Undecidable problem

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

Two-Way Counter Machines and Diophantine Equations

Abstract: Let Q be the class of deterministic two-way one-counter machines accepting only bounded languages. Each machine in Q has the property that in every accepting computation, the counter makes at most a fixed number of reversals. We show that the emptiness problem for Q is decidable. When the counter is unrestricted or when the machine is provided with two reversal-bounded counters, the emptiness problem becomes undecidable. The decidability of the emptiness problem for Q is useful in proving the solvability of some numbertheoretic problems. It can also be used to prove that the language L = {u1iu2i2|i≥0} cannot be accepted by any machine in Q (u1 and u2 are distinct symbols). The proof technique is new in that it does not employ the usual "pumping", "counting", or "diagonal" argument. Note that L can be accepted by a deterministic two-way machine with two counters, each of which makes exactly one reversal.

On the Equivalence of Two Systems of Affine Recurrence Equations (Research Note)

Abstract: This paper deals with the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not. A solution to this problem would be a step toward algorithm recognition, an important tool in program analysis, optimization and parallelization. We first prove that in the general case, the problem is undecidable. We then show that there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. Many partial solutions are known. We then report on a pilot implementation of the algorithm, describe its limitations, and point to unsolved problems.

Conservative ambiguity detection in context-free grammars

Abstract: The ability to detect ambiguities in context-free grammars is vital for their use in several fields, but the problem is undecidable in the general case. We present a safe, conservative approach, where the approximations cannot result in overlooked ambiguous cases. We analyze the complexity of the verification, and provide formal comparisons with several other ambiguity detection methods.

Partial polymorphic type inference is undecidable

Abstract: Polymorphic type systems combine the reliability and efficiency of static type-checking with the flexibility of dynamic type checking. Unfortunately, such languages tend to be unwieldy unless they accommodate omission of much of the information necessary to perform type checking. The automatic inference of omitted type information has emerged as one of the fundamental new implementation problems of these languages. We show here that a natural formalization of the problem is undecidable. The proof is directly applicable to some practical situations, and provides a partial explanation of the difficulties encountered in other cases.

Modal Logics of Topological Relations

Abstract: Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.