Showing papers on "Undecidable problem published in 2021"

A Temporal Logic for Asynchronous Hyperproperties.

Abstract: Hyperproperties are properties of computational systems that require more than one trace to evaluate, e.g., many information-flow security and concurrency requirements. Where a trace property defines a set of traces, a hyperproperty defines a set of sets of traces. The temporal logics HyperLTL and HyperCTL* have been proposed to express hyperproperties. However, their semantics are synchronous in the sense that all traces proceed at the same speed and are evaluated at the same position. This precludes the use of these logics to analyze systems whose traces can proceed at different speeds and allow that different traces take stuttering steps independently. To solve this problem in this paper, we propose an asynchronous variant of HyperLTL. On the negative side, we show that the model-checking problem for this variant is undecidable. On the positive side, we identify a decidable fragment which covers a rich set of formulas with practical applications. We also propose two model-checking algorithms that reduce our problem to the HyperLTL model-checking problem in the synchronous semantics.

Verifying Safety of Synchronous Fault-Tolerant Algorithms by Bounded Model Checking

Abstract: Many fault-tolerant distributed algorithms are designed for synchronous or round-based semantics. In this paper, we introduce the synchronous variant of threshold automata, and study their applicability and limitations for the verification of synchronous distributed algorithms. We show that in general, the reachability problem is undecidable for synchronous threshold automata. Still, we show that many synchronous fault-tolerant distributed algorithms have a bounded diameter, although the algorithms are parameterized by the number of processes. Hence, we use bounded model checking for verifying these algorithms.

Asynchronous extensions of HyperLTL

Abstract: Hyperproperties are a modern specification paradigm that extends trace properties to express properties of sets of traces. Temporal logics for hyperproperties studied in the literature, including HyperLTL, assume a synchronous semantics and enjoy a decidable model checking problem. In this paper, we introduce two asynchronous and orthogonal extensions of HyperLTL, namely Stuttering HyperLTL (HyperLTL S ) and Context HyperLTL (HyperLTL C ). Both of these extensions are useful, for instance, to formulate asynchronous variants of information-flow security properties. We show that for these logics, model checking is in general undecidable. On the positive side, for each of them, we identify a fragment with a decidable model checking that subsumes HyperLTL and that can express meaningful asynchronous requirements. Moreover, we provide the exact computational complexity of model checking for these two fragments which, for the HyperLTL S fragment, coincides with that of the strictly less expressive logic HyperLTL.

The membership problem for constant-sized quantum correlations is undecidable

Abstract: When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations -- that is, correlations for which the number of measurements and number of measurement outcomes are fixed -- such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.

Freezing, Bounded-Change and Convergent Cellular Automata

Abstract: This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

A sound algorithm for asynchronous session subtyping and its implementation

Abstract: Session types, types for structuring communication between endpoints in distributed systems, are recently being integrated into mainstream programming languages. In practice, a very important notion for dealing with such types is that of subtyping, since it allows for typing larger classes of system, where a program has not precisely the expected behaviour but a similar one. Unfortunately, recent work has shown that subtyping for session types in an asynchronous setting is undecidable. To cope with this negative result, the only approaches we are aware of either restrict the syntax of session types or limit communication (by considering forms of bounded asynchrony). Both approaches are too restrictive in practice, hence we proceed differently by presenting an algorithm for checking subtyping which is sound, but not complete (in some cases it terminates without returning a decisive verdict). The algorithm is based on a tree representation of the coinductive definition of asynchronous subtyping; this tree could be infinite, and the algorithm checks for the presence of finite witnesses of infinite successful subtrees. Furthermore, we provide a tool that implements our algorithm. We use this tool to test our algorithm on many examples that cannot be managed with the previous approaches, and to provide an empirical evaluation of the time and space cost of the algorithm.

