Undecidability Results Research Articles

Descent on elliptic curves and Hilbert’s tenth problem

Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert’s Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.

Reachability analysis of fragments of mobile ambients in AC term rewriting

Abstract In this paper, we investigate the connection between fragments of associative-commutative Term Rewriting and fragments of Mobile Ambients, a powerful model for mobile and distributed computations. The connection can be used to transfer decidability and undecidability results for important computational properties like reachability from one formalism to the other. Furthermore, it can be viewed as a vehicle to apply tools based on rewriting for the simulation and validation of specifications given in Mobile Ambients.

WORD EQUATIONS OVER GRAPH PRODUCTS

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.

Some Decidability and Undecidability Results on Green's Relations for Automatic Monoids

We consider the problem of deciding Green's relations for various types of finitely presented automatic monoids. We establish some decidability as well as some undecidability results.

Improved matrix pair undecidability results

We improve the undecidability bounds for problems involving two integer matrices by showing that Scalar Reachability, Zero in the Right Upper Corner, Vector Reachability, and Zero in the Left Upper Corner are undecidable for dimensions of 9, 10, 11, and 13, respectively. Problems Scalar Reachability, Zero in the Right Upper Corner, and Vector Reachability were previously known undecidable for dimensions 18, 18, and 16, respectively.

A statistical model for undecidable viral detection

This paper presents a statistical model of the malware detection problem. Where Chess and White (An undetectable computer virus. In: Virus Bulletin Conference, 2000) just partially addressed this issue and gave only existence results, we give here constructive results of undetectable malware. We show that any existing detection techniques can be modelled by one or more statistical tests. Consequently, the concepts of false positive and non detection are precisely defined. The concept of test simulability is then presented and enables us to gives constructive results how undetectable malware could be developped by an attacker. Practical applications of this statistical model are proposed. Finally, we give a statistical variant of Cohen’s undecidability results of virus detection.

Real-Time Model-Checking: Parameters everywhere

In this paper, we study the model-checking and parameter synthesis problems of the logic TCTL over discrete-timed automata where parameters are allowed both in the model (timed automaton) and in the property (temporal formula). Our results are as follows. On the negative side, we show that the model-checking problem of TCTL extended with parameters is undecidable over discrete-timed automata with only one parametric clock. The undecidability result needs equality in the logic. On the positive side, we show that the model-checking and the parameter synthesis problems become decidable for a fragment of the logic where equality is not allowed. Our method is based on automata theoretic principles and an extension of our method to express durations of runs in timed automata using Presburger arithmetic.

On the freeze quantifier in Constraint LTL: Decidability and complexity

Constraint LTL, a generalisation of LTL over Presburger constraints, is often used as a formal language to specify the behavior of operational models with constraints. The freeze quantifier can be part of the language, as in some real-time logics, but this variable-binding mechanism is quite general and ubiquitous in many logical languages (first-order temporal logics, hybrid logics, logics for sequence diagrams, navigation logics, logics with λ-abstraction, etc.). We show that Constraint LTL over the simple domain 〈N,=〉 augmented with the freeze quantifier is undecidable which is a surprising result in view of the poor language for constraints (only equality tests). Many versions of freeze-free Constraint LTL are decidable over domains with qualitative predicates and our undecidability result actually establishes Σ11-completeness. On the positive side, we provide complexity results when the domain is finite (ExpSpace-completeness) or when the formulae are flat in a sense introduced in the paper. Our undecidability results are sharp (i.e. with restrictions on the number of variables) and all our complexity characterisations ensure completeness with respect to some complexity class (mainly PSpace and ExpSpace).

Stability Criteria for Switched and Hybrid Systems

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

Diophantine definability and decidability in extensions of degree 2 of totally real fields

We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q . Among other results we prove the following. The “big” subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for all extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of “small” and “big” subrings of number fields in the infinite algebraic extensions of Q , previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes.

Robust parametric reachability for timed automata

We review the known decidability and undecidability results for reachability in parametric timed automata. Then, we present a new proof of undecidability in dense time for open timed automata that avoids equalities in clock constraints. Our result shows that the undecidability of parametric timed automata does not follow from their ability to specify punctual constraints in a dense time domain.

An undecidability result for AGh

We present an undecidability result for the verification of security protocols. Since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties, several recent works relax this assumption, allowing the intruder to exploit these properties. We are interested in the Abelian groups theory in combination with the homomorphism axiom. We show that the security problem for a bounded number of sessions (expressed by satisfiability of symbolic deductibility constraints) is undecidable, obtaining in this way the first undecidability result concerning a theory for which unification is known to be decidable [F. Baader, Unification in commutative theories, Hilbert's basis theorem, and Gröbner bases, J. ACM 40(3) (1993) 477–503].

Modal Logics of Topological Relations

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.

Undecidability Results for Bisimilarity on Prefix Rewrite Systems

We answer an open question related to bisimilarity checking on labelled transition systems generated by prefix rewrite rules on words. Stirling (1996, 1998) proved the decidability of bisimilarity for normed pushdown processes. This result was substantially extended by Sénizergues (1998, 2005) who showed the decidability for regular (or equational) graphs of finite out-degree (which include unnormed pushdown processes). The question of decidability of bisimilarity for a more general class of so called Type -1 systems (generated by prefix rewrite rules of the form R ---> w where R is a regular language) was left open; this was repeatedly indicated by both Stirling and Sénizergues. Here we answer the question negatively, i.e., we show undecidability of bisimilarity on Type -1 systems, even in the normed case. We complete the picture by considering classes of systems that use rewrite rules of the form w ---> R and R_1 ---> R_2 and show when they yield low undecidability ( Pi^0_1-completeness) and when high undecidability ( Sigma^1_1-completeness), all with and without the assumption of normedness.

Improved undecidability results on weighted timed automata

In this paper, we strengthen two recent undecidability results about weighted timed automata, an extension of timed automata with cost variables. More precisely, we propose new encodings of a Minsky machine that only require three clocks and one stopwatch cost, while previous reductions required five clocks and one stopwatch cost.

A scenario-matching approach to the description and model checking of real-time properties

A major obstacle in the technology transfer agenda of behavioral analysis and design methods is the need for logics or automata to express properties for control-intensive systems. Interaction-modeling notations may offer a replacement or a complement, with a practitioner-appealing and lightweight flavor, due partly to the sub specification of intended behavior by means of scenarios. We propose a novel approach consisting of engineering a new formal notation of this sort based on a simple compact declarative semantics: VTS (visual timed event scenarios). Scenarios represent event patterns, graphically depicting conditions over traces. They predicate general system events and provide features to describe complex properties not expressible with MSC-like notations. The underlying formalism supports partial orders and real-time constraints. The problem of checking whether a timed-automaton model has a matching trace is proven decidable. On top of this kernel, we introduce a notation to state properties over all system traces: conditional scenarios, allowing engineers to describe uniquely rich connections between antecedent and consequent portions of the scenario. An undecidability result is presented for the general case of the model-checking problem over dense-time domains, to later identify a decidable-yet practically relevant-subclass, where verification is solvable by generating antiscenarios expressed in the VTS-kernel notation.

Polymorphic Systems with Arrays, 2-Counter Machines and Multiset Rewriting

Polymorphic systems with arrays (PSAs) is a general class of nondeterministic reactive systems. A PSA is polymorphic in the sense that it depends on a signature, which consists of a number of type variables, and a number of symbols whose types can be built from the type variables. Some of the state variables of a PSA can be arrays, which are functions from one type to another. We present several new decidability and undecidability results for parameterised control-state reacha- bility problems on subclasses of PSAs.

RESTRICTED SETS OF TRAJECTORIES AND DECIDABILITY OF SHUFFLE DECOMPOSITIONS

The decidability of the shuffle decomposition problem for regular languages is a long standing open question. We consider decompositions of regular languages with respect to shuffle along a regular set of trajectories and obtain positive decidability results for restricted classes of trajectories. Also we consider decompositions of unary regular languages. Finally, we establish in the spirit of the Dassow-Hinz undecidability result an undecidability result for regular languages shuffled along a fixed linear context-free set of trajectories.

Computing with locally effective matrices

In this work, we start from the naive notion of integer infinite matrix (i.e., the functions of the set ℤℕ × ℕ = {f: ℕ × ℕ → ℤ} ). Then, several undecidability results are established, leading to a convenient data structure for effective machine computations. We call this data structure a locally effective matrix. We study when (and how) the standard matrix calculus (Ker and CoKer computations) can be extended to the infinite case. We find again several undecidability barriers. When these limitations are overcome, we describe effective procedures for computing in the locally effective case. Finally, the role played by these data structures in the development of real symbolic computation systems for algebraic topology (based on the effective homology notion) is illustrated.

Deciding Global Partial-Order Properties

Model checking of asynchronous systems is traditionally based on the interleaving model, where an execution is modeled by a total order between atomic events. Recently, the use of partial order semantics, representing the causal order between events, is becoming popular. This paper considers the model checking problem for partial-order temporal logics. Solutions to this problem exist for partial order logics over local states. For the more general global logics that are interpreted over global states, only undecidability results have been proved. In this paper, we present a decision procedure for a partial order temporal logic over global states. We also sharpen the undecidability results by showing that a single until operator is sufficient for undecidability.