Topic
Undecidable problem
About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.
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TL;DR: It is shown, under some assumption on the signature, that the *This formula not viewable on a Text-Browser* fragment of the theory of any lexicographic path ordering is undecidable.
30 citations
22 Oct 1990
TL;DR: The results show that, depending on the optical model, ray tracing is sometimes undecidable, sometimes PSPACE-hard, and sometimes in PSPACE.
Abstract: The ray-tracing problem is considered for optical systems consisting of a set of refractive or reflective surfaces. It is assumed that the position and the tangent of the incident angle of the initial light ray are rational. The computability and complexity of the ray-tracing problems are investigated for various optical models. The results show that, depending on the optical model, ray tracing is sometimes undecidable, sometimes PSPACE-hard, and sometimes in PSPACE. >
30 citations
01 Jan 2008
TL;DR: An extension of the controller synthesis problem to nondeterministic processes is studied and it is shown that nondeterminism of the plant can be handled at no extra cost, both for centralized and decentralized control.
Abstract: The controller synthesis problem as formalized by Ramadge and Wonham consists of finding a finite controller that when synchronized with a given plant results in a system satisfying a required property. In this setting, both a plant and a controller are deterministic finite automata, while synchronization is modelled by a synchronous product. Originally, the framework was developed only for safety and some deadlock properties. More recently, Arnold et. al. have extended the setting to all mu-calculus expressible properties and proposed a reduction of the synthesis problem to the satisfiability problem of the mu-calculus. They have also presented some results on decidability of distributed synthesis problem where one requires to find several controllers that control the plant at the same time. The additional difficulty in this case is that each controller is aware of a different part of the whole system. In this paper, an extension of the setting to nondeterministic processes is studied. In other words, the case when both a system and a controller can be presented by a nondeterministic automaton. This extension is motivated by examples in control where a continuous quantity is measured and digitized thereby introducing imprecision and uncertainty. It is shown that nondeterminism of the plant can be handled at no extra cost, both for centralized and decentralized control. Centralized synthesis remains decidable even for nondeterministic controllers. In contrast, very few cases of decentralized control are decidable when controllers are allowed to be nondeterministic. A classification of decidable/undecidable variants of this problem is given.
29 citations
08 Jun 2019
TL;DR: This work proposes a novel notion, called view-switching, and provides a code-to-code translation from an input program under RA to a program under SC, which leads to a reduction, in polynomial time, of the bounded view- Switching reachability problem underRA to the bounded context-switched problem under SC.
Abstract: We address the verification of concurrent programs running under the release-acquire (RA) semantics. We show that the reachability problem is undecidable even in the case where the input program is finite-state. Given this undecidability, we follow the spirit of the work on context-bounded analysis for detecting bugs in programs under the classical SC model, and propose an under-approximate reachability analysis for the case of RA. To this end, we propose a novel notion, called view-switching, and provide a code-to-code translation from an input program under RA to a program under SC. This leads to a reduction, in polynomial time, of the bounded view-switching reachability problem under RA to the bounded context-switching problem under SC. We have implemented a prototype tool VBMC and tested it on a set of benchmarks, demonstrating that many bugs in programs can be found using a small number of view switches.
29 citations
TL;DR: It is shown that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability.
Abstract: We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧i-1n pi, where n is a parameter). The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus (called STAB) allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability
29 citations