Showing papers on "Undecidable problem published in 2007"

Bounded synthesis

Abstract: The bounded synthesis problem is to construct an implementation that satisfies a given temporal specification and a given bound on the number of states. We present a solution to the bounded synthesis problem for linear-time temporal logic (LTL), based on a novel emptiness-preserving translation from LTL to safety tree automata. For distributed architectures, where standard unbounded synthesis is in general undecidable, we show that bounded synthesis can be reduced to a SAT problem. As a result, we obtain an effective algorithm for the bounded synthesis from LTL specifications in arbitrary architectures. By iteratively increasing the bound, our construction can also be used as a semi-decision procedure for the unbounded synthesis problem.

Conservative extensions in expressive description logics

Abstract: The notion of a conservative extension plays a central role in ontology design and integration: it can be used to formalize ontology refinements, safe mergings of two ontologies, and independent modules inside an ontology. Regarding reasoning support, the most basic task is to decide whether one ontology is a conservative extension of another. It has recently been proved that this problem is decidable and 2ExpTime-complete if ontologies are formulated in the basic description logic ALC. We consider more expressive description logics and begin to map out the boundary between logics for which conservativity is decidable and those for which it is not. We prove that conservative extensions are 2ExpTime-complete in ALCQI, but undecidable in ALCQIO. We also show that if conservative extensions are defined model-theoretically rather than in terms of the consequence relation, they are undecidable already in ALC.

On data mining, compression, and Kolmogorov complexity

Abstract: Will we ever have a theory of data mining analogous to the relational algebra in databases? Why do we have so many clearly different clustering algorithms? Could data mining be automated? We show that the answer to all these questions is negative, because data mining is closely related to compression and Kolmogorov complexity; and the latter is undecidable. Therefore, data mining will always be an art, where our goal will be to find better models (patterns) that fit our datasets as best as possible.

Temporalising Tractable Description Logics

Abstract: It is known that for temporal languages, such as first-order LTL, reasoning about constant (time-independent) relations is almost always undecidable. This applies to temporal description logics as well: constant binary relations together with general concept subsumptions in combinations of LTL and the basic description logic ALC cause undecidability. In this paper, we explore temporal extensions of two recently introduced families of 'weak' description logics known as DL-Lite and EL. Our results are twofold: temporalisations of even rather expressive variants of DL-Lite turn out to be decidable, while the temporalisation of EL with general concept subsumptions and constant relations is undecidable.

CWA-solutions for data exchange settings with target dependencies

Abstract: Data exchange deals with the following problem: given an instance over a source schema, a specification of the relationship between the source and the target,and dependencies on the target, construct an instance over a target schema that satisfies the given relationships and dependencies. Recently - for data exchange settings without target dependencies - Libkin (PODS'06) introduced a new concept of solutions based on the closed world assumption (so calledCWA-solutions), and showed that, in some respects, this new notion behaves better than the standard notion of solutions considered in previous papers on data exchange. The present paper extends Libkin's notion of CWA-solutions to data exchange settings with target dependencies. We show that, when restricting attention to data exchange settings with weakly acyclic target dependencies, this new notion behaves similarly as before: the core is the unique "minimal" CWA-solution, and computing CWA-solutions as well as certain answers to positive queries is possible in polynomial time and can be PTIME-hard. However, there may be more than one "maximal" CWA-solution. And going beyond the class of positive queries, we obtain that there are conjunctive queries with (just) one inequality, for which evaluating the certain answers is coNP-hard. Finally, we consider the EXISTENCE-OF-CWA-SOLUTIONS problem: while the problem is tractable for data exchange settings with weakly acyclic target dependencies, it turns out to be undecidable for general data exchange settings. As a consequence, we obtain that also the EXISTENCE-OF-UNIVERSAL-SOLUTIONS problem is undecidable in genera.

