Anti-unification and Generalization: A Survey

Anti-unification (AU), also known as generalization, is a fundamental operation used for inductive inference and is the dual operation to unification, an operation at the foundation of theorem proving. Interest in AU from the AI and related communities is growing, but without a systematic study of the concept, nor surveys of existing work, investigations7 often resort to developing application-specific methods that may be covered by existing approaches. We provide the first survey of AU research and its applications, together with a general framework for categorizing existing and future developments.Comment: Accepted at IJCAI 2023 - Survey Trac

E-Generalization Using Grammars

We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.Comment: 49 pages, 16 figures, author address given in header is meanwhile outdated, full version of an article in the "Artificial Intelligence Journal", appeared as technical report in 2003. An open-source C implementation and some examples are found at the Ancillary file

Anti-Pattern Matching Modulo

International audienceNegation is intrinsic to human thinking and most of the time when searching for something, we base our patterns on both positive and negative conditions. In a previous work, we have extended the notion of term to the one of anti-term that may contain complement symbols. Matching such anti-terms against terms has the nice property of being unitary. Here we generalize the syntactic anti-pattern matching to anti-pattern matching modulo an arbitrary equational theory E, and we study the specific and practically very useful case of associativity, possibly with a unity (AU). To this end, based on the syntacticness of associativity, we present a rule-based associative matching algorithm, and we extend it to AU. This algorithm is then used to solve AU anti-pattern matching problems. This allows us to be generic enough so that for instance, the AllDiff standard predicate of constraint programming becomes simply expressible in this framework. AU anti-patterns are implemented in the Tom language and we show some examples of their usage

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

Rewriting Logic Techniques for Program Analysis and Optimization

Esta tesis propone una metodología de análisis dinámico que mejora el diagnóstico de programas erróneos escritos en el lenguaje Maude. La idea clave es combinar técnicas de verificación de aserciones en tiempo de ejecución con la fragmentación dinámica de trazas de ejecución para detectar automáticamente errores en tiempo de ejecución, al tiempo que se reduce el tamaño y la complejidad de las trazas a analizar. En el caso de violarse una aserción, se infiere automáticamente el criterio de fragmentación, lo que facilita al usuario identificar rápidamente la fuente del error. En primer lugar, la tesis formaliza una técnica destinada a detectar automáticamente eventuales desviaciones del comportamiento deseado del programa (síntomas de error). Esta técnica soporta dos tipos de aserciones definidas por el usuario: aserciones funcionales (que restringen llamadas a funciones deterministas) y aserciones de sistema (que especifican los invariantes de estado del sistema). La técnica de verificación dinámica propuesta es demostrablemente correcta en el sentido de que todos los errores señalados definitivamente delatan la violación de las aserciones. Tras eventuales violaciones de aserciones, se generan automáticamente trazas fragmentadas (es decir, trazas simplificadas pero igualmente precisas) que ayudan a identificar la causa del error. Además, la técnica también sugiere una posible reparación para las reglas implicadas en la generación de los estados erróneos. La metodología propuesta se basa en (i) una notación lógica para especificar las aserciones que se imponen a la ejecución; (ii) una técnica de verificación aplicable en tiempo de ejecución que comprueba dinámicamente las aserciones; y (iii) un mecanismo basado en la generalización (ecuacional) menos general que automáticamente obtiene criterios precisos para fragmentar trazas de ejecución a partir de aserciones falsificadas. Por último, se presenta una implementación de la técnica propuesta en la herramienta de análisis dinámico basado en aserciones ABETS, que muestra cómo es posible combinar el trazado de las propiedades asertadas del programa para obtener un algoritmo preciso de análisis de trazas que resulta útil para el diagnóstico y la depuración de programas.This thesis proposes a dynamic analysis methodology for improving the diagnosis of erroneous Maude programs. The key idea is to combine runtime assertion checking and dynamic trace slicing for automatically catching errors at runtime while reducing the size and complexity of the erroneous traces to be analyzed (i.e., those leading to states that fail to satisfy the assertions). In the event of an assertion violation, the slicing criterion is automatically inferred, which facilitates the user to rapidly pinpoint the source of the error. First, a technique is formalized that aims at automatically detecting anomalous deviations of the intended program behavior (error symptoms) by using assertions that are checked at runtime. This technique supports two types of user-defined assertions: functional assertions (which constrain deterministic function calls) and system assertions (which specify system state invariants). The proposed dynamic checking is provably sound in the sense that all errors flagged definitely signal a violation of the specifications. Then, upon eventual assertion violations, accurate trace slices (i.e., simplified yet precise execution traces) are generated automatically, which help identify the cause of the error. Moreover, the technique also suggests a possible repair for the rules involved in the generation of the erroneous states. The proposed methodology is based on (i) a logical notation for specifying assertions that are imposed on execution runs; (ii) a runtime checking technique that dynamically tests the assertions; and (iii) a mechanism based on (equational) least general generalization that automatically derives accurate criteria for slicing from falsified assertions. Finally, an implementation of the proposed technique is presented in the assertion-based, dynamic analyzer ABETS, which shows how the forward and backward tracking of asserted program properties leads to a thorough trace analysis algorithm that can be used for program diagnosis and debugging.Esta tesi proposa una metodologia d'anàlisi dinàmica que millora el diagnòstic de programes erronis escrits en el llenguatge Maude. La idea clau és combinar tècniques de verificació d'assercions en temps d'execució amb la fragmentació dinàmica de traces d'execució per a detectar automàticament errors en temps d'execució, alhora que es reduïx la grandària i la complexitat de les traces a analitzar. En el cas de violar-se una asserció, s'inferix automàticament el criteri de fragmentació, la qual cosa facilita a l'usuari identificar ràpidament la font de l'error. En primer lloc, la tesi formalitza una tècnica destinada a detectar automàticament eventuals desviacions del comportament desitjat del programa (símptomes d'error). Esta tècnica suporta dos tipus d'assercions definides per l'usuari: assercions funcionals (que restringixen crides a funcions deterministes) i assercions de sistema (que especifiquen els invariants d'estat del sistema). La tècnica de verificació dinàmica proposta és demostrablement correcta en el sentit que tots els errors assenyalats definitivament delaten la violació de les assercions. Davant eventuals violacions d'assercions, es generen automàticament traces fragmentades (és a dir, traces simplificades però igualment precises) que ajuden a identificar la causa de l'error. A més, la tècnica també suggerix una possible reparació de les regles implicades en la generació dels estats erronis. La metodologia proposada es basa en (i) una notació lògica per a especificar les assercions que s'imposen a l'execució; (ii) una tècnica de verificació aplicable en temps d'execució que comprova dinàmicament les assercions; i (iii) un mecanisme basat en la generalització (ecuacional) menys general que automàticament obté criteris precisos per a fragmentar traces d'execució a partir d'assercions falsificades. Finalment, es presenta una implementació de la tècnica proposta en la ferramenta d'anàlisi dinàmica basat en assercions ABETS, que mostra com és possible combinar el traçat cap avant i cap arrere de les propietats assertades del programa per a obtindre un algoritme precís d'anàlisi de traces que resulta útil per al diagnòstic i la depuració de programes.Sapiña Sanchis, J. (2017). Rewriting Logic Techniques for Program Analysis and Optimization [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/94044TESI

