Undecidable problem

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

Generic complexity of the Conjugacy Problem in HNN-extensions and algorithmic stratification of Miller's groups

Abstract: We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may have be undecidable. Our results show that "hard" instances of the problem comprise a negligibly small part of the group.

Detecting Unrealizable Specifications of Distributed Systems

Abstract: Writing formal specifications for distributed systems is difficult. Even simple consistency requirements often turn out to be unrealizable because of the complicated information flow in the distributed system: not every information is available in every component, and information transmitted from other components may arrive with a delay or not at all, especially in the presence of faults. The problem of checking the distributed realizability of a temporal specification is, in general, undecidable. Semi-algorithms for synthesis, such as bounded synthesis, are only useful in the positive case, where they construct an implementation for a realizable specification, but not in the negative case: if the specification is unrealizable, the search for the implementation never terminates. In this paper, we introduce counterexamples to distributed realizability and present a method for the detection of such counterexamples for specifications given in linear-time temporal logic (LTL). A counterexample consists of a set of paths, each representing a different sequence of inputs from the environment, such that, no matter how the components are implemented, the specification is violated on at least one of these paths. We present a method for finding such counterexamples both for the classic distributed realizability problem and for the distributed realizability problem with faulty nodes. Our method considers, incrementally, larger and larger sets of paths until a counterexample is found. While counterexamples for full LTL may consist of infinitely many paths, we give a semantic characterization such that the required number of paths can be bounded. For this fragment, we thus obtain a decision procedure. Experimental results, obtained with a QBF-based prototype implementation, show that our method finds simple errors very quickly, and even problems with high combinatorial complexity, like the Byzantine Generals’ Problem, are tractable.

Scenario-Based Verification of Uncertain MDPs.

Abstract: We consider Markov decision processes (MDPs) in which the transition probabilities and rewards belong to an uncertainty set parametrized by a collection of random variables. The probability distributions for these random parameters are unknown. The problem is to compute the probability to satisfy a temporal logic specification within any MDP that corresponds to a sample from these unknown distributions. In general, this problem is undecidable, and we resort to techniques from so-called scenario optimization. Based on a finite number of samples of the uncertain parameters, each of which induces an MDP, the proposed method estimates the probability of satisfying the specification by solving a finite-dimensional convex optimization problem. The number of samples required to obtain a high confidence on this estimate is independent from the number of states and the number of random parameters. Experiments on a large set of benchmarks show that a few thousand samples suffice to obtain high-quality confidence bounds with a high probability.

A Note on the Emptiness of Semigroup Intersections

Abstract: We consider decidability questions for the emptiness problem of intersections of matrix semigroups. This problem was studied by A. Markov [7] and more recently by V. Halava and T. Harju [5]. We give slightly strengthened results of their theorems by using a different matrix encoding. In particular, we show that given two finitely generated semigroups of non-singular upper triangular 3 × 3 matrices over the natural numbers, checking the emptiness of their intersections is undecidable. We also show that the problem is undecidable even for unimodular matrices over 3 × 3 rational matrices.

Comments on `Two Undecidable Problems of Analysis'

Abstract: We first discuss some technical questions which arise in connection with the construction of undecidable propositions in analysis, in particular in connection with the notion of the normal form of a function representing a predicate. Then it is stressed that while a function f(x) may be computable in the sense of recursive function theory, it may nevertheless have undecidable properties in the realm of Fourier analysis. This has an implication for a conjecture of Penrose's which states that classical physics is computable.