Undecidable problem

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

Formal Specification of Real-time Systems

Abstract: This paper presents the formal syntax and semantics of Real Time Logic (RTL), a logic for the specification of real-time systems. An example illustrating the specification of a system in RTL is presented, and natural deduction is used to verify that the system satisfies a given safety property. RTL is shown to be undecidable by a reduction from the acceptance problem for two-counter machines. Decidable subclasses of the logic are also discussed.

Universality and language inclusion for open and closed timed automata

Abstract: The algorithmic analysis of timed automata is fundamentally limited by the undecidability of the universality problem. For this reason and others, there has been considerable interest in restricted classes of timed automata. In this paper we study the universality problem for two prominent such subclasses: open and closed timed automata. This problem is described as open in [6,8] in the case of open timed automata. We show here that the problem is undecidable for open timed automata over strongly monotonic time (no two events are allowed to occur at the same time), and decidable over weakly monotonic time. For closed timed automata, we show that the problem is undecidable regardless of the monotonicity assumptions on time. As a corollary, we settle the various language inclusion problems over these classes of timed automata.

Polynomial Size Analysis of First-Order Shapely Functions

Abstract: We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be implementations of matrix multiplication and the Cartesian product of two lists. The type system is proved to be sound w.r.t. the operational semantics of the language. The type checking problem is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic. Furthermore, we have shown that the type-inference problem is at least semi-decidable (under this restriction). We have implemented a procedure that combines run-time testing and type-checking to automatically obtain size dependencies. It terminates on total typable function definitions.