Showing papers on "Undecidable problem published in 2004"

Multiset rewriting and the complexity of bounded security protocols

Abstract: We formalize the Dolev-Yao model of security protocols, using a notation based on multiset rewriting with existentials. The goals are to provide a simple formal notation for describing security protocols, to formalize the assumptions of the Dolev-Yao model using this notation, and to analyze the complexity of the secrecy problem under various restrictions. We prove that, even for the case where we restrict the size of messages and the depth of message encryption, the secrecy problem is undecidable for the case of an unrestricted number of protocol roles and an unbounded number of new nonces. We also identify several decidable classes, including a DEXP-complete class when the number of nonces is restricted, and an NP-complete class when both the number of nonces and the number of roles is restricted. We point out a remaining open complexity problem, and discuss the implications these results have on the general topic of protocol analysis.

A Survey of Regular Model Checking

Abstract: Regular model checking is being developed for algorithmic verification of several classes of infinite-state systems whose configurations can be modeled as words over a finite alphabet. Examples include parameterized systems consisting of an arbitrary number of homogeneous finite-state processes connected in a linear or ring-formed topology, and systems that operate on queues, stacks, integers, and other linear data structures. The main idea is to use regular languages as the representation of sets of configurations, and finite-state transducers to describe transition relations. In general, the verification problems considered are all undecidable, so the work has consisted in developing semi-algorithms, and decidability results for restricted cases. This paper provides a survey of the work that has been performed so far, and some of its applications.

Finite Presentations of Infinite Structures:Automata and Interpretations

Abstract: We study definability problems and algorithmic issues for infinite structures that are finitely presented After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations These two ways of presenting a structure are related Indeed, a structure is automatic if, and only if, it is first-order interpretable in an appropriate expansion of Presburger arithmetic or, equivalently, in the infinite binary tree with prefix order and equal length predicate Similar results hold for ω-automatic structures and appropriate expansions of the real ordered group We also discuss the relationship to automatic groups The model checking problem for FO(∃ω), first-order logic extended by the quantifier “there are infinitely many”, is proved to be decidable for automatic and ω-automatic structures Further, the complexity for various fragments of first-order logic is determined On the other hand, several important properties not expressible in FO, such as isomorphism or connectedness, turn out to be undecidable for automatic structures Finally, we investigate methods for proving that a structure does not admit an automatic presentation, and we establish that the class of automatic structures is closed under Feferman–Vaught-like products

A survey of regular model checking

Abstract: Regular model checking is being developed for algorithmic verification of several classes of infinite-state systems whose configurations can be modeled as words over a finite alphabet. Examples include parameterized systems consisting of an arbitrary number of homogeneous finite-state processes connected in a linear or ring-formed topology, and systems that operate on queues, stacks, integers, and other linear data structures. The main idea is to use regular languages as the representation of sets of configurations, and finite-state transducers to describe transition relations. In general, the verification problems considered are all undecidable, so the work has consisted in developing semi-algorithms, and decidability results for restricted cases. This paper provides a survey of the work that has been performed so far, and some of its applications.

The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions

Abstract: Many cyberinformaticians would agree that, had it not been for amphibious epistemologies, the refinement of randomized algorithms might never have occurred [114, 114, 188, 62, 114, 62, 70, 179, 68, 95, 54, 188, 152, 95, 191, 59, 168, 148, 99, 152]. In this work, we disprove the deployment of cache coherence [58, 129, 128, 106, 154, 51, 176, 164, 76, 59, 134, 203, 193, 116, 65, 24, 123, 109, 48, 177]. Leat, our new system for reliable models, is the solution to all of these issues.

On the language inclusion problem for timed automata: closing a decidability gap

Abstract: We consider the language inclusion problem for timed automata: given two timed automata A and B, are all the timed traces accepted by B also accepted by A? While this problem is known to be undecidable, we show here that it becomes decidable if A is restricted to having at most one clock. This is somewhat surprising, since it is well-known that there exist timed automata with a single clock that cannot be complemented. The crux of our proof consists in reducing the language inclusion problem to a reachability question on an infinite graph; we then construct a suitable well-quasi-order on the nodes of this graph, which ensures the termination of our search algorithm. We also show that the language inclusion problem is decidable if the only constant appearing among the clock constraints of A is zero. Moreover, these two cases are essentially the only decidable instances of language inclusion, in terms of restricting the various resources of timed automata.

