Davis- Prehistory and Early History of Automated Deduction

Chapter 1

The Early History of Automated

Dedu tion

Dedi ated to the memory of Hao Wang

Martin Davis

Se ond readers: Peter Andrews, Wolfgang Bibel, Alan Robinson, and

J�org Siekmann.

Contents

1 Presburger's Pro edure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Newell, Shaw & Simon, and H. Gelernter . . . . . . . . . . . . . . . . . . . . . . . . 6

3 First-Order Logi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

HANDBOOK OF AUTOMATED REASONING

Edited by Alan Robinson and Andrei Voronkov

2001 Elsevier S ien e Publishers B.V. All rights reserved

The Early History of Automated Dedu tion 5

With the ready availability of serious omputer power, dedu tive reasoning, espe-

ially as embodied in mathemati s, presented an ideal target for those interested in

experimenting with omputer programs that purported to implement the \higher"

human fa ulties. This was be ause mathemati al reasoning ombines obje tivity

with reativity in a way diÆ ult to �nd in other domains. For this endeavor, two

paths presented themselves. One was to try to understand what people do when

they reate proofs and to write programs emulating that pro ess. The other was

to make use of the systemati work of the logi ians in redu ing logi al reasoning

to standard anoni al forms on whi h algorithms ould be based. Ea h path on-

fronted daunting obsta les. The diÆ ulty with the �rst approa h was that available

information about how reative mathemati ians go about their business was and re-

mains vague and ane dotal. On the other hand, the well-known unsolvability results

of Chur h and Turing showed that the kind of algorithm on whi h a programmer

might want to base a theorem-proving program simply did not exist. Moreover, it

was all too obvious that an attempt to generate a proof of something non-trivial

by beginning with the axioms of some logi al system and systemati ally apply-

ing the rules of inferen e in all possible dire tions was sure to lead to a giganti

ombinatorial explosion.

Ea h of these approa hes has led to important and interesting work. Unfortu-

nately, for many years the proponents of the two approa hes saw themselves as

opponents and engaged in polemi s in whi h they largely spoke past ea h other.

One problem was that whereas they appeared to be working on the same problems,

they tended to di�er not only in their approa hes, but also in their fundamental

goals. Those whose method was the emulation of the human mathemati ian tended

to see their resear h as part of an e�ort to help understand human thought. Those

who proposed to use the methods of mathemati al logi tended to see the goal as

the development of useful systems of automated dedu tion. Ultimately, the most

su essful developments in orporated insights deriving from both approa hes.

For a brief a ount of the history of the developments in logi that provided the

ba kground for resear h in this �eld see [Davis 1983 ℄. An interesting a ount of

the two approa hes and their mutual intera tion an be found in [Ma Kenzie 1995℄.

The volume [Siekmann and Wrightson 1983℄ is a useful anthology of the prin ipal

arti les on automated dedu tion to appear in the years through 1966.

1. Presburger's Pro edure

In 1929, M. Presburger had shown that the �rst-order theory of addition in the

arithmeti of integers is de idable, that is he had provided an algorithm whi h

would be able to determine for a given senten e of that language, whether or not

it is true. In 1954, Martin Davis programmed this algorithm for the va uum tube

omputer at the Institute for Advan ed Study in Prin eton. As was stated by Davis

[1983 ℄

Sin e it is now known that Presburger's pro edure has worse than exponential

omplexity, it is not surprising that this program did not perform very well. Its

6 Martin Davis

great triumph was to prove that the sum of two even numbers is even.

