Showing papers on "Undecidable problem published in 2013"

Reasoning About Strategies

Abstract: In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between open entities and express that the system is correct no matter how the environment behaves. An important contribution in this context is given by the modal logics for strategic ability, in the setting of multi-agent games, such as Atl, Atl \(^{*}\), and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which we denote here by CHP-Sl, with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-Sl is obtained by using first-order quantifications over strategies and it has been investigated in the specific setting of two-agents turned-based game structures where a non-elementary model-checking algorithm has been provided. While CHP-Sl is a very expressive logic, we claim that it does not fully capture the strategic aspects of multi-agent systems. In this work, we introduce and study a more general strategy logic, denoted Sl, for reasoning about strategies in multi-agent concurrent systems. We prove that Sl strictly includes CHP-Sl, while maintaining a decidable model-checking problem. Indeed, we show that it is 2ExpTime-complete under a reasonable semantics, thus not harder than that for Atl \(^{*}\) and a remarkable improvement of the same problem for CHP-Sl. We also consider the satisfiability problem and show that it is undecidable already for the sub-logic CHP-Sl under the concurrent game semantics.

Undecidability in epistemic planning

Abstract: Dynamic epistemic logic (DEL) provides a very expressive framework for multi-agent planning that can deal with nondeterminism, partial observability, sensing actions, and arbitrary nesting of beliefs about other agents' beliefs. However, as we show in this paper, this expressiveness comes at a price. The planning framework is undecidable, even if we allow only purely epistemic actions (actions that change only beliefs, not ontic facts). Undecidability holds already in the S5 setting with at least 2 agents, and even with 1 agent in S4. It shows that multi-agent planning is robustly undecidable if we assume that agents can reason with an arbitrary nesting of beliefs about beliefs. We also prove a corollary showing undecidability of the DEL model checking problem with the star operator on actions (iteration).

Regression verification: proving the equivalence of similar programs

Abstract: SUMMARY Proving the equivalence of successive, closely related versions of a program has the potential of being easier in practice than functional verification, although both problems are undecidable. There are three main reasons for this claim: (i) it circumvents the problem of specifying what the program should do; (ii) the problem can be naturally decomposed and hence is computationally easier; and (iii) there is an automatic invariant that enables to prove equivalence of loops and recursive functions in most practical cases. Theoretical and practical aspects of this problem are considered. Copyright © 2012 John Wiley & Sons, Ltd.

From propositional to first-order monitoring

Abstract: The main purpose of this paper is to introduce a first-order temporal logic, LTLFO, and a corresponding monitor construction based on a new type of automaton, called spawning automaton. Specifically, we show that monitoring a specification in LTLFO boils down to an undecidable decision problem. The proof of this result revolves around specific ideas on what we consider a "proper" monitor. As these ideas are general, we outline them first in the setting of standard LTL, before lifting them to the setting of first-order logic and LTLFO. Although due to the above result one cannot hope to obtain a complete monitor for LTLFO, we prove the soundness of our automata-based construction and give experimental results from an implementation. These seem to substantiate our hypothesis that the automata-based construction leads to efficient runtime monitors whose size does not grow with increasing trace lengths (as is often observed in similar approaches). However, we also discuss formulae for which growth is unavoidable, irrespective of the chosen monitoring approach.

