The Emptiness Problem for Tree Automata with at Least One Disequality Constraint is NP-hard

The model of tree automata with equality and disequality constraints was introduced in 2007 by Filiot, Talbot and Tison. In this paper we show that if there is at least one disequality constraint, the emptiness problem is NP-hard

The HOM problem is EXPTIME-complete

We define a new class of tree automata with constraints and prove decidability of the emptiness problem for this class in exponential time. As a consequence, we obtain several EXPTIME-completeness results for problems on images of regular tree languages under tree homomorphisms, like set inclusion, regularity (HOM problem), and finiteness of set difference. Our result also has implications in term rewriting, since the set of reducible terms of a term rewrite system can be described as the image of a tree homomorphism. In particular, we prove that inclusion of sets of normal forms of term rewrite systems can be decided in exponential time. Analogous consequences arise in the context of XML typechecking, since types are defined by tree automata and some type transformations are homomorphic.Peer ReviewedPostprint (published version

Random Generation of Positive TAGEDs wrt. the Emptiness Problem

Tree automata are a widely used formalism in Computer Science. Since their creation in the fifties, numerous more expressive extensions have been proposed. Unfortunately, the decision problems associated with these extensions are quite often undecidable or in prohibitive classes of algorithmic complexity (NP-complete or worse), and little work has gone into finding efficient heuristics for them. Beyond the inherent difficulty of those problems, a common hitch in this line of research is the experimental evaluation of new algorithms. As those extensions of tree automata have remained in chiefly theoretical spheres, there are no established testbeds from the "real world" against which to quantify the efficiency (or lack thereof) of new algorithms. Failing that, there is a need to generate suitable testbeds at random. Regrettably, there is little material in the literature regarding random generation of tree automata, and none at all regarding extensions such as Tree Automata with Global Equality and Disequality Constraints (TAGEDs). It should also be noted that what little material there is does not concern itself with the interest of the generated automata wrt. specific decision problems. In this report we present a scheme for random generation of positive TAGEDs, with a focus on making them interesting wrt. the Emptiness problem

Automata-based Analysis of Recursive Cryptographic Protocols

Cryptographic protocols can be divided into (1) protocols where the protocol steps are simple from a computational point of view and can thus be modeled by simple means, for instance, single rewrite rules---we call these protocols non-looping---and (2) protocols, such as group protocols, where the protocol steps are complex and typically involve an iterative or recursive computation---we call them recursive. While many results on the decidability of security are known for non-looping protocols, only little is known for recursive protocols. In this paper, we prove decidability of security (w.r.t.~the standard Dolev-Yao intruder) for a core class of recursive protocols and undecidability for several extensions. The key ingredient of our protocol model are specifically designed tree transducers which work over infinite signatures and have the ability to generate new constants (which allow us to mimic key generation). The decidability result is based on an automata-theoretic construction which involves a new notion of regularity, designed to work well with the infinite signatures we use

An approximation and refinement approach to first-order automated reasoning

With the goal of lifting model-based guidance from the propositional setting to first order logic, I have developed an approximation theorem proving approach based on counterexample-guided abstraction refinement. A given clause set is transformed into a simplified form where satisfiability is decidable. This approximation extends the signature and preserves unsatisfiability: if the simplified clause set is satisfiable, so is the original clause set. A resolution refutation generated by a decision procedure on the simplified clause set can then either be lifted to a refutation in the original clause set, or it guides a refinement excluding the previously found unliftable refutation. This way the approach is refutationally complete. The monadic shallow linear Horn fragment, which is the initial target of the approximation, is well-known to be decidable. It was a long standing open problem how to extend the fragment to the non-Horn case, preserving decidability, that would, e.g., enable to express non-determinism in protocols. I have now proven decidability of the non-Horn monadic shallow linear fragment via ordered resolution. I further extend the clause language with a new type of constraints, called straight dismatching constraints. The extended clause language is motivated by an improved refinement of the approximation-refinement framework. All needed operations on straight dismatching constraints take linear or linear logarithmic time in the size of the constraint. Ordered resolution with straight dismatching constraints is sound and complete and the monadic shallow linear fragment with straight dismatching constraints is decidable. I have implemented my approach based on the SPASS theorem prover. On certain satisfiable problems, the implementation shows the ability to beat established provers such as SPASS, iProver, and Vampire.Mit dem Ziel die Modell-basierten Methoden der Aussagenlogik auf die Logik erster Stufe anzuwenden habe ich ein approximations Beweis-System entwickelt, das auf der Idee der ’Gegenbeispiel-gelenkten Abstraktions-Verfeinerung’ beruht. Eine gegebene Klausel-Menge wird zunächst in eine vereinfachte Form übersetzt, in der die Erfüllbarkeit entscheidbar ist. Diese sogenannte Approximation erweitert die Signatur, aber erhält Unerfüllbarkeit: Falls die approximierte Klauseln erfüllbar sind, so ist es auch die ursprüngliche Menge. Ein Resolutions-Beweis, der von einer Entscheidungs-Prozedur auf der Approximation erzeugt wurde, kann dann entweder als Basis eines Unerfüllbarkeits Beweises der ursprünglichen Menge dienen oder aber eine Verfeinerung der Approximation aufzeigen, welche den gefundenen Beweis davon ausschließt noch einmal gefunden zu werden. Damit ist der Ansatz widerlegungs vollständig. Das monadisch flache lineare Horn Fragment, das als anfängliches Ziel der Approximation dient, ist bereits seit längerem als entscheidbar bekannt. Es war ein lange offenes Problem, wie man das Fragment auf den nicht-Horn Fall erweitern kann ohne Entscheidbarkeit zu verlieren. Damit lassen sich unter anderem nicht-deterministische Protokolle ausdrücken. Ich habe nun die Entscheidbarkeit des nicht-Horn monadisch flachen linearen Fragments mittels geordneter Resolution bewiesen. Zusätzlich habe ich die Klausel-Sprache durch eine neue Art von Constraints erweitert, die ich als ’straight dismatching constraints’ bezeichne. Diese Erweiterung ist dadurch motiviert dass sie eine Verbesserung der Approximations-Verfeinerung des vorgestellten Systems erlaubt. Alle benötigten Operationen auf diesen Constraints nehmen lediglich lineare oder linear-logarithmische Zeit und Platz in Anspruch. Ich zeige, dass geordnete Resolution mit Constraints korrekt und vollständig ist und dass das monadische flache lineare Fragment mit Constraints entscheidbar ist. Ich habe meinen Ansatz auf dem Theorem-Beweiser SPASS basierend implementiert. Auf bestimmten erfüllbaren Problem schlägt meine Implementierung sogar etablierte Beweiser wie SPASS, iProver und Vampire

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator