Mechanical Theorem Proving in Computational Geometry

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Mechanical Theorem Proving in Computational
Geometry

• ARW 2005
• Saturday 30th August
• Laura I. Meikle
• Jacques D. Fleuriot

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Why formally verify algorithms?

• Does our confidence in geometric algorithms rely
  on intuition?
• Seems many algorithms are not verified thoroughly
• Diagrams are heavily relied upon
• Confidence in algorithms is sometimes suspect
• Correctness should be a higher priority
• Formal verification can give more confidence

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Formalising Grahams Scan

• Grahams Scan computes 2D convex hull
• Formalised algorithm in Isabelle/HOL
• Used development of Floyd-Hoare logic
• Inspired by Knuths Axioms and Hulls
• Reasons about signed areas
• Our work differs in many regards
• No axioms
• Formalism built on theory of vectors
• Different algorithm verified
• Degenerate cases permitted

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On Automation

• How could verification be made easier?
• Loop invariant discovery
• How applicable are current techniques?
• Geometry Theorem Proving
• Need to deal with ordered geometry
• Arithmetic Decision Procedures
• Integrate Maple or QEPCAD?
• Integrate Maple or QEPCAD?