The Boundary Between Decidability and Undecidability for Transitive-Closure Logics

Abstract: To reason effectively about programs, it is important to have some version of a transitive-closure operator so that we can describe such notions as the set of nodes reachable from a program’s variables. On the other hand, with a few notable exceptions, adding transitive closure to even very tame logics makes them undecidable.

The groups of richard thompson and complexity

Abstract: We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We give a faithful representation in the Cuntz C⋆-algebra. For the finitely presented simple group V we show that the word-length and the table size satisfy an n log n relation. We show that the word problem of V belongs to the parallel complexity class AC1 (a subclass of P), whereas the generalized word problem of V is undecidable. We study the distortion functions of V and show that V contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of V, the set of Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

Parameterized Model Checking of Ring-Based Message Passing Systems

Abstract: The Parameterized Model Checking Problem (PMCP) is to decide whether a temporal property holds for a uniform family of systems, Un, comprised of finite, but arbitrarily many, copies of a template process U. Unfortunately, it is undecidable in general [3]. In this paper, we consider the PMCP for systems comprised of processes arranged in a ring that communicate by passing messages via tokens whose values can be updated at most a bounded number of times. Correctness properties are expressed using the stuttering-insensitive linear time logic LTL∖X. For bidirectional rings we show how to reduce reasoning about rings with an arbitrary number of processes to rings with up to a certain finite cutoff number of processes. This immediately yields decidability of the PMCP at hand. We go on to show that for unidirectional rings small cutoffs can be achieved, making the decision procedure provably efficient. As example applications, we consider protocols for the leader election problem.

Properties of Object Petri Nets

Abstract: In this presentation the structure of formalisms are studied that allow Petri nets as tokens. The relationship towards common Petri net models and decidability issues are studied. Especially for ”elementary object-net systems” defined by Valk [x] the decidability of the reachability and the boundedness problem is considered. It is shown that reachability becomes undecidable while boundedness remains decidable for elementary object-net systems. Furthermore it is shown that even for minimal extensions the formalism obtains the power of Turing machines.

Consistency of a counterexample to Naimark's problem.

Abstract: We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assuming that these axioms are consistent). We prove that the statement “there exists a counterexample to Naimark's problem which is generated by elements” is undecidable in standard set theory.

Distributed games with causal memory are decidable for series-parallel systems

Abstract: This paper deals with distributed control problems by means of distributed games played on Mazurkiewicz traces. The main difference with other notions of distributed games recently introduced is that, instead of having a local view, strategies and controllers are able to use a more accurate memory, based on their causal view. Our main result states that using the causal view makes the control synthesis problem decidable for series-parallel systems for all recognizable winning conditions on finite behaviors, while this problem with local view was proved undecidable even for reachability conditions.

Fragments of Language

Abstract: By a fragment of a natural language we mean a subset of that language equipped with semantics which translate its sentences into some formal system such as first-order logic. The familiar concepts of satisfiability and entailment can be defined for any such fragment in a natural way. The question therefore arises, for any given fragment of a natural language, as to the computational complexity of determining satisfiability and entailment within that fragment. We present a series of fragments of English for which the satisfiability problem is polynomial, NP-complete, EXPTIME-complete, NEXPTIME-complete and undecidable. Thus, this paper represents a case study in how to approach the problem of determining the logical complexity of various natural language constructions. In addition, we draw some general conclusions about the relationship between natural language and formal logic.

A verification technique using term rewriting systems and abstract interpretation

Abstract: Verifying the safety property of a transition system given by a term rewriting system is an undecidable problem. In this paper, we give an abstraction for the problem which is automatically generated from a given TRS by using abstract interpretation. Then we show that there are some cases in which the problem can be decided. Also we show a new decidable subclass of term rewriting systems which effectively preserves recognizability.

Captive Cellular Automata

Abstract: We introduce a natural class of cellular automata characterised by a property of the local transition law without any assumption on the states set. We investigate some algebraic properties of the class and show that it contains intrinsically universal cellular automata. In addition we show that Rice’s theorem for limit sets is no longer true for that class, although infinitely many properties of limit sets are still undecidable.

On Logics of Aliasing

Abstract: In this paper we investigate the existence of a deductive verification method based on a logic that describes pointer aliasing. The main idea of such a method is that the user has to annotate the program with loop invariants, pre- and post-conditions. The annotations are then automatically checked for validity by propagating weakest preconditions and verifying a number of induced implications. Such a method requires an underlying logic which is decidable and has a sound and complete weakest precondition calculus. We start by presenting a powerful logic (wAL) which can describe the shapes of most recursively defined data structures (lists, trees, etc.) has a complete weakest precondition calculus but is undecidable. Next, we identify a decidable subset (pAL) for which we show closure under the weakest precondition operators. In the latter logic one loses the ability of describing unbounded heap structures, yet bounded structures can be characterized up to isomorphism. For this logic two sound and complete proof systems are given, one based on natural deduction, and another based on the effective method of analytic tableaux. The two logics presented in this paper can be seen as extreme values in a framework which attempts to reconcile the naturally oposite goals of expressiveness and decidability.

Counter-example generation in symbolic abstract model-checking

Abstract: The boundaries of model-checking have been extended through the use of abstraction. These techniques are conservative, in the following sense: when the verification succeeds, the verified property is guaranteed to hold; but when it fails, it may result either from the non satisfaction of the property, or from a too rough abstraction. In case of failure, it is, in general, undecidable whether an abstract trace corresponding to a counter-example has any concrete counterparts. For debugging purposes, one usually desires to go further than giving a “yes/no” answer (actually, a “yes/don’t know” answer!), and look for such concrete counter-examples. We propose a solution in which we apply standard test-pattern generation technology to search for concrete instances of abstract traces.

A Verification Technique Using Term Rewriting Systems and Abstract Interpretation

Abstract: Verifying the safety property of a transition system given by a term rewriting system is an undecidable problem. In this paper, we give an abstraction for the problem which is automatically generated from a given TRS by using abstract interpretation. Then we show that there are some cases in which the problem can be decided. Also we show a new decidable subclass of term rewriting systems which effectively preserves recognizability.

A Note on Join and Auto-Intersection of n-ary Rational Relations

Abstract: A finite-state machine with n tapes describes a rational (or regular) relation on n strings. It is more expressive than a relational database table with n columns, which can only describe a finite relation. We describe some basic operations on n-ary rational relations and propose notation for them. (For generality we give the semiring-weighted case in which each tuple has a weight.) Unfortunately, the join operation is problematic: if two rational relations are joined on more than one tape, it can lead to non-rational relations with undecidable properties. We recast join in terms of “auto-intersection” and illustrate some cases in which difficulties arise. We close with the hope that partial or restricted algorithms may be found that are still powerful enough to have practical use.

Model checking synchronized products of infinite transition systems

Abstract: Formal verification using the model-checking paradigm has to deal with two aspects. The systems models are structured, often as products of components, and the specification logic has to be expressive enough to allow the formalization of reachability properties. The present paper is a study on what can be achieved for infinite transition systems under these premises. As models, we consider products of infinite transition systems with different synchronization constraints. We introduce finitely synchronized transition systems, i.e. product systems which contain only finitely many synchronized transitions, and show that the decidability of FO(R), first-order logic extended by reachability predicates, of the product system can be reduced to the decidability of FO(R) of the components in a Feferman-Vaught like style. This result is optimal in the following sense. (1) If we allow semifinite synchronization, i.e. just in one component infinitely many transitions are synchronized, the FO(R)-theory of the product system is in general undecidable. (2) We cannot extend the expressive power of the logic under consideration. Already a weak extension of first-order logic with transitive closure, where we restrict the transitive closure operators to arity one and nesting depth two, is undecidable for an asynchronous (and hence finitely synchronized) product, namely for the infinite grid.

Some Open Problems in Decidability of Brick (Labelled Polyomino) Codes

Abstract: Bricks are polyominoes with labelled cells. The problem of whether a given set of bricks is a code is undecidable in general. It is open for two-element sets. Here we consider sets consisting of square bricks only. We show that in this setting, the codicity of small sets (two bricks) is decidable, but 15 bricks are enough to make the problem undecidable. Thus the frontier between decidability and undecidability lies somewhere between these two numbers. Additionally we know that the codicity problem is decidable for sets with keys of size n when n = 1 and, under obvious constraints, for every n. We prove that it is undecidable in the general case of sets with keys of size n when n≥ 6.

Hilbert's tenth problem for function fields of varieties over ℂ

Abstract: Let K be the function field of a variety of dimension greater than or equal to 2 over an algebraically closed field of characteristic zero. Then Hilbert's tenth problem for K is undecidable. This generalizes the result by Kim and Roush, 1992, that Hilbert's tenth problem for the purely transcendental function field ℂ(t 1 ,t 2 ) is undecidable.