Verifying concurrent programs against sequential specifications

Abstract: We investigate the algorithmic feasibility of checking whether concurrent implementations of shared-memory objects adhere to their given sequential specifications; sequential consistency, linearizability, and conflict serializability are the canonical variations of this problem. While verifying sequential consistency of systems with unbounded concurrency is known to be undecidable, we demonstrate that conflict serializability, and linearizability with fixed linearization points are EXPSPACE-complete, while the general linearizability problem is undecidable. Our (un)decidability proofs, besides bestowing novel theoretical results, also reveal novel program explorations strategies. For instance, we show that every violation to conflict serializability is captured by a conflict cycle whose length is bounded independently from the number of concurrent operations. This suggests an incomplete detection algorithm which only remembers a small subset of conflict edges, which can be made complete by increasing the number of remembered edges to the cycle-length bound. Similarly, our undecidability proof for linearizability suggests an incomplete detection algorithm which limits the number of "barriers" bisecting non-overlapping operations. Our decidability proof of bounded-barrier linearizability is interesting on its own, as it reduces the consideration of all possible operation serializations to numerical constraint solving. The literature seems to confirm that most violations are detectable by considering very few conflict edges or barriers.

Independence in Database Relations

Abstract: We investigate the implication problem for independence atoms $X \bot Y$ of disjoint attribute sets X and Y on database schemata. A relation satisfies $X \bot Y$ if for every X-value and every Y-value that occurs in the relation there is some tuple in the relation in which the X-value occurs together with the Y-value. We establish an axiomatization by a finite set of Horn rules, and derive an algorithm for deciding the implication problem in low-degree polynomial time in the input. We show how to construct Armstrong relations which satisfy an arbitrarily given set of independence atoms and violate every independence atom not implied by the given set. Our results establish independence atoms as an efficient subclass of embedded multivalued data dependencies which are not axiomatizable by a finite set of Horn rules, and whose implication problem is undecidable.

Graph Logics with Rational Relations

Abstract: We investigate some basic questions about the interaction of regular and ra- tional relations on words. The primary motivation comes from the study of logics for querying graph topology, which have recently found numerous applications. Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword or subsequence. Evaluating formulae in such extended graph logics boils down to checking nonemptiness of the intersection of rational relations with regular or recognizable relations (or, more generally, to the gener- alized intersection problem, asking whether some projections of a regular relation have a nonempty intersection with a given rational relation). We prove that for several basic and commonly used rational relations, the intersec- tion problem with regular relations is either undecidable (e.g., for subword or sux, and some generalizations), or decidable with non-primitive-recursive complexity (e.g., for sub- sequence and its generalizations). These results are used to rule out many classes of graph logics that freely combine regular and rational relations, as well as to provide the sim- plest problem related to verifying lossy channel systems that has non-primitive-recursive complexity. We then prove a dichotomy result for logics combining regular conditions on individual paths and rational relations on paths, by showing that the syntactic form of formulae classies them into either eciently checkable or undecidable cases. We also give examples of rational relations for which such logics are decidable even without syntactic restrictions.

Asynchronous games over tree architectures

Abstract: We consider the distributed control problem in the setting of Zielonka asynchronous automata Such automata are compositions of finite processes communicating via shared actions and evolving asynchronously Most importantly, processes participating in a shared action can exchange complete information about their causal past This gives more power to controllers, and avoids simple pathological undecidable cases as in the setting of Pnueli and Rosner We show the decidability of the control problem for Zielonka automata over acyclic communication architectures We provide also a matching lower bound, which is l-fold exponential, l being the height of the architecture tree

Distributed synthesis for LTL fragments

Abstract: We consider the distributed synthesis problem for temporal logic specifications. Traditionally, the problem has been studied for LTL, and the previous results show that the problem is decidable iff there is no information fork in the architecture. We consider the problem for fragments of LTL and our main results are as follows: (1) We show that the problem is undecidable for architectures with information forks even for the fragment of LTL with temporal operators restricted to next and eventually. (2) For specifications restricted to globally along with non-nested next operators, we establish decidability (in EXPSPACE) for star architectures where the processes receive disjoint inputs, whereas we establish undecidability for architectures containing an information fork-meet structure. (3) Finally, we consider LTL without the next operator, and establish decidability (NEXPTIME-complete) for all architectures for a fragment that consists of a set of safety assumptions, and a set of guarantees where each guarantee is a safety, reachability, or liveness condition.

