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Combining algebraic computing and term-rewriting for geometry theorem proving [chapter]

Stéphane Fèvre, Dongming Wang
<span title="">1998</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
This note reports some of our investigations on combining algebraic computing and term-rewriting techniques for automated geometry theorem proving.  ...  A general approach is proposed that requires both Clifford algebraic reduction and term-rewriting.  ...  Implementing a geometry theorem prover based on our approach thus requires tools for both symbolic algebraic computation and term-rewriting.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190218140850/http://pdfs.semanticscholar.org/0c30/3f25a7742dc133fdfde196cb2d421bdaa507.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0c/30/0c303f25a7742dc133fdfde196cb2d421bdaa507.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bfb0055909"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Page 438 of Mathematical Reviews Vol. , Issue 92a [page]

<span title="">1992</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The author pre- sents examples of how theorems of affine geometry may be proved by equational reasoning. {For the entire collection see MR 91b:68009.}  ...  The author proposes a Grassmann algebra formalism as the ba- sis of a programming language for geometry.  ... 
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Index

<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
Constraint contextual rewriting, 193 Automated short proof generation for projective geometric theorems with Cayley and bracket algebras I.  ...  Incidence geometry, 717 Automated short proof generation for projective geometric theorems with Cayley and bracket algebras II.  ... 
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Page 3371 of Mathematical Reviews Vol. , Issue 88g [page]

<span title="">1988</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Asser (Greifswald) 88g:03020 03B35 68T15 Hsiang, Jieh (1-SUNYS-C) Rewrite method for theorem proving in first order theory with equality. Rewriting techniques and applications (Dijon, 1985). J.  ...  Summary: “It was proved in a paper by L. Henschen and L. Wos [J. Assoc. Comput.  ... 
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System Description: GCLCprover + GeoThms [chapter]

Predrag Janičić, Pedro Quaresma
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
, isometric transformations, parametric curves, but also for symbolic expressions, and program loops.  ...  2 GCLC (originally a tool for producing geometrical illustrations for L A T E X, hence its name -Geometrical Constructions → L A T E X Converter) [3, 4] provides support for a range of geometrical constructions  ...  GCLCprover Automated theorem proving in geometry has two major lines of research: synthetic proof style and algebraic proof style (see, for instance, [5] for a survey).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11814771_13">doi:10.1007/11814771_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dcv2jhlb5rhtddurbjskvflfki">fatcat:dcv2jhlb5rhtddurbjskvflfki</a> </span>
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ZSS: Zipper logic revisited

Marius Buliga
<span title="2021-06-11">2021</span> <i title="figshare"> figshare.com </i> &nbsp;
of zipper logic arXiv:1405.6095, and prove that it allows universal computation using tangles and zippers, via directed Interaction Combinators.  ...  Can we compute with tangles and Reidemeister moves?  ...  An e.a. is a combination of algebraic and topological information, see Figure 2 . See also [6] arXiv:1807.02058 for an associated lambda calculus style term rewriting system.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.6084/m9.figshare.14769666.v1">doi:10.6084/m9.figshare.14769666.v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oxiwb7hhkbdnnck7kf7uzopt2u">fatcat:oxiwb7hhkbdnnck7kf7uzopt2u</a> </span>
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Equational Logic and Rewriting [chapter]

Claude Kirchner, Hélène Kirchner
<span title="">2014</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3htcvojut5de5ct6zf42o424vi" style="color: black;">Handbook of the History of Logic</a> </i> &nbsp;
Handbook-History-Logic-Computation-EquationalLogic Handbook-History-Logic-Computation-Rewriting  ...  In addition to the validity problem of equational logic, it has also been applied to inductive theorem proving, first-order theorem proving, unification theory, geometry theorem proving.  ...  In this section, we focus on the contribution of the term rewriting techniques to first-order theorem proving. The term rewriting approach to theorem proving is unique in several aspects.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/b978-0-444-51624-4.50006-x">doi:10.1016/b978-0-444-51624-4.50006-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eqtvr66m65eula7otl7fr6ppaa">fatcat:eqtvr66m65eula7otl7fr6ppaa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190430174838/https://hal.inria.fr/hal-01183817/file/eqLogicRw.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0c/be/0cbe44fc89b21bdfb2984c3d770540d1fabc436f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/b978-0-444-51624-4.50006-x"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

A Survey of Automated Deduction [chapter]

Alan Bundy
<span title="">1999</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Machine, a theorem prover for Euclidean Geometry, Gelernter, 1963b; Gelernter, 1963a ; a description of Slagle's symbolic integrator, Saint.  ...  These four were: a description of a theorem prover for propositional logic, called the Logic Theory Machine, written by Newell, Shaw and Simon, Newell et al., 1957 ; two papers on Gelernter's Geometry  ...  The combination of much faster computers and more e cient coding techniques has enabled resolution theorem provers to prove non-trivial theorems | even open conjectures.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-48317-9_6">doi:10.1007/3-540-48317-9_6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r7kti6fn2bcjbodyatqdq64wu4">fatcat:r7kti6fn2bcjbodyatqdq64wu4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170814212628/https://www.era.lib.ed.ac.uk/bitstream/handle/1842/3392/0001.pdf?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ab/07/ab074dd97e4b9a2ed2cbcae7377015ac91f8030e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-48317-9_6"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Page 5909 of Mathematical Reviews Vol. , Issue 92k [page]

<span title="">1992</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
A coordinate-free algebraic language for expressing properties of projective geometry is described; the extent of its expressiveness for these properties is explored; and a technique for analytically proving  ...  output of future symbolic computer algorithms for analytic projective geometry.  ... 
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Page 4755 of Mathematical Reviews Vol. , Issue 94h [page]

<span title="">1994</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This method has a number of applications, such as solving systems of polynomial equations and inequalities, mechanical theorem proving in geometry, irre- ducible decomposition of algebraic varieties and  ...  (English summary) Conditional term rewriting systems (Pont-a-Mousson, 1992), 483-497, Lecture Notes in Comput. Sci., 656, Springer, Berlin, 1993.  ... 
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Proof Certificates for Algebra and Their Application to Automatic Geometry Theorem Proving [chapter]

Benjamin Grégoire, Loïc Pottier, Laurent Théry
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
automatically produces formal proofs of theorems in geometry.  ...  In this paper, we describe how, starting from Hilbert's Nullstellensatz theorem, we combine a modified version of Buchberger's algorithm and some reflexive techniques to get an effective procedure that  ...  We thanks the referees for their constructive comments on the first version of this paper.  ... 
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Page 1644 of Mathematical Reviews Vol. , Issue 85d [page]

<span title="">1985</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Ching Chou, Proving elementary geometry theorems using Wu’s algorithm (pp. 243-286); Douglas B.  ...  (pp. 47-48); Hao Wang, Computer theorem proving and artificial intelligence (pp. 49-70); Nils J.  ... 
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Page 6028 of Mathematical Reviews Vol. , Issue 88k [page]

<span title="">1988</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Dershowitz, Inference rules for rewrite-based first-order theorem proving (pp. 331-337); J. H. Gallier, S. Raatz and W.  ...  Heuillard and P. Lescanne, Decidability of the confluence of ground term rewriting systems (pp. 353-359).  ... 
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Lambda-calculus and Reversible Automatic Combinators [article]

Alberto Ciaffaglione, Furio Honsell, Marina Lenisa, Ivan Scagnetto
<span title="2018-08-30">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The gist of our approach is that the GoI interpretation of a term based on involutions is dual to the principal type of the term, w.r.t. the type discipline for a linear/affine lambda-calculus.  ...  algebra and the full affine combinatory algebra are not.  ...  The Abstraction Operation in Definition 4 and the Abstraction Theorem 1 can then be extended to the full affine case. A counterpart to Theorem 2 can be proved for full affine λ-algebras.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1806.06759v2">arXiv:1806.06759v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/puey7lrl3zaqho5inklm6hw7he">fatcat:puey7lrl3zaqho5inklm6hw7he</a> </span>
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Editorial

Jean-Pierre Jouannaud
<span title="">1987</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
Several other special issues (on computational geometry, on rewrite techniques for automated theorem proving, on logic programming, on algebraic complexity, etc.) are in the planning stage.  ...  This justifies the claim that term rewriting is as powerful as paramodulation and resolution combined.  ... 
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