Undecidable problem

About: Undecidable problem is a(n) research topic. Over the lifetime, 3135 publication(s) have been published within this topic receiving 71238 citation(s).

A theory of timed automata

Abstract: Alur, R. and D.L. Dill, A theory of timed automata, Theoretical Computer Science 126 (1994) 183-235. We propose timed (j&e) automata to model the behavior of real-time systems over time. Our definition provides a simple, and yet powerful, way to annotate state-transition graphs with timing constraints using finitely many real-valued clocks. A timed automaton accepts timed words-infinite sequences in which a real-valued time of occurrence is associated with each symbol. We study timed automata from the perspective of formal language theory: we consider closure properties, decision problems, and subclasses. We consider both nondeterministic and deterministic transition structures, and both Biichi and Muller acceptance conditions. We show that nondeterministic timed automata are closed under union and intersection, but not under complementation, whereas deterministic timed Muller automata are closed under all Boolean operations. The main construction of the paper is an (PSPACE) algorithm for checking the emptiness of the language of a (nondeterministic) timed automaton. We also prove that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable (II i-hard) in the nondeterministic case and PSPACE-complete in the deterministic case. Finally, we discuss the application of this theory to automatic verification of real-time requirements of finite-state systems.

Automata on infinite objects

Abstract: Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.

Model-Checking in Dense Real-Time

Abstract: Model-checking is a method of verifying concurrent systems in which a state-transition graph model of the system behavior is compared with a temporal logic formula. This paper extends model-checking for the branching-time logic CTL to the analysis of real-time systems, whose correctness depends on the magnitudes of the timing delays. For specifications, we extend the syntax of CTL to allow quantitative temporal operators such as ?? <5, meaning "possibly within 5 time units." The formulas of the resulting logic, Timed CTL (TCTL), are interpreted over continuous computation trees, trees in which paths are maps from the set of nonnegative reals to system states. To model finite-state systems we introduce timed graphs-state-transition graphs annotated with timing constraints. As our main result, we develop an algorithm for model-checking, for determining the truth of a TCTL-formula with respect to a timed graph. We argue that choosing a dense domain instead of a discrete domain to model time does not significantly blow up the complexity of the model-checking problem. On the negative side, we show that the denseness of the underlying time domain makes the validity problem for TCTL ?11-hard. The question of deciding whether there exists a timed graph satisfying a TCTL-formula is also undecidable.

Model-checking for real-time systems

Abstract: This research extends CTL model-checking to the analysis of real-time systems, whose correctness depends on the magnitudes of the timing delays. For specifications, the syntax of CTL is extended to allow quantitative temporal operators. The formulas of the resulting logic, TCTL, are interpretation over continuous computation trees, trees in which paths are maps from the set of nonnegative reals to system states. To model finite-state systems the notion of timed graphs is introduced-state-transition graphs extended with a mechanism that allows the expression of constant bounds on the delays between the state transition. As the main result, an algorithm is developed for model checking, that is, for determining the truth of a TCTL formula with respect to a timed graph. It is argued that choosing a dense domain, instead of a discrete domain, to model time does not blow up the complexity of the model-checking problem. On the negative side, it is shown that the denseness of the underlying time domain makes TCTL II/sub 1//sup 1/-hard. The question of deciding whether a given TCTL formula is implementable by a timed graph is also undecidable. >

