2
1. History, Signi cance, and Terminology Vanquishing the XCB Question: The Metho dological
Discovery of the Last Shortest Single Axiom for the
Equivalential Calculus
With the use of an e ective metho dology,we answer the nal op en
question concerning p ossible shortest single axioms for the classical
equivalential calculus. Sp eci cally, when detachment (that is, modus
Larry Wos
ponens) and substitution are the inference rules in use, is the formula
Mathematics and Computer Science Division, Argonne National Laboratory,
Argonne, IL 60439, U.S.A. E-mail: [email protected]
e(x; e(e(e(x; y );e(z; y));z)) (XCB)
Dolph Ulrich
a single axiom for that calculus, or is it to o weak? This question, p osed
Department of Philosophy, Purdue University, West Lafayette, IN 47907-1360,
byJ.Peterson [9], remained op en for a quarter of a century, eluding the
U.S.A. E-mail: [email protected]
e orts of various researchers. Here we tell the full story of the discovery
Branden Fitelson
of the main result rep orted in our companion article \XCB, The Last
Philosophy Department, San Jose State University, San Jose, CA 95192, U.S.A.
of the Shortest Single Axioms for the Classical Equivalential Calculus"
E-mail: branden@ telson.org
[16], where we also treat a simpler op en question p osed by K. Ho dgson
[2]: Is XCB a single axiom for classical equivalential calculus in the
Abstract. With the inclusion of an e ective metho dology, this article answers in
presence of substitution, detachment, and reverse detachment?
detail a question that, for a quarter of a century, remained op en despite intense
For a formula to b e a single axiom for a formal system, all theorems
study byvarious 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 that system must b e deducible from the formula. Wemake this
of detachment(modus ponens) and substitution? Where the function e represents
p erhaps obvious observation b ecause, rather than relying on the com-
equivalence, this calculus can b e axiomatized quite naturally with the formulas
bination of detachment and substitution, we study XCB here in the
e(x; x), e(e(x; y );e(y; x)), and e(e(x; y );e(e(y; z );e(x; z ))), which corresp ond to re-
presence of but one inference rule, condensed detachment (discussed
exivity, symmetry, and transitivity, resp ectively.(We note that e(x; x) is dep endent
more fully shortly), a rule that captures detachment coupled with a
on the other two axioms.) Heretofore, thirteen shortest single axioms for classical
equivalence of length eleven had b een discovered, and XCB was the only remaining
constrained form of substitution. To answer Peterson's question, we
formula of that length whose status was undetermined. ToshowthatXCB is indeed
study the sp eci c question of whether one can rely solely on condensed
such a single axiom, we fo cus on the rule of condensed detachment, a rule that
detachment to deduce from the formula XCB some known basis (axiom
captures detachment together with an appropriately general, but restricted, form of
system) for the equivalential calculus. If that could b e proved imp ossi-
substitution. The pro of we present in this pap er consists of twenty- ve applications
ble, then XCB would b e to o weaktoserve as a single axiom (see, for
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
example, [5]). If, on the other hand, such a deduction can b e found,
why XCB resisted relinqui shi ng its treasure for so long. Our approach relied on
then XCB must itself also b e such a basis, and the question op en for
diverse strategies applied by the automated reasoning program OTTER. Thus ends
two and one-half decades is answered in the armative.
the search for shortest single axioms for the equivalential calculus.
We do in fact answer Peterson's question armatively by presenting
a pro of (in Section 3) obtained with invaluable assistance from W. Mc-
Keywords: equivalence, equivalential calculus, single axioms, shortest single
axioms, detachment, condensed detachment, XCB, OTTER, automated reasoning
Cune's automated reasoning program OTTER [7]. Reliance on sucha
program naturally suggests an attack featuring condensed detachment
rather than detachment coupled with substitution. The former rule is
(as will b e shown) easily implemented in OTTER through the use of
hyperresolution, whereas the latter pair of rules is far less attractive
b ecause of the lack of e ective strategies for choosing from among
the myriad instances obtainable with substitution. In addition, pro ofs
The work of the rst and third authors was supp orted by the Mathematical,
based on condensed detachment are often more elegant. Indeed, when
Sciences Division subprogram of the Oce of Ad- Information, and Computational
one is faced with reading a formula (obtained, say,by substitution)
vanced Scienti c Computing Research, U.S. Department of Energy, under Contract
The pro of long, one nds the task daunting. of symbols thousands W-31-109-Eng-38.
c
Netherlands. the in Printed Publishers. Kluwer Academic 2002
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Vanquishing the XCB Question 3 4
the equivalential calculus instead in terms of the sole inference rule wegive inSection 3would, if presented in terms of detachmentand
condensed detachment. Brie y (see [5] for a detailed presentation), this substitution, include twosuchformulas.
rule takes as premisses twoformulas e(s; t)and r (assumed to share no The discovery of this pro of, deducing from XCB a known 2-basis
variables in common) and, when s and r are uni able, p ermits the consisting of formulas that corresp ond to symmetry and transitivity,
deduction of the formula obtained by applying to t the most general marks a victory b oth for logic and for automated reasoning [14]. Thus
substitution unifying s and r . Condensed detachment can b e easily im- ends the search for shortest single axioms for the equivalential calculus.
plemented in OTTER with the use of hyperresolution and the following Indeed, exactly fourteen such axioms exist, and no others can b e found.
clause as its nucleus, where \ j " denotes logical or and \ - " logical The use of metho dology and strategy (discussed in Section 2) shows
not. that automated reasoning played a vital role. Before we detail the
techniques crucial to our success, we brie y review the equivalential
-P(e(x,y)) | -P(x) | P(y).
calculus and discuss the relevantcontributions made by earlier logicians
to the study of shortest single axioms for this eld.
