Showing papers on "Undecidable problem published in 1996"

Symbolic Verification of Communication Protocols with Infinite State Spaces Using QDDs (Extended Abstract)

Abstract: We study the verification of properties of communication protocols modeled by a finite set of finite-state machines that communicate by exchanging messages via unbounded FIFO queues. It is well-known that most interesting verification problems, such as deadlock detection, are undecidable for this class of systems. However, in practice, these verification problems may very well turn out to be decidable for a subclass containing most “real” protocols.

Decision problems for semi-Thue systems with a few rules

Abstract: For several decision problems about semi-Thue systems, we try to locate the frontier between the decidable and the undecidable from the point of view of the number of rules. We show that the the Termination Problem, the U-Termination Problem, the Accessibility Problem and the Common-Descendant Problem are undecidable for 3 rules semi-Thue systems. As a corollary we obtain the undecidability of the Post-Correspondence Problem for 7 pairs of words.

Undecidable Verification Problems for Programs with Unreliable Channels

Abstract: We consider the class of finite-state systems communicating through unbounded butlossyFIFO channels (calledlossy channel systems) These systems have infinite state spaces due to the unboundedness of the channels In an earlier paper, we showed that the problems of checking reachability, safety properties, and eventuality properties are decidable for lossy channel systems In this paper, we show that the following problems are undecidable:?The model checking problem in propositional temporal logics such as propositional linear time temporal logic (PTL) and computation tree logic (CTL)?The problem of deciding eventuality properties with fair channels: do all computations eventually reach a given set of states if the unreliable channels satisfy fairness assumptions ?The results are obtained through reduction from a variant of the Post correspondence problem

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

Undecidable fragments of elementary theories

Abstract: We introduce a general framework to prove undecidability of fragments. This is applied to fragments of theories arising in algebra and recursion theory. For instance, the V3V-theories of the class of finite distributive lattices and of the p.o. of recursively enumerable many-one degrees are shown to be undecidable.

The subtyping problem for second-order types is undecidable

Abstract: We prove that the subtyping problem induced by Mitchell's containment relation (1988) for second-order polymorphic types is undecidable. It follows that type-checking is undecidable for the polymorphic lambda-calculus extended by an appropriate subsumption rule.

Solving linear equations over polynomial semirings

Abstract: We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only non-negative coefficients is shown to be undecidable. Applications to undecidability proofs of several unification problems are illustrated, one of which, unification modulo one associative-commutative function and one endomorphism, has been a long-standing open problem. The problem of solving multiset constraints is also shown to be undecidable.

Simultaneous rigid E-unification and related algorithmic problems

Abstract: The notion of simultaneous rigid E-unification was introduced in 1987 in the area of automated theorem proving with equality in sequent-based methods, for example the connection method or the tableau method. Recently, simultaneous rigid E-unification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid E-unification and show the connections between this notion and some algorithmic problems of logic and computer science.

A Formal Basis for Dynamic Schema Integration

Abstract: We construct a framework for determining when two schemata can be meaningfully integrated. Intuitively, two schemata can be integrated if the rules of one of the schemata together with a set of integration assertions do not restrict the models of the other schema. We formalise this concept using the notion of conflictfreeness and shows how it can be used to ensure that the merging of two schemata results in an integrated schema with the same information capacity as the original ones. The problem of conflictfreeness is undecidable, and we outline how it can be addressed for finite domains and determine its complexity properties in this case. Our approach takes into account static as well as dynamic aspects of a schema, where the dynamics is modelled by means of the event concept. We introduce correspondence assertions for events that can be used to specify the equivalence of event combinations.

An Algorithm for the Analysis of Termination of Large Trigger Sets in an OODBMS

Abstract: In this paper we describe an algorithm for the analysis of termination of a large set of triggers in an OODBMS. It is quite clear that, if the trigger mechanism is of sufficient complexity, the problem is undecidable. Yet, by the extensive use of object-oriented concepts, like derived classes, and lattice theory, we are able to give some sufficient conditions for termination which yield satisfying results. Another advantage of our approach is the uniform treatment of generic update operations on the one hand, and methods and abstract data types on the other.

The First-Order Theory of One-Step Rewriting is Undecidable

Abstract: The theory of one-step rewriting for a given rewrite system R and signature e is the first-order theory of the following structure: Its universe consists of all e-ground terms, and its only predicate is the relation “x rewrites to y in one step by R”. The structure contains no function symbols and no equality. We show that there is no algorithm deciding the ∃*∀*-fragment of this theory for an arbitrary rewrite system. The proof uses both non-linear and non-shallow rewrite rules.