The classical decision problem

Abstract: 1. Introduction: The Classical Decision Problem.- 1.1 The Original Problem.- 1.2 The Transformation of the Classical Decision Problem.- 1.3 What Is and What Isn't in this Book.- I. Undecidable Classes.- 2. Reductions.- 2.1 Undecidability and Conservative Reduction.- 2.1.1 The Church-Turing Theorem and Reduction Classes.- 2.1.2 Trakhtenbrot's Theorem and Conservative Reductions.- 2.1.3 Inseparability and Model Complexity.- 2.2 Logic and Complexity.- 2.2.1 Propositional Satisfiability.- 2.2.2 The Spectrum Problem and Fagin's Theorem.- 2.2.3 Capturing Complexity Classes.- 2.2.4 A Decidable Prefix-Vocabulary Class.- 2.3 The Classifiability Problem.- 2.3.1 The Problem.- 2.3.2 Well Partially Ordered Sets.- 2.3.3 The Well Quasi Ordering of Prefix Sets.- 2.3.4 The Well Quasi Ordering of Arity Sequences.- 2.3.5 The Classifiability of Prefix-Vocabulary Sets.- 2.4 Historical Remarks.- 3. Undecidable Standard Classes for Pure Predicate Logic.- 3.1 The Kahr Class.- 3.1.1 Domino Problems.- 3.1.2 Formalization of Domino Problems by $$[\forall \exists \forall , (0,\omega )]$$-Formulae.- 3.1.3 Graph Interpretation of $$[\forall \exists \forall , (0,\omega )]$$-Formulae.- 3.1.4 The Remaining Cases Without $$\exists *$$.- 3.2 Existential Interpretation for $$[{{\forall }^{3}}\exists *, (0,1)]$$.- 3.3 The Gurevich Class.- 3.3.1 The Proof Strategy.- 3.3.2 Reduction to Diagonal-Freeness.- 3.3.3 Reduction to Shift-Reduced Form.- 3.3.4 Reduction toFi-Elimination Form.- 3.3.5 Elimination of MonadicFi.- 3.3.6 The Kostyrko-Genenz and Suranyi Classes.- 3.4 Historical Remarks.- 4. Undecidable Standard Classes with Functions or Equality.- 4.1 Classes with Functions and Equality.- 4.2 Classes with Functions but Without Equality.- 4.3 Classes with Equality but Without Functions: the Goldfarb Classes 161 4.3.1 Formalization of Natural Numbers in $$[{{\forall }^{3}}\exists *, (\omega ,\omega ),(0)]$$=.- 4.3.2 Using Only One Existential Quantifiers.- 4.3.3 Encoding the Non-Auxiliary Binary Predicates.- 4.3.4 Encoding the Auxiliary Binary Predicates of NUM*.- 4.4 Historical Remarks.- 5. Other Undecidable Cases.- 5.1 Krom and Horn Formulae.- 5.1.1 Krom Prefix Classes Without Functions or Equality.- 5.1.2 Krom Prefix Classes with Functions or Equality.- 5.2 Few Atomic Subformulae.- 5.2.1 Few Function and Equality Free Atoms.- 5.2.2 Few Equalities and Inequalities.- 5.2.3 Horn Clause Programs With One Krom Rule.- 5.3 Undecidable Logics with Two Variables.- 5.3.1 First-Order Logic with the Choice Operator.- 5.3.2 Two-Variable Logic with Cardinality Comparison.- 5.4 Conjunctions of Prefix-Vocabulary Classes.- 5.4.1 Reduction to the Case of Conjunctions.- 5.4.2 Another Classifiability Theorem.- 5.4.3 Some Results and Open Problems.- 5.5 Historical Remarks.- II. Decidable Classes and Their Complexity.- 6. Standard Classes with the Finite Model Property.- 6.1 Techniques for Proving Complexity Results.- 6.1.1 Domino Problems Revisited.- 6.1.2 Succinct Descriptions of Inputs.- 6.2 The Classical Solvable Cases.- 6.2.1 Monadic Formulae.- 6.2.2 The Bernays-Schonfinkel-Ramsey Class.- 6.2.3 The Godel-Kalmar-Schutte Class: a Probabilistic Proof.- 6.3 Formulae with One ?.- 6.3.1 A Satisfiability Test for [?*??*, all, all].- 6.3.2 The Ackermann Class.- 6.3.3 The Ackermann Class with Equality.- 6.4 Standard Classes of Modest Complexity.- 6.4.1 The Relational Classes in P, NP and Co-NP.- 6.4.2 Fragments of the Theory of One Unary Function.- 6.4.3 Other Functional Classes.- 6.5 Finite Model Property vs. Infinity Axioms.- 6.6 Historical Remarks.- 7. Monadic Theories and Decidable Standard Classes with Infinity Axioms.- 7.1 Automata, Games and Decidability of Monadic Theories.- 7.1.1 Monadic Theories.- 7.1.2 Automata on Infinite Words and the Monadic Theory of One Successor.- 7.1.3 Tree Automata, Rabin's Theorem and Forgetful De terminacy.- 7.1.4 The Forgetful Determinacy Theorem for Graph Games.- 7.2 The Monadic Second-Order Theory of One Unary Function.- 7.2.1 Decidability Results for One Unary Function.- 7.2.2 The Theory of One Unary Function is not Elementary Recursive.- 7.3 The Shelah Class.- 7.3.1 Algebras with One Unary Operation.- 7.3.2 Canonic Sentences.- 7.3.3 Terminology and Notation.- 7.3.4 1-Satisfiability.- 7.3.5 2-Satisfiability.- 7.3.6 Refinements.- 7.3.7 Villages.- 7.3.8 Contraction.- 7.3.9 Towns.- 7.3.10 The Final Reduction.- 7.4 Historical Remarks.- 8. Other Decidable Cases.- 8.1 First-Order Logic with Two Variables.- 8.2 Unification and Applications to the Decision Problem.- 8.2.1 Unification.- 8.2.2 Herbrand Formulae.- 8.2.3 Positive First-Order Logic.- 8.3 Decidable Classes of Krom Formulae.- 8.3.1 The Chain Criterion.- 8.3.2 The Aanderaa-Lewis Class.- 8.3.3 The Maslov Class.- 8.4 Historical Remarks.- A. Appendix: Tiling Problems.- A.1 Introduction.- A.2 The Origin Constrained Domino Problem.- A.3 Robinson's Aperiodic Tile Set.- A.4 The Unconstrained Domino Problem.- A.5 The Periodic Problem and the Inseparability Result.- Annotated Bibliography.