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Missing Proofs Found Missing Pro ofs Found Branden Fitelson University of Wisconsin Department of Philosophy Madison WI and Mathematics and Computer Science Division Argonne National Lab oratory Argonne IL email telsonfacstawiscedu Larry Wos Mathematics and Computer Science Division Argonne National Lab oratory Argonne IL email wosmcsanlgov Abstract For close to a century despite the eorts of ne minds that include Hilb ert and Ackermann Lukasiewicz and Rose and Rosser various pro ofs of a number of signi cant theorems have remained missingat least not rep orted in the literatureamply demonstrating the depth of the corresp onding problems The types of such missing pro ofs are indeed diverse For one example a result may b e guaranteed provable b e cause of b eing valid and yet no pro of has b een found For a second example a theorem may have b een proved via metaargument but the desired axiomatic pro of based solely on the use of a given inference rule may have eluded the exp erts For a third example a theorem may have b een announced by a master but no pro of was supplied The nding of missing pro ofs of the cited types as well as of other types is the fo cus of this article The means to nding such pro ofs rests with heavy use of McCunes automated reasoning program OTTER reliance on a variety of p owerful strategies this program oers and employment of diverse metho dologies Here we present some of our successes and b ecause it may prove useful for circuit design and program synthesis as well as in the context of mathematics and logic detail our approach to nding missing pro ofs Welldened and unmet challenges are included This work was supp orted by the Mathematical Information and Computational Sciences Division subprogram of the Oce of Advanced Scientic Computing Research US Department of Energy under Contract WEng Keywords automated reasoning missing pro ofs termavoidance pro ofs Challenging Problems Unaided Researchers and Auto mated Reasoning Programs Known to many and esp ecially to researchers is the continual seeking by mathematicians and logicians of sound and rigorous pro ofs Although success is frequently the result some times the quarry eludes even great minds such as Hilb ert and Ackermann Tarski and Bernays Lukasiewicz and Rose and Rosser Indeed rather than the type of axiomatic pro of in fo cus in this article often such masters have recourse to equality to substitution and to other means In such cases one cannot say whether the desired axiomatic pro of was always in hand When that o ccurs a challenging problem is presenteda desired pro of is missing Less well known but still vital is access to sound pro ofs in the context of verication b oth of chips and circuits and of computer programs Indeed with the fo cus on chips free of bugs such rms as Intel and AMD currently here in the year are most concerned with pro ofs of various theorems The consideration of mo dels is often found lacking which explains why such rms are now seriously interested in theorem proving When an appropriate pro of is not in hand again a desired pro of is missing Pro ofs come in many forms constructive forward backward direct indirect by con tradiction rstorder higherorder and preferred by many among the other types ax iomatic In addition to their aesthetically pleasing prop erties axiomatic pro ofs oer various advantages not the least of which is their clarity We cannot say with certainty what was the primary app eal of such pro ofs for Hilb ert known to some as Mr Axiom However where automated reasoning is involved the very nature of axiomatic pro ofs makes them particularly attractive esp ecially when one or more designated inference rules are to b e used In this article we are not concerned with disproving some conjecture by nding an appropriate mo del such a dispro of can of course b e considered a pro of a pro of that a result do es not hold Therefore when the hunt for a missing pro of b egins in earnest and an automated reasoning program such as William McCunes OTTER McCune is part of the team the target is indeed an axiomatic pro of In contrast to the unaided researcher who has knowledge exp erience and intuition to draw up on the mere decision to seek an axiomatic pro of that is missing is in general of little use to a reasoning program the problem is to o broad Indeed in place of what a researcher brings to a problem the program must b e given some guidelines to direct its attack and the researcher must select the metho dology to b e employed Sucient for nding various pro ofs is OTTERs autonomous mode which removes much of the decision making from the researcher however here we are concerned with the challenge of nding pro ofs that have b een missing for decades and with sp ecic guidelines and diverse metho dologies that led to diverse successes The choice of the approach aimed at nding a missing pro of is often sharply inuenced by the type of missing pro of In other words the problem of seeking a missing pro of is replaced by a subproblem in part delineated by the type of missing pro of b eing sought In this article we fo cus on various classes of missing pro of given in Section and we discuss how some of the classes naturally suggest a feasible approach to take The most familiar class of missing pro ofs is encountered when studying an op en ques tion oered by some mathematician or logician Such was the case for more than six decades where the question concerned the p ossibility that every Robbins algebra is a Bo olean alge bra When that question was answered in the armative by McCune McCune using one of his reasoning programs EQP it marked a monumental success for automated rea soning For a second example we turn to Epsteins recent b o ok Epstein chapter where he oers two op en questions concerning the indep endence of two sets of axioms for classical prop ositional logic Both of these op en questions have b een answered using OTTER unpublished Indeed several other op en questions in Epsteins b o ok have b een partially resolved using OTTER as well To prove that no dep endency exists one can sup ply appropriate mo dels On the other hand to show that one of the axioms is dep endent on the remaining one can supply a pro of that completes with the deduction of the dep endent member OTTER serves well in either capacity As long as a question is op en a pro of is missing For additional op en questions to consider see Chapter of Wos In addition to the area fo cusing on op en questions pro ofs may b e missing for a variety of other reasons In the worst case the purp orted theorem may not in fact b e a theorem Of a sharply dierent nature and a case less highlighted than that for op en questions a pro of may exist based on metaargument but the more preferred axiomatic pro of remains missing one say based on the use of the inference rule condensed detachment The pro ofs we feature here all rely on that inference rule alone Then there is the case in which the pro of may b e incomplete indeed such is often the case with pro ofs supplied by even wellrespected researchers Or although various pro ofs exist each may b e considered unsatisfactory for one or more reasons For example one may desire a pro of that totally avoids the use of some type of term or that is shorter than all of those oered by the literature In this article we show how missing pro ofs of diverse types including those already cited can b eand indeed have b eenfound The key is reliance on McCunes automated reasoning program OTTER The pro ofs provided by OTTER have two imp ortant prop erties First the pro ofs are free of error In mathematics and in logic ideally no doubt must exist concerning the accuracy of a pro of indeed no question must remain concerning the assertion that a given result has in fact b een proved Fortunately OTTERs pro ofs are awless at least those many we have thoroughly pro of checked Second the pro ofs oered by OTTER can b e completely veried by using one of its features setbuild pro of ob ject followed by the use of Ivy written by McCune in the BoyerMoore logic In particular the pro ofs can b e appropriately detailed including history and termsubstitution for variables by using the option setbuild pro of ob ject This article shows that we have had considerable success in using OTTER to nd pro ofs that had either eluded the great minds of some of the masters in logic or at least had not b een rep orted by them in the literature In large part our success rests with the reliance on diverse and p owerful strategies this program oers and on the use of various metho dologies Although clearly not an algorithmalmost never for deep problems do es an algorithm existthe metho dologies we oer can easily b e applied by other researchers Clearly the nding of missing pro ofs is of substantial interest to mathematicians and logicians Less obvious is the fact that pro ofs are of substantial interest to other disciplines such as circuit design and computer programming Indeed an unexp ectedly elegant con structive pro of can p oint the way to the design of a far more ecient circuit or present computer co de far more eective than was already in hand For example a pro of that avoids the use of some type of term may provide the key to avoiding the use of an exp ensive comp onent or the key to sharply reducing the CPU time for a subroutine To set the stage we rst briey discuss some of the types of missing pro of OTTER nds in Section We then in Sections and provide much detail concerning strategy and metho dology give actual successes and fo cus on what we b elieve are startling
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