The Universal Fragment of Presburger Arithmetic with Unary Uninterpreted Predicates is Undecidable

The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the known boundary between decidable and undecidable in that we show that the purely universal fragment of the extended theory is already undecidable. Our proof is based on a reduction of the halting problem for two-counter machines to unsatisfiability of sentences in the extended language of Presburger arithmetic that does not use existential quantification. On the other hand, we argue that a single $\forall\exists$ quantifier alternation turns the set of satisfiable sentences of the extended language into a $\Sigma^1_1$-complete set. Some of the mentioned results can be transfered to the realm of linear arithmetic over the ordered real numbers. This concerns the undecidability of the purely universal fragment and the $\Sigma^1_1$-hardness for sentences with at least one quantifier alternation. Finally, we discuss the relevance of our results to verification. In particular, we derive undecidability results for quantified fragments of separation logic, the theory of arrays, and combinations of the theory of equality over uninterpreted functions with restricted forms of integer arithmetic. In certain cases our results even imply the absence of sound and complete deductive calculi

Decidability of Difference Logic over the Reals with Uninterpreted Unary Predicates

First-order logic fragments mixing quantifiers, arithmetic, and uninterpreted predicates are often undecidable, as is, for instance, Presburger arithmetic extended with a single uninterpreted unary predicate. In the SMT world, difference logic is a quite popular fragment of linear arithmetic which is less expressive than Presburger arithmetic. Difference logic on integers with uninterpreted unary predicates is known to be decidable, even in the presence of quantifiers. We here show that (quantified) difference logic on real numbers with a single uninterpreted unary predicate is undecidable, quite surprisingly. Moreover, we prove that difference logic on integers, together with order on reals, combined with uninterpreted unary predicates, remains decidable.Comment: This is the preprint for the submission published in CADE-29. It also includes an additional detailed proof in the appendix. The Version of Record of this contribution will be published in CADE-2

Decidability of graph neural networks via logical characterizations

We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving ``Presburger quantifiers''. We show how to use these logics to measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems

The First-Order Theory of Sets with Cardinality Constraints is Decidable

We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is undecidable. Our language allows relating the cardinalities of sets to the values of integer variables, and can distinguish finite and infinite sets. We use quantifier elimination to show the decidability and obtain an elementary upper bound on the complexity. Precise program analyses can use our decidability result to verify representation invariants of data structures that use an integer field to represent the number of stored elements.Comment: 18 page

Decidable fragments of first-order logic and of first-order linear arithmetic with uninterpreted predicates

First-order logic is one of the most prominent formalisms in computer science and mathematics. Since there is no algorithm capable of solving its satisfiability problem, first-order logic is said to be undecidable. The classical decision problem is the quest for a delineation between the decidable and the undecidable parts. The results presented in this thesis shed more light on the boundary and open new perspectives on the landscape of known decidable fragments. In the first part we focus on the new concept of separateness of variables and explore its applicability to the classical decision problem and beyond. Two disjoint sets of first-order variables are separated in a given formula if none of its atoms contains variables from both sets. This notion facilitates the definition of decidable extensions of many well-known decidable first-order fragments. We demonstrate this for several prefix fragments, several guarded fragments, the two-variable fragment, and for the fluted fragment. Although the extensions exhibit the same expressive power as the respective originals, certain logical properties can be expressed much more succinctly. In two cases the succinctness gap cannot be bounded using elementary functions. This fact already hints at computationally hard satisfiability problems. Indeed, we derive non-elementary lower bounds for the separated fragment, an extension of the Bernays-Schönfinkel-Ramsey fragment (E*A*-prefix sentences). On the semantic level, separateness of quantified variables may lead to weaker dependences than we encounter in general. We investigate this property in the context of model-checking games. The focus of the second part of the thesis is on linear arithmetic with uninterpreted predicates. Two novel decidable fragments are presented, both based on the Bernays-Schönfinkel-Ramsey fragment. On the negative side, we identify several small fragments of the language for which satisfiability is undecidable.Untersuchungen der Logik erster Stufe blicken auf eine lange Tradition zurück. Es ist allgemein bekannt, dass das zugehörige Erfüllbarkeitsproblem im Allgemeinen nicht algorithmisch gelöst werden kann - man spricht daher von einer unentscheidbaren Logik. Diese Beobachtung wirft ein Schlaglicht auf die prinzipiellen Grenzen der Fähigkeiten von Computern im Allgemeinen aber auch des automatischen Schließens im Besonderen. Das Hilbertsche Entscheidungsproblem wird heute als die Erforschung der Grenze zwischen entscheidbaren und unentscheidbaren Teilen der Logik erster Stufe verstanden, wobei die untersuchten Fragmente der Logik mithilfe klar zu erfassender und berechenbarer syntaktischer Eigenschaften beschrieben werden. Viele Forscher haben bereits zu dieser Untersuchung beigetragen und zahlreiche entscheidbare und unentscheidbare Fragmente entdeckt und erforscht. Die vorliegende Dissertation setzt diese Tradition mit einer Reihe vornehmlich positiver Resultate fort und eröffnet neue Blickwinkel auf eine Reihe von Fragmenten, die im Laufe der letzten einhundert Jahre untersucht wurden. Im ersten Teil der Arbeit steht das syntaktische Konzept der Separiertheit von Variablen im Mittelpunkt, und dessen Anwendbarkeit auf das Entscheidungsproblem und darüber hinaus wird erforscht. Zwei Mengen von Individuenvariablen gelten bezüglich einer gegebenen Formel als separiert, falls in jedem Atom der Formel die Variablen aus höchstens einer der beiden Mengen vorkommen. Mithilfe dieses leicht verständlichen Begriffs lassen sich viele wohlbekannte entscheidbare Fragmente der Logik erster Stufe zu größeren Klassen von Formeln erweitern, die dennoch entscheidbar sind. Dieser Ansatz wird für neun Fragmente im Detail dargelegt, darunter mehrere Präfix-Fragmente, das Zwei-Variablen-Fragment und sogenannte "guarded" und " uted" Fragmente. Dabei stellt sich heraus, dass alle erweiterten Fragmente ebenfalls das monadische Fragment erster Stufe ohne Gleichheit enthalten. Obwohl die erweiterte Syntax in den betrachteten Fällen nicht mit einer erhöhten Ausdrucksstärke einhergeht, können bestimmte Zusammenhänge mithilfe der erweiterten Syntax deutlich kürzer formuliert werden. Zumindest in zwei Fällen ist diese Diskrepanz nicht durch eine elementare Funktion zu beschränken. Dies liefert einen ersten Hinweis darauf, dass die algorithmische Lösung des Erfüllbarkeitsproblems für die erweiterten Fragmente mit sehr hohem Rechenaufwand verbunden ist. Tatsächlich wird eine nicht-elementare untere Schranke für den entsprechenden Zeitbedarf beim sogenannten separierten Fragment, einer Erweiterung des bekannten Bernays-Schönfinkel-Ramsey-Fragments, abgeleitet. Darüber hinaus wird der Ein uss der Separiertheit von Individuenvariablen auf der semantischen Ebene untersucht, wo Abhängigkeiten zwischen quantifizierten Variablen durch deren Separiertheit stark abgeschwächt werden können. Für die genauere formale Betrachtung solcher als schwach bezeichneten Abhängigkeiten wird auf sogenannte Hintikka-Spiele zurückgegriffen. Den Schwerpunkt des zweiten Teils der vorliegenden Arbeit bildet das Entscheidungsproblem für die lineare Arithmetik über den rationalen Zahlen in Verbindung mit uninterpretierten Prädikaten. Es werden zwei bislang unbekannte entscheidbare Fragmente dieser Sprache vorgestellt, die beide auf dem Bernays-Schönfinkel-Ramsey-Fragment aufbauen. Ferner werden neue negative Resultate entwickelt und mehrere unentscheidbare Fragmente vorgestellt, die lediglich einen sehr eingeschränkten Teil der Sprache benötigen