2. Newell, Shaw & Simon, and H. Gelernter

The propositional al ulus is the most elementary part of mathemati al logi , deal-

ing as it does with the onne tives : ^ _ � . Its treatment onstitutes Se tion A

of Part I (37 pages) of Whitehead and Russell's Prin ipia Mathemati a, their mon-

umental three volume e�ort purporting to demonstrate that all of mathemati s an

be viewed as a part of logi . Their treatment pro eeds from �ve parti ular formu-

las, that may be alled axioms or primitive propositions to whi h are applied the

expli itly stated rule of modus ponens or deta hment

1

and impli it rules permitting

substitutions for propositional variables and repla ing de�ned symbols using their

de�nitions. Newell, Shaw and Simon set themselves the problem of produ ing a

omputer program that emulates the pro ess by whi h a person might seek proofs

in the propositional al ulus of Prin ipia. Although the formalism is simple enough

so that su h a program would be feasible, the pro ess requires enough ingenuity

that the problem was hardly trivial.

In Newell, Shaw and Simon's [1957℄ report on experiments with their \Logi

Theory Ma hine," (developed around the same time as Davis's Presburger program)

the authors are very expli it about their goals:

Our explorations . . . represent a step in a program of resear h . . . aimed at devel-

oping a theory . . . and applying [it℄ to su h �elds as omputer programming and

human learning and problem-solving

Although it would be diÆ ult to laim that this work has helped very mu h with

su h an ambitious agenda, it did provide a paradigm employed by many theorem-

provers developed later, and this was surely its lasting in uen e. Among the te h-

niques made expli it were forward and ba kward haining, the generation of useful

subproblems, and seeking substitutions that produ e desired mat hes.

The authors emphasize that their program is \heuristi " rather than \algorith-

mi ," and this purported distin tion has given rise to mu h dissension and on-

fusion. In this ontext, \heuristi " seems to mean little more than the la k of a

guarantee that the pro ess will always work (given suÆ ient spa e and time). The

algorithm they ontrast with their own pro edure is the \British Museum algo-

rithm" by whi h all possible proofs are generated until one leading to the desired

result is rea hed. Indeed, the authors seem to have been unaware that Post's proof

of the ompleteness of the Prin ipia propositional al ulus using truth tables had,

in e�e t supplied a simple algorithm by means of whi h a demonstration by truth

tables ould be onverted into a proof in Prin ipia [Post 1921℄.

Wang and Gao [1987℄ presented a Gentzen-style proof system for the proposi-

tional al ulus designed for eÆ ien y. Unlike the program of Newell et al, Wang's

system is omplete: for any input, pro essing eventually halts, yielding either a

1

A tually Whitehead and Russell's tenden y to onfuse obje t and meta-language led them to

state this onfusingly as \Anything implied by a true proposition is true." But this lapse is not

important for the present dis ussion.


proof or a disproof. The simple examples that are expli itly listed in Prin ipia,

in luding those that stumped the Logi Theory Ma hine, were easily disposed of.

Although Wang seems not to have quite understood that produ ing an eÆ ient

generator of proofs in the propositional al ulus was not what the Logi Theory

Ma hine designers were after, they did leave themselves open to Wang's riti ism

by giving the impression that the absurd British Museum algorithm was the only

possible \non-heuristi " proof-generating system for the propositional al ulus.

Like the propositional al ulus, the elementary geometry of the plane an be

spe i�ed by a formal system for whi h an algorithmi de ision pro edure is available.

This is seen by introdu ing a oordinate system and relying on the redu tion of

geometry to algebra and Tarski's de ision pro edure for the algebra of the real

numbers. However, unlike the ase of truth table methods for the propositional

al ulus, this method is utterly unfeasible. Although theoreti al on�rmation of this

did not ome until mu h later, it was already apparent from Davis's experien e with

the mu h simpler Presburger pro edure. Herbert Gelernter's Geometry Ma hine is

very mu h in the spirit of Newell at al. A lue to Gelernter's orientation is provided

by his statement [Gelernter 1959℄:

. . . geometry provides illustrative material in treatises and experiments in hu-

man problem-solving. It was felt that we ould ex hange valuable insights with

behavioral s ientists . . .

Te hni ally, in addition to the repertoire of The Logi Ma hine (ba kward hain-

ing, the use of subproblems), the geometry ma hine introdu ed two interesting

innovations: the systemati use of symmetries to abbreviate proofs and the use of a

oordinate system to simulate the arefully drawn diagram a student of geometry

might employ. This last was used to tip o� the prover to the fa t that ertain pairs

of line segments and of angles \appeared" to be equal to one another, and thereby

to guide the sear h for a proof.