On Forward Closure and the Finite Variant Property

Abstract: Equational unification is an important research area with many applications, such as cryptographic protocol analysis. Unification modulo a convergent term rewrite system is undecidable, even with just a single rule. To identify decidable (and tractable) cases, two paradigms have been developed — Basic Syntactic Mutation [14] and the Finite Variant Property [6]. Inspired by the Basic Syntactic Mutation approach, we investigate the notion of forward closure along with suitable redundancy constraints. We show that a convergent term rewriting system R has a finite forward closure if and only if R has the finite variant property. We also show the undecidability of the finiteness of forward closure, therefore determining if a system has the finite variant property is undecidable.

Context unification is in PSPACE

Abstract: Contexts are terms with one `hole', i.e. a place in which we can substitute an argument. In context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. Context unification is a natural subvariant of second-order unification, which is undecidable, and a generalization of word equations, which are decidable, at the same time. It is the unique problem between those two whose decidability is uncertain (for already almost two decades). In this paper we show that the context unification is in PSPACE. The result holds under a (usual) assumption that the first-order signature is finite. This result is obtained by an extension of the recompression technique, recently developed by the author and used in particular to obtain a new PSPACE algorithm for satisfiability of word equations, to context unification. The recompression is based on performing simple compression rules (replacing pairs of neighbouring function symbols), which are (conceptually) applied on the solution of the context equation and modifying the equation in a way so that such compression steps can be in fact performed directly on the equation, without the knowledge of the actual solution.

Verification of Golog Programs over Description Logic Actions

Abstract: High-level action programming languages such as Golog have successfully been used to model the behavior of autonomous agents. In addition to a logic-based action formalism for describing the environment and the effects of basic actions, they enable the construction of complex actions using typical programming language constructs. To ensure that the execution of such complex actions leads to the desired behavior of the agent, one needs to specify the required properties in a formal way, and then verify that these requirements are met by any execution of the program. Due to the expressiveness of the action formalism underlying Golog (Situation Calculus), the verification problem for Golog programs is in general undecidable. Action formalisms based on Description Logic (DL) try to achieve decidability of inference problems such as the projection problem by restricting the expressiveness of the underlying base logic. However, until now these formalisms have not been used within Golog programs. In the present paper, we introduce a variant of Golog where basic actions are defined using such a DL-based formalism, and show that the verification problem for such programs is decidable. This improves on our previous work on verifying properties of infinite sequences of DL actions in that it considers (finite and infinite) sequences of DL actions that correspond to (terminating and non-terminating) runs of a Golog program rather than just infinite sequences accepted by a Buchi automaton abstracting the program.

Tiling Problems on Baumslag-Solitar groups

Abstract: We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.

Extendability of continuous maps is undecidable

Abstract: We consider two basic problems of algebraic topology, the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity. The extension problem is the following: Given topological spaces X and Y, a subspace A\subseteq X, and a (continuous) map f:A->Y, decide whether f can be extended to a continuous map \bar{f}:X->Y. All spaces are given as finite simplicial complexes and the map f is simplicial. Recent positive algorithmic results, proved in a series of companion papers, show that for (k-1)-connected Y, k>=2, the extension problem is algorithmically solvable if the dimension of X is at most 2k-1, and even in polynomial time when k is fixed. Here we show that the condition \dim X<=2k-1 cannot be relaxed: for \dim X=2k, the extension problem with (k-1)-connected Y becomes undecidable. Moreover, either the target space Y or the pair (X,A) can be fixed in such a way that the problem remains undecidable. Our second result, a strengthening of a result of Anick, says that the computation of \pi_k(Y) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.

Decision Problems for Additive Regular Functions

Abstract: Additive Cost Register Automata (ACRA) map strings to integers using a finite set of registers that are updated using assignments of the form "x := y + c" at every step. The corresponding class of additive regular functions has multiple equivalent characterizations, appealing closure properties, and a decidable equivalence problem. In this paper, we solve two decision problems for this model. First, we define the register complexity of an additive regular function to be the minimum number of registers that an ACRA needs to compute it. We characterize the register complexity by a necessary and sufficient condition regarding the largest subset of registers whose values can be made far apart from one another. We then use this condition to design a PSPACE algorithm to compute the register complexity of a given ACRA, and establish a matching lower bound. Our results also lead to a machine-independent characterization of the register complexity of additive regular functions. Second, we consider two-player games over ACRAs, where the objective of one of the players is to reach a target set while minimizing the cost. We show the corresponding decision problem to be EXPTIME-complete when costs are non-negative integers, but undecidable when costs are integers.

Hyperplane separation technique for multidimensional mean-payoff games

Abstract: Two-player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems. In this work, we consider both finite-state game graphs, and recursive game graphs (or pushdown game graphs) that model the control flow of sequential programs with recursion. The objectives we study are multidimensional mean-payoff objectives, where the goal of player 1 is to ensure that the mean-payoff is non-negative in all dimensions. In pushdown games two types of strategies are relevant: (1) global strategies, that depend on the entire global history; and (2) modular strategies, that have only local memory and thus do not depend on the context of invocation. Our main contributions are as follows: (1) We show that finite-state multidimensional mean-payoff games can be solved in polynomial time if the number of dimensions and the maximal absolute value of the weights are fixed; whereas if the number of dimensions is arbitrary, then the problem is known to be coNP-complete. (2) We show that pushdown graphs with multidimensional mean-payoff objectives can be solved in polynomial time. For both (1) and (2) our algorithms are based on hyperplane separation technique. (3) For pushdown games under global strategies both one and multidimensional mean-payoff objectives problems are known to be undecidable, and we show that under modular strategies the multidimensional problem is also undecidable; under modular strategies the one-dimensional problem is NP-complete. We show that if the number of modules, the number of exits, and the maximal absolute value of the weights are fixed, then pushdown games under modular strategies with one-dimensional mean-payoff objectives can be solved in polynomial time, and if either the number of exits or the number of modules is unbounded, then the problem is NP-hard. (4) Finally we show that a fixed parameter tractable algorithm for finite-state multidimensional mean-payoff games or pushdown games under modular strategies with one-dimensional mean-payoff objectives would imply the fixed parameter tractability of parity games.

Rapidly Growing Ramsey Functions

Abstract: Godel’s paper on formally undecidable propositions in first order Peano arithmetic (Godel 1931) showed that any recursive axiomatic system containing Peano arithmetic still admits propositions which are not decidable. Godel’s original example of such a proposition was not that illuminating. It was merely a kind of formalization of the well known antinomy of the liar. This raised the problem to look for intuitively meaningful propositions which are independent of Peano arithmetic.

Type classes for efficient exact real arithmetic in coq

Abstract: Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.

Analysis of Recursively Parallel Programs

Abstract: We propose a general formal model of isolated hierarchical parallel computations, and identify several fragments to match the concurrency constructs present in real-world programming languages such as Cilk and X10. By associating fundamental formal models (vector addition systems with recursive transitions) to each fragment, we provide a common platform for exposing the relative difficulties of algorithmic reasoning. For each case we measure the complexity of deciding state reachability for finite-data recursive programs, and propose algorithms for the decidable cases. The complexities which include PTIME, NP, EXPSPACE, and 2EXPTIME contrast with undecidable state reachability for recursive multithreaded programs.

Conjunctive queries with negation over DL-Lite: a closer look

Abstract: While conjunctive query (CQ) answering over DL-Lite has been studied extensively, there have been few attempts to analyse CQs with negated atoms. This paper deepens the study of the problem. Answering CQs with safe negation and CQs with a single inequality over DL-Lite with role inclusions is shown to be undecidable, even for a fixed TBox and query.Without role inclusions, answering CQs with one inequality is P-hard and with two inequalities coNP-hard in data complexity.