From the viewp oint of substitution and detachment, when attempt-
ing to apply condensed detachment, one rst seeks a most general
1.1. The Equivalential Calculus
substitution of terms for variables that (if such exists) yields, when
applied to r and s, a common term. One next applies that substitution
Formulas of the classical equivalential calculus are constructed from
(a most general uni er) to b oth r and e(s; t)toobtaintwo(so-to-
sentential variables and the two-place function symbol e (for \equiv-
sp eak) new hypotheses. Finally, to the new pair of hypotheses, one
alence"). The theorems of this logic are precisely the formulas
applies detachment. Clearly,anyformula obtainable from an axiom
in whicheachvariable o ccurs an even number of times|e(x; x),
set by condensed detachment alone can b e obtained from that set by
e(e(x; y ),e(y; x)), e(e(y; y), e(y; y)), and the like. Those theorems in
detachment together with substitution. Conversely, it can b e shown [5]
whicheachvariable present o ccurs exactly twice are said to have the
that every formula obtainable with detachment and substitution is an
2-prop erty.An interesting and useful fact is that every theorem of the
instance of at least one formula obtainable with condensed detachment
calculus either has the 2-prop erty or is a substitution instance of a
alone.
theorem with the 2-prop erty.For example, e(e(y; y);e(y; y)) do es not
have the 2-prop erty, but this formula is an instance of e(x; x), which
1.2. History
do es. Also known [1] (compare [4]) is the fact that, when condensed
detachment is successfully applied to twoformulas each of which has
In 1933, J. Luk asiewicz found the rst three eleven-symbol formulas
the 2-prop erty, the formula that results from that application always
that could eachserve as a single axiom for the equivalential calculus [6].
has the 2-prop erty.
Expressed as clauses, they are the following. (For ease of reference, we
As one might guess from its name, the equivalential calculus can
adopt the useful naming convention for formulas of this length devised
b e axiomatized (see, for example, [13]) with formulas corresp onding to
later by Kalman and presented in the app endixes to [9] and [2].)
re exivity, symmetry, and transitivity, expressed here as clauses.
P(e(e(x,y),e(e(z,y),e(x,z)))). %YCL
P(e(x,x)).
P(e(e(x,y),e(e(x,z),e(z,y)))). %YQF
P(e(e(x,y),e(e(y,z),e(x,z)))).
P(e(e(x,y),e(e(z,x),e(y,z)))). %YQJ
P(e(e(x,y),e(y,x))).
In the same pap er, Luk asiewicz shows that no shorter formula can serve
Unexp ectedly, p erhaps, re exivity is provably dep endent on the 2-basis
as a single axiom.
consisting of symmetry and transitivity.
Meredith later discovered the following seven additional shortest
In the early years, b efore C. A. Meredith entered the game, equiv-
single axioms [8].
alential calculus was studied exclusively in terms of two rules of
%UM P(e(e(e(x,y),z),e(y,e(z,x)))). p ermits the de- Detachment and substitution. inference, detachment
%XGF P(e(x,e(e(y,e(x,z)),e(z,y)))). duction of t from the twohyp otheses (premisses) e(s; t) and s. Thanks
%WN P(e(e(x,e(y,z)),e(z,e(x,y)))). to study practice standard [8], however, it has b ecome to Meredith
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Vanquishing the XCB Question 5 6
For the rst of the two, Winker's pro of has length 159 (applications of P(e(e(x,y),e(z,e(e(y,z),x)))). %YRM
condensed detachment), for the second, length 83. (Wenowhavefar P(e(e(x,y),e(z,e(e(z,y),x)))). %YRO
shorter pro ofs, of lengths 19 and 21, resp ectively.) P(e(e(e(x,e(y,z)),z),e(y,x))). %PYO
One formula remained unclassi ed, as it had since 1977: XCB. With P(e(e(e(x,e(y,z)),y),e(z,x))). %PYM
the goal of proving that this recalcitrant formula is in fact a single
In the mid-1970s, J. Kalman and his studentPeterson underto ok a
axiom, we soughttoshow that one could deduce from it either the
computer-assisted investigation of all 630 eleven-symbol equivalential
3-basis cited earlier or one of the thirteen previously known shortest
theses (distinct up to alphab etical variance). Kalman found another
single axioms. (Indeed, b ecause of the dep endence of re exivity, the
shortest single axiom among these [3], the eleventh. (Wenotethat
indep endent 2-basis would haveserved well as a target; but, as a matter
Kalman himself graciously regards this eleventh axiom as simply
of historical fact, the 3-basis was one of the targets.)
correcting a misprint in [8].)
Precisely how the mystery was solved to reveal XCB as the four-
teenth and nal shortest single axiom is the fo cus of Section 2. There
P(e(x,e(e(y,e(z,x)),e(z,y)))). %XGK
we detail the metho dology and strategy we used to obtain the original
71-step pro of. In Section 3, we discuss pro of re nement and howwe
Peterson showed that 612 of the eleven-symbol theses were to o weak [9],
eventually obtained from the 71-step pro of a far shorter pro of, one of
and p osed as op en questions the status of the remaining seven formulas
length 25. There we present that 25-step pro of, the shortest weknow
of length eleven.
of. The pro of has b een indep endently veri ed with two other programs.
P(e(x,e(y,e(e(e(z,y),x),z)))). %XJL
In Section 4, we consider why XCB remained unclassi ed for so many
P(e(x,e(y,e(e(x,e(z,y)),z)))). %XKE
years and present additional observations on some unusual asp ects of
P(e(x,e(e(e(e(y,z),x),z),y))). %XAK
XCB. Section 5 provides a summary.
P(e(e(e(e(x,e(y,z)),z),y),x)). %BXO
P(e(x,e(e(y,z),e(e(x,z),y)))). %XHK
P(e(x,e(e(y,z),e(e(z,x),y)))). %XHN
2. An E ective Metho dology
P(e(x,e(e(e(x,y),e(z,y)),z))). %XCB
Here wepresent the highlights of our successful attackon the XCB
Kalman's place in the history of equivalential calculus, and certainly
question. In fact, a set of pro ofs showing that XCB is a single axiom
in the history of this pap er, is by no means limited to the preceding
for the equivalential calculus was discovered, many of them complet-
observations. He was apparently the rst to introduce automated rea-
ing with the deduction of one of the previously known shortest single
soning and the equivalential calculus to each other, writing his own
axioms. The shortest pro of we found (given in Section 3) uses 25 appli-
theorem-proving program to study p ossible shortest single axioms early
cations of condensed detachmenttoreach, rather than a single axiom,
on and using it to discover his pro of for the eleventh shortest single
the indep endent 2-basis consisting of e(e(x; y );e(e(y; z);e(x; z ))) and
axiom. Later, having learned of the Argonne e ort fo cusing on the au-
e(e(x; y );e(y; x)). Additional details concerning our approach, includ-
tomation of reasoning, he visited Argonne (in the late 1970s, if memory
ing some input les, summaries of output les, and useful commentary,
serves). He brought a ne gift, the op en questions concerning the status
are available on the Web page www.mcs.anl.gov/wos/XCB/. Perhaps
of those seven unclassi ed formulas of length eleven.
