A Decision Method for Elementary Algebra and Geometry

Abstract

Prepared for publication with the assistance of J. C. C. McKinsey This work has a long history. Its main results were found in 1930 and first mentioned in print a year later. It took nine years, however, before the material in its full development was prepared for publication. In fact, a monograph embodying those results was scheduled to appear in 1939 under the title The completeness of elementary algebm and geometry in the collection Actualites scientifiques et industrielles, Hermann & Cie, Paris. As a result of the war, the publication did not materialize; two existing sets of page proofs are probably the only trace of this venture.Naturally, I did not abandon the hope of seeing the results in print. However, as often happens with authors, the original version ceased to satisfy me. The specific character of the results seemed to call for a more formal presentation, and the addition of new observations and conclusions seemed to be desirable. The perplexity of the war and postwar periods resulted in postponing the revision from year to year. Hence I was happy when, in the beginning of 1948, the RAND Corporation, Santa Monica, California, became interested in my results and offered to publish them. It was especially fortunate that Professor J. C. C. McKinsey (Stanford University), who was at the time working with the RAND Corporation, was entrusted with the task of preparing the work for publication. With my collaboration, he actually prepared a new draft of the monograph and, aside from his competent editorial work, he contributed to a simplification of the development. Within a few months the monograph was published. As was to be expected, it reflected the specific interests which 25the RAND Corporation found in the results. The decision method for elementary algebra and geometry -which is one of the main results of the workwas presented in a systematic and detailed way, thus bringing to the fore the possibility of constructing an actual decision machine. Other, more theoretical aspects of the problems discussed were treated less thoroughly, and only in notes.The present edition is a photographic reprint of that published by the RAND Corporation, and hence no extensive changes could be introduced in the text. However, all known errors have been corrected, the introduction has been partly changed, and some supplementary notes have been added. In addition to new bibliographical references (mostly to recent literature), these supplementary notes contain elaborations of the original text which, in at least one case, result in enriching the content of the monograph by new theoretical material.I should like to thank the readers and reviewers who drew my attention to misprints and minor

Hybrid Systems with Finite Bisimulations

“…Even though decision methods exist for (R, +, −, ×, <, 0, 1) [22], it is not known if the theory of R exp is decidable, although in [16] it was shown that it would be a consequence of Schanuel's conjecture in number theory. The results we obtained in this paper suggest how to find some restricted classes of vector fields for which the algorithm is constructive.…”

“…Tarski [22] was interested in the extension of the theory of the real numbers by the exponential function, (R, +, −, ×, <, 0, 1, exp) (i.e., there is an additional symbol in the language for the exponential function). We denote this structure by R exp .…”

Counting special points: Logic, diophantine geometry, and transcendence theory

“…While the finiteness and uniformity properties of PL-geometry and tropical geometry are easy enough to demonstrate directly, the o-minimality of this underlying structure puts these results into context. That (R, +, ×, <, 0, 1), the set of real numbers considered as an ordered field, is o-minimal is a consequence of Tarski's theorem on elementary geometry (first proved in 1929, but only published in 1948 [41]) and is the basis of semi-algebraic geometry.…”

A New Class of Decidable Hybrid Systems

“…Tarski [10] showed that OF(R) is decidable, i.e., there is a computational procedure that, given any R-sentence φ, decides whether φ belongs to OF(R). The decision procedure is based on the elimination of the quantifiers.…”

“…The main computational tool for symbolic set manipulation in this context is quantifier elimination. Since quantifier elimination is possible for the theory of reals with addition and multiplication [10,15], we either find or transform subclasses of o-minimal hybrid systems which are definable in this theory. This immediately leads to new decidability results.…”