Metamorphism, Formal Grammars and Undecidable Code Mutation

Abstract: This paper presents a formalisation of the different existing code mutation techniques (polymorphism and metamor- phism) by means of formal grammars. While very few theoretical results are known about the detection complexity of viral mutation techniques, we exhaustively address this critical issue by considering the Chomsky classification of formal grammars. This enables us to determine which family of code mutation techniques are likely to be detected or on the contrary are bound to remain undetected. As an illustration we then present, on a formal basis, a proof-of-concept metamorphic mutation engine denoted PB MOT, whose detection has been proven to be undecidable. Keywords—Polymorphism, Metamorphism, Formal Grammars, Formal Languages, Language Decision, Code Mutation, Word Prob- lem. that the set Di of polymorphic viruses with an infinite number of forms is a Σ3-complete set. Unfortunately, no results is known for other classes of polymorphic viruses and for the general case of metamorphism. Many open problems still remain. Up to now, only very few examples of metamorphic codes are known to exist. The most sophisticated one is the MetaPHOR engine whose essential feature is a certain amount of non-determinism. Experiments in our laboratory showed that existing antivirus software can be very easily defeated by MetaPHOR-like technology. However, the analysis of this engine (9, Chap. 4) has proved that its metamorphic techniques still belong to trivial classes. Our research thus focused on the formalisation of metamor- phism by means of formal grammar and languages. We aimed at identifying the different possible classes of possible code mutation techniques. The first results, which are presented in this paper, enable to assert that detection complexity of code mutation techniques can be far higher that NP-complete and that for some well-chosen classes, detection is an undecidable problem.

Undecidability Bounds for Integer Matrices using Claus Instances

Abstract: There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ4×4 whether or not kI4 ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

Conservative ambiguity detection in context-free grammars

Abstract: The ability to detect ambiguities in context-free grammars is vital for their use in several fields, but the problem is undecidable in the general case. We present a safe, conservative approach, where the approximations cannot result in overlooked ambiguous cases. We analyze the complexity of the verification, and provide formal comparisons with several other ambiguity detection methods.

Polynomial size analysis of first-order functions

Abstract: We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be matrix multiplication and the Cartesian product of two lists. The type checking problem for the type system is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic. Furthermore, a method that infers polynomial size dependencies for a non-trivial class of function definitions is suggested.

Improved Undecidability Results on the Emptiness Problem of Probabilistic and Quantum Cut-Point Languages

Abstract: We give constructions of small probabilistic and MO-type quantum automata that have undecidable emptiness problem for the cut-point languages.

Conservative Extensions in the Lightweight Description Logic $\mathcal{EL}$

Abstract: We bring together two recent trends in description logic (DL): lightweight DLsin which the subsumption problem is tractable and conservative extensionsas a central tool for formalizing notions of ontology design such as refinement and modularity. Our aim is to investigate conservative extensions as an automated reasoning problem for the basic tractable DL $\mathcal{EL}$. The main result is that deciding (deductive) conservative extensions is ExpTime -complete, thus more difficult than subsumption in $\mathcal{EL}$, but not more difficult than subsumption in expressive DLs. We also show that if conservative extensions are defined model-theoretically, the associated decision problem for $\mathcal{EL}$ is undecidable.

Real-Time Model-Checking: Parameters Everywhere

Abstract: 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.

Model-checking one-clock priced timed automata

Abstract: We consider the model of priced (a.k.a. weighted) timed automata, an extension of timed automata with cost information on both locations and transitions. We prove that model-checking this class of models against the logic WCTL, CTL with cost-constrained modalities, is PSPACE-complete under the "single-clock" assumption. In contrast, it has been recently proved that the model-checking problem is undecidable for this model as soon as the system has three clocks. We also prove that the model-checking of WCTL* becomes undecidable, even under this "single-clock" assumption.

On the Kolmogorov−Chaitin Complexity for short sequences

Abstract: This is a presentation about joint work between Hector Zenil and Jean-Paul Delahaye. Zenil presents Experimental Algorithmic Theory as Algorithmic Information Theory and NKS, put together in a mixer. Algorithmic Complexity Theory defines the algorithmic complexity k(s) as the length of the shortest program that produces s. But since finding this short program is in general an undecidable question, the only way to approach k(s) is to use compression algorithms. He shows how to use the Compress function in Mathematica to give an idea about the compressibility of various sequences. However, the idea of applying a compression algorithm breaks down for very short sequences. This is true not only for the Compress function, but also for any other compression algorithm. Zenil's approach is to construct a metric of algorithmic complexity for short sequences from scratch. He defines the algorithmic probability as the probability that an arbitrary program produces a sequence. The basic idea is to run a whole class of computational devices such as Turing Machines or Cellular Automata, and compute the distributions of the sequences they generate. Zenil presents a comparison of frequency distributions of sequences generated by 2-state 3-color Turing Machines and 2-color radius 1 Cellular Automata. He also compared these distributions to distributions found in data from the real world, and found that not only there is correlation across different systems, but also that the distributions are rather stable, and the difference between the distributions in abstract systems and real-world data can be attributed to noise. In his paper Zenil elaborates on the nature of the noise he has encountered. Zenil conjectures that the correlation distances between different systems decreases with a larger number of steps, and converge in the infinite limit case.

On the (High) Undecidability of Distributed Synthesis Problems

Abstract: The distributed synthesis problem [11] is known to be undecidable. Our purpose here is to study further this undecidability. For this, we consider distributed games [8], an infinite variant of Peterson and Reif multiplayer games with partial information [10], in which Pnueli and Rosner's distributed synthesis problem can be encoded and, when decidable [11,6,7], uniformly solved [8]. We first prove that even the simple problem of solving 2-process distributed game with reachability conditions is undecidable ($\Sigma^0_1$-complete). This decision problem, equivalent to two process distributed synthesis with fairly restricted FO-specification was left open [8]. We prove then that the safety case is $\Pi^0_1$-complete. More generally, we establish a correspondence between 2-process distributed game with Mostowski's weak parity conditions [9] and levels of the arithmetical hierarchy. finally, distributed games with general i¾?-regular infinitary conditions are shown to be highly undecidable ($\Sigma^1_1$-complete).

Generic complexity of the conjugacy problem in hnn-extensions and algorithmic stratification of miller's groups

Abstract: We discuss time complexity of the Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may be undecidable. Our results show that "hard" instances of the problem comprise a negligibly small part of the group.

Distributed synthesis for alternating-time logics

Abstract: We generalize the distributed synthesis problem to the setting of alternating-time temporal logics. Alternating-time logics specify the game-like interaction between processes in a distributed system, which may cooperate on some objectives and compete on others. Our synthesis algorithm works for hierarchical architectures (in any two processes there is one that can see all inputs of the other process) and specifications in the temporal logics ATL, ATL*, and the alternating-time µ-calculus. Given an architecture and a specification, the algorithm constructs a distributed system that is guaranteed to satisfy the specification. We show that the synthesis problem for non-hierarchical architectures is undecidable, even for CTL specifications. Our algorithm is therefore a comprehensive solution for the entire range of specification languages from CTL to the alternating-time µ-calculus.

Selecting theories and nonce generation for recursive protocols

Abstract: Truderung's selecting theory model is one of the few models of cryptographic protocols which allows to model iterative (recursive) computations of principals and, at the same time, an automatic analysis in the following sense: there exists a procedure that checks whether in all runs of a given protocol a certain message is not revealed to an intruder. A major drawback of Truderung's model is that it allows only a finite number of constants, that is, there is no mechanism by which a principal can generate an unbounded number of fresh tokens such as nonces or session keys. We extend Truderung's model by such a mechanism and show that the extended model still allows an automatic analysis. We also demonstrate that the extended model is more appropriate to model the Recursive Authentication Protocol.It is important to know that after publication of Truderung's result it became clear that his proof only works in a restricted setting. We show that there is no hope to find a remedy to this by proving that the secrecy problem in the unrestricted setting is undecidable. In light of this, the aforementioned result about extending Truderung's model is to be read as follows. Adding anonymous constants in the restricted setting leads to a model of cryptographic protocols where insecurity is decidable.

Reachability problems in quaternion matrix and rotation semigroups

Abstract: We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are undecidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers.

Model Checking Synchronized Products of Infinite Transition Systems

Abstract: Formal verification using the model checking paradigm has to deal with two aspects: The system 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 (parameterized) 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. 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.

A Stochastic Complexity Perspective of Induction in Economics and Inference in Dynamics

Abstract: Rissanen's fertile and pioneering minimum description length principle (MDL) has been viewed from the point of view of statistical estimation theory, information theory, as stochastic complexity theory - i.e., a computable approximation of Kolomogorov Complexity - or Solomonoff's recursion theoretic induction principle or as analogous to Kolmogorov's sufficient statistics. All these - and many more - interpretations are valid, interesting and fertile. In this paper I view it from two points of view: those of an algorithmic economist and a dynamical system theorist. From these points of view I suggest, first, a recasting of Jevon's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics

AVACS -- Automatic Verification and Analysis of Complex Systems.

Abstract: The recently presented constraint solver HySAT tackles the in general undecidable problem of solving mixed Boolean and non-linear arithmetic constraint formulae over the reals involving transcendental functions. The algorithmic core of HySAT is the iSAT algorithm, a tight integration of the Davis-Putnam-Logemann-Loveland algorithm with interval constraint propagation enriched by enhancements like conflict-driven clause learning and non-chronological backtracking. However, it was an open question whether HySAT’s conflict analysis scheme, an adapted first unique implication point technique as applied in most modern SAT solvers, performs favorably compared to other schemes. In this paper, we compare several conflict analysis heuristics to answer this question. Furthermore, we consider the integration of restarts into the iSAT algorithm and investigate their impact. For our empirical results we use benchmarks from the hybrid systems domain.

Well-definedness and semantic type-checking for the nested relational calculus

Abstract: The well-definedness problem for a programming language consists of checking, given an expression and an input type, whether the semantics of the expression is defined for all inputs adhering to the input type. A related problem is the semantic type-checking problem which consists of checking, given an expression, an input type, and an output type whether the expression always returns outputs adhering to the output type on inputs adhering to the input type. Both problems are undecidable for general-purpose programming languages. In this paper we study these problems for the Nested Relational Calculus, a specific-purpose database query language. We also investigate how these problems behave in the presence of programming language features such as singleton coercion and type tests.

Verifying compensating transactions

Abstract: We study the safety verification problem for business-process orchestration languages with respect to regular properties. Business transactions involve long-running distributed interactions between multiple partners which must appear as a single atomic action. This illusion of atomicity is maintained through programmer-specified compensation actions that get run to undo previous actions when certain parts of the transaction fail to finish. Programming languages for business process orchestration provide constructs for declaring compensation actions, which are co-ordinated by the run time system to provide the desired transactional semantics. The safety verification problem for business processes asks, given a program with programmer specified compensation actions and a regular language specifying "good" behaviors of the system, whether all observable action sequences produced by the program are contained in the set of good behaviors. We show that the usual trace-based semantics for business process languages leads to an undecidable verification problem, but a tree-based semantics gives an algorithm that runs in time exponential in the size of the business process. Our constructions translate programs with compensations to tree automata with one memory.

On flat programs with lists

Abstract: In this paper we analyze the complexity of checking safety and termination properties, for a very simple, yet non-trivial, class of programs with singly-linked list data structures. Since, in general, programs with lists are known to have the power of Turing machines, we restrict the control structure, by forbidding nested loops and destructive updates. Surprisingly, even with these simplifying conditions, verifying safety and termination for programs working on heaps with more than one cycle are undecidable, whereas decidability can be established when the input heap may have at most one loop. The proofs for both the undecidability and the decidability results rely on non-trivial number-theoretic results.