one mighthave preferred to b e shown a fully automated approach, but
Not long after Kalman's visit, the Argonne group b egan an intense
in fact no e ective algorithm for answering such deep questions has yet
study of the seven unclassi ed formulas, conquering six of them. It was
b een found.
proved that the rst four of those formulas (XJL, XKE, XAK,and
In essence, the original attackinvolved three phases: the deduction
BX O) are to o weak [12]. Of the remaining three formulas, S. Winker
of re exivity, the deduction of transitivity presuming the availabilityof
proved (as rep orted in [12]) that XHN and XHK are in fact new
symmetry, and the deduction of symmetry itself. Of these three goals,
shortest single axioms, the twelfth and thirteenth.
the the third proved by far the most dicult to reach. Historically,
rst and second goals were reached more than one year ago. The recent % XHN P(e(x,e(e(y,z),e(e(z,x),y)))).
a in the end) to obtain (successful to an attempt e ort was devoted % XHK P(e(x,e(e(y,z),e(e(x,z),y)))).
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Vanquishing the XCB Question 7 8
application. In particular, when the program is ready to choose a new pro of of symmetry from the formula XCB alone. The interim to ok the
clause on which to fo cus, it prefers (if available) one that matches a form of a long pause that witnessed essentially no researchdevoted to
hint. We selected the option of choosing a matching clause or one that this intriguing formula. Throughout, wewere continually aware that
subsumes a hint. We note that hints are used only to guide the pro- re exivity is easily proved to b e dep endent on symmetry together
gram's reasoning; they are not used as hypotheses in the application of with transitivity. Nevertheless, the 3-basis was originally one of our
inference rules. Rather, eachis viewed as an attractive symbol pattern targets for pro of completion. Also included as targets were the thirteen
to b e matched, if p ossible. previously known shortest single axioms for the equivalential calculus.
The hints that were used grew in number, each corresp onding to a The attack relied on a number of strategies and features o ered by
pro of step of a target lemma proved in an earlier run. That is, as the OTTER. Among these, the rst chosen was lemma adjunction,which
size of the initial set of supp ort grew from run to run and exp eriment involves adjoining to XCB, in successive runs, the nal steps of various
to exp eriment, so did the size of the list of hints. Toeach hint, the intermediate targets proved in earlier runs or even the entire pro ofs
researcher can assign a value; the smaller the value, the higher the pri- of such targets. The so-called lemmas that are adjoined are placed
ority for b eing chosen as the fo cus of attention. By way of illustration, in the initial set of supp ort, and the program is instructed to fo cus
an included hint that corresp onds to a formula of length 47 (measured on each to initiate applications of the inference rule or rules in use
in symbols) can in e ect b e treated as having length 1. Indeed, various b efore fo cusing on a newly deduced and retained conclusion. These
47-symbol formulas o ccur in pro ofs of intermediate targets; among such lemmas, rather than necessarily app earing as steps in the nal sought-
formulas are the two found in our 25-step pro of. When the program is after pro of, are intended to direct the program toward imp ortant steps
instructed to rely on complexity preference rather than on breadth rst, that will app ear in the nal pro of. (The set of supp ort strategy,forward
the highest priority (for directing its reasoning) is given to formulas subsumption, and usually back subsumption are almost always featured
of least complexity. The program computes complexity based on the in our research with OTTER.) For example, the inclusion initially of
assigned values to included resonators and hints, otherwise based on the eleven steps of our original pro of of re exivity aslemmaswas not
symbol count. coupled with the exp ectation that anyorall ofthose stepswould app ear
A second direction strategy was also employed, McCune's ratio strat- in the nal pro of, if suchwere found. In fact, as it turned out, its most
egy [14]. That strategy blends the choosing of fo cal clauses based on complex step (of length 47, not counting the predicate symbol) do es not
complexity (which, as indicated, can b e in uenced by included hints) app ear in the pro of we o er in Section 3, nor do es it playmuch of a role
and rst-come rst-serve. If, for example, the value assigned to the throughout. In contrast, the pro of in Section 3 relies on two 47-symbol
pick formulas, b oth of which o ccurred in a 52-step pro of of e(x; e(y; e(x; y ))), given ratio is 2, then the program chooses two clauses by com-
completed in a breadth- rst search, a pro of obtained in approximately plexity, one by rst-come rst-serve, two, one, and so on. Inclusion
78 CPU-hours. The twoformulas are, resp ectively, the 38th and 41st of the ratio strategy with a low assigned value to the pick given ratio
deduced steps of the 52-step pro of. p ermits the program to regularly fo cus on conclusions retained early,
In the search for a pro of of symmetry from XCB,wechose as inter- some of whichmaybevery long and would otherwise b e delayed in the
mediate targets all fteen of the 7-symbol theorems of the equivalential context of inference rule initiation, p erhaps forever.
calculus with the 2-prop erty. (Hereafter, we shall suppress the phrase weight parameter o ered byOT- To restrict the reasoning, the max
\with the 2-prop erty".) This choice was motivatedinpartby the fact TER proves most useful. New conclusions whose complexity (that is,
that symmetry is such a theorem and in part b ecause short formulas their weight, as determined by assigned values to resonators or hints
are, in our exp erience, often easier to prove than longer ones. Of course, weightare or by symbol count) exceeds the value assigned to max
if all fteen were proved, then symmetry (b eing sucha7-symb ol the- can restrict the program's reasoning discarded. A small assigned value
orem) would b e proved, and the centerpiece of our work would b e in so severely that all pro ofs are blo cked, while to o large an assigned value
hand. can drown the program in newly deduced and retained information.
For directing the program's reasoning, R. Vero 's hints strategy One additional restriction strategy was employed, at least in the
[11] played the prominent role. To use that strategy, the researcher early stages. The strategy is a version of term avoidance (sometimes
chooses one or more formulas or equations that the program treats strategy). In early runs, we instructed the referred to as a subtautology
rule inference than other such items for initiating as more imp ortant that conclusion deduced any newly immediately to discard program
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Vanquishing the XCB Question 9 10
rst search. For the other, wechose McCune's ratio strategy, with an contains as a prop er subformula a formula of the form e(t; t) for some
assigned value of 2 to the pick term t. This action was taken b ecause we feared that, otherwise, the given ratio. In b oth approaches, the
space of conclusions to b e explored would grow far to o rapidly and value of 48 was assigned to the max weight. In b oth, we also included
prevent the program from reaching the goal of a pro of of symmetry. The nineteen hints corresp onding to the steps of a pro of we had in hand that
danger of its inclusion (whichwas in fact eventually exp erienced) is that the single axiom XHN implies the single axiom UM. This action was
key information on the path to a needed conclusion may b e discarded, taken b ecause we had abundant evidence of the diculty of attempting
preventing the program from completing an assignment. Later in the to answer the op en question concerning the axiomatic status of XCB
study, on the path that pro duced a pro of of symmetry and more, the and b ecause settling the question for XHN had initially proved so
useoftermavoidance was abandoned. dicult.
Finally, at the outermost level of directing the program's reasoning, Both approaches b egan by employing the version of the term-
we considered the choice b etween instructing the program to conduct avoidance strategy mentioned earlier. Although the program could have
its searchby complexity preference (within the context of the ratio b een instructed to rely on the weighting strategy, instead weused de-
strategy) and by breadth rst (that is, level saturation, in which the mo dulation to rewrite unwanted new conclusions to junk. As noted, we
ratio strategy has no function). Although b oth choices may require were motivated by exp erience gleaned from much exp erimentation that
consideration of a space of conclusions that can grow exp onentially,far teaches one that the space of deducible conclusions is sharply reduced
more e ective metho dologies exist for coping with this p ossible growth with this strategy.
with the rst choice than with the second. Nevertheless, the use of We placed in the passive list the negations of the intermediate tar-
breadth rst (level saturation) eventually did provide key results for gets, namely, the full set of fteen 7-symbol theorems. Also included
b oth the lemma-adjunction phase and for the growing hints list. We were the negations of the known thirteen shortest single axioms, as
simultaneously and, as it turned out, pro tably employed b oth of these well as the negation of transitivity.Memb ers of the passive list are
global strategies. used mainly to signal pro of completion (by unit con ict) and for sub-
Consistent with the plan of targeting the fourteen cited bases, the sumption. In general, we also use the passive list to monitor progress,
program commenced its attack. In the spirit of lemma adjunction, we viewing the pro of of any of its members as a go o d sign even if wedo
b egan (as noted) by relying on the year-old 11-step pro of of re ex- not intend to use the result. In this study, of course, the intention was
ivity from XCB, using its steps b oth as lemmas and as hints. (The to add as lemmas in a later run the deduced steps of all pro ofs of those
single 47-symbol formula found among these eleven steps motivated intermediate 7-symbol targets.
us to instruct the program throughout our attack to retain formulas In the input usable list (whose members never initiate inference rule
of this complexity. The inclusion of the cited eleven lemmas enabled application), we placed the clause that captures (with hyperresolution)
the program to prob e far deep er levels than it would have b een able condensed detachment.
to otherwise, providing us with additional pro of steps to b e used in
-P(e(x,y)) | -P(x) | P(y).
later runs b oth as hints and as lemmas.) We instructed the program
to treat any formula that was identical to one of the eleven steps or
As noted, the key 7-symbol formula is symmetry, b ecause our prior
that subsumed one of them as b eing of length 1. Clearly,wewere
studies had shown that, if we had in hand a pro of of symmetry deduced
instructing the program to direct its reasoning with much preference
from XCB alone, we could complete a pro of that XCB is indeed a
for newly deduced clauses that matched or subsumed a hint. (The hints
shortest single axiom. We of course had no way of knowing whether
strategy is generally preferred over the resonance strategy for studies
the nal pro of, if obtained, would include any other proved 7-symbol
in equivalential calculus b ecause to o many conclusions matcha given
formulas or their pro of steps. Their adjunction was intended merely to
resonator.)
aid the search in general, such adjunction of lemmas having proved to
OTTER's arsenal encourages attacks that rest on multiple ap-
be a powerful metho dology in manycontexts in the past.
proaches applied in separate but simultaneous runs. Wechose a
Another of the 7-symbol formulas, expressed in clausal form as
attack. The sole di erence b etween the two approaches two-pronged
had b een proved with some diculty in an earlier P(e(e(e(x,y),x),y)),
that were chosen involved the means used to direct the program at the
exp eriment with a di erent program. We strongly susp ect that the
wechose breadth- For one approach, level of its reasoning. outermost
all to include the key role in our decision of that pro of played discovery
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Vanquishing the XCB Question 11 12
7-symbol theorems with the two-propertyas intermediate targets for In that run, nally, a proofofsymmetrywas completed. An 18-step
eventual use in lemma adjunction. The exp eriment in question would pro of was found in approximately 45 CPU-minutes, a pro of relying
prove almost immediately to b e valuable in evaluating the two-pronged on nine of the pro of-step lemmas that had b een adjoined during the
approach, since it had not b ene ted from reliance on the strategies attack. Deciding at this p oint not to rely on much earlier derivations,
presented earlier. Indeed, wewere encouraged to continue pursuing we next sought a pro of of transitivity (where the target was the familiar
b oth approaches b ecause each obtained a pro of of that clause. The 3-basis) or a pro of of one of the known thirteen shortest single axioms.
approach relying on the ratio strategy completed its pro of in less than Of course, our intention was to have OTTER prove directly from XCB
one CPU-minute; the breadth- rst approach required approximately alone a known basis without reliance on any lemmas.
eleven CPU-minutes. (In addition to any p ossible contributions from Wechose another two-pronged approach. On one branch, we relied
the 11-step pro of of re exivity, the length of the rst pro of is 13 on the far, far earlier set of hints that corresp onded to a pro of of tran-
and of the second is 17. Since wewere not seeking shorter pro ofs at sitivity from symmetry. On the other branch, we ignored such earlier
that p oint, we note that a shorter pro of may exist with XCB as the discoveries. The second branch o ered more app eal in that it corre-
sole hypothesis.) Our success in proving the apparently dicult clause sp onded more closely to a type of attackwe enjoy.For that approach,
P(e(e(e(x,y),x),y)) o ered a promise that would indeed b e ful lled. we relied on the hints used in the preceding run together with thirty
The approach based on the use of the ratio strategy yielded no hints corresp onding to the join of the new pro ofs obtained in that run,
additional pro ofs and was temp orarily discontinued. The breadth- rst among whichwas symmetry. The only hypothesis that was used was
approach, however, was p ermitted to continue its search (which, as XCB; the pick given ratio was assigned the value 2; the max weight
will b e seen shortly,was most fortunate). While waiting for more was assigned the value 64, in case longer formulas might b e useful; no
pro ofs from the breadth- rst search|if such could b e found|the ratio- term avoidance was employed. McCune's program succeeded in nding
strategy approachwas resumed, but with the addition (as lemmas) of (in approximately 7 CPU-seconds) a 61-step pro of of symmetry and
the thirteen deduced steps of the pro of just cited; our ob jectivewas to (in approximately 15 CPU-minutes) a 71-step pro of of transitivity.As
seek pro ofs of additional targets. This e ort failed. exp ected, the latter do es in fact dep end on symmetry, its 66th step.
An analysis suggested that, just p erhaps, the use of the term- Our attackhad vanquished XCB.
avoidance strategy might b e blo cking progress. Therefore, we ceased As for the rst of the two approaches, it also succeeded and nished
using it but otherwise continued as in the preceding case. The use of even earlier. In approximately 4 CPU-seconds, a 66-step pro of of tran-
term avoidance was indeed the problem, at least temp orarily. Without sitivitywas found, and in roughly 1 additional CPU-second symmetry
it, four additional theorems of the fteen were proved, making a total was proved in 64 applications of condensed detachment. The former,
of six proved (b ecause one of the fteen is an instance of re exivity), contrary to exp ectation, clearly do es not dep end on the latter. In other
with nine yet to prove. Of course, symmetry was still the only one of words, aided byhints corresp onding to results from more than one year
them crucial to the attack, the pro ofs of the others b eing of interest earlier showing that the use of symmetry can lead to a pro of of transitiv-
primarily for lemma adjunction and for supplying additional hints. ity, a pro of of the latter indep endent of symmetry was found. Whereas
At this p oint, the decision to allow the breadth- rst branchto the approach that relied up on hints proving transitivity from symmetry
continue broughtriches. Sp eci cally, the rst of the following twofor- proved ve of the previously known shortest single axioms, the other
mulas was proved in approximately 45 CPU-hours and the second in approachproved in approximately 45 CPU-minutes all thirteen.
approximately 78 CPU-hours. Neither approach pro duced a complete pro of of the 3-basis as a
single pro of, although eachmemb er of that basis was proved in each
approach. The type of pro of we preferred would nd a contradiction
P(e(e(x,e(y,x)),y)).
with the denial of the conjunction of the three members, thus providing
P(e(x,e(y,e(x,y)))).
a pro of with no duplicate steps. Our failure to nd such a pro of is
explained by our failure to include a clause corresp onding to the denial
Completion of pro ofs for these twoformulas signaled progress. Of far
of the members of the 3-basis. We simply reached of the conjunction
the steps in those pro ofs turned out to playa greater imp ortance,
the pro of of the three members so oner than we had exp ected. Based
crucial role. The join of the two pro ofs (of resp ective lengths 33 and
the type of pro of wehave in mind would approach, on our preferred
for the next run. to use as lemmas formulas 71 additional 52) provided
xcb-jar.tex; 8/10/2002; 10:12; p.11 xcb-jar.tex; 8/10/2002; 10:12; p.12
Vanquishing the XCB Question 13 14
found was, in general, used as a new target with its steps included b e at least length 72 b ecause re exivity had not b een proved in the
as hints. Throughout our attack, wecontinued to rely on ancestor 71-step pro of, though symmetry had b een proved. (Had the denial of
subsumption. In addition, we employed a technique called demodulation the 3-basis b een included in the input, say, in the second approach, an
blocking to blo ck the use of the steps of a given pro of one at a time, exp eriment shows that indeed a 72-step pro of would have b een found.)
with the ob jective of nding an abridgment. This technique has proved Therefore, with the goal of pro ducing a pro of of the 3-basis as a
e ectivein manycontexts, whether the fo cus is on formulas or on single pro of rather than as three separate pro ofs (and with no duplicate
equations, and in the presence of various inference rules. steps), and with the additional goal of b eginning a re nement study
When wereachedapoint at which neither ancestor subsumption fo cusing on pro of length, we turned to yet another run. We simply to ok
nor demo dulation blo cking yielded an abridgment, we turned to the the input le for the second of the two approaches just describ ed, added
cramming strategy [15]. The ob ject of (one incarnation of ) the cram- the denial of the 3-basis to the usable list, and added the command
ming strategy is to cram, or force, so manystepsofachosen subpro of set(ancestor subsume). (For those who are curious, the inclusion of this
into new subpro ofs of the remaining members of a conjunction that command ordinarily has a dramatic e ect on CPU time, slowing the
the length of the new pro of of the whole is strictly less than that which program sharply in many cases.) That command instructs OTTER
prompted the e ort. Intuitively, the ob ject is to have the program fo cus to compare derivation lengths with the same conclusion and prefer the
on a subpro of and have its steps do double duty, triple duty,ormore. shorter; it automatically seeks shorter pro ofs. (We emphasize, however,
For the target of the 3-basis, cramming was successfully used in two that shorter subpro ofs do not necessarily a shorter total pro of make.)
cases: to cram on the subpro of of transitivity and later (with a shorter In fact, ve pro ofs of the desired type were found of (in order)
pro of of the 3-basis in hand) to cram on the subpro of of symmetry. length 49, 46, 48, 47, and 42. The rst was completed in approximately
During the re nement phase, OTTER eventually discovered a 26- 24 CPU-seconds and the fth in approximately 15 CPU-minutes. Re-
step pro of of the indep endent 2-basis. OTTER also discovered a 27-step markable to us and even startling, in approximately 14 CPU-hours the
pro of of the previously known single axiom YQF and one of that length previously known thirteen shortest single axioms were eachproved in
for the previously known single axiom WN. With some thought, we less than 50 applications of condensed detachment.
were then able to shorten each of the three pro ofs by one step. We next
observed that, with an appropriate use of condensed detachment, the
25-step pro of of the 2-basis could b e extended to a 26-step pro of of the
3. Pro of and Re nement
3-basis (including re exivity).
