Undecidable problem

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

Computability and complexity of ray tracing

Abstract: The ray-tracing problem is, given an optical system and the position and direction of an initial light ray, to decide if the light ray reaches some given final position. For many years ray tracing has been used for designing and analyzing optical systems. Ray tracing is now used extensively in computer graphics to render scenes with complex curved objects under global illumination. We show that ray-tracing problems in some three-dimensional simple optical systems (purely geometrical optics) are undecidable. These systems may consist of either reflective objects that are represented by rational quadratic equations, or refractive objects that are represented by rational linear equations. Some problems in more restricted models are shown to be PSPACE-hard or sometimes in PSPACE.

An undecidable case of lineability in R^R

Abstract: Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$}) one has that the set of Sierpi\'nski-Zygmund functions is (2^{\mathfrak{c}})-strongly algebrable (and, thus, (2^{\mathfrak{c}})-lineable). Here we prove that these two statements are actually equivalent and, moreover, they both are undecidable. This would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability.

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Abstract: We consider first-order logic over the subword ordering on finite words where each word is available as a constant. Our first result is that the Σ 1 theory is undecidable (already over two letters). We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ 1 fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ 2 fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

On the Complexity of Model Checking Counter Automata

Abstract: Theoretical and practical aspects of the verification of infinite-state systems have attracted a lot of interest in the verification community throughout the last 30 years. One goal is to identify classes of infinite-state systems that admit decidable decision problems on the one hand, and which are sufficiently general to model systems, programs or protocols with unbounded data or recursion depth on the other hand. The first part of this thesis is concerned with the computational complexity of verifying counter automata, which are a fundamental and widely studied class of infinite-state systems. Counter automata consist of a finite-state controller manipulating a finite number of counters ranging over the naturals. A classic result by Minsky states that reachability in counter automata is undecidable already for two counters. One restriction that makes reachability decidable and that this thesis primarily focuses on is the restriction to one counter. A main result of this thesis is to show that reachability in one-counter automata with counter updates encoded in binary is NP-complete, which solves a problem left open by Rosier and Yen in 1986. We also consider parametric one-counter automata, in which counter updates can be parameters ranging over the naturals. Reachability for this class asks whether there are values of the parameters such that a target configuration can be reached from an initial configuration. This problem is also shown to be NP-complete. Subsequently, we establish decidability and complexity results of model checking problems for one-counter automata with and without parameters for specifications written in EF, CTL and LTL. The second part of this thesis is about the verification of programs with pointers and linked lists in the framework of separation logic. We consider the fragment of separation logic introduced by Berdine, Calcagno and O'Hearn in 2004 and the problem of deciding entailment between formulae of this fragment. We improve the known coNP upper bound and show that this problem can actually be solved in polynomial time. This result is based on a novel approach in which we represent separation logic formulae as graphs and decide entailment between them by checking for the existence of a graph homomorphism. We complement this result by considering various natural extensions of this fragment which make entailment coNP-hard.

Computing Knowledge in Security Protocols Under Convergent Equational Theories

Abstract: The analysis of security protocols requires reasoning about the knowledge an attacker acquires by eavesdropping on network traffic. In formal approaches, the messages exchanged over the network are modelled by a term algebra equipped with an equational theory axiomatising the properties of the cryptographic primitives (e.g. encryption, signature). In this context, two classical notions of knowledge, deducibility and indistinguishability, yield corresponding decision problems. We propose a procedure for both problems under arbitrary convergent equational theories. Since the underlying problems are undecidable we cannot guarantee termination. Nevertheless, our procedure terminates on a wide range of equational theories. In particular, we obtain a new decidability result for a theory we encountered when studying electronic voting protocols. We also provide a prototype implementation.