Beta-Reduction as Unification

Abstract: We define a unification problem ^UP with the property that, given a pure lambda-term M, we can derive an instance Gamma(M) of ^UP from M such that Gamma(M) has a solution if and only if M is beta-strongly normalizable. There is a type discipline for pure lambda-terms that characterizes beta-strong normalization; this is the system of intersection types (without a ``top'''' type that can be assigned to every lambda-term). In this report, we use a lean version LAMBDA of the usual system of intersection types. Hence, ^UP is also an appropriate unification problem to characterize typability of lambda-terms in LAMBDA. It also follows that ^UP is an undecidable problem, which can in turn be related to semi-unification and second-order unification (both known to be undecidable).

Remarks on generalized Post Correspondence problem

Abstract: It is shown that Post Correspondence Problem remains undecidable even in the case where one of the morphisms is fixed. Accordingly the generalized PCP is undecidable even in the case where both of the morphisms are fixed, and, moreover, the cardinality of their domain alphabet is 7. In particular, GPCP(7 is undecidable. On the other hand, GPCP(2) is not only decidable, but, as we shall show here, its all solutions can be effectively found.

Relative Undecidability in Term Rewriting

Abstract: For two hierarchies of properties of term rewriting systems related to confluence and termination, respectively, we prove relative undecidability: for implications X⇒Y in the hierarchies the property X is undecidable for term rewriting systems satisfying Y.

DATALOG SIRUPs uniform boundedness is undecidable

Abstract: DATALOG is the paradigmatic database query language If it is possible to eliminate recursion from a DATALOG program then it is uniformly bounded Since uniformly bounded programs can be executed in parallel constant time, the possibility of automated boundedness detection is an important issue, and has been studied in many papers In this paper we solve one of the most famous open problems in the theory of deductive databases (see eg PC Kanellakis, Elements of Relational Database Theory in Handbook of Theoretical Computer Science) showing that uniform boundedness is undecidable for single rule programs (called also sirups)

Pseudorecursive varieties of semigroups — ii

Abstract: Constructions that yield pseudorecursiveness in [I] (Int. J. Algebra Comput.6 (1996) 457–510) are extended in this article. Finitely based varieties of semigroups with increasingly strict expansions by additional unary operation symbols or individual constants are shown to have the pseudorecursive property: the equational theory is undecidable, but the subsets obtained by bounding the number of distinct variables are all recursive. The most stringent case considered here is the single unary operation or distinguished element. New techniques of stratified reducibility and interpretation via rewriting rules are employed to show the property inherits along a chain of theories. Pure semigroup varieties that are both finitely based and pseudorecursive will be discussed in a later paper.

Undecidability and intuitionistic incompleteness

Abstract: Let S be a deductive system such that S-derivability (⊢s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and ⊢s, it follows constructively that the K-completeness of ⊢s implies MP(S), a form of Markov's Principle. If ⊢s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if ⊢s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when ⊢s is many-one complete, MP(S) implies the usual Markov's Principle MP.

Smallest horn clause programs

Abstract: The simplest nontrivial program pattern in logic programming is the following where fact, goal, left , and right are arbitrary terms. Because the well-known append program matches this pattern, we will denote such programs “ append -like.” In spite of their simple appearance, we prove in this paper that termination and satisfiability (i.e., the existence of answer-substitutions, called the emptiness problem) for append -like programs are undecidable. We also study some subcases depending on the number of occurrences of variables in fact, goal, left , or right . Moreover, we prove that the computational power of append -like programs is equivalent to the one of Turing machines; we show that there exists an append -like universal program. Thus, we propose an equivalent of the Bohm-Jacopini theorem for logic programming. This result confirms the expressiveness of logic programming. The proofs are based on program transformations and encoding of problems, unpredictable iterations within number theory defined by J. H. Conway, or the Post correspondence problem.

Uniform representation of recursively enumerable sets with simultaneous rigid E-unification

Abstract: Recently it was proved that the problem of simultaneous rigid E­unification (SREU) is undecidable. Here we perform an in­depth investigation of this matter and obtain that one can use SREU to uniformly represent any recursively enumerable set. From the exact form of this representation follows that SREU is undecidable already for 6 rigid equations with ground left hand sides and 2 variables. There is a close correspondence between solvability of SREU problems and provability of the corresponding formulas in intuitionistic first order logic with equality. Due to this correspondence we obtain a new (uniform) representation of the recursively enumerable sets in intuitionistic first order logic with equality with one binary function symbol and a countable set of constants. From this result follows the undecidability of the EE­fragment of intuitionistic logic with equality. This is an improvement of a recent result regarding the undecidability of the E*­fragment in general.