Wenow present our 25-step pro of of transitivity and then symmetry
Had Lukasiewicz, Meredith, or Prior, for example, b een presented with
from XCB.To aid one in reading OTTER's pro ofs, we included the
the set of pro ofs that included the 71-step pro of of transitivity,or
command set(order history), a command that instructs the program
b etter yet, the 42-step pro of of the join that was found with ances-
to list the hypotheses (bynumber) of a deduced step in the order
tor subsumption, he would (we conjecture with virtual certainty) have
i; j; k , where i is the clause for condensed detachment, j the clause
embarked on a search for an abridgment, a shorter pro of. The discovery
corresp onding to the ma jor premiss, and k the clause corresp onding to
of shorter and simpler pro ofs was clearly also of interest to Hilb ert and
the minor premiss.
was the sub ject of his recently discovered twenty-fourth problem [10].
Long b efore learning of the Hilb ert problem, much researchbymemb ers
A 25-Step Pro of from XCB
of the Argonne group had b een devoted to developing metho dology for
The command was "otter". The processID is 24362. OTTER to apply in the context of pro of re nement.
Naturally, therefore, next in order was the pursuit of a pro of of
-----> EMPTY CLAUSE at 0.15 sec ----> length strictly less than 42 showing that XCB is in fact a single ax-
134 [hyper,52,132,128] $ANSWER(all_s_t_indep). iom for the equivalential calculus. Although no constraintwas placed
on the target|for example, anymemb er of the thirteen previously
Length of proof is 25. Level of proof is 19. known shortest single axioms would have b een more than acceptable|
we mainly continued to fo cus, p erhaps for reasons of momentum, on the
------PROOF ------pro of e. Each new and shorter hwas again iterativ Our approac 3-basis.
xcb-jar.tex; 8/10/2002; 10:12; p.14 xcb-jar.tex; 8/10/2002; 10:12; p.13
Vanquishing the XCB Question 15 16
z)),e(e(e(u,v),e(w,v)),w)),u)).
51 [] -P(e(x,y)) | -P(x) | P(y). 123 [hyper,51,106,122] P(e(e(e(e(x,y),e(z,y)),z),x)).
52 [] -P(e(e(a,b),e(b,a))) | -P(e(e(a,b),e(e(b,c),e(a,c)))) | 124 [hyper,51,53,123] P(e(e(e(e(e(e(e(x,y),e(z,y)),
ANSWER(all_s_t_indep). z),x),u),e(v,u)),v)).
53 [] P(e(x,e(e(e(x,y),e(z,y)),z))). 125 [hyper,51,124,123] P(e(e(e(x,y),x),y)).
105 [hyper,51,53,53] P(e(e(e(e(x,e(e(e(x,y),e(z,y)),z)), 127 [hyper,51,124,108] P(e(e(e(e(x,e(e(e(x,y),
u),e(v,u)),v)). e(z,y)),z)),e(e(e(e(e(u,v),e(w,v)),w),u),
106 [hyper,51,105,53] P(e(e(e(e(x,e(e(e(x,y),e(z,y)),z)), v6)),v7),e(v6,v7))).
u),v),e(u,v))). 128 [hyper,51,127,123] P(e(e(x,y),e(e(y,z),e(x,z)))).
107 [hyper,51,53,106] P(e(e(e(e(e(e(e(x,e(e(e(x,y),e(z,y)), 130 [hyper,51,128,125] P(e(e(x,y),e(e(e(z,x),z),y))).
z)),u),v),e(u,v)),w),e(v6,w)),v6)). 131 [hyper,51,128,130] P(e(e(e(e(e(x,y),x),z),u),
108 [hyper,51,106,53] P(e(x,e(e(e(e(e(y,e(e(e(y,z),e(u,z)), e(e(y,z),u))).
u)),x),v),e(w,v)),w))). 132 [hyper,51,131,123] P(e(e(x,y),e(y,x))).
109 [hyper,51,107,53] P(e(e(e(e(e(e(e(x,e(e(e(x,y),e(z,y)),
Op en Question. Do es there exist a pro of of length 24 or less show-
z)),u),v),e(u,v)),w),v6),e(w,v6))).
ing that XCB is a single axiom, where the target is any of the other
110 [hyper,51,106,108] P(e(x,e(e(e(e(e(y,e(e(e(y,z),e(u,z)),
thirteen shortest single axioms for the equivalential calculus or is the
u)),e(e(v,e(e(e(v,w),e(v6,w)),v6)),x)),v7),e(v8,v7)),v8))).
2-basis consisting of symmetry and transitivity?
111 [hyper,51,53,108] P(e(e(e(e(x,e(e(e(e(e(y,e(e(e(y,z),
e(u,z)),u)),x),v),e(w,v)),w)),v6),e(v7,v6)),v7)).
With this pro of in hand|but twenty- ve steps in length|one might
112 [hyper,51,105,110] P(e(e(e(e(x,e(e(e(x,y),e(z,y)),z)),
naturally wonder why it to ok so long to answer the XCB question. One
e(e(u,e(e(e(u,v),e(w,v)),w)),e(e(v6,e(e(e(v6,v7),
answer rests in part with the fact that the iterative approachgiven in
e(v8,v7)),v8)),v9))),v10),e(v9,v10))).
