Papers by Christoph Benzmüller
Calculemus-ii | systems for computer-supported mathematical knowledge evolution
DIALOG: Dialog In Natural Language on Mathematical Proofs
DIALOG, 2003
Page 1. Draft 18/3/2003 DIALOG: Dialog in Natural Language on Mathematical Proofs Christoph Benzm... more Page 1. Draft 18/3/2003 DIALOG: Dialog in Natural Language on Mathematical Proofs Christoph Benzmüller , Armin Fiedler , Malte Gabsdil ¡ , Helmut Horacek , Ivana Kruijff-Korbayová ¡ , Jörg Siekmann , Dimitra Tsovaltzi ¡ , Bao Quoc Vo , Magdalena Wolska ¡ ...
In this paper we address assertion retrieval and application in theorem proving systems or proof ... more In this paper we address assertion retrieval and application in theorem proving systems or proof planning systems for classical first-order logic. Due to Huang the notion of assertion comprises mathematical knowledge such as definitions, theorems, and axioms. We propose a distributed mediator module between a mathematical knowledge base 7 98 and a theorem proving system @ BA which is independent of the particular proof representation format of @ BA and which applies generalised resolution in order to analyze the logical consequences of arbitrary assertions for a proof context at hand. Our approach is applicable also to the assumptions which are dynamically created during a proof search process. It therefore realises a crucial first step towards full automation of assertion level reasoning. We discuss the benefits and connection of our approach to proof planning and motivate an application in a project aiming at a tutorial dialogue system for mathematics.
… JOINT CONFERENCE ON …, 2003
Our work addresses assertion retrieval and application in theorem proving systems or proof planni... more Our work addresses assertion retrieval and application in theorem proving systems or proof planning systems for classical first-order logic. We propose a distributed mediator M between a mathemat-ical knowledge base KB and a theorem proving system TP which is ...
… and Evaluation (LREC …, 2004
Our goal is to develop a flexible dialog system for tutoring mathematical problem solving. Empiri... more Our goal is to develop a flexible dialog system for tutoring mathematical problem solving. Empirical findings in the area of intelligent tutoring show that flexible natural language dialog supports active learning. Therefore, we focus on the development of solutions allowing flexible dialog. However, little is known about the use of natural language in dialog settings in formal domains, such as mathematics, due to the lack of empirical data. We designed and performed an experiment with a simulated tutorial dialog system for teaching proofs in naive set theory. To investigate the correlations between (i) domain-specific content and its linguistic realization, and (ii) the use, distribution, and linguistic realization of dialog moves, we are annotating the corpus with (i) dependency-based semantic relations that build up the linguistic meaning of the utterances and (ii) with dialog moves. 1 The DIALOG project is a collaboration between the Computer Science and Computational Linguistics departments of University of the Saarland as part of the Collaborative Research Center on Resource-Adaptive Cognitive Processes, SFB 378 (www.coli. uni-sb.de/sfb378).
Springer eBooks, 2015
Implementing proof reconstruction is difficult because it involves symbolic manipulations of form... more Implementing proof reconstruction is difficult because it involves symbolic manipulations of formal objects whose representation varies between different systems. It requires significant knowledge of the source and target systems. One cannot simply re-target to another logic. We present a modular proof reconstruction system with separate components, specifying their behaviour and describing how they interact. This system is demonstrated and evaluated through an implementation to reconstruct proofs generated by Leo-II and Satallax in Isabelle/HOL, and is shown to work better than the current method of rediscovering proofs using a select set of provers.
Logic Journal of the IGPL, Jan 25, 2023
This paper reports on an exploration of Boolos' Curious Inference, using higher-order automated t... more This paper reports on an exploration of Boolos' Curious Inference, using higher-order automated theorem provers (ATPs). Surprisingly, only suitable shorthand notations had to be provided by hand for ATPs to find a short proof. The higher-order lemmas required for constructing a short proof are automatically discovered by the ATPs. Given the observations and suggestions in this paper, full proof automation of Boolos' and related examples now seems to be within reach of higher-order ATPs.
arXiv (Cornell University), Aug 14, 2022
This paper reports on an exploration of Boolos' Curious Inference, using higher-order automated t... more This paper reports on an exploration of Boolos' Curious Inference, using higher-order automated theorem provers (ATPs). Surprisingly, only suitable shorthand notations had to be provided by hand for ATPs to find a short proof. The higher-order lemmas required for constructing a short proof are automatically discovered by the ATPs. Given the observations and suggestions in this paper, full proof automation of Boolos' and related examples now seems to be within reach of higher-order ATPs.
This is a joint application by two partner universities: Saarland University (1 researcher) and I... more This is a joint application by two partner universities: Saarland University (1 researcher) and International University Bremen (1 researcher). It is also based on substantial international cooperation with american and english universities (hence we use English for this document).
Interactive theorem proving systems for mathematics require user interfaces which can present pro... more Interactive theorem proving systems for mathematics require user interfaces which can present proof states in a human understandable way. Often the underlying calculi of interactive theorem proving systems are problematic for comprehensible presentations since they are not optimally suited for practical, human oriented reasoning in mathematical domains. The recently developed CORE theorem proving framework [Aut03] is an improvement of traditional calculi and facilitates flexible reasoning at the assertion level. We make use of COREs reasoning power and develop a communication layer on top of it, called the task layer. For this layer we define a set of manipulation rules that are implemented via COREs calculus rules. We thereby obtain a human oriented interaction layer that improves and combines ideas underlying the window inference technique [RS93], the proof by pointing approach [BKT94], and the focus windows of [PB02].