On equations and first-order theory of one-relator monoids

Abstract: We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family F of one-relator monoids of the form 〈 A | w = 1 〉 where for each monoid M in F , the longstanding open problem of decidability of word equations with length constraints reduces to the Diophantine problem (i.e. decidability of systems of equations) in M. We achieve this result by finding an interpretation in M of a free monoid, using only systems of equations together with length relations. It follows that each monoid in F has undecidable positive AE-theory, hence in particular it has undecidable first-order theory. The family F includes many one-relator monoids with torsion 〈 A | w n = 1 〉 ( n > 1 ). In contrast, all one-relator groups with torsion are hyperbolic, and all hyperbolic groups are known to have decidable Diophantine problem. We further describe a different class of one-relator monoids with decidable Diophantine problem.

Temporal Stream Logic modulo Theories.

Abstract: Temporal Stream Logic (TSL) is a temporal logic that extends LTL with updates and predicates over arbitrary function terms. This allows for specifying data-intensive systems for which LTL is not expressive enough. In TSL, functions and predicates are uninterpreted. In this paper, we investigate the satisfiability problem of TSL both with respect to the standard underlying theory of uninterpreted functions and with respect to other theories such as Presburger arithmetic. We present an algorithm for checking the satisfiability of a TSL formula in the theory of uninterpreted functions and evaluate it on different benchmarks: It scales well and is able to validate assumptions in a real-world system design. The algorithm is not guaranteed to terminate. In fact, we show that TSL satisfiability is highly undecidable in the theories of uninterpreted functions, equality, and Presburger arithmetic, proving that no complete algorithm exists. However, we identify three fragments of TSL for which the satisfiability problem is (semi-)decidable in the theory of uninterpreted functions.

Algorithmic Problems in the Symbolic Approach to the Verification of Automatically Synthesized Cryptosystems

Abstract: Automated methods can be used to generate cryptosystems by combining the primitives in an arbitrary fashion, to weed out insecure cryptosystems, and to prove the security of those that survive. In this paper, we study several algorithmic problems arising from the verification of automatically synthesized cryptosystems built from block ciphers, in a theory that includes ACUN. One of these is static equivalence to an algorithm that produces a sequence of random terms. The other is invertibility, the problem of determining whether, given an automatically synthesized cryptosystem, built from block ciphers, and the ability to compute inverses, is it always possible to compute the original plaintext from the ciphertext? We show that static equivalence to random in this theory is undecidable in general. In addition, we identify a reasonable special case for which there is a decidable condition implying security, along with an algorithm for verifying it. For invertibility, we identify a reasonable class of cryptosystems for which invertibility is equivalent to a simple syntactic condition that can be easily verified.

Quantum logic is undecidable

Abstract: We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature $$(\vee ,\perp ,0,1)$$ , where ‘ $$\perp $$ ’ is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent result of Slofstra in combinatorial group theory. It follows upon reinterpreting that result in terms of the hypergraph approach to quantum contextuality, for which it constitutes a proof of the inverse sandwich conjecture. It can also be interpreted as stating that a certain quantum satisfiability problem is undecidable.

Decidable ∃∗∀∗ First-Order Fragments of Linear Rational Arithmetic with Uninterpreted Predicates

Abstract: First-order linear rational arithmetic enriched with uninterpreted predicates yields an interesting and very expressive modeling language. However, already the presence of a single uninterpreted predicate symbol of arity one or greater renders the associated satisfiability problem undecidable. We identify two decidable fragments, both based on the Bernays–Schonfinkel–Ramsey prefix class. Due to the inherent infiniteness of the underlying domain, a finite model property in the usual sense cannot be established. Nevertheless, we show that satisfiable sentences always have a model that can be described by finite means. To this end, we restrict the syntax of arithmetic atoms. In the first fragment that is presented, arithmetic operations are only allowed over terms without universally quantified variables. In the second fragment, arithmetic atoms are essentially confined to difference constraints over universally quantified variables with bounded range. We will call such atoms bounded difference constraints. As bounded intervals over the rationals still comprise infinitely many values, a trivial instantiation procedure is not sufficient to solve the satisfiability problem. After a slight shift of perspective, the positive decidability result for the first fragment can be restated in the framework of combinations of theories over non-disjoint vocabularies. More precisely, we combine first-order theories that share a dense total order without endpoints.