Section 2 was only recently formulated. Perhaps a b etter answer rests
113 [hyper,51,109,111] P(e(e(x,e(y,e(e(e(e(e(z,e(e(e(z,u),
with the b ehavior and, more imp ortant, the apparent b ehavior of XCB,
e(v,u)),v)),e(e(w,e(e(e(w,v6),e(v7,v6)),v7)),y)),v8),
which is the fo cus of next section.
e(v9,v8)),v9))),x)).
114 [hyper,51,107,112] P(e(e(e(e(e(x,e(e(e(x,y),e(z,y)),
z)),e(u,e(e(e(u,v),e(w,v)),w))),v6),v7),e(v6,v7))).
4. Close Insp ection of XCB
115 [hyper,51,53,113] P(e(e(e(e(e(x,e(y,e(e(e(e(e(z,
e(e(e(z,u),e(v,u)),v)),e(e(w,e(e(e(w,v6),e(v7,v6)),
Insight, understanding, and ideas can sometimes b e gained by asking
v7)),y)),v8),e(v9,v8)),v9))),x),v10),e(v11,v10)),v11)).
why a question remained op en for manyyears. In this section, we
116 [hyper,51,114,106] P(e(x,e(e(y,e(e(e(y,z),e(u,z)),
consider various factors that may explain whyintense e ort and study
u)),x))).
failed to reveal the true status of XCB.We also discuss some of the
117 [hyper,51,53,116] P(e(e(e(e(x,e(e(y,e(e(e(y,z),
behavior and p ower of XCB, p ossibly providing insight that will aid
e(u,z)),u)),x)),v),e(w,v)),w)).
in answering additional questions concerning axiomatizations of other
118 [hyper,51,112,117] P(e(e(e(e(e(x,
formal systems.
e(e(y,e(e(e(y,z),e(u,z)),u)),x)),v),w),e(v,w)),
e(v6,e(e(e(v6,v7),e(v8,v7)),v8)))).
4.1. The Resistance of the XCB Question
), 119 [hyper,51,112,118] P(e(e(e(e(e(e(x,e(e(y,e(e(e(y,z
e(u,z)),u)),x)),e(v,e(e(e(v,w),e(v6,w)),v6))),v7),
Our study of XCB suggests several p ossible factors that mayhave
). v8),e(v7,v8)),e(v9,e(e(e(v9,v10),e(v11,v10)),v11)))
contributed to its resistance to classi cation. One such factor concerns
P(e(e(e(x,e(y,e(e(e(y,z), 120 [hyper,51,115,119]
a prop erty that one might easily guess (though incorrectly, as it turns
e(u,z)),u))),v),e(x,v))).
out) is shared by all of the theorems deducible from XCB. Indeed,
P(e(e(e(x,e(e(e(x,y),e(z,y)), 122 [hyper,51,120,105]
using our usual from XCB deducible that are readily the theorems
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Vanquishing the XCB Question 17 18
strategies contain|as do the rst seventeen steps of the pro of presented e(v9,e(e(e(v9,v10),e(v11,v10)),v11)))). %76
in Section 3|at least one alphab etical variantofXCB as a subformula.
One can easily imagine howdaunting would b e the prosp ect of
Therefore, it is not surprising that those researchers credited with the
applying by hand condensed detachment to this pair of formulas. As
dispatching of XJL and its three equally weak cousins asserted that
measured in symbol count, the most general common instance of the
XCB was also to o weak, that all theorems deducible from it must
antecedent of clause 72 and of all of clause 76 has complexity2919.
contain a variantofXCB. (That claim, made in [12], was corrected in
The rst completed (42-step) pro of of the 3-basis contained these
[14].) In fact, as some of our later exp eriments using breadth- rst search
two formulas and one other of the same complexity (47). Weknow
show, all of the 1,494 theorems obtainable from XCB through the rst
of no pro of that is free of 47-symbol formulas, although wedo have
ve levels contain such an alphab etic variant. Moreover, with the single
in hand a pro of with but one 47-symbol formula, di erent from the
exception of the formula e(e(x; y ),e(x; y ))|which of course refutes the
two just displayed. By comparison, a pro of for XHN |the last of the
conjecture|so do the additional 319,493 theorems obtainable at level
previously known single axioms to b e found|can b e completed with
six.
complexity limited to 35.
Had the conjecture held, the theorem e(x; x)would have b een out of
reach (not deducible) with XCB as hyp othesis and condensed detach-
ment as the sole rule of inference. Instead, a simple exp eriment relying
4.2. The Behavior of XCB
on a breadth- rst search and consideration of formulas of complexity
less than or equal to 35 (not counting the predicate symbol) shows that
The b ehavior of the formula XCB is dramatically di erent from that
re exivity is in fact provable from XCB with but eleven applications of
of the other shortest single axioms in a number of ways. For one dif-
condensed detachment, a pro of obviously di erent from the one found
ference, with XCB as the sole hypothesis, the range of the lengths of
more than a year ago (whichcontained a 47-symbol formula). Further,
the pro ofs of the remaining thirteen axioms is rather small, from 26 to
a limit on complexityof31even suces, pro ducing a 17-step pro of.
30 steps. Our exp eriences with other areas of logic, with other areas
This simple formula eluded capture, however, in part b ecause (some
of mathematics, and (most imp ortant) with the other shortest single
of ) the authors and their colleagues overlo oked for far to o long the
axioms had never b efore revealed such clustering of pro of lengths. The
p ossible value of such a search and in part b ecause of their conjecture
fourteen single axioms can thus b e arranged so that XCB is at the
that re exivitywas not provable. Among the truths ab out research
center of a ring with the other thirteen shortest single axioms roughly
are: The knowledge that a result holds seems to make its pro of easier
equidistant from it. In contrast, the shortest path lengths wehave found
to complete; the b elief that it do es not disp oses one to follow the wrong
from the single axiom UM to the other shortest single axioms vary
path if that b elief is mistaken.
sharply.Inparticular,XGF follows in a single application of condensed
The discovery of a pro of of re exivitywas encouraging. But, even af-
detachment, while, at the other end of the sp ectrum, XCB requires
ter this discovery, XCB continued to resist classi cation. Other factors
twenty-three.
