Undecidable problem

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

On priorities of parallelism: Petri nets under the maximum firing strategy

Abstract: The computational power of Petri nets is extended up to the power of counter machines by realizing certain priorities of parallelism. Hence certain concurrent computations can not exactly be reflected by the sets of all sequentialized computations in related systems. Moreover, the reachability, boundedness and liveness problems are undecidable under the modified firing rule.

On the decidability of reachability in linear time-invariant systems

Abstract: We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which control sets are affine subspaces of Rn is a fundamental result in control theory. Our first result is that reachability is undecidable if the set of controls is a finite union of affine subspaces. We also consider versions of the reachability problem in which (i) the set of controls consists of a single affine subspace together with the origin and (ii) the set of controls is a convex polytope. In these two cases we respectively show that the reachability problem is as hard as Skolem's Problem and the Positivity Problem for linear recurrence sequences (whose decidability has been open for several decades). Our main contribution is to show decidability of a version of the reachability problem in which control sets are convex polytopes, under certain spectral assumptions on the transition matrix.

Parameterized synthesis

Abstract: We study the synthesis problem for distributed architectures with a parametric number of finite-state components. Parameterized specifications arise naturally in a synthesis setting, but thus far it was unclear how to decide realizability and how to perform synthesis. Using a classical result from verification, we show that for specifications in LTL\X, parameterized synthesis of token ring networks is equivalent to distributed synthesis of a network consisting of a few copies of a single process. Adapting a result from distributed synthesis, we show that the latter problem is undecidable. We then describe a semi-decision procedure based on bounded synthesis and show applicability on a simple case study. Finally, we sketch a general framework for parameterized synthesis based on cut-off results for verification.

Parameterised Pushdown Systems with Non-Atomic Writes

Abstract: We consider the master/slave parameterised reachability problem for networks of pushdown systems, where communication is via a global store using only non-atomic reads and writes. We show that the control-state reachability problem is decidable. As part of the result, we provide a constructive extension of a theorem by Ehrenfeucht and Rozenberg to produce an NFA equivalent to certain kinds of CFG. Finally, we show that the non-parameterised version is undecidable.

Two variables and two successors

Abstract: We look at the finite satisfiability problem of the two variable fragment of first order logic with the successors of two linear orders. While the logic with both the successors and their transitive closures remains undecidable, we prove that the logic with only the successors is decidable.