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Finding Proofs in Tarskian Geometry
Jun 22, 2016
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.
XCB, the Last of the Shortest Single Axioms for the Classical Equivalential Calculus
Nov 13, 2002It has long been an open question whether the formula XCB = EpEEEpqErqr is, with the rules of substitution and detachment, a single axiom for the classical equivalential calculus. This paper answers that question affirmatively, thus completing a search for all such eleven-symbol single axioms that began seventy years ago.
Vanquishing the XCB Question: The Methodology Discovery of the Last Shortest Single Axiom for the Equivalential Calculus
Nov 13, 2002With the inclusion of an effective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study by various researchers. Is the formula XCB = e(x,e(e(e(x,y),e(z,y)),z)) a single axiom for the classical equivalential calculus when the rules of inference consist of detachment (modus ponens) and substitution? Where the function e represents equivalence, this calculus can be axiomatized quite naturally with the formulas e(x,x), e(e(x,y),e(y,x)), and e(e(x,y),e(e(y,z),e(x,z))), which correspond to reflexivity, symmetry, and transitivity, respectively. (We note that e(x,x) is dependent on the other two axioms.) Heretofore, thirteen shortest single axioms for classical equivalence of length eleven had been discovered, and XCB was the only remaining formula of that length whose status was undetermined. To show that XCB is indeed such a single axiom, we focus on the rule of condensed detachment, a rule that captures detachment together with an appropriately general, but restricted, form of substitution. The proof we present in this paper consists of twenty-five applications of condensed detachment, completing with the deduction of transitivity followed by a deduction of symmetry. We also discuss some factors that may explain in part why XCB resisted relinquishing its treasure for so long. Our approach relied on diverse strategies applied by the automated reasoning program OTTER. Thus ends the search for shortest single axioms for the equivalential calculus.
A Spectrum of Applications of Automated Reasoning
May 30, 2002The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it offers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's program OTTER, discussing widely varied successes in answering open questions, and touching on some of the strategies and methodologies that played a key role. The applications include finding a first proof, discovering single axioms, locating improved axiom systems, and simplifying existing proofs. The last application is directly pertinent to the recently found (by R. Thiele) Hilbert's twenty-fourth problem--which is extremely amenable to attack with the appropriate automated reasoning program--a problem concerned with proof simplification. The methodologies include those for seeking shorter proofs and for finding proofs that avoid unwanted lemmas or classes of term, a specific option for seeking proofs with smaller equational or formula complexity, and a different option to address the variable richness of a proof. The type of proof one obtains with the use of OTTER is Hilbert-style axiomatic, including details that permit one sometimes to gain new insights. We include questions still open and challenges that merit consideration.