Alternating-time temporal logic with finite-memory strategies

Abstract: Model-checking the alternating-time temporal logics ATL and ATL* with incomplete information is undecidable for perfect recall semantics However, when restricting to memoryless strategies the model-checking problem becomes decidable In this paper we consider two other types of semantics based on finite-memory strategies One where the memory size allowed is bounded and one where the memory size is unbounded (but must be finite) This is motivated by the high complexity of model-checking with perfect recall semantics and the severe limitations of memoryless strategies We show that both types of semantics introduced are different from perfect recall and memoryless semantics and next focus on the decidability and complexity of model-checking in both complete and incomplete information games for ATL/ATL* In particular, we show that the complexity of model-checking with bounded-memory semantics is Delta_2p-complete for ATL and PSPACE-complete for ATL* in incomplete information games just as in the memoryless case We also present a proof that ATL and ATL* model-checking is undecidable for n >= 3 players with finite-memory semantics in incomplete information games

Bounded programs: a new decidable class of logic programs with function symbols

Abstract: While function symbols are widely acknowledged as an important feature in logic programming, they make common inference tasks undecidable. To cope with this problem, recent research has focused on identifying classes of logic programs imposing restrictions on the use of function symbols, but guaranteeing decidability of common inference tasks. This has led to several criteria, called termination criteria, providing sufficient conditions for a program to have finitely many stable models, each of finite size. This paper introduces the new class of bounded programs which guarantees the aforementioned property and strictly includes the classes of programs determined by current termination criteria. Different results on the correctness, the expressiveness, and the complexity of the class of bounded programs are presented.

Best-response dynamics out of sync: complexity and characterization

Abstract: In many computational and economic models of multi-agent interaction, each participant repeatedly "best-responds" to the others' actions. Game theory research on the prominent "best-response dynamics" model typically relies on the premise that the interaction between agents is somehow synchronized. However, in many real-life settings, e.g., internet protocols and large-scale markets, the interaction between participants is asynchronous. We tackle the following important questions: (1) When are best-response dynamics guaranteed to converge to an equilibrium even under asynchrony? (2) What is the (computational and communication) complexity of verifying guaranteed convergence? We show that, in general, verifying guaranteed convergence is intractable. In fact, our main negative result establishes that this task is undecidable. We exhibit, in contrast, positive results for several environments of interest, including complete, computationally-tractable, characterizations of convergent systems. We discuss the algorithmic implications of our results, which extend beyond best-response dynamics to applications such as asynchronous Boolean circuits.

Recursive queries on trees and data trees

Abstract: The analysis of datalog programs over relational structures has been studied in depth, most notably the problem of containment. The analysis problems that have been considered were shown to be undecidable with the exception of (i) containment of arbitrary programs in nonrecursive ones, (ii) containment of monadic programs, and (iii) emptiness. In this paper, we are concerned with a much less studied problem, the analysis of datalog programs over data trees. We show that the analysis of datalog programs is more complex for data trees than for arbitrary structures. In particular, we prove that the three aforementioned problems are undecidable for data trees. But in practice, data trees (e.g., XML trees) are often of bounded depth. We prove that all three problems are decidable over bounded depth data trees.Another contribution of the paper is the study of a new form of automata called pattern automata, that are essentially equivalent to linear datalog programs. We use pattern automata to show that the emptiness problem for linear monadic datalog programs with data value inequalities is decidable over arbitrary data trees.