were contributing to its resistance. For an example of one such factor,
To test whether this is the case for other shortest single axioms,
recall that condensed detachment pro ceeds by unifying the antecedent
we fo cused on XHN .We used as hints the 20 steps of our pro of that
(leftmost ma jor argument) of the ma jor premiss with the minor pre-
UM can b e deduced from XHN . Simultaneously,withXCB as the
miss. Common to the ma jorityof proofs wehave discovered is the
hypothesis, we used as hints the 26 steps of our pro of of the 3-basis.
application of condensed detachment to the following twoformulas
Each exp eriment ran for more than twenty-three CPU-hours on a 400
(expressed as clauses) of complexity 47, with the rst as ma jor and
MHz computer. When XCB was the sole hypothesis, the pro ofs of the
the second as minor premiss.
other thirteen shortest single axioms ranged in length (as noted) from
26 to 30. In contrast, when XHN was the sole hypothesis, the pro of
P(e(e(e(e(e(x,e(y,e(e(e(e(e(z,e(e(e(z,u),e(v,u)),v)), lengths of the other thirteen single axioms ranged from 19 to 37. In
e(e(w,e(e(e(w,v6),e(v7,v6)),v7)),y)),v8),e(v9,v8)), the two exp eriments under discussion, ancestor subsumption was used
)). %72 v9))),x),v10),e(v11,v10)),v11 with the goal of nding \short" pro ofs. We remark that the length of
)), P(e(e(e(e(e(e(x,e(e(y,e(e(e(y,z),e(u,z)),u)),x a pro of in no way suggests how deep is the theorem that is proved nor
e(v,e(e(e(v,w),e(v6,w)),v6))),v7),v8),e(v7,v8)), of diculty indication a b etter howhardit isto nd aproof.Perhaps
xcb-jar.tex; 8/10/2002; 10:12; p.17 xcb-jar.tex; 8/10/2002; 10:12; p.18
Vanquishing the XCB Question 19 20
3. Kalman, J. A., \A shortest single axiom for the classical equivalential is o ered by the number of years a question remains op en: less than
calculus", Notre Dame J. Formal Logic 19 (1978) 141{144. four years suced for dispatching XHN ,twenty- veyears for XCB.
For XCB, the pro ofs wehave found so far require twelve distinct
4. Kalman, J. A., \The two-property and condensed detachment",
variables, whereas nine suce for seeking pro ofs relying on XHN .
Studia Logica 41 (1982), 173{179.
Whether a pro of for XCB exists relying on strictly fewer than twelve
distinct variables is not yet known. In contrast, we strongly conjecture
5. Kalman, J. A., \Condensed detachment as a rule of inference", Studia
that for XHN the lower b ound is nine.
Logica 42 (1983), 443{451.
We conclude this section by noting that wehave also discovered
6. Luk asiewicz, J., SelectedWorks, edited byL.Borokowski, North
twointriguing 27-step pro ofs relying solely on XCB (and, of course,
Holland, Amsterdam, 1970.
condensed detachment), the rst completing with YQF and the second
with WN,each a shortest single axiom for equivalential calculus. The
7. McCune, W., OTTER 3.0 Reference Manual and Guide, Tech.
two pro ofs agree on their rst twenty-six steps. Moreover, the last step
Rep ort ANL-94/6, Argonne National Lab oratory, Argonne, Illinois,
of each of the two pro ofs is obtained by applying condensed detachment
1994.
to the same pair of formulas, which indeed implies that the role of
8. Meredith, C. A., and Prior, A., \Notes on the axiomatics of the
ma jor and minor premiss is exchanged. This striking phenomenon was
prop ositional calculus", NotreDame J.Formal Logic 4, no. 3 (1963)
certainly new to us.
171{187.
9. Peterson, J. G., The Possible Shortest Single Axioms for EC-
5. Summary
Tautologies, Rep ort 105, Department of Mathematics, Universityof
Auckland, 1977.
In logic and in mathematics, once axioms have b een found for an area
of interest, a natural question concerns the existence of a single axiom.
10. Thiele, R., and Wos, L., \Hilb ert's twenty-fourth problem", J.
If such is found, one might then seek to nd a \short" single axiom. If
AutomatedReasoning, to app ear.
success again o ccurs, next comes the question of the existence of one
11. Vero , R., \Using hints to increase the e ectiveness of an automated
or more shortest single axioms.
reasoning program: Case studies", J. AutomatedReasoning 16, no. 3
With the main result of this article in hand, the set of answers to
(1996) 223{239.
those questions ab out equivalential calculus is complete. There indeed
do exist single axioms for that calculus. The shortest have length eleven
12. Wos, L., Winker, S., Vero , R., Smith, B., and Henschen, L.,
(measured in symbols), and exactly fourteen such axioms exist. Thus
\Questions concerning p ossible shortest single axioms for the equivalen-
the search for shortest single axioms for the equivalential calculus is at
tial calculus: An application of automated theorem proving to in nite
an end.
domains", NotreDame J.Formal Logic 24 (1983) 205{223.
Perhaps the story of these formulas unfolded in an appropriate way:
13. Wos, L., \Meeting the challenge of ftyyears of logic", J. Automated XCB, the fourteenth and last of the shortest single axioms to b e found,
Reasoning 6 (1990) 213{232. is unique among the fourteen in various ways and app ears to have b een
the most dicult to study.
14. Wos, L., and Piep er, G. W., AFascinating Country in the World
of Computing: Your Guide to AutomatedReasoning, World Scienti c,
Singap ore, 1999.
References
15. Wos, L., \The strategy of cramming", Preprint ANL/MCS-P898-
0801, Mathematics and Computer Science Division, Argonne National 1. Belnap, N., \The two-property", RelevanceLogic Newsletter 1 (1976)
Lab oratory, Argonne, Illinois, 2002. 173{180.
calculus 2. Ho dgson, K., \Shortest single axioms for the equivalential
283{316. 20 (1998) dReasoning J. Automate with CD and RCD",
xcb-jar.tex; 8/10/2002; 10:12; p.20 xcb-jar.tex; 8/10/2002; 10:12; p.19
Vanquishing the XCB Question 21
16. Wos, L., Ulrich, D., and Fitelson, B., \XCB, the last of the shortest
single axioms for the classical equivalential calculus", Bul letin of the
Section on Logic, accepted.
xcb-jar.tex; 8/10/2002; 10:12; p.21