# Logical Inference and Term Rewriting

This approach applies general logical reasoning techniques to the geometric problem of constraint solving. This approach has been taken by Aldefeld [1] and Bruderlin [10], among others. As an example of this method, consider the system described by Bruderlin. Geometric entities are restricted to points, lines, vectors, and triangles, and the constraints allowed are distances between points, angles between lines, or two angles of a triangle. These geometries and constraints are incorporated into a set of predicates for the system. A set of allowable congruence relationships for the geometries used are established, and then rules of Euclidean geometry for ruler and compass constructions are applied. These rules are set up in Prolog, and Prolog rewrite-rules are used to solve the system. The result is a construction technique for solving the input constraint system. All possible physical solutions can be found using the Prolog backtracking mechanism.

While this approach has the potential to be a generic solver, as implemented the system of Bruderlin is an instance solver, since the predicates and rules use the actual constraint values throughout the deductive process. The major advantage of this method is that it avoids translation of the system into complex algebraic equations. The limitations are that only constructive geometries can be handled, and that the method is not very efficient for large systems of constraints. These disadvantages are common among this type of solver, and hence they are not often applied in commercial constraint solvers.

## References

• A. Aldefeld [1]
• A. Borning, M. Maher, A. Martindale, and M. Wilson [4]
• B. Bruderlin [10], [11]
• A. Bundy and B. Welham [13]
• J. Cohen [24]
• W. Sohrt [97]
• H. Suzuki, H. Ando, and F. Kimura [106]
• Y. Yamaguchi and F. Kimura [122]