Undecidable problem

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

Tiling Problems on Baumslag-Solitar groups

Abstract: We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.

Linear-Time model-checking for multithreaded programs under scope-bounding

Abstract: We address the model checking problem of omega-regular linear-time properties for shared memory concurrent programs modeled as multi-pushdown systems. We consider here boolean programs with a finite number of threads and recursive procedures. It is well-known that the model checking problem is undecidable for this class of programs. In this paper, we investigate the decidability and the complexity of this problem under the assumption of scope-boundedness defined recently by La Torre and Napoli in [24]. A computation is scope-bounded if each pair of call and return events of a procedure executed by some thread must be separated by a bounded number of context-switches of that thread. The concept of scope-bounding generalizes the one of context-bounding [31] since it allows an unbounded number of context switches. Moreover, while context-bounding is adequate for reasoning about safety properties, scope-bounding is more suitable for reasoning about liveness properties that must be checked over infinite computations. It has been shown in [24] that the reachability problem for multi-pushdown systems under scope-bounding is PSPACE-complete. We prove in this paper that model-checking linear-time properties under scope-bounding is also decidable and is EXPTIME-complete.

The undecidability of the ambiguity problem for minimal linear grammars

Abstract: The ambiguity problem for general context-free phrase structure grammars has been shown undecidable. This paper proves the undecidability of a new form of Post's correspondence problem. Using this result, the undecidability of the ambiguity problem for minimal linear grammars is obtained.

Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

Abstract: Is undecidability a requirement for open-ended evolution (OEE)? Using methods derived from algorithmic complexity theory, we propose robust computational definitions of open-ended evolution and the adaptability of computable dynamical systems. Within this framework, we show that decidability imposes absolute limits on the stable growth of complexity in computable dynamical systems. Conversely, systems that exhibit (strong) open-ended evolution must be undecidable, establishing undecidability as a requirement for such systems. Complexity is assessed in terms of three measures: sophistication, coarse sophistication, and busy beaver logical depth. These three complexity measures assign low complexity values to random (incompressible) objects. As time grows, the stated complexity measures allow for the existence of complex states during the evolution of a computable dynamical system. We show, however, that finding these states involves undecidable computations. We conjecture that for similar complexity measures that assign low complexity values, decidability imposes comparable limits on the stable growth of complexity, and that such behavior is necessary for nontrivial evolutionary systems. We show that the undecidability of adapted states imposes novel and unpredictable behavior on the individuals or populations being modeled. Such behavior is irreducible. Finally, we offer an example of a system, first proposed by Chaitin, that exhibits strong OEE.