Matching Logic

This paper presents matching logic, a first-order logic (FOL) variant for specifying and reasoning about structure by means of patterns and pattern matching. Its sentences, the patterns, are constructed using variables, symbols, connectives and quantifiers, but no difference is made between function and predicate symbols. In models, a pattern evaluates into a power-set domain (the set of values that match it), in contrast to FOL where functions and predicates map into a regular domain. Matching logic uniformly generalizes several logical frameworks important for program analysis, such as: propositional logic, algebraic specification, FOL with equality, modal logic, and separation logic. Patterns can specify separation requirements at any level in any program configuration, not only in the heaps or stores, without any special logical constructs for that: the very nature of pattern matching is that if two structures are matched as part of a pattern, then they can only be spatially separated. Like FOL, matching logic can also be translated into pure predicate logic with equality, at the same time admitting its own sound and complete proof system. A practical aspect of matching logic is that FOL reasoning with equality remains sound, so off-the-shelf provers and SMT solvers can be used for matching logic reasoning. Matching logic is particularly well-suited for reasoning about programs in programming languages that have an operational semantics, but it is not limited to this

A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic

Frex: dependently-typed algebraic simplification

We present an extensible, mathematically-structured algebraic simplification library design. We structure the library using universal algebraic concepts: a free algebra -- fral -- and a free extension -- frex -- of an algebra by a set of variables. The library's dependently-typed API guarantees simplification modules, even user-defined ones, are terminating, sound, and complete with respect to a well-specified class of equations. Completeness offers intangible benefits in practice -- our main contribution is the novel design. Cleanly separating between the interface and implementation of simplification modules provides two new modularity axes. First, simplification modules share thousands of lines of infrastructure code dealing with term-representation, pretty-printing, certification, and macros/reflection. Second, new simplification modules can reuse existing ones. We demonstrate this design by developing simplification modules for monoid varieties: ordinary, commutative, and involutive. We implemented this design in the new Idris2 dependently-typed programming language, and in Agda