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Information retrieval and rendering with MML Query
 Proc. of MKM 2006, Lecture Notes in Artificial Intelligence 4108
, 2006
"... Abstract. Mizar, a proofchecking system, is used to build the Mizar Mathematical Library (MML). MML Query is a semanticsbased tool for searching, browsing and presentation of the evolving MML content. The tool is becoming widely used as an aid for Mizar authors and plays an essential role in the o ..."
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Abstract. Mizar, a proofchecking system, is used to build the Mizar Mathematical Library (MML). MML Query is a semanticsbased tool for searching, browsing and presentation of the evolving MML content. The tool is becoming widely used as an aid for Mizar authors and plays an essential role in the ongoing reorganization of MML. We present new features of MML Query implemented in the third release and describe the possibilities offered by them. 1
Semantic Selection of Premisses for Automated Theorem Proving
"... We develop and implement a novel algorithm for discovering the optimal sets of premisses for proving and disproving conjectures in firstorder logic. The algorithm uses interpretations to semantically analyze the conjectures and the set of premisses of the given theory to find the optimal subsets of ..."
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We develop and implement a novel algorithm for discovering the optimal sets of premisses for proving and disproving conjectures in firstorder logic. The algorithm uses interpretations to semantically analyze the conjectures and the set of premisses of the given theory to find the optimal subsets of the premisses. For each given conjecture the algorithm repeatedly constructs interpretations using an automated model finder, uses the interpretations to compute the optimal subset of premisses (based on the knowledge it has at the point) and tries to prove the conjecture using an automated theorem prover. 1 Importance of selecting appropriate premisses in automated theorem proving A proper set of premisses 1 can be essential for proving a conjecture by an automated theorem prover. Clearly, the larger the number of the initial premisses the larger the number of the inferred formulae. And as for the most proving techniques the number of inferred formulae is in general superexponential in the number of input formulae,
Automated Reasoning for Mizar: Artificial Intelligence through Knowledge Exchange
"... This paper gives an overview of the existing link between the Mizar project for formalization of mathematics and Automated Reasoning tools (mainly the Automated Theorem Provers (ATPs)). It explains the motivation for this work, gives an overview of the translation method, discusses the projects and ..."
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This paper gives an overview of the existing link between the Mizar project for formalization of mathematics and Automated Reasoning tools (mainly the Automated Theorem Provers (ATPs)). It explains the motivation for this work, gives an overview of the translation method, discusses the projects and works that are based on it, and possible future projects and directions. 1
Managing mathematical texts with OWL and their graphical representation
"... Mathematical knowledge contained in scientific digital publications poses a challenge for intelligent retrieval mechanisms. Many current approaches use statistical (e.g. Google) or natural language processing methods to find correlations in texts and annotate texts semantically. However both kinds o ..."
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Mathematical knowledge contained in scientific digital publications poses a challenge for intelligent retrieval mechanisms. Many current approaches use statistical (e.g. Google) or natural language processing methods to find correlations in texts and annotate texts semantically. However both kinds of approaches face the problem of extracting and processing knowledge from mathematical equations. The presented system is based on natural language processing techniques, and benefits from characteristic linguistic structures defined by the language used in mathematical texts. It accumulates extracted information snippets from texts, symbols, and equations in knowledge bases. These knowledge bases provide the foundation for the information retrieval. This article describes the concepts and the prototypical technical implementation.
mArachna – Ontology Engineering for Mathematical Natural Language Texts
"... The knowledge contained in the growing number of scientific digital publications, particularly over the internet creates new demands for intelligent retrieval mechanisms. One basic approach in support of such retrieval mechanisms is the generation of semantic annotation, based on ontologies describi ..."
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The knowledge contained in the growing number of scientific digital publications, particularly over the internet creates new demands for intelligent retrieval mechanisms. One basic approach in support of such retrieval mechanisms is the generation of semantic annotation, based on ontologies describing both the field and the structure of the texts themselves. Many current approaches use statistical methods similar to the ones employed by Google to find correlations within the texts. This approach neglects the additional information provided in the upper ontology used by the author. mArachna, however, is based on natural language processing techniques, taking advantage of characteristic linguistic structures defined by the language used in mathematical texts. It stores the extracted knowledge in a knowledge base, creating a lowlevel ontology of mathematics and mapping this ontology onto the structure of the knowledge base. The following article gives an overview over the concepts and technical implementation of the mArachna prototype. 1
A Qualitative Comparison of the Suitability of Four Theorem Provers for Basic Auction Theory ⋆
"... Abstract Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them m ..."
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Abstract Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machinechecked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/ TPTP theorem provers for reproducing a key result of auction theory: Vickrey’s 1961 theorem on the properties of secondprice auctions. Based on our formalisation experience, taking an auction designer’s perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox. 1
UITP 2010 Pollackinconsistency
"... For interactive theorem provers a very desirable property is consistency: it should not be possible to prove false theorems. However, this is not enough: it also should not be possible to think that a theorem that actually is false has been proved. More precisely: the user should be able to know wha ..."
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For interactive theorem provers a very desirable property is consistency: it should not be possible to prove false theorems. However, this is not enough: it also should not be possible to think that a theorem that actually is false has been proved. More precisely: the user should be able to know what it is that the interactive theorem prover is proving. To make these issues concrete we introduce the notion of Pollackconsistency. This property is related to a system being able to correctly parse formulas that it printed itself. In current systems it happens regularly that this fails. We argue that a good interactive theorem prover should be Pollackconsistent. We show with examples that many interactive theorem provers currently are not Pollackconsistent. Finally we describe a simple approach for making a system Pollackconsistent, which only consists of a small modification to the printing code of the system. The most intelligent creature in the universe is a rock. None would know it because they have lousy I/O. — quote from the Internet
A SYNTHESIS OF THE PROCEDURAL AND DECLARATIVE STYLES OF INTERACTIVE THEOREM PROVING
"... Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the adva ..."
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Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style – the possibility to write formal proofs like normal mathematical text – and the procedural style – strong automation and help with shaping the proofs, including determining the statements of intermediate steps. Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations. To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has ‘industrial strength’. Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers. 1.
New developments in parsing mizar
"... The mizar language aims to capture mathematical vernacular by providing a rich language for mathematics. From the perspective of a user, the richness of the language is welcome because it makes writing texts more “natural”. But for the developer, the richness leads to syntactic complexity, such as d ..."
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The mizar language aims to capture mathematical vernacular by providing a rich language for mathematics. From the perspective of a user, the richness of the language is welcome because it makes writing texts more “natural”. But for the developer, the richness leads to syntactic complexity, such as dealing with overloading. As part of the larger project of opensourcing the mizar codebase, the mizar team is opening up the mizar parser. The result has led them to consider afresh the problems of parsing the mizar language and making it accessible to users and other developers. In this paper we describe these new parsing efforts and some applications thereof, such as largescale text refactorings, prettyprinting, HTTP parsing services, and normalizations of mizar texts. 1