3. First-Order Logi

Unlike the ases of propositional logi and elementary geometry, there is no general

de ision pro edure for �rst-order logi . On the other hand, given appropriate axioms

as premises, all mathemati al reasoning an be expressed in �rst- order logi , and

that is why so mu h attention has been paid to proof pro edures for this domain.

Investigations by Skolem and Herbrand in the 1920s and early 1930s provided the

basi tools needed for theorem-proving programs for �rst-order logi [Davis 1983 ℄.

In 1957 a �ve week Summer Institute for Symboli Logi held at Cornell Uni-

versity was attended by almost every logi ian working in the United States. Many

of the more theoreti ally in lined resear hers from the nearby IBM fa ilities were

also present; FORTRAN, a brand-new innovation in programming pra ti e was un-

veiled. After dis ussions with Gelernter, the logi ian Abraham Robinson was led to

give a short talk [Robinson 1957℄ in whi h he pointed to Skolem fun tions and \Her-

brand's theorem" as useful tools for general purpose theorem-provers. He also made

the provo ative remark that the auxilliary points, lines, or ir les \ onstru ted" as

8 Martin Davis

part of the solution to a geometry problem an be thought of as being elements of

what is now alled the Herbrand universe for the problem.

The �rst theorem-provers for �rst-order logi to be implemented based on Her-

brand's theorem employed a ompletely unguided sear h of the Herbrand universe.

Instead of using Skolem fun tions to deal with instantiations, variables were re-

pla ed by parameters; so the program had to provide a apability for keeping tra k

of dependen ies among these parameters. Tests for truth-fun tional satis�ability

used either simple truth table al ulations or expansion into disjun tive normal

form. Not surprisingly, these programs were apable of proving only the simplest

theorems. Among the �rst of these programs, that by Gilmore [1960℄ served as a

parti ularly useful stimulus for further investigations.

In his later ommentary, Prawitz [1983a℄ explained that the development of new

proof pro edures and ompleteness proofs for �rst- order logi together with the

availability of omputational resour es tempted him to be ome involved in imple-

menting su h a pro edure. He adopted a modi�ed form of the method of semanti

tableaux, and formulated his own high level algorithmi language in whi h the pro-

edure ould be written. The detailed implementation was a omplished by Prawitz,

Prawitz and Voghera [1960℄. Despite being based on an up-to-date underlying log-

i al system, this program su�ered from the same limitations as Gilmore's.

Martin Davis and Hilary Putnam noted that Gilmore's program failed on some

rather simple examples be ause of its relian e on expansions into disjun tive normal

form for satis�ability testing. This led them to the optimisti (and in retrospe t

rather naive) on lusion that the la k of e�e tive methods for testing large for-

mulas of the propositional al ulus for satis�ability was the main obsta le to be

surmounted. Although their interest in algorithms for what ame to be known as

the satis�ability problem was only be ause they wanted to use su h methods as part

of a proof pro edure for �rst-order logi , they se ured support from the National

Se urity Agen y, to spend the summer of 1958 working on this problem. In their

unpublished report to the NSA [Davis and Putnam 1958℄, they emphasized the use

of onjun tive normal form for satis�ability testing. The spe i� redu tion methods

whose use together have been linked to the names Davis-Putnam are all present in

this report. These are:

1. The one literal rule also known as the unit rule.

2. The aÆrmative-negative rule also known as the pure literal rule.

3. The rule for eliminating atomi formulas

4. The splitting rule, alled in the report, the rule of ase analysis

The Davis-Putnam paper usually ited [Davis and Putnam 1960℄ was written a

year later. The proposed pro edure would a ept as input a formula that had been

prepro essed by �rst using Skolem fun tions to eliminate existential quanti�ers and

then expanding the formula into onjun tive normal form. Many theorem-provers

(in luding some that have been very su essful) have used this approa h. Satis�a-

bility testing was to be arried out using rules 1,2,3 above, and it was noted that

an example that stumped Gilmore's program ould easily be done by hand om-

putation. When George Logemann and Donald Loveland attempted to implement


the program they found that the rule for eliminating atomi formulas (later alled

ground resolution) whi h repla ed a formula

(p _ A) ^ (:p _B) ^ C

by

(A _B) ^ C

used too mu h RAM. So it was proposed to use instead the splitting rule whi h

generates the pair of formulas

A ^ C B ^ C

The idea was that a sta k for formulas to be tested ould be kept in external

storage (in fa t a tape drive) so that formulas in RAM never be ame too large.

2

Although testing for satis�ability was performed very eÆ iently, it soon be ame

lear that no very interesting results ould be obtained without �rst devising a

method for avoiding the generation of spurious elements of the Herbrand universe

[Davis, Logemann and Loveland 1962℄.

During the same years, Hao Wang was attempting to apply some of the more

sophisti ated work that had been done in proof theory and on solvable ases of

Hilbert's Ents heidungsproblem to automati dedu tion programs. He announ ed

a omputer program that proved all of the theorems (about 400) of Whitehead

and Russell's Prin ipia Mathemati a of �rst-order logi with equality [Wang and

Zhi 1998, Wang and Zhi 1998℄. However, this apparently momentous a hievement

in automating dedu tion was (as Wang himself pointed out) possible only be-

ause all of these theorems an be brought into prenex form with the simple pre�x

8 : : :89 : : :9. Wang on luded that:

The most interesting lesson from these results is perhaps that even in a fairly

ri h domain, the theorems a tually proved are mostly ones whi h all on a very

small portion of the available resour es of the domain. ([Wang 1963 ℄ p. 32)

Prawitz's [1960℄ in uential paper taught the growing automated dedu tion om-

munity that unne essary terms in the Herbrand expansion ould be avoided by using

algorithms that did not generate elements of the Herbrand universe until needed.

Most later progress was based on this key insight. Prawitz's pro edure worked by

obtaining expansions into disjun tive normal form before repla ing variables by

2

Unfortunately both pro edures using rules 1,2, and 3 and pro edures using rules 1,2, and 4

have been alled the \Davis-Putnam pro edure" in the literature; the �rst is generally onsidered

for worst ase analysis while it is the se ond that is ordinarily implemented.

Wolfgang Bibel has kindly pointed out to me that the \rule for eliminating atomi formulas"

otherwise known as \ground resolution" was �rst proposed in A. Blake's dissertation in 1937 and

(in its dual form) was also mentioned by W.V. Quine in 1955 under the name \ onsensus rule".

For further information, see [Bibel 1993℄. Otherwise, as far as I know, the other rules mentioned

o urred for the �rst time in [Davis and Putnam 1958℄.

It should also be mentioned that rules 2 and 4 were found independently by Dunham, Fridsal

and Sward [1959℄. They emphasized that a program based on these rules performs very e�e tively

without using \heuristi " devi es.

10 Martin Davis

elements of the Herbrand universe. The algorithm thus generated disjun tive nor-

mal forms of in reasing length seeking one with the property that some substitution

from the Herbrand universe would yield a truth-fun tionally unsatis�able formula.

3

Sin e this ondition amounts to ea h disjun tive lause in luding a pair of literals

of the form `;:`, it an be formulated as the need to satisfy a system of equations

in the parameters of the expansion.

4

Prawitz's pro edure was a great improvement over what had been done previously

be ause no spurious elements of the Herbrand universe were generated. Unfortu-

nately, the huge expansions into disjun tive normal form that would be generated

by all but the simplest problems made it lear that, at least as presented, this

was still an unsatisfa tory pro edure. However, it ontained the seminal idea of

sear hing for substitutions that would transform pairs of literals into negations of

one another. Moreover if existential quanti�ers are eliminated in favor of Skolem

fun tions at the outset, instead of systems of equations, one has the simple problem

of unifying pairs of terms.

