Detecting truth, just on parts

Abstract

We introduce and discuss, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test on the dynamic geometry program GeoGebra.Keywords automatic deduction in geometry, automatic geometry theorem proving · automatic geometry theorem discovery · elementary geometry · Gröbner basis · zero divisor · true on parts, false on parts · true on components · dynamic geometry Partially supported by the Spanish Research

Dealing with Degeneracies in Automated Theorem Proving in Geometry

“…Of course, the equality g 2 + h • i = 0 cannot hold for positive values of g, h, i and, thus, (g 2 − h • i) • (g 2 + h • i) = 0 is equivalent to g 2 = h • i for real positive values; however, working over the real numbers and involving inequalities requires replacing Gröbner-based elimination methods by real quantifier elimination algorithms, with weaker performance (see [14] for some very promising but preliminary results in this direction). Thus, a different operating alternative has been developed in [15,16], observing that g 2 = h • i holds over some primary components of the variety described by the complex solutions of…”

“…, yields the zero ideal with eliminating all except the free variables. Thus the statement is 'true on parts' [15,16], that is, neither true nor false, because point B-the intersection of a line and a circle-is not uniquely defined and yields different primary components on the hypotheses ideal, some of them holding true, some false, the given thesis.…”

“…We refer, for precise details and examples to illustrate the difficulties and the adopted technical solutions, to the basic document [12] and, for recent updates, to [13,15] or [17]. The algorithms described in these papers are implemented in GeoGebra through the embedded computer algebra system Giac [18,19].…”

Automated Detection of Interesting Properties in Regular Polygons

“…This result helps in finding theorems that are true on parts. See [20] for more details. Now let us assume that C(RS k ) is non-degenerate and let I = t .…”

“…We conclude that from the hypotheses either t 1 or t 2 follows in the algebraic approach. In other words, t 1 (as well t 2 ) may be true just on parts (see [20] for more details). Finally we present a result which can be proven by exhaustion: • In a regular 11-gon the only rational lengths in S 2 are 1, 2, and the only quadratic surd is √ 3.…”