From Tarski to Hilbert

Braun, Gabriel; Narboux, Julien

doi:10.1007/978-3-642-40672-0_7

Cited by 16 publications

(18 citation statements)

References 16 publications

(21 reference statements)

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“…Our goal is to produce readable and machine verifiable proofs of theorems from SST automatically. Just like in the Coq formalization [11], we will focus only on the theorems found in the first 12 chapters of SST that belong to plane geometry. Those chapters, covering 120 pages of the book, contain 203 theorems, while 179 of them belong to plane geometry.…”

Section: Case Study: Tarski's Geometrymentioning

confidence: 99%

“…-The set of theorems in the book is well-rounded, starting from easy ones to very complex ones, forming a non-trivial, important mathematical knowledge base. -Computer proofs of SST, using both automated theorem proving [24,2,3] and interactive theorem proving [22,11], have already been developed. These developments can be used as a reference point but also for guiding the search within our framework.…”

Section: Introductionmentioning

confidence: 99%

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Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

Đurđević

Narboux

Janicic

2015

Ann Math Artif Intell

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The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of nontrivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics-a book on Tarski's geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37% of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42%. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge.

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Section: Case Study: Tarski's Geometrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

Đurđević

Narboux

Janicic

2015

Ann Math Artif Intell

Self Cite

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“…The choice of the topic is not accidental -recent code available in Coq [7] and the use of automated equational provers caught an eye of researchers and, as a by-product, some results, which shed some new light on the axiomatization of geometry, were published. One of the bright milestones was also the publication of the new issue of the classical textbook Metamathematische Methoden in der Geometrie by Wolfram Schwabhäuser, Wanda Szmielew and Alfred Tarski [40] (to which we refer by the acronym SST) with the foreword of Michael Beeson.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we are not focused on any of geometric challenges known in the community, as proving Hilbert axioms from Tarski [7], or formalizing full SST in Mizar, although it can be definitely good starting point as in [38] we prove some Hilbert's axioms. What we were trying to do was to increase of the integrity of (the geometrical part of) the MML as pointed out in the paper of Piotr Rudnicki and Andrzej Trybulec [39] and this could be a kind of partial realization of their postulates.…”

Section: Introductionmentioning

confidence: 99%

Tarski's geometry modelled in Mizar computerized proof assistant

Grabowski¹

2016

Annals of Computer Science and Information Systems

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Abstract-In the paper, we discuss the formal approach to Tarski geometry axioms modelled with the help of the Mizar computerized proof assistant system. Although our basic development was inspired by Julien Narboux's Coq pseudo-code and is dated back to 2014, there are significant steps in the formalization of geometry done in the last decade of the previous century. Taking this into account, we will propose the reuse of existing results within this new framework (including Hilbert's axiomatic approach), with the ultimate future goal to encode the textbook Metamathematische Methoden in der Geometrie by Schwabhäuser, Szmielew and Tarski. We try however to go much further from the use of simple predicates in the direction of the use of structures with their inheritance, attributes as a tool of more human-friendly namespaces for axioms, and registrations of clusters to obtain more automation (with the possible use of external equational theorem provers like Otter/Prover9).

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“…This includes the proof that Hilbert's axiomatic system (without continuity) can be derived from Tarski's axioms [BN12], some equivalences between different versions of the parallel postulate [BNS15b], some decidability properties of geometric predicates [BNSB14a] and the synthetic proof of some popular high-school geometry theorems. The proofs are mainly manual, but we used the tactics presented in [BNSB14b,BNS15a].…”

Section: Introductionmentioning

confidence: 99%

From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry

Boutry

Braun

Narboux

EPiC Series in Computing

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This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of [SST83]. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method.

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From Tarski to Hilbert

Cited by 16 publications

References 16 publications

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

Tarski's geometry modelled in Mizar computerized proof assistant

From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry

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