A superexponential lower bound for Gro¨bner bases and church-Rosser Commutative Thue systems
Information and Control
Mechanical geometry theorem proving
Mechanical geometry theorem proving
Information Sciences: an International Journal
A refutational approach to geometry theorem proving
Geometric reasoning
The revised Fundamental Theorem of Moment Invariants
IEEE Transactions on Pattern Analysis and Machine Intelligence
Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Exponential lower bounds for restricted monotone circuits
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Artificial Intelligence Review
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Shapes such as triangles or rectangles can be defined in terms of geometric properties invariant under a group of transformations. Complex shapes can be described by logic formulae with simpler shapes as the atoms. A standard technique for computing invariant properties of simple shapes is the method of moment invariants, known since the early sixties. We generalize this technique to shapes described by arbitrary monotone formulae (formulae in propositional logic without negation). Our technique produces a reduced Grobner basis for approximate shape descriptions. We show how to use this representation to solve decision problems related to shapes. Examples include determining if a figure has a particular shape, if one description of a shape is more general than another, and whether a specific geometric property is really necessary for characterizing a shape. Unlike geometry theorem proving, our approach does not require the shapes to be explicitly defined. Instead, logic formulae combined with measurements performed on actual shape instances are used to compute well characterized least squares approximations to the shapes. Our results provide a proof that decision problems stated in terms of these approximations can be solved in a finite number of steps.