Analysis of Message Passing Programs Using SMT-Solvers

Abstract: We consider message passing programs where processes communicate asynchronously over unbounded channels. The reachability problem for such systems are either undecidable or have very high complexity. In order to achieve efficiency, we consider the phase-bounded reachability problem, where each process is allowed to perform a bounded number of phases during a run of the system. In a given phase, the process is allowed to perform send or receive transitions (but not both). We present a uniform framework where the channels are assigned different types of semantics such as lossy, stuttering, or unordered. We show that the framework allows a uniform translation of bounded-phase reachability for each of the above mentioned semantics to the satisfiability of quantifierfree Presburger formulas. This means that we can use the full power of modern smt-solvers for efficient analysis of our systems. Furthermore, we show that the translation implies that bounded-phase reachability is np-complete. Finally, we prove that the problem becomes undecidable if we allow perfect channels or push-down processes communicating through (stuttering) lossy channels. We report on the result of applying the prototype on a number of non-trivial examples.

Composition Problems for Braids

Abstract: In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with 3 strands, B_3, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-hard for braids with only 3 strands. The membership problem is decidable for B_3, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least 5 strands, but decidability of these problems for B_4 remains open. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms.

Solving Partial-Information Stochastic Parity Games

Abstract: We study one-sided partial-information 2-player concurrent stochastic games with parity objectives. In such a game, one of the players has only partial visibility of the state of the game, while the other player has complete knowledge. In general, such games are known to be undecidable, even for the case of a single player (POMDP). These undecidability results depend crucially on player strategies that exploit an infinite amount of memory. However, in many applications of games, one is usually more interested in finding a finite-memory strategy. We consider the problem of whether the player with partial information has a finite-memory winning strategy when the player with complete information is allowed to use an arbitrary amount of memory. We show that this problem is decidable.

Expressive Reasoning on Tree Structures: Recursion, Inverse Programs, Presburger Constraints and Nominals

Abstract: The Semantic Web lays its foundations on the study of graph and tree logics. One of the most expressive graph logics is the fully enriched μ-calculus, which is a modal logic equipped with least and greatest fixed-points, nominals, inverse programs and graded modalities. Although it is well-known that the fully enriched μ-calculus is undecidable, it was recently shown that this logic is decidable when its models are finite trees. In the present work, we study the fully-enriched μ-calculus for trees extended with Presburger constraints. These constraints generalize graded modalities by restricting the number of children nodes with respect to Presburger arithmetic expressions. We show that the logic is decidable in EXPTIME. This is achieved by the introduction of a satisfiability algorithm based on a Fischer-Ladner model construction that is able to handle binary encodings of Presburger constraints.

Tiling Problems on Baumslag-Solitar groups

Abstract: Wang tilings are colorings of the Euclidean plane that respect some local constraints. They can be viewed both as dynamical systems and computational models, but they were first introduced by Hao Wang to study decision problems on some classes of logical formulas [18]. The concept of Wang tiles may be generalized to define tilings on the Cayley graph of a finitely generated group. The domino problem – deciding whether a finite tile set can tile the group or not – is undecidable on Z2 [2]. A more recent result [16, 10] characterizes groups having a decidable monadic second order logic. A consequence of this result states that if the group is virtually free, then it has a decidable domino problem. The reciprocal statement is still an open question, and very few examples of effective constructions to prove the (un-)decidability of the domino problem on finitely presented groups are known [9, 14]. Obviously the domino problem cannot be decidable on groups with undecidable word problem, so it is relevant to investigate non virtually free groups with decidable word problem. A famous theorem by Magnus [13] states that provided a finitely presented group possess a one-relator presentation, it has decidable word problem. In this article we investigate tilings on Baumslag-Solitar groups, that are two generators and onerelator non virtually free groups. This paper is organized as follows. Section 1 is devoted to the definition of tilings on finitely presented groups, which relies on the Cayley graph representation of such a group. In Section 2 we present the class of Baumslag-Solitar groups, and define Wang tiles on these groups. Following the ideas in [8] and [9], we exhibit a weakly aperiodic tile set (see Section 3), and we prove that the domino problem is undecidable on these groups (see Section 4). Analogously to [6], we conclude that there exist arecursive tile sets on the Baumslag-Solitar groups, that is, tile sets that admit valid tilings but all valid tilings are non-recursive.