The 3 Frenchmen Method Proves Undecidability of the Uniform Boundedness for Single Recursive Rule Ternary DATALOG Programs

Abstract: DATALOG is the language of logic programs without function symbols. It is considered to be the paradigmatic database query language. If it is possible to eliminate the recursion from the program then it is uniformly bounded. We show that the uniform boundedness is undecidable for ternary DATALOG programs containing only one recursive rule, and for linear programs of arity 3. The proof is based on the discovery of, how we call it, Achilles-Turtle machine. It computes the subsequent iterations of a Conway function and is, up to our knowledge, the simplest known universal machine.

1996 Kluwer Academic Publishers. Printed in the Netherlands.

Abstract: We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: **point of reference - reference pointer" which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions.

Meta-variables in logic programming, or in praise of ambivalent syntax

Abstract: We show here that meta-variables of Prolog admit a simple declarative interpretation. This allows us to extend the usual theory of SLD-resolution to the case of logic programs with meta-variables, and to establish soundness and strong completeness of the corresponding extension of the SLD-resolution. The key idea is the use of ambivalent syntax which allows us to use the same symbols as function and relation symbols. We also study the problem of absence of run-time errors in presence of meta-variables. We prove that this problem is undecidable. However, we also provide some sufficient and polynomial-time-decidable conditions which imply absence of run-time errors.

Translating the hypergame paradox: Remarks on the set of founded elements of a relation

Abstract: In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct an undecidable formula.

The spectral radius of a pair of matrices is hard to compute

Abstract: We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalised spectral radii of two integer matrices are not approximable in polynomial time, and that, two related quantities-the lower spectral radius and the largest Lyapunov exponent-are not algorithmically approximable. As a corollary of these results we show that: 1) the problem of deciding if all possible products of two given matrices are stable is NP-hard, 2) the problem of deciding if at, least one product of two given matrices is stable is undecidable.

Lower Bounds for Geometrical and Physical Problems

Abstract: Motion planning involving arbitrarily many degrees of freedom is known to be PSPACE-hard. In this paper, we examine the complexity of generalized motion-planning problems for planar mechanisms consisting of independently movable objects. Our constructions constitute a general framework for reducing problems in information processing to motion planning, leading to easy proofs of known PSPACE-hardness results and to exponential lower bounds for geometrical problems related to motion planning. Particulalrly, we show that the problem of deciding whether a given mechanism $A$ can always avoid a collision with another mechanism $B$ is EXPSPACE-hard. New lower bounds are also obtained for the problem of planning under given physical side conditions. We consider the case that certain motions require forces, e.g., to subdue friction, and ask for motions that stay under a given energy limit. Within our framework, we show that such shortest-path problems are EXPTIME-hard if we use number representations by mantissa and exponent, and even undecidable if we allow that some motions require no force or an infinite amount. The proof consists of a simulation of Turing machines with infinite tape and shows that the notion of Turing computability can be interpreted in purely geometrical terms. The geometrical model obtained is capable of expressing a variety of physical-planning problems.

Unification in Pseudo-Linear Sort Theories is Decidable

Abstract: In this paper the decidability of unification in pseudo-linear sort theories is shown. In contrast to standard unification, variables range over subsets of the domain described by sorts. The denotations of sorts are fixed by declarations in a sort theory. Then two terms are unifiable, if they are unifiable in the standard sense and the assignments of the unifier satisfy the restrictions on the domain variables with respect to the sort theory. This problem is known to be undecidable in general, but is known to be decidable for elementary, weak-elementary, linear and semi-linear sort theories. So-called pseudo-linear sort theories properly include all these sort theories. We show that the problem of whether two terms are unifiable with respect to a pseudo-linear sort theory is decidable and of unification type infinitary.

Unique Deciperability for Partially Commutative Alphabet (Extended Abstract)

Abstract: We consider the unique decipherability problem for partially commutative alphabet. It is shown that the complexity of this problem depends heavily on the size of the alphabet and the structure of the commutativity relation graph. In particular, for alphabets with ≤3 letters the problem is decidable and for alphabets with ≥4 letters the problem is undecidable.