Undecidable problem

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

Undecidable boundedness problems for datalog programs

Abstract: A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. [7]. We introduce new techniques for proving the undecidability of (various kinds of) boundedness, which allow us to considerably strengthen the results of Gaifman et al. [7]. In particular, (1) we use a new generic reduction technique to show that program boundedness is undecidable for arity 2 predicates, even with linear rules; (2) we use the mortality problem of Turing machines to show that uniform boundedness is undecidable for arity 3 predicates and for arity 1 predicates when ≠ is also allowed; (3) by encoding all possible transitions of a two-counter machine in a single rule, we show that program (resp., predicate) boundedness is undecidable for two linear rules (resp., one rule and a projection) and one initialization rule, where all predicates have small arities (6 or 7).

Verification and control of partially observable probabilistic systems

Abstract: We present automated techniques for the verification and control of partially observable, probabilistic systems for both discrete and dense models of time. For the discrete-time case, we formally model these systems using partially observable Markov decision processes; for dense time, we propose an extension of probabilistic timed automata in which local states are partially visible to an observer or controller. We give probabilistic temporal logics that can express a range of quantitative properties of these models, relating to the probability of an event's occurrence or the expected value of a reward measure. We then propose techniques to either verify that such a property holds or synthesise a controller for the model which makes it true. Our approach is based on a grid-based abstraction of the uncountable belief space induced by partial observability and, for dense-time models, an integer discretisation of real-time behaviour. The former is necessarily approximate since the underlying problem is undecidable, however we show how both lower and upper bounds on numerical results can be generated. We illustrate the effectiveness of the approach by implementing it in the PRISM model checker and applying it to several case studies from the domains of task and network scheduling, computer security and planning.

Degrees of Unsolvability

Abstract: Modern computability theory began with Turing [Turing, 1936], where he introduced the notion of a function computable by a Turing machine. Soon after, it was shown that this definition was equivalent to several others that had been proposed previously and the Church-Turing thesis that Turing computability captured precisely the informal notion of computability was commonly accepted. This isolation of the concept of computable function was one of the greatest advances of twentieth century mathematics and gave rise to the field of computability theory. Among the first results in computability theory was Church and Turing’s work on the unsolvability of the decision problem for first-order logic. Computability theory to a great extent deals with noncomputable problems. Relativized computation, which also originated with Turing, in [Turing, 1939], allows the comparison of the complexity of unsolvable problems. Turing formalized relative computation with oracle Turing machines. If a set A is computable relative to a set B, we say that A is Turing reducible to B. By identifying sets that are reducible to each other, we are led to the notion of degree of unsolvability first introduced by Post in [Post, 1944]. The degrees form a partially ordered set whose study is called degree theory. Most of the unsolvable problems that have arisen outside of computability theory are computably enumerable (c.e.). The c.e. sets can intuitively be viewed as unbounded search problems, a typical example being those formulas provable in some effectively given formal system. Reducibility allows us to isolate the most difficult c.e. problems, the complete problems. The standard method for showing that a c.e. problem is undecidable is to show that it is complete. Post [Post, 1944] asked if this technique always works, i.e., whether there is a noncomputable, incomplete c.e. set. This problem came to be known as Post’s Problem and it was origin of degree theory. Degree theory became one of the core areas of computability theory and attracted some of the most brilliant logicians of the second half of the twentieth century. The fascination with the field stems from the quite sophisticated techniques needed to solve the problems that arose, many of which are quite easy to state. The hallmark of the field is the priority method introduced by

Checking Consistency of Database Constraints: a Logical Basis

Abstract: This paper addresses the problem of consistency of a set of integrity con4raints itself, independent from any state. 11 is pointed out that database constraints have not only to ,be consistent. but in addition to be finitely .+atisfiablc. Thb stronger property reflects that the constraints have to admit a finite set of [stored ps well as derivqble) facts. As opposed .tu consistency. being undecidable, linite satisfiability is semidecidable. For effickncy purposes WC investigate methods that check both linite s&i&ability as ,well as unratisfiability. Two different methods are proposed which extend two ajternative approaches to refutation. ventional databases, constraints have to admit finite models as every state consists of a finite ‘number of facts. In definile drdubtivr databases (as defined in 191) the set of deduction ryle: always has a linite minimal model, which is intended to be a model of the constraint set as well. Sqtisfiability does not necessarily imply finite sat+xbiUy, i.e., the existence of a finite model. There are ‘satisfiabk sets of for&n&s -’ called ‘axioms of, infinity’ that have ‘onI?: infinite models. Consider, e.g., a se1 of integrity constraints for a managerial database containing (among others) the following constraints: l Everybody works for somebody. . Nobody works for himself. ’ . If x works for y and F works for t, then x works for L.