Undecidable problem

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

On Arnold's Hilbert Symposium Problems

Abstract: We prove that stability is undecidable for dynamical systems whose right-hand side is explicitly written in the language of elementary analysis.

The Complexity of Width Minimization for Existential Positive Queries

Abstract: Existential positive queries are logical sentences built from conjunction, disjunction, and existential quantication, and are also known as select-project-join-union queries in database theory, where they are recognized as a basic and fundamental class of queries. It is known that the number of variables needed to express an existential positive query is the crucial parameter determining the complexity of evaluating it on a database, and is hence a natural measure from the perspective of query optimization and rewriting. In this article, we study the complexity of the natural decision problem associated to this measure, which we call the expressibility problem: Given an existential positive query and a number k, can the query be expressed using k (or fewer) variables? We precisely determine the complexity of the expressibility problem, showing that it is complete for the level p of the polynomial hierarchy. Moreover, we prove that the expressibility problem is undecidable in positive logic (that is, existential positive logic plus universal quantication), thus establishing existential positive logic as a maximal syntactic fragment where expressibility is decidable.

A fresh look at testing for asynchronous communication

Abstract: Testing is one of the fundamental techniques for verifying if a computing system conforms to its specification. We take a fresh look at the theory of testing for message-passing systems based on a natural notion of observability in terms of input-output relations. We propose two notions of test equivalence: one which corresponds to presenting all test inputs up front and the other which corresponds to interactively feeding inputs to the system under test. We compare our notions with those studied earlier, notably the equivalence proposed by Tretmans. In Tretmans' framework, asynchrony is modelled using synchronous communication by augmenting the state space of the system with queues. We show that the first equivalence we consider is strictly weaker than Tretmans' equivalence and undecidable, whereas the second notion is incomparable. We also establish (un)decidability results for these equivalences.