Undecidable problem

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

Detecting Equalities of Variables: Combining Efficiency with Precision

Abstract: Detecting whether different variables have the same value at a program point is generally undecidable. Though the subclass of equalities, whose validity holds independently from the interpretation of operators (Herbrand-equivalences), is decidable, the technique which is most widely implemented in compilers, value numbering, is restricted to basic blocks. Basically, there are two groups of algorithms aiming at globalizations of value numbering: first, a group of algorithms based on the algorithm of Kildall, which uses data flow analysis to gather information on value equalities. These algorithms are complete in detecting Herbrand-equivalences, however, expensive in terms of computational complexity. Second, a group of algorithms influenced by the algorithm of Alpern, Wegman and Zadeck. They do not fully interpret the control flow, which allows them to be particularly efficient, however, at the price of being significantly less precise than their Kildall-like counterparts. In this article we discuss how to combine the best features of both groups by aiming at a fair balance between computational complexity and precision. We propose an algorithm, which extends the one of Alpern, Wegman and Zadeck. The new algorithm is polynomial and, in practice, expected to be almost as efficient as the original one. Moreover, for acyclic control flow it is as precise as Kildall's one, i. e. it detects all Herbrand-equivalences.

I: A General Method in Proofs of Undecidability

Abstract: Publisher Summary A theory T is called decidable or undecidable depending on whether the solution of the decision problem is positive or negative. The decision problem is one of the central problems of contemporary metamathematics. Because only few theories turn out to be decidable, most endeavors are directed toward a negative solution. The aim of this chapter is to set up theoretical foundations for the general method, which is referred to as theories with standard formalization. They can be briefly characterized as theories that are formalized within the first-order predicate logic. The symbols that occur in expressions of a given theory T are divided into variables and constant. With every predicate and every operation symbol, a positive integer is correlated, which is called the rank of the symbol. The identity symbol, though regarded as a logical constant, is included in the set of binary predicates. In practice, in addition to variables and constants, the so-called technical symbols, like parentheses and commas, are also used in constructing expressions; theoretically, however, these technical symbols can be dispensed with.

Constrained Properties, Semilinear Systems, and Petri Nets

Abstract: We investigate the verification problem of two classes of infinite state systems wrt nonregular properties (ie, nondefinable by finite-state Ω-automata) The systems we consider are Petri nets as well as semilinear systems including pushdown systems and PA processes On the other hand, we consider properties expressible in the logic CLTL which is an extension of the linear-time temporal logic LTL allowing two kinds of constraints: pattern constraints using finite-state automata and counting constraints using Presburger arithmetics formulas While the verification problem of CLTL is undecidable even for finite-state systems, we identify a fragment called CLTL◊ for which the verification problem is decidable for pushdown systems as well as for Petri nets This fragment is strictly more expressive than finite-state Ω-automata We show that, however, the verification problem of semilinear systems (PA processes in particular) is undecidable even wrt LTL formulas Therefore, we identify another fragment (a restriction of LTL extended with counting constraints) covering a significant class of properties and for which the verification problem is decidable for all PA processes

On a Theorem of Cobham Concerning Undecidable Theories

Abstract: Publisher Summary This introductory chapter discusses the general method for establishing the undecidability of theories. The chapter presents a new proof of Cobham's theorem that may be called as “existential interpretation.” Cobham's theorem can be derived from a theorem of Trahtenbrot. The chapter also discusses several other ways in which Cobham's theorem can be improved or generalized along with a new proof of Cobham's theorem. Any axiomatizable theory whose undecidability follows from Trahtenbrots theorem also fulfills the hypothesis of Tarski's condition. The chapter shows a way to proof that the similar conclusion also applies to axiomatizable theories whose undecidability follows from Cobham's Theorem. Cobham proves the theory of groups (G) of finite groups in hereditarily and undecidable manner (and, hence, not axiomatizable).The chapter also illustrates several related problems to Cobham's theorem.

Discrete-Time Control for Rectangular Hybrid Automata

Abstract: Rectangular hybrid automata model digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in real-numbered time, and the controller samples the plant state and changes the control state discretely, only at the integer points in time. We prove that rectangular hybrid automata have finite bisimilarity quotients when all control transitions happen at integer times, even if the constraints on the derivatives of the variables vary between control states. This is sharply in contrast with the conventional model where control transitions may happen at any real time, and already the reachability problem is undecidable. Based on the finite bisimilarity quotients, we give an exponential algorithm for the symbolic sampling-controller synthesis of rectangular automata. We show our algorithm to be optimal by proving the problem to be EXPTIME-hard. We also show that rectangular automata form a maximal class of systems for which the sampling-controller synthesis problem can be solved algorithmically.