Freie Universität Berlin

Leo III - A Massively Parallel Higher-Order Theorem Prover

Open Theses

Please contact Christoph Benzmüller if you are interested in one of the following topics. See also further topics.

Bachelors theses

  1. Kodierung von logischen Einbettungen (evtl. auch als Masterarbeit)

    Semantische Einbettungen erlauben es, verschiedene spezielle Logiken in klassische Logik höherer Stufe zu „simulieren“ (siehe z.B. [6]). Es soll ein Mechanismus entwickelt werden, der Eingaben einer speziellen Logik (semi-)automatisch in TPTP-Konforme Eingaben [4] der klassischen Logik höherer Stufe übersetzt.

  2. Generating Proof Outputs/First-Order Proof Reconstruction

    Proofs act as witnesses for the correctness and eligibility of propositions. It is therefore desirable to have an ATP system return not only the final proof result, but also to output a proof object of some kind that implements this witness role for the prover's result. In this work, the proof output mechanism of LEO-II is to be enhanced in order to robustly reconstruct first-order proofs of the included E prover and integrate them in the final proof output. Additionally, certain proof minimalization techniques can be employed.

Masters theses

  1. Automated Analysis of Correspondence Properties in First-Order Conditional Logics (Masters)

    In non-classical logics, e.g. modal logics, correspondence theorems link certain axiom schemata with corresponding semantic conditions. For example, the modal T axiom (schema) " [r] p => p " (where r is the accessibility relation underlying the box [] and diamond <> operators) corresponds to the semantic condition that the accessibility relation r is serial, i.e., " Forall x. Exists y. r(x,y) ". Previous work (see [5]) has shown that by exploiting the logic embeddings approach (see [6]) such correspondence results can very efficiently be automatically verified with higher-order automated theorem provers. The proposal for this thesis project is to apply and extend this kind of automated meta-level analysis to the area of (first-order) conditional logics. Respective embeddings for conditional logics have been studied in [7].

Ongoing Theses

  1. Preprocessing Techniques for Faster Reasoning with Boolean Types in Higher-Order Theorem Proving (M.Sc. Thesis)

    Hans-Jörg Schurr

  2. Proof minimization of LEO-II/Leo-III proofs (B.Sc. Thesis)

    Adrian Miloradovic

Completed Theses

  1. Agent-based Blackboard Architecture for a Higher-Order Theorem Prover (M.Sc. Thesis)

    Max Wisniewski

    Current competitive theorem provers sequentially manipulate sets of formulas in a global loop to obtain a proof. Nowadays in almost every area in computer science, concurrent and parallel approaches are increasingly used. Although some research towards parallel theorem proving has been done and even some systems were implemented, most modern theorem provers do not use any form of parallelism.
    In this thesis we present an architecture for Leo-III that use parallelism in its very core. To this end an agent-based blackboard architecture is employed. Agents denote independent programs which can act on their own. In comparison to classical theorem prover archiectures, the global loop is broken down to a set of tasks that can be computed in parallel. The results produced by all agents will be stored in a blackboard, a globally shared datastructure, thus visible to all other agents. For a proof of concept example agents are given demonstrating an agent-based approach can be used to implement a higher-order theorem prover.

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  3. Efficient Data Structures for Automated Theorem Proving in Expressive Higher-Order Logics (M.Sc. Thesis)

    Alexander Steen

    In this thesis, a higher-order term representation based upon the polymorphically typed λ-calculus is presented. This term representation employs spine notation, explicit substitutions and perfect term sharing for efficient term traversal, fast β-normalization and reuse of already constructed terms, respectively. An evaluation of the term representation is performed on the basis of a heterogeneous benchmark set. It shows that while the presented term data structure performs quite well in general, the normalization results indicate that a context dependent choice of reduction strategies is beneficial.
    A term indexing data structure for fast term retrieval based on various low-level criteria is presented and discussed. It supports symbol-based term retrieval, indexing of terms via structural properties, and subterm indexing.

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  4. Improved Computation of CNF in Higher Order Logics (B.Sc. Thesis)

    Kim Kern

    This work will provide a basic implementation of the CNF com- putation in higher order logics. Also it will present some combinable improving mechanisms. With regard to these, measures to evaluate them in terms of complexity of both the calculation and the resulting clauses as well as their impact on different reasoning systems.

  5. Benutzer-adaptives zeitkritisches Empfehlungssystem für digitale Zeitungsartikel (Diploma thesis)

    Benjamin Eckstein

    In cooperation with Niiu publishing GmbH

    In dieser Masterarbeit wird ein Empfehlungssystem für digitale Zeitungsartikel vorgestellt. Die kooperierende Firma Niiu publishing GmbH empfiehlt täglich für über 3.000 aktive Kunden über 12.000 tagesaktuelle Zeitungsartikel. Eine für Niiu große Problematik ist, dass für die täglich neuen Artikel noch keine Benutzerdaten vorliegen. Die Arbeit zeigt ein Verfahren diese als “Sparsity Problem” bekannte Hürde zu umgehen und realisiert einen neuen Ansatz Empfehlungen benutzer­adaptiv zu generieren. Die Arbeit beschreibt den Aufbau eines kompletten Empfehlungssystems. Dabei werden Artikel semantisch aufbereitet, Empfehlungen generiert und Benutzerverhalten interpretiert und ausgewertet. Die zeitliche Dauer der Berechnungen für das Aufbereiten von Artikeln und das Generieren von Empfehlungen ist dabei stark eingeschränkt. Zu dieser Arbeit wurde ein Prototyp entwickelt, der die Annahmen realisiert und mit echten Teilnehmern evaluiert. Die Ergebnisse der Evaluation werden präsentiert und mögliche Erkenntnisse diskutiert. Abschließend wird die Arbeit kritisch betrachtet und einige Ausblicke für aufbauende Arbeiten vorgestellt.