Semi-Oblivious Chase Termination: The Sticky Case

Abstract: The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A key problem concerning the chase procedure is all-instances termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is undecidable, it is natural to ask whether known well-behaved classes of TGDs, introduced in different contexts such as ontological reasoning, ensure decidability. We consider a prominent paradigm that led to a robust TGD-based formalism, called stickiness. We show that for sticky sets of TGDs, all-instances chase termination is decidable if we focus on the (semi-)oblivious chase, and we pinpoint its exact complexity: PSpace-complete in general, and NLogSpace-complete for predicates of bounded arity. These complexity results are obtained via a graph-based syntactic characterization of chase termination that is of independent interest.

Action Logic is Undecidable

Abstract: Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely, -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.

The Decidability of Verification under PS 2.0

Abstract: We consider the reachability problem for finite-state multi-threaded programs under the promising semantics (PS 2.0) of Lee et al., which captures most common program transformations. Since reachability is already known to be undecidable in the fragment of PS 2.0 with only release-acquire accesses (PS 2.0-ra), we consider the fragment with only relaxed accesses and promises (PS 2.0-rlx). We show that reachability under PS 2.0-rlx is undecidable in general and that it becomes decidable, albeit non-primitive recursive, if we bound the number of promises.

Reasoning about strategic voting in modal logic quickly becomes undecidable

Abstract: In this paper, we show that modal logics for reasoning about social choice quickly become undecidable. In particular, we study modal logics that can be used to reason about situations involving both actual and claimed preferences in the context of a social choice function and argue that reasoning on this level often occurs in social choice. We formally define a particular logic, interpreted in such situations, that can express the properties involved in the Gibbard–Satterthwaite theorem. We then, however, demonstrate that any modal logic interpreted in such situations having a certain natural expressive power, in particular a modality quantifying over all possible claimed preferences, becomes undecidable when there are enough agents in the system. We also discuss a decidable special case and provide a complete axiomatization of fragment of the language.

The Decidability and Complexity of Interleaved Bidirected Dyck Reachability

Abstract: Dyck reachability is the standard formulation of a large domain of static analyses, as it achieves the sweet spot between precision and efficiency, and has thus been studied extensively. Interleaved Dyck reachability (denoted $D_k\odot D_k$) uses two Dyck languages for increased precision (e.g., context and field sensitivity) but is well-known to be undecidable. As many static analyses yield a certain type of bidirected graphs, they give rise to interleaved bidirected Dyck reachability problems. Although these problems have seen numerous applications, their decidability and complexity has largely remained open. In a recent work, Li et al. made the first steps in this direction, showing that (i) $D_1\odot D_1$ reachability (i.e., when both Dyck languages are over a single parenthesis and act as counters) is computable in $O(n^7)$ time, while (ii) $D_k\odot D_k$ reachability is NP-hard. In this work we address the decidability and complexity of all variants of interleaved bidirected Dyck reachability. First, we show that $D_1\odot D_1$ reachability can be computed in $O(n^3\cdot \alpha(n))$ time, significantly improving over the existing $O(n^7)$ bound. Second, we show that $D_k\odot D_1$ reachability (i.e., when one language acts as a counter) is decidable, in contrast to the non-bidirected case where decidability is open. We further consider $D_k\odot D_1$ reachability where the counter remains linearly bounded. Our third result shows that this bounded variant can be solved in $O(n^2\cdot \alpha(n))$ time, while our fourth result shows that the problem has a (conditional) quadratic lower bound, and thus our upper bound is essentially optimal. Fifth, we show that full $D_k\odot D_k$ reachability is undecidable. This improves the recent NP-hardness lower-bound, and shows that the problem is equivalent to the non-bidirected case.

