal logics and automata on multi-attributed data words. Recently, BD-LTL was introduced as a temporal logic on data words extending LTL by navigation along positions of single data values. As allowing for navigation wrt. tuples of data values renders the logic undecidable, we introduce ND-LTL, an extension of BD-LTL by a restricted form of tuple-navigation. While complete ND-LTL is still undecidable, the two natural fragments allowing for either future or past navigation along data values are shown to be Ackermann-hard, yet decidability is obtained by reduction to nested multi-counter systems. To this end, we introduce and study nested variants of data automata as an intermediate model simplifying the constructions. To complement these results we show that imposing the same restrictions on BD-LTL yields two 2ExpSpace-complete fragments while satisfiability for the full logic is known to be as hard as reachability in Petri nets.

Restricted one-counter machines with undecidable universe problems

Abstract: We show the undecidability of the universe problem for two restricted classes of nondeterministic one-counter machines These classes are among the simplest known for which the universe problem can be shown unsolvable

Game arguments in computability theory and algorithmic information theory

Abstract: We provide some examples showing how game-theoretic arguments can be used in computability theory and algorithmic information theory: unique numbering theorem (Friedberg), the gap between conditional complexity and total conditional complexity, Epstein--Levin theorem and some (yet unpublished) result of Muchnik and Vyugin

SPeeDI: A verification tool for polygonal hybrid systems

Abstract: Hybrid systems combining discrete and continuous dynamics arise as mathematical models of various artificial and natural systems, and as an approximation to complex continuous systems. A very important problem in the analysis of the behavior of hybrid systems is reachability. It is well-known that for most non-trivial subclasses of hybrid systems this and all interesting verification problems are undecidable. Most of the proved decidability results rely on stringent hypothesis that lead to the existence of a finite and computable partition of the state space into classes of states which are equivalent with respect to reachability. This is the case for classes of rectangular automata [4] and hybrid automata with linear vector fields [9]. Most implemented computational procedures resort to (forward or backward) propagation of constraints, typically (unions of convex) polyhedra or ellipsoids [1, 6, 8]. In general, these techniques provide semi-decision procedures, that is, if the given final set of states is reachable, they will terminate, otherwise they may fail to. Maybe the major drawback of set-propagation, reach-set approximation procedures is that they pay little attention to the geometric properties of the specific (class of) systems under analysis. An interesting and still decidable class of hybrid system are the (2-dimensional) polygonal differential inclusions (or SPDI for short).