In his survey paper, Davis [1963℄ proposed

. . . a new kind of pro edure whi h seeks to ombine the virtues of the Prawitz

pro edure and those of the Davis-Putnam pro edure.

The idea, also noted by Dunham and North [1962℄, was that by the \pure literal

rule" from the Davis-Putnam pro edure (Rule 2, above), substitutions an help to

render a onjun tive set of disjun tive lauses unsatis�able only if they su eed in

transforming a literal from one of these lauses into the negation of a literal in

another lause. A theorem-proving program based on these ideas was written by

D. M Ilroy at Bell Laboratories and was improved and orre ted by Peter Hinman.

The program in luded an implementation of the ordinary uni� ation algorithm

[Chinlund, Davis, Hinman and M Ilroy 1964℄.

Merely the existen e of this volume makes it abundantly lear that automated

reasoning is a thriving �eld with a huge literature. The bimonthly publi ation The

Journal of Automated Reasoning is devoted entirely to this �eld. If one event an be

pinpointed as marking its emergen e as a mature subje t, it would be the publi a-

tion [Robinson 1965b℄ in whi h J.A. Robinson announ ed the resolution prin iple.

[Robinson 1965b℄ was Robinson's se ond paper in the area, and it is helpful in tra -

ing his thought to begin with the �rst [Robinson 1963℄. He began with the basi

framework of Davis-Putnam: existential quanti�ers eliminated in favor of Skolem

fun tions and onjun tive normal form. He noted Prawitz's te hnique for avoiding

spurious elements of the Herbrand universe and Davis's survey paper. Evidently

Davis's sket h of his proposed pro edure was insuÆ iently lear, and Robinson

wrote:

5

3

This a ount is not quite a urate be ause in Prawitz's paper matters are expressed in terms

of �nding a proof rather than a refutation. So what he a tually did is pre isely the dual of what

is stated above.

4

As pointed out to the author by G�erard Huet, this same use of equations o urs already in

Herbrand's [1930, p. 145℄ thesis.

5

In the interest of larity, the referen e numbers in this quote were repla ed by the numbers in

the present bibliography orresponding to the same papers.


Davis [1963℄ has therefore proposed a way of exploiting Prawitz' powerful idea

while avoiding Prawitz' disasterous use of normal forms{in mu h the same way

that the te hniques of Davis and Putnam [1960℄ avoid the use of normal forms

whi h aused Gilmore's [1960℄ program to be unable even to solve [an easy prob-

lem℄. From the few remarks at the end of [Davis 1963℄ it does not yet seem

lear just how Davis will pro eed, and one waits with great interest his further

resear hes along these lines.

The rest of the paper has a number of interesting omputer proofs generated by

using the Davis-Putnam \one literal lause" rule, and, when that fails, requiring

the user to pre-spe ify the elements of the Herbrand universe needed to obtain a

proof. Finding these elements was onje tured to be \the really ` reative' part of

the art of proof- onstru tion."

Robinson's method of resolution introdu ed in his highly in uential [1965b℄ revo-

lutionized the subje t. Robinson found a single rule of inferen e, easily performable

by omputer, that was omplete for �rst- order logi . Using resolution required no

separate pro edure for dealing with propositional al ulus. Starting with the usual

pre-pro essed onjun tive set of disjun tive lauses, Robinson's te hnique was to

seek all possible \uni� ations" that would make it possible to express the set of

lauses as

(` _ A) ^ (:` _ B) ^ C

where ` is a literal that doesn't o ur in C. This yields the \resolvent"

(A _B) ^ C

whi h after (A _ B) is \multiplied out" yields a new set of lauses that is un-

satis�able just in ase the original set was. This was similar to Davis's proposal

[Davis 1963, Chinlund et al. 1964℄, in seeking uni� ations that generate omplemen-

tary literals. It di�ers not only in not requiring separate truth fun tional testing,

but also in not requiring, as part of the input, spe i� ation of the number of in-

stan es of ea h lause to parti ipate in the �nal proof. [Robinson 1965b℄ is striking

for its ombinatorial simpli ity, as well as for the sheer mathemati al elegan e of the

presentation. Unfortunately, as soon be ame apparent, the bare resolution method

ould easily produ e many thousands of lauses without rea hing a proof. Finding

a proof using resolution be omes the problem of providing riteria for the order in

whi h resolutions are to be sought. Early attempts to ut down the sear h spa e

were Robinson's own elegant hyperresolution [Robinson 1965a℄, and the strategies

of unit preferen e [Wos, Carson and Robinson 1964℄ and set of support [Wos, Robin-

son and Carson 1965℄.

The three de ades sin e the �rst implementations of resolution have seen an

outpouring of resear h devoted to automated reasoning systems. While some of

the most su essful are based on resolution, others have pro eeded in di�erent

dire tions. For further information, the reader is referred to the other arti les in

this volume.

12 Martin Davis

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Index

A

aÆrmative-negative rule . . . . . . . . . . . . . . . . 8

B

Bibel, Wolfgang . . . . . . . . . . . . . . . . . . . . . . . . 9

Blake, A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

C

haining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6, 7

Chur h, Alonzo . . . . . . . . . . . . . . . . . . . . . . . . . 5

onjun tive normal form . . . . . . . . . . . . . . . . 8

D

Davis, Martin . . . . . . . . . . . . . . . . . 5, 8, 10{12

Davis-Putnam pro edure . . . . . . . . . . . . 8{10

disjun tive normal form . . . . . . . . . . . . . . 8, 9

Dunham and North . . . . . . . . . . . . . . . . . . . .10

Dunham, Fridsal, and Sward . . . . . . . . . . . .9

E

Ents heidungsproblem . . . . . . . . . . . . . . . . . . 9

F

�rst-order logi . . . . . . . . . . . . . . . . . . . . . . . . . 7

G

G�erard Huet . . . . . . . . . . . . . . . . . . . . . . . . . . .10

Gelernter, H. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Geometry Ma hine . . . . . . . . . . . . . . . . . . . . . 7

Gilmore, P.C. . . . . . . . . . . . . . . . . . . . . . . . . . . .8

ground resolution . . . . . . . . . . . . . . . . . . . . . . . 9

H

Herbrand universe . . . . . . . . . . . . . . . . . . .8{11

Herbrand's theorem . . . . . . . . . . . . . . . . . . 7, 8

Herbrand, J. . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Hinman, Peter . . . . . . . . . . . . . . . . . . . . . . . . .10

hyperresolution . . . . . . . . . . . . . . . . . . . . . . . . 11

L

Logemann, George . . . . . . . . . . . . . . . . . . . . . .8

Logi Theory Ma hine . . . . . . . . . . . . . . . . . . 6

Loveland, Donald . . . . . . . . . . . . . . . . . . . . . . . 8

M

M Ilroy, D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

N

Newell, Shaw and Simon . . . . . . . . . . . . . . . . 6

O

one literal rule . . . . . . . . . . . . . . . . . . . . . . 8, 11

P

Post, E.L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Prawitz, Dag . . . . . . . . . . . . . . . . . . . . . . . . 8{10

Prawitz, H�akan . . . . . . . . . . . . . . . . . . . . . . . . . 8

Presburger, M. . . . . . . . . . . . . . . . . . . . . . . . . . 5

Prin ipia Mathemati a . . . . . . . . . . . . . . . 6, 9

propositional al ulus . . . . . . . . . . . . . . . . . . . 6

pure literal rule . . . . . . . . . . . . . . . . . . . . . 8, 10

Putnam, Hilary . . . . . . . . . . . . . . . . . . . . . . . . . 8

Q

Quine, W.V. . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

R

resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 11

Robinson, Abraham . . . . . . . . . . . . . . . . . . . . 7

Robinson, J.A. . . . . . . . . . . . . . . . . . . . . . 10, 11

rule for eliminating atