Undecidable problem

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

Logics with counting and equivalence

Abstract: We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a single distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NExpTime-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable.

Sufficient-completeness, ground-reducibility and their complexity

Abstract: The sufficient-completeness property of equational algebraic specifications has been found useful in providing guidelines for designing abstract data type specifications as well as in proving inductive properties using the induction-less-induction method. The sufficient-completeness property is known to be undecidable in general. In an earlier paper, it was shown to be decidable for constructor-preserving, complete (canonical) term rewriting systems, even when there are relations among constructor symbols. In this paper, the complexity of the sufficient-completeness property is analyzed for different classes of term rewriting systems. A number of results about the complexity of the sufficient-completeness property for complete (canonical) term rewriting systems are proved: (i) The problem is co-NP-complete for term rewriting systems with free constructors (i.e., no relations among constructors are allowed), (ii) the problem remains co-NP-complete for term rewriting systems with unary and nullary constructors, even when there are relations among constructors, (iii) the problem is provably in “almost” exponential time for left-linear term rewriting systems with relations among constructors, and (iv) for left-linear complete constructor-preserving rewriting systems, the problem can be decided in steps exponential innlogn wheren is the size of the rewriting system. No better lower-bound for the complexity of the sufficient-completeness property for complete (canonical) term rewriting system with nonlinear left-hand sides is known. An algorithm for left-linear complete constructor-preserving rewriting systems is also discussed. Finally, the sufficient-completeness property is shown to be undecidable for non-linear complete term rewriting systems with associative functions. These complexity results also apply to the ground-reducibility property (also called inductive-reducibility) which is known to be directly related to the sufficient-completeness property.

Applying an update method to a set of receivers

Abstract: In the context of object databases, we study the application of an update method to a collection of receivers rather than to a single one. The obvious strategy of applying the update to the receivers one after the other, in some arbitrary order, brings up the problem of order independence. On a very general level, we investigate how update behavior can be analyzed in terms of certain schema annotations, called colorings. We are able to characterize those colorings that always describe order-independedent updates. We also consider a more specific model of update methods implemented in the relational algebra. Order-independence of such algebraic methods is undecidable in general, but decidable if the expressions used are positive. Finally, we consider an alternative parallel strategy for set-oriented applications of algebraic update methods and compare and relate it to the sequential strategy.

Tilings and Submonoids of Metabelian Groups

Abstract: In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product ℤ≀(ℤ×ℤ). We also show that subsemimodule membership is undecidable for finite rank free (ℤ×ℤ)-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.