Computer supported mathematics with Ωmega

AbstractClassical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces—often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques.The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ωmega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Mathematical applications of inductive logic programming

Accepted versio

Searching and retrieving in content-based repositories of formal mathematical knowledge

In this thesis, the author presents a query language for an RDF (Resource Description Framework) database and discusses its applications in the context of the HELM project (the Hypertextual Electronic Library of Mathematics). This language aims at meeting the main requirements coming from the RDF community. in particular it includes: a human readable textual syntax and a machine-processable XML (Extensible Markup Language) syntax both for queries and for query results, a rigorously exposed formal semantics, a graph-oriented RDF data access model capable of exploring an entire RDF graph (including both RDF Models and RDF Schemata), a full set of Boolean operators to compose the query constraints, fully customizable and highly structured query results having a 4-dimensional geometry, some constructions taken from ordinary programming languages that simplify the formulation of complex queries. The HELM project aims at integrating the modern tools for the automation of formal reasoning with the most recent electronic publishing technologies, in order create and maintain a hypertextual, distributed virtual library of formal mathematical knowledge. In the spirit of the Semantic Web, the documents of this library include RDF metadata describing their structure and content in a machine-understandable form. Using the author's query engine, HELM exploits this information to implement some functionalities allowing the interactive and automatic retrieval of documents on the basis of content-aware requests that take into account the mathematical nature of these documents

A global workspace framework for combined reasoning

Artificial Intelligence research has produced many effective techniques for solving a wide range of problems. Practitioners tend to concentrate their efforts in one particular problem solving paradigm and, in the main, AI research describes new methods for solving particular types of problems or improvements in existing approaches. By contrast, much less research has considered how to fruitfully combine different problem solving techniques. Numerous studies have demonstrated how a combination of reasoning approaches can improve the effectiveness of one of those methods. Others have demonstrated how, by using several different reasoning techniques, a system or method can be developed to accomplish a novel task, that none of the individual techniques could perform. Combined reasoning systems, i.e., systems which apply disparate reasoning techniques in concert, can be more than the sum of their parts. In addition, they gain leverage from advances in the individual methods they encompass. However, the benefits of combined reasoning systems are not easily accessible, and systems have been hand-crafted to very specific tasks in certain domains. This approach means those systems often suffer from a lack of clarity of design and are inflexible to extension. In order for the field of combined reasoning to advance, we need to determine best practice and identify effective general approaches. By developing useful frameworks, we can empower researchers to explore the potential of combined reasoning, and AI in general. We present here a framework for developing combined reasoning systems, based upon Baars’ Global Workspace Theory. The architecture describes a collection of processes, embodying individual reasoning techniques, which communicate via a global workspace. We present, also, a software toolkit which allows users to implement systems according to the framework. We describe how, despite the restrictions of the framework, we have used it to create systems to perform a number of combined reasoning tasks. As well as being as effective as previous implementations, the simplicity of the underlying framework means they are structured in a straightforward and comprehensible manner. It also makes the systems easy to extend to new capabilities, which we demonstrate in a number of case studies. Furthermore, the framework and toolkit we describe allow developers to harness the parallel nature of the underlying theory by enabling them to readily convert their implementations into distributed systems. We have experimented with the framework in a number of application domains and, through these applications, we have contributed to constraint satisfaction problem solving and automated theory formation

