A Decidable Class of Nested Iterated Schemata (extended version)

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /\i=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /\i=1..n (/\j=1..n ...).Comment: 43 pages, extended version of "A Decidable Class of Nested Iterated Schemata", submitted to IJCAR 200

Programming Using Automata and Transducers

Automata, the simplest model of computation, have proven to be an effective tool in reasoning about programs that operate over strings. Transducers augment automata to produce outputs and have been used to model string and tree transformations such as natural language translations. The success of these models is primarily due to their closure properties and decidable procedures, but good properties come at the price of limited expressiveness. Concretely, most models only support finite alphabets and can only represent small classes of languages and transformations. We focus on addressing these limitations and bridge the gap between the theory of automata and transducers and complex real-world applications: Can we extend automata and transducer models to operate over structured and infinite alphabets? Can we design languages that hide the complexity of these formalisms? Can we define executable models that can process the input efficiently? First, we introduce succinct models of transducers that can operate over large alphabets and design BEX, a language for analysing string coders. We use BEX to prove the correctness of UTF and BASE64 encoders and decoders. Next, we develop a theory of tree transducers over infinite alphabets and design FAST, a language for analysing tree-manipulating programs. We use FAST to detect vulnerabilities in HTML sanitizers, check whether augmented reality taggers conflict, and optimize and analyze functional programs that operate over lists and trees. Finally, we focus on laying the foundations of stream processing of hierarchical data such as XML files and program traces. We introduce two new efficient and executable models that can process the input in a left-to-right linear pass: symbolic visibly pushdown automata and streaming tree transducers. Symbolic visibly pushdown automata are closed under Boolean operations and can specify and efficiently monitor complex properties for hierarchical structures over infinite alphabets. Streaming tree transducers can express and efficiently process complex XML transformations while enjoying decidable procedures

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Tree Automata Techniques and Applications

International audienc

Saturation-based decision procedures for extensions of the guarded fragment

We apply the framework of Bachmair and Ganzinger for saturation-based theorem proving to derive a range of decision procedures for logical formalisms, starting with a simple terminological language EL, which allows for conjunction and existential restrictions only, and ending with extensions of the guarded fragment with equality, constants, functionality, number restrictions and compositional axioms of form S ◦ T ⊆ H. Our procedures are derived in a uniform way using standard saturation-based calculi enhanced with simplification rules based on the general notion of redundancy. We argue that such decision procedures can be applied for reasoning in expressive description logics, where they have certain advantages over traditionally used tableau procedures, such as optimal worst-case complexity and direct correctness proofs.Wir wenden das Framework von Bachmair und Ganzinger für saturierungsbasiertes Theorembeweisen an, um eine Reihe von Entscheidungsverfahren für logische Formalismen abzuleiten, angefangen von einer simplen terminologischen Sprache EL, die nur Konjunktionen und existentielle Restriktionen erlaubt, bis zu Erweiterungen des Guarded Fragment mit Gleichheit, Konstanten, Funktionalität, Zahlenrestriktionen und Kompositionsaxiomen der Form S ◦ T ⊆ H. Unsere Verfahren sind einheitlich abgeleitet unter Benutzung herkömmlicher saturierungsbasierter Kalküle, verbessert durch Simplifikationsregeln, die auf dem Konzept der Redundanz basieren. Wir argumentieren, daß solche Entscheidungsprozeduren für das Beweisen in ausdrucksvollen Beschreibungslogiken angewendet werden können, wo sie gewisse Vorteile gegenüber traditionell benutzten Tableauverfahren besitzen, wie z.B. optimale worst-case Komplexität und direkte Korrektheitsbeweise

Subtype satisfiability and entailment

Subtype constraints were introduced in advanced programming language research for designing subtype systems and program analysis algorithms. Two logical problems arise in this context: subtype satisfiability and subtype entailment. Subtype satisfiability underlies subtype inference; subtype entailment is for simplifying subtyping constraints in the same application. In this thesis, we investigate both problems systematically for a number of dialects of subtyping constraint languages that may vary in the following dimensions: types may be simple (finite) or recursive (infinite), type constants may be ordered in lattices or in general partially ordered sets, subtyping can be structural or non-structural, depending on whether least and greatest types are permitted. We use and develop new formal reasoning techniques based on automata, unification, and modal logic. Subtype satisfiability is well understood for all dialects with constants ordered in a lattice. Although cubic time algorithms are given by Palsberg and O\u27Keefe (1995), Pottier (1996), and Palsberg, Wand, and O\u27Keefe (1997), little is known about dialects where constants belong to arbitrary partially ordered sets. We present a uniform treatment to determine the complexities of all these classes. As a consequence, we settle a problem left open by Tiuryn and Wand in 1993 and also subsume complexity bounds given by Wand and Tiuryn (1993), Tiuryn (1992), and Frey (2002). Our results are based on a new connection between modal logic and subtype constraints that we present. Subtype entailment is known to be hard even for simple subtype constraint languages. Rehof and Henglein determined the complexity of structural subtype entailment with type constants ordered in a lattice. They proved coNP-completeness for simple types (1997) and PSPACE-completeness for recursive types (1998). Furthermore, they showed that non-structural subtype entailment is PSPACE-hard and is conjectured PSPACE-complete for the case with only two type constants for the least and greatest types respectively (1998). Yet the problem still remains open today. We argue that the difficulty occurs due to e ects linked to non-regular word languages. In order to do so, we precisely characterize subtype entailment by finite word automata with word equations. This characterization induces new results on non-structural subtype entailment, constituting a promising starting point for future investigation on decidability.Diese Arbeit untersucht zwei logische Probleme der programmiersprachlichen Typinferenz: Erfüllbarkeit und Subsumption von Teiltyp-Constraints. Wir untersuchen diese Probleme systematisch für eine Reihe von Constraintsprachen. Dabei greifen wir auf Methoden der computationalen Logik, Unifikations- und Automatentheorie zurück. Teiltyp-Erfüllbarkeit ist für den Fall wohl verstanden, dass die Typkonstanten in einem Verband angeordnet sind (Palsberg und O\u27Keefe (1995), Pottier (1996), Palsberg, Wand und O\u27Keefe (1997)). Der allgemeinere Fall mit beliebig angeordneten Konstanten wurde bislang weniger untersucht. Wir stellen einen ersten universellen Ansatz vor, indem wir erstmals einen Zusammenhang zwischen Teiltyp-Constraints und Modallogik aufzeigen. Dadurch lösen wir unter Anderem ein seit 1993 offenes Komplexitätsproblem von Wand und Tiuryn. Teiltyp-Subsumption ist selbst für einfachste Constraintsprachen von hoher Komplexität. Rehof und Henglein zeigten dies für den strukturellen Verbandsfall (mit zwei Typkonstanten 1997, 1998), ließen jedoch den nicht-strukturellen Fall offen. In dieser Arbeit betrachten wir den einfachsten nicht-strukturellen Fall. Hier zeigen wir, dass versteckte Wortgleichungen neue Schwierigkeiten verursachen. Hierzu charakterisieren wir Teiltyp-Subsumption durch spezielle endliche Automaten mit Wortgleichungen. Unsere Charakterisierung liefert partielle Entscheidbarkeitsresulte zur nichtstrukturellen Teiltyp-Subsumption und kann als Grundlage für künftige Untersuchungen dienen