Automating Diagrammatic Proofs of Arithmetic Arguments

256 Pages·2011·1.54 MB·English

Automating Diagrammatic Proofs of Arithmetic
Arguments
Mateja Jamnik
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Ph.D.
University of Edinburgh
1999

Abstract
This thesis is on the automation of diagrammatic proofs, a novel approach to mech-
anised mathematical reasoning. Theorems in automated theorem proving are usually
proved by formal logical proofs. However, there are some conjectures which humans
can prove by the use of geometric operations on diagrams that somehow represent
these conjectures, so called diagrammatic proofs. Insight is often more clearly per-
ceived in these diagrammatic proofs than in the algebraic proofs. We are investigating
and automating such diagrammatic reasoning about mathematical theorems.
Concrete rather than general diagrams are used to prove ground instances of a univer-
sallyquanti(cid:12)edtheorem. Thediagrammaticproofisconstructedbyapplyinggeometric
operations to the diagram. These operations are the inference steps of the proof. A
general schematic proof is extracted from the ground instances of a proof. It is rep-
resented as a recursive program that consists of a general number of applications of
geometric operations. When given a particular diagram, a schematic proof generates
a proof for that diagram. To verify that the schematic proof produces a correct proof
of the conjecture for each ground instance we check its correctness in a theory of dia-
grams. We use the constructive !-rule and schematic proofs to make a transitionfrom
concrete instances to a general argument about the diagrammatic proof.
The realisation of our ideas is a diagrammatic reasoning system Diamond. Diamond
allows a user to interactively construct instances of a diagrammatic proof. It then
automaticallyabstractstheseintoageneralschematic proofandchecks thecorrectness
of this proof using an inductive theorem prover. Unlike other existing systems which
usediagramstoconstructessentiallysymbolicproofs,Diamondreasonswithdiagrams
directly, so all the inference rules of a proof are diagrammatic.
Despite a popular view of logicians from the past century that diagrams cannot be
used in formal proofs, we show the contrary. The general diagrammatic proof frame-
work presented in this thesis is a formalisation of diagrammatic reasoning. Diamond
provides an environment in which formal diagrammatic proofs of mathematical theor-
ems can be constructed.
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Acknowledgements
First and foremost, I should like to thank my supervisors Alan Bundy and Ian Green
who gave me invaluable guidance and encouragement throughout the process of my
research. Alan'ssenseofdirectionandgeneralunderstandingofhowthings(cid:12)ttogether
in the big picture of science has been an inspiration for me. Ian's countless questions
and comments helped me convey my ideas in a more coherent manner.
Thanks to Toby Walsh for commenting on the drafts of this thesis, to Predrag Jani(cid:20)ci(cid:19)c
for endless discussions which inspired many of my ideas, and to the members of the
Dream group in Edinburgh for many enjoyable discussions and useful comments.
I especially thank Gavin Bierman for his love, understanding and encouragement
throughout this time.
Lastly, najlep(cid:20)sa hvala to my family for their continuous love and support.
This research was supported by a studentship from the Department of Arti(cid:12)cial Intel-
ligence at the University of Edinburgh Studentship, a supplementary grant from the
Slovenian Scienti(cid:12)cFoundation, and a studentshipfrom the BritishOverseas Research
Scheme.
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Declaration
I hereby declare that I composed this thesis entirely myself and that it describes my
own research.
Mateja Jamnik
Edinburgh
February 22, 1999
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Publications
Partsofthisthesishavealreadyappearedinprint,havebeensubmittedforpublication,
or have been made publicly available:
(cid:15) Portions of Chapters 3, 7 and 8 are to appear in Journal of Logic, Language and
Information [Jamnik et al 99].
(cid:15) Theoriginalproposalforthisthesis,thuspartsofChapter3andChapter4, were
published in Perspectives on Cognitive Science [Jamnik et al 97a].
(cid:15) The main body of Chapter 7 appeared in Proceedings of the 15th IJCAI
[Jamnik et al 97b].
(cid:15) Portions of Chapter8 were publishedinProceedings of the 1998 AAAI Fall Sym-
posium onFormalising Reasoning withVisualandDiagrammatic Representations
[Jamnik et al 98].
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