Reasoning in the Bernays-Schönfinkel-Ramsey Fragment of Separation Logic

International audienceSeparation Logic (SL) is a well-known assertion language used in Hoare-style modular proof systems for programs with dynamically allocated data structures. In this paper we investigate the fragment of first-order SL restricted to the Bernays-Schönfinkel-Ramsey quantifier prefix ∃ * ∀ * , where the quantified variables range over the set of memory locations. When this set is uninterpreted (has no associated theory) the fragment is PSPACE-complete, which matches the complexity of the quantifier-free fragment [7]. However, SL becomes undecid-able when the quantifier prefix belongs to ∃ * ∀ * ∃ * instead, or when the memory locations are interpreted as integers with linear arithmetic constraints, thus setting a sharp boundary for decidability within SL. We have implemented a decision procedure for the decidable fragment of ∃ * ∀ * SL as a specialized solver inside a DPLL(T) architecture, within the CVC4 SMT solver. The evaluation of our implementation was carried out using two sets of verification conditions, produced by (i) unfolding inductive predicates, and (ii) a weakest precondition-based verification condition generator. Experimental data shows that automated quantifier instantiation has little overhead, compared to manual model-based instantiation

Temporal Stream Logic modulo Theories

Temporal stream logic (TSL) 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 the semantics of TSL, functions and predicates are left uninterpreted. In this paper, we extend TSL with first-order theories, enabling us to specify systems using interpreted functions and predicates such as incrementation or equality. We investigate the satisfiability problem of TSL modulo the standard underlying theory of uninterpreted functions as well as with respect to Presburger arithmetic and the theory of equality: For all three theories, TSL satisfiability is neither semi-decidable nor co-semi-decidable. Nevertheless, we identify three fragments of TSL for which the satisfiability problem is (semi-)decidable in the theory of uninterpreted functions. Despite the undecidability, we present an algorithm – which is not guaranteed to terminate – for checking the satisfiability of a TSL formula in the theory of uninterpreted functions and evaluate it: It scales well and is able to validate assumptions in a real-world system design

The Well Structured Problem for Presburger Counter Machines

International audienceWe introduce the well structured problem as the question of whether a model (here a counter machine) is well structured (here for the usual ordering on integers). We show that it is undecidable for most of the (Presburger-defined) counter machines except for Affine VASS of dimension one. However, the strong well structured problem is decidable for all Presburger counter machines. While Affine VASS of dimension one are not, in general, well structured, we give an algorithm that computes the set of predecessors of a configuration; as a consequence this allows to decide the well structured problem for 1-Affine VASS