Theorem Proving in Higher Order Logics, 2007
Leo-II, a resolution based theorem prover for classical higherorder logic, is currently being dev... more Leo-II, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year research project at the University of Cambridge, UK, with support from Saarland University, Germany. We report on the current stage of development of Leo-II. In particular, we sketch some main aspects of Leo-II's automated proof search procedure, discuss its cooperation with first-order specialist provers, show that Leo-II is also an interactive proof assistant, and explain its shared term data structure and its term indexing mechanism.
This is a joint application by two partner universities: Saarland University (1 researcher) and I... more This is a joint application by two partner universities: Saarland University (1 researcher) and International University Bremen (1 researcher). It is also based on substantial international cooperation with american and english universities (hence we use English for this document).
With Thanks to
LEO-II is a standalone, resolution-based higher-order theorem prover that is designed for fruitfu... more LEO-II is a standalone, resolution-based higher-order theorem prover that is designed for fruitful cooperation with specialist provers for first-order and propositional logic. The idea is to combine the strengths of the different systems. On the other hand, LEO-II itself, as an external reasoner, wants to support interactive proof assistants such as Isabelle/HOL, HOL, and OMEGA by efficiently automating subproblems and thereby reducing user effort. LEO-II predominantly addresses higher-order aspects in its reasoning process with the aim to quickly remove higher-order clauses from the search space and to turn them into essentially first-order clauses which can then be refuted with a first-order prover. For this LEO-II cooperates with the first-order theorem provers E, Spass or Vampire. LEO-II’s Data Structures LEO-II provides efficient term data structures based on a perfectly shared term graph, i.e., syntactically equal terms are represented by a single instance. Ideas from first-or...
Lecture Notes in Computer Science
One of the keys to the success of the Thousands of Problems for Theorem Provers (TPTP) problem li... more One of the keys to the success of the Thousands of Problems for Theorem Provers (TPTP) problem library and related infrastructure is the consistent use of the TPTP language. This paper introduces the core of the TPTP language for higher-order logic -THF0, based on Church's simple type theory. THF0 is a syntactically conservative extension of the untyped first-order TPTP language.
Cognitive Technologies, 2010
This paper summarises and gives an overview of the development of the ΩMEGA system during the 12 ... more This paper summarises and gives an overview of the development of the ΩMEGA system during the 12 years of funding by the SFB 378. The research objective of the ΩMEGA project has been to lay the foundation complex, heterogenous, but well integrated assistance systems for mathematics, which support the wide range of typical research, publication and knowledge management activities of a working mathematician. Examples are computing (for instance algebraic and numeric problems), proving (lemmas or theorems) , solving (for instance equations), modelling (by axiomatic definitions), verifying (typically a proof), structuring (for instance the new theory and knowledge base), maintaining (the knowledge base), searching (in a very large mathematical knowledge base), inventing (your new theorems), paper writing, explaining and illustrating in natural language and diagrams. Clearly, some of them require a high amount of human ingenuity
Lecture Notes in Computer Science, 2009
The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well estab... more The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well established infrastructure supporting research, development, and deployment of first-order Automated Theorem Proving (ATP) systems. Recently, the TPTP has been extended to include problems in higher-order logic, with corresponding infrastructure and resources. This paper describes the practical progress that has been made towards the goal of TPTP support for higher-order ATP systems.
Mathematics in Computer Science, 2008
Mathematical assistance systems and proof assistance systems in general have traditionally been d... more Mathematical assistance systems and proof assistance systems in general have traditionally been developed as large, monolithic systems which are often hard to maintain and extend. In this article we propose a component network architecture as a means to design and implement such systems. Under this view a mathematical assistance system is an integrated knowledge-based system composed as a network of individual, specialized components. These components manipulate and mutually exchange different kinds of mathematical knowledge encoded within different document formats. Consequently, several units of mathematical knowledge coexist throughout the system within these components and this knowledge changes non-monotonically over time. Our approach has resulted in a lean and maintainable system code and makes the system open for extensions. Moreover, it naturally decomposes the global and complex reasoning and truth maintenance task into local reasoning and truth maintenance tasks inside the system components. The interplay between neighboring components in the network is thereby realized by nonmonotonic updates over agreed interface representations encoding different kinds of mathematical knowledge.
Electronic Notes in Theoretical Computer Science, 2007
We present a generic mediator, called PLATΩ, between text-editors and proof assistants. PLATΩ aim... more We present a generic mediator, called PLATΩ, between text-editors and proof assistants. PLATΩ aims at integrated support for the development, publication, formalization, and verification of mathematical documents in a natural way as possible: The user authors his mathematical documents with a scientific WYSIWYG text-editor in the informal language he is used to, that is a mixture of natural language and formulas. These documents are then semantically annotated preserving the textual structure by using the flexible, parameterized proof language which we present. From this informal semantic representation PLATΩ automatically generates the corresponding formal representation for a proof assistant, in our case ΩMEGA. The primary task of PLATΩ is the maintenance of consistent formal and informal representations during the interactive development of the document.
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Papers by Christoph Benzmüller