  6. Software Model Checking With Higher-Order Automated Theorem Provers: A Logic Embedding Approach (M.Sc. Thesis)

    Maximilian Claus

    Research in automating classical higher-order logic (HOL) has been on the rise for several years now and as a result, automated theorem provers for HOL, such as Satallax or LEO-II, are readily available. The areas of application for these new tools is vast: The expressivity of the logic makes it easy to translate deduction problems of many facades into equivalent problems in higher-order logic. This work is particulary concerned with solving problems in software verification, which can be adressed by, among others, the technique of model check- ing. In model checking, the state space of a program is inspected and exhaustively tested to whether it fulfills a certain property of interest or not. Such a property is commonly expressed in a verification logic, mostly temporal logics like LTL or CTL. We show how both the state space inspection and the verification logic can be embedded in HOL, and therefore how problems in model checking can be converted to problems for HOL theorem provers. The result is a verification soft- ware which is not optimised for one specific verification logic, but can interpret program properties in an arbitrary logic, given an appropri- ate embedding in HOL. Furthermore, we present improvements for the current automated provers, which are required to treat such generated problems efficiently. These additions might also be highly relevant for problems in different domains and therefore throughly discussed.

  7. Effizienter Parser für logische Daten (B.Sc. Thesis)

    Samuel Gfrörer

    Es soll ein Parser für verschiedene Formate für logische Eingabedaten (Formeln, ...) der TPTP-Spezifikation [4] entwickelt werden. Dabei soll ein besonderer Augenmerk auf die Effizienz gelegt werden.

  8. Automatisierung von Freier Logik in Logik höherer Stufe (B.Sc. Thesis)

    Irina Makarenko

    (English version) Free logic extends logic by including the aspect of non-existence of objects and creates space for undefinedness. Reasoning about such non-classical logics can be done using unconventional embeddings in classical higher-order logic. Automated theorem provers for higher-order logic rely on the TPTP language THF, a standardized encoding for formulae of higher-order logic. In this thesis, a TPTP compliant encoding will be formulated for non-classical free logic, the embedding of free logic in higher-order logic will be discussed and an automated translation of free logic into higher-order logic will be implemented. The goal of this thesis was to enable native formulation of free logic formulae – for their evaluation, however, the capabilities of notable higher-order theorem provers such as Leo-II(I) were used. The effectiveness of the translation was investigated based on exemplary forma- lizations. The starting point was the category theory book Categories, Allegories by Freyd und Scedrov (1990)

    (German version) Freie Logik erweitert Logik um den Aspekt der Nichtexistenz von Objekten und schafft Raum für Undefiniertheit. Logische Schlussfolgerungen über solche nicht- klassischen Logiken können über unkonventionelle Einbettungen in die klassische Logik höherer Stufe gezogen werden. Automatische Beweiser für höherstufige Logik setzen auf die TPTP-Sprache THF, eine standardisierte Kodierung für Formeln der Logik höherer Stufe. In dieser Arbeit wurde eine TPTP-konforme Kodierung für die nicht-klassische freie Logik entworfen sowie die Einbettung von freier Logik in Logik höherer Stufe diskutiert und eine automatisierte Übersetzung von freier Logik in die Logik höherer Stufe implementiert. Ziel der Arbeit war es, die native Formulierung von Formeln der freien Logik zu ermöglichen, für deren Auswertung aber trotzdem auf die Fertigkeiten von namhaften höherstufigen Theorembeweisern wie Leo-II(I) ausgewichen wird. Die Effektivität der Übersetzung wurde anhand beispielhafter Formalisierungen untersucht. Ansatzpunkt dafür bildete das kategorientheoretische Buch Catego- ries, Allegories von Freyd und Scedrov (1990).

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  9. Converting Higher-Order Modal Logic Problems into Classical Higher-Order Logic (B.Sc. Thesis)

    Tobias Gleißner

    Higher-order modal logic is a framework for modeling a large vari- ety of real world problems and automating reasoning in this framework would be very beneficial. Unfortunateley there exists no oftware which achieves this but such software does exist for higher-order logic. There is also a method which allows expressing higher-order modal logic in terms of higher-order logic which enables automatic reasoning in this logic. The development of tool which converts a higher-order modal problem into a higher-order problem applying this method is the sub- ject of this thesis.

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  1. http://www.mi.fu-berlin.de/inf/groups/ag-ki/
  2. http://page.mi.fu-berlin.de/lex/leo3/
  3. http://leoprover.org
  4. http://tptp.org/
  5. http://christoph-benzmueller.de/papers/J21.pdf
  6. http://christoph-benzmueller.de/papers/J23.pdf
  7. http://christoph-benzmueller.de/papers/C37.pdf