Making Theory Reasoning Simpler

Abstract: Reasoning with quantifiers and theories is at the core of many applications in program analysis and verification. Whilst the problem is undecidable in general and hard in practice, we have been making large pragmatic steps forward. Our previous work proposed an instantiation rule for theory reasoning that produced pragmatically useful instances. Whilst this led to an increase in performance, it had its limitations as the rule produces ground instances which (i) can be overly specific, thus not useful in proof search, and (ii) contribute to the already problematic search space explosion as many new instances are introduced. This paper begins by introducing that specifically addresses these two concerns as it produces general solutions and it is a simplification rule, i.e. it replaces an existing clause by a ‘simpler’ one. Encouraged by initial success with this new rule, we performed an experiment to identify further common cases where the complex structure of theory terms blocked existing methods. This resulted in four further simplification rules for theory reasoning. The resulting extensions are implemented in the Vampire theorem prover and evaluated on SMT-LIB, showing that the new extensions result in a considerable increase in the number of problems solved, including 90 problems unsolved by state-of-the-art SMT solvers.

Model Checking Algorithms for Hyperproperties.

Abstract: Hyperproperties generalize trace properties by expressing relations between multiple computations. Hyperpropertes include policies from information-flow security, like observational determinism or non-interference, and many other system properties including promptness and knowledge. In this paper, we give an overview on the model checking problem for temporal hyperlogics. Our starting point is the model checking algorithm for HyperLTL, a reduction to Buchi automata emptiness. This basic construction can be extended with propositional quantification, resulting in an algorithm for HyperQPTL. It can also be extended with branching time, resulting in an algorithm for HyperCTL*. However, it is not possible to have both extensions at the same time: the model checking problem of HyperQCTL* is undecidable. An attractive compromise is offered by MPL[E], i.e., monadic path logic extended with the equal-level predicate. The expressiveness of MPL[E] falls strictly between that of HyperCTL* and HyperQCTL*. MPL[E] subsumes both HyperCTL* and HyperKCTL*, the extension of HyperCTL* with the knowledge operator. We show that the model checking problem for MPL[E] is still decidable.

Adding the Relation Meets to the Temporal Logic of Prefixes and Infixes makes it EXPSPACE-Complete

Abstract: The choice of the right trade-off between expressiveness and complexity is the main issue in interval temporal logic. In their seminal paper, Halpern and Shoham showed that the satisfiability problem for HS (the temporal logic of Allen's relations) is highly undecidable over any reasonable class of linear orders. In order to recover decidability, one can restrict the set of temporal modalities and/or the class of models. In the following, we focus on the satisfiability problem for HS fragments under the homogeneity assumption, according to which any proposition letter holds over an interval if only if it holds at all its points. The problem for full HS with homogeneity has been shown to be non-elementarily decidable, but its only known lower bound is EXPSPACE (in fact, EXPSPACE-hardness has been shown for the logic of prefixes and suffixes BE, which is a very small fragment of it. The logic of prefixes and infixes BD has been recently shown to be PSPACE-complete. In this paper, we prove that the addition of the Allen relation Meets to BD makes it EXPSPACE-complete.

String Theories Involving Regular Membership Predicates: From Practice to Theory and Back

Abstract: Widespread use of string solvers in formal analysis of string-heavy programs has led to a growing demand for more efficient and reliable techniques which can be applied in this context, especially for real-world cases. Designing an algorithm for the (generally undecidable) satisfiability problem for systems of string constraints requires a thorough understanding of the structure of constraints present in the targeted cases. In this paper, we investigate benchmarks presented in the literature containing regular expression membership predicates, extract different first order logic theories, and prove their decidability, resp. undecidability. Notably, the most common theories in real-world benchmarks are \(\mathsf {PSPACE}\)-complete and directly lead to the implementation of a more efficient algorithm to solving string constraints.