Interactive Theorem Proving with Indexed Formulas

Since more than two decades research in interactive theorem proving (ITP) has attracted growing interest. The primary application domains for ITPs range from hard- and software verification tools to mathematical tutor systems. To support communication with the user in an adequate way these systems depend on calculi that allow for the construction of human understandable and readable proofs. However, most calculi that are used in current ITPs fall still short of supporting the user in an optimal way. The reason is that they enforce the user to construct proofs at a level that is far more detailed than the one that can be found in human constructed proofs. Autexier [Aut03] has recently proposed a new theorem proving framework that allows to model different logics and calculi in an uniform way. In CORE, a proof-state is always represented as a single formula that can be manipulated by the application of replacement rules that are generated from the logical context of the subformula under transformation. This approach also facilitates proof construction at the assertion level which is considered as more closely matching the level at which humans construct proofs (see for instance [Hua94]). Together with COREs window inference technique this makes CORE a potentially well suited basis for interactive theorem proving. This thesis tries to excerpt COREs potential for interactive theorem proving by mapping important concepts of the established proof system ΩMEGA to CORE. A task structure is developed to present the context of a subformula in an intuitive way to the user and to assist him in structuring proofs. The development of a method interpreter makes it possible to specify abstract inference steps declaratively and to encode proof strategies for the use in CORE. The adaptation of ΩMEGAS agent-based suggestion mechanism ΩANTS to CORE helps the user with the identification of applicable methods and replacement rules.Interaktives Theorembeweisen hat in den letzten zwei Jahrzehnten zunehmend an Bedeutung gewonnen. Die Anwendungsbereiche von Systemen mit denen sich Beweise interaktiv führen lassen reichen von der Hard- und Software Verifikation bis zu mathematischen Tutor-Systemen. Die genannten Anwendungsgebiete machen es erforderlich das der Anwender bei der Beweisführung adequat untersützt wird. Um eine entsprechende Kommunikation mit dem Benutzer zu ermöglichen verwenden interaktive Beweissysteme Kalküle, in denen Beweise in einer für den Benutzer nachvollziehbaren Art und Weise, geführt werden können. Trotzdem kann man noch nicht davon sprechen, dass interaktive Beweiser den Benutzer optimal unterstützen. Der Hauptrgund hierfür ist, dass die eingesetzten Kalküle automatisch dazu führen, dass Beweise auf einer viel detailliertern Ebene geführt werden müssen, als man typischerweise in einem mathematischen Beweis finden würde. Autexier [Aut03]hat kürzlich eine neue logische Umgebung für die Beweissuche entwickelt, welche es ermöglicht verschiedene Logiken und Kalküle einheitlich zu modellieren. In CORE ist ein Beweiszustand immer als eine einzige Formel repräsentiert, welche durch das Anwenden von Ersetzungsregeln, die aus dem Kontext einer Teilformel abgeleitet werden, transformiert werden kann. Diese Herangehensweise erleichtert es auch, Beweise auf der sogenannten Assertion-Ebene (vgl. [Hua94]) zu führen, welche allgemein als eine natürlichere Ebene für die Beweisführung angesehen wird. Zusammen mit der Window-Inferenz Technik, die von CORE unterstützt, wird stellt CORE ein System dar das potentiell als eine verbesserte Grundlage für die interaktive Beweissuche angesehen werden kann. In dieser Arbeit geht es darum, das Potential von CORE im Hinblick auf die interaktive Beweiskonstruktion auszunutzen. Dieses geschieht zum einen dadurch, dass etablierte Konzepte im Bereich des interaktiven Beweisens, wie sie auch im Beweis-System ΩMEGA verwendet werden, auf das CORE-System abgebildet werden. Desweiteren wird eine Task-Srukture entwickelt, die es zum einen ermöglicht, den logischen Kontext einer Teilformel für den Benutzer verständlich aufzubereiten und darzustellen; zum anderen untersützt sie den Anwender auch darin, Beweise strukturiert zu führen. Der Entwurf eines Methoden-Interpreters ermöglicht es, abstrakte Beweisschritte zu kodieren und im System anzuwenden. Die Anpassung des Vorschlags-Mechanismus ΩANTS an CORE stellt eine weitere Unterstützung für den Benutzer bereit, indem sie automatisch Vorschläge über mögliche Fortsetzungen eines Beweises generiert

Implementation methodology for using concurrent and collaborative approaches for theorem provers, with case studies of SAT and LCF style provers

Theorem provers are faced with the challenges of size and complexity, fueled by the increasing range of applications. The use of concurrent/ distributed programming paradigms to engineer better theorem provers merits serious investigation, as it provides: more processing power and opportunities for implementing novel approaches to address theorem proving tasks hitherto infeasible in a sequential setting. Investigation of these opportunities for two diverse theorem prover settings with an emphasis on desirable implementation criteria is the core focus of this thesis. Concurrent programming is notoriously error prone, hard to debug and evaluate. Thus, implementation approaches which promote easy prototyping, portability, incremental development and effective isolation of design and implementation can greatly aid the enterprise of experimentation with the application of concurrent techniques to address specific theorem proving tasks. In this thesis, we have explored one such approach by using Alice ML, a functional programming language with support for concurrency and distribution, to implement the prototypes and have used programming abstractions to encapsulate the implementations of the concurrent techniques used. The utility of this approach is illustrated via proof-of-concept prototypes of concurrent systems for two diverse case studies of theorem proving: the propositional satisfiability problem (SAT) and LCF style (first-order) theorem proving, addressing some previously unexplored parallelisation opportunities for each, as follows:. SAT: We have developed a novel hybrid approach for SAT and implemented a prototype for the same: DPLL-Stalmarck. It uses two complementary algorithms for SAT, DPLL and Stalmarck’s. The two solvers run asynchronously and dynamic information exchange is used for co-operative solving. Interaction of the solvers has been encapsulated as a programming abstraction. Compared to the standalone DPLL solver, DPLL-Stalmarck shows significant performance gains for two of the three problem classes considered and comparable behaviour otherwise. As an exploratory research effort, we have developed a novel algorithm, Concurrent Stalmarck, by applying concurrent techniques to the Stalmarck algorithm. A proof-of-concept prototype for the same has been implemented. Implementation of the saturation technique of the Stalmarck algorithm in a parallel setting, as implemented in Concurrent Stalmarck, has been encapsulated as a programming abstraction. LCF: Provision of programmable concurrent primitives enables customisation of concurrent techniques to specific theorem proving scenarios. In this case study, we have developed a multilayered approach to support programmable, sound extensions for an LCF prover: use programming abstractions to implement the concurrent techniques; use these to develop novel tacticals (control structures to apply tactics), incorporating concurrent techniques; and use these to develop novel proof search procedures. This approach has been implemented in a prototypical LCF style first-order prover, using Alice ML. New tacticals developed are: fastest-first; distributed composition; crossTalk: a novel tactic which uses dynamic, collaborative information exchange to handle unification across multiple sub-goals, with shared meta-variables; a new tactic, performing simultaneous proof-refutation attempts on propositional (sub- )goals, by invoking an external SAT solver (SAT case study), as a counter-example finder. Examples of concrete theorem proving scenarios are provided, demonstrating the utility of these extensions. Synthesis of a variety of automatic proof search procedures has been demonstrated, illustrating the scope of programmability and customisation, enabled by our multilayered approach