Automated Deduction in Geometry: 10th International by Francisco Botana, Pedro Quaresma

By Francisco Botana, Pedro Quaresma

This ebook constitutes the completely refereed post-workshop court cases of the tenth overseas Workshop on automatic Deduction in Geometry, ADG 2014, held in Coimbra, Portugal, in July 2014. The eleven revised complete papers provided during this quantity have been rigorously chosen from 20 submissions. The papers express the craze set of present examine in computerized reasoning in geometry.

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Additional resources for Automated Deduction in Geometry: 10th International Workshop, ADG 2014, Coimbra, Portugal, July 9-11, 2014, Revised Selected Papers

Example text

Xk ) : ψ ∈ LRCF . (7) In particular, k = 0 is a decision problem: is Φ true? If we have a CAD D(n) of Rn (noting that the xi must be ordered in the same way in Definition 1 and formula (6)) sign-invariant for the polynomials of Φ, then constructing Ψ is conceptually easy. 1. The truth of φ in a cell Di of D(n) is that of φ at the sample point si . 2. D(n) projects to a CAD D(k) of Rk . H. Davenport and M. England 3. ) Boolean combination of the truth of φ in the cells of D that project to Dj .

X + y = 1 x= 2 2 Geometrically, this means that the parallel to edge (1, 2) through vertex 4 meets the parallel to edge (23) through vertex 5 on the line (13). We’ll mark this point as 2’. Since infinitesimal flexibility is not affected by cyclic permutations of the indices, the above considerations yield the following: Condition ( 5 ): For all indices i ∈ Z5 , taken mod 5, the parallel to edge (i, i+1) through vertex i + 3 meets the parallel to edge (i + 1, i + 2) through vertex i + 4 on the line (i, i + 2).

The converse questions are essentially questions of refutation, see [JdM12]. Questions involving a mixture of quantifiers are much harder. Almost all current implementations of CAD are based on computer algebra systems, which are generally unverified. We can at least compare, on a fairly level playing field, the implementations in Maple of four algorithms: see Table 1. The classification of the amount of mathematics involved is subjective, but we note that [McC84], and hence [BDEW13], relies on [Zar65,Zar75] to justify the smaller projection set compared with [Col75].

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