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Finding Proofs in Tarskian Geometry

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Abstract

We report on a project to use a theorem prover to find proofs of the theorems in Tarskian geometry. These theorems start with fundamental properties of betweenness, proceed through the derivations of several famous theorems due to Gupta and end with the derivation from Tarski’s axioms of Hilbert’s 1899 axioms for geometry. They include the four challenge problems left unsolved by Quaife, who two decades ago found some OTTER proofs in Tarskian geometry (solving challenges issued in Wos’s 1998 book). There are 212 theorems in this collection. We were able to find OTTER proofs of all these theorems. We developed a methodology for the automated preparation and checking of the input files for those theorems, to ensure that no human error has corrupted the formal development of an entire theory as embodied in two hundred input files and proofs. We distinguish between proofs that were found completely mechanically (without reference to the steps of a book proof) and proofs that were constructed by some technique that involved a human knowing the steps of a book proof. Proofs of length 40–100, roughly speaking, are difficult exercises for a human, and proofs of 100–250 steps belong in a Ph.D. thesis or publication. 29 of the proofs in our collection are longer than 40 steps, and ten are longer than 90 steps. We were able to derive completely mechanically all but 26 of the 183 theorems that have “short” proofs (40 or fewer deduction steps). We found proofs of the rest, as well as the 29 “hard” theorems, using a method that requires consulting the book proof at the outset. Our “subformula strategy” enabled us to prove four of the 29 hard theorems completely mechanically. These are Ph.D. level proofs, of length up to 108.

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Notes

  1. There is also a long tradition, going back to Descartes, of reducing geometric problems to algebra calculations by introducing coordinates. Algorithms for carrying out such calculations by computer have been extensively studied, including special methods intended for geometry and Tarski’s general decision procedure for real closed fields. We mention these only to emphasize that such methods are irrelevant to this paper, which is concerned with proofs in an axiomatic system for geometry.

  2. There is a similar theorem, Satz 10.5, about reflection in a line, but that theorem is not quite equational, because it requires that the two points determining the line be distinct.

  3. Therefore we could not satisfy a request that the constants have names unique across all files. Researchers with that wish can write PHP code to append a number to each constant. Then, however, they won’t be able to use hints extracted from our proofs.

  4. There are 214 theorems in our Master List; but the last three are all the Hilbert parallel axiom, in two cases and in a combined statement. So really, there are 212 theorems, of which 29 are hard and 183 are easy.

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Acknowledgements

This material was based in part on work supported by the U.S. Department of Energy, Office of Science, under contract DE-ACO2-06CH11357.

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Authors and Affiliations

  1. San José State University, San Jose, CA, USA

    Michael Beeson

  2. Argonne National Laboratory, Lemont, IL, USA

    Larry Wos

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  1. Michael Beeson
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  2. Larry Wos
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Correspondence to Michael Beeson.

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Beeson, M., Wos, L. Finding Proofs in Tarskian Geometry. J Autom Reasoning 58, 181–207 (2017). https://doi.org/10.1007/s10817-016-9392-2

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