Undecidable problem

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

Deciding Properties of Integral Relational Automata

Abstract: This paper investigates automated model checking possibilities for CTL* formulae over infinite transition systems represented by relational automata (RA). The general model checking problem for CTL* formulae over RA is shown undecidable, the undecidability being observed already on the class of Restricted CTL formulae. The decidability result, however, is obtained for another substantial subset of the logic, called A-CTL*+, which includes all ”linear time” formulae.

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.

On the expressivity of feature logics with negation, functional uncertainty, and sort equations

Abstract: Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.

The decision problem for standard classes

Abstract: The standard classes of a first-order theory T are certain classes of prenex T -sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T , (1) the decision problem for any class of prenex T -sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K 1 , …, K n such that any undecidable standard class contains one of K 1 , …, K n . These results give direction to the study of the decision problem. Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

One More Decidable Class of Finitely Ground Programs

Abstract: When a logic program is processed by an answer set solver, the first task is to generate its instantiation In a recent paper, Calimeri et el made the idea of efficient instantiation precise for the case of disjunctive programs with function symbols, and introduced the class of "finitely ground" programs that can be efficiently instantiated Since that class is undecidable, it is important to find its large decidable subsets In this paper, we introduce such a subset--the class of argument-restricted programs It includes, in particular, all finite domain programs, *** -restricted programs, and *** -restricted programs