Positive first-order logic on words

Abstract: We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is an FO-definable language that is monotone in monadic predicates but not definable in FO+. This provides a simple proof that Lyndon’s preservation theorem fails on finite structures. We additionally show that given a regular language, it is undecidable whether it is definable in FO+.

Adding the Relation Meets to the Temporal Logic of Prefixes and Infixes makes it EXPSPACE-Complete

Abstract: The choice of the right trade-off between expressiveness and complexity is the main issue in interval temporal logic. In their seminal paper, Halpern and Shoham showed that the satisfiability problem for HS (the temporal logic of Allen's relations) is highly undecidable over any reasonable class of linear orders. In order to recover decidability, one can restrict the set of temporal modalities and/or the class of models. In the following, we focus on the satisfiability problem for HS fragments under the homogeneity assumption, according to which any proposition letter holds over an interval if only if it holds at all its points. The problem for full HS with homogeneity has been shown to be non-elementarily decidable, but its only known lower bound is EXPSPACE (in fact, EXPSPACE-hardness has been shown for the logic of prefixes and suffixes BE, which is a very small fragment of it. The logic of prefixes and infixes BD has been recently shown to be PSPACE-complete. In this paper, we prove that the addition of the Allen relation Meets to BD makes it EXPSPACE-complete.

On the Subtle Nature of a Simple Logic of the Hide and Seek Game

Abstract: We discuss a simple logic to describe one of our favourite games from childhood, hide and seek, and show how a simple addition of an equality constant to describe the winning condition of the seeker makes our logic undecidable. There are certain decidable fragments of first-order logic which behave in a similar fashion and we add a new modal variant to that class of logics. We also discuss the relative expressive power of the proposed logic in comparison to the standard modal counterparts.

Demonic lattices and semilattices in relational semigroups with ordinary composition

Abstract: Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of nondeterminism, has also provided us with an extension of that reasoning to total correctness. We formalise the framework for relational reasoning about total correctness in nondeterministic programs using semigroups with ordinary composition and demonic lattice operations. We show that the class of representable demonic join semigroups is not finitely axiomatisable and that the representation class of demonic meet semigroups does not have the finite representation property for its finite members. For lattice semigroups (with composition, demonic join and demonic meet) we show that the representation problem for finite algebras is undecidable, moreover the finite representation problem is also undecidable. It follows that the representation class is not finitely axiomatisable, furthermore the finite representation property fails.

Satisfiability and Containment of Recursive SHACL.

Abstract: The Shapes Constraint Language (SHACL) is the recent W3C recommendation language for validating RDF data, by verifying certain shapes on graphs. Previous work has largely focused on the validation problem and the standard decision problems of satisfiability and containment, crucial for design and optimisation purposes, have only been investigated for simplified versions of SHACL. Moreover, the SHACL specification does not define the semantics of recursively-defined constraints, which led to several alternative recursive semantics being proposed in the literature. The interaction between these different semantics and important decision problems has not been investigated yet. In this article we provide a comprehensive study of the different features of SHACL, by providing a translation to a new first-order language, called SCL, that precisely captures the semantics of SHACL. We also present MSCL, a second-order extension of SCL, which allows us to define, in a single formal logic framework, the main recursive semantics of SHACL. Within this language we also provide an effective treatment of filter constraints which are often neglected in the related literature. Using this logic we provide a detailed map of (un)decidability and complexity results for the satisfiability and containment decision problems for different SHACL fragments. Notably, we prove that both problems are undecidable for the full language, but we present decidable combinations of interesting features, even in the face of recursion.

Computability and Beltrami fields in Euclidean space.

Abstract: In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of $\mathbb R^3$, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on $\mathbb T^3$ (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) arbitrarily high computational complexity. We also quantify the energetic cost for a Beltrami field on $\mathbb T^3$ to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability $1$. Finally, our proof also yields Turing complete flows and maps on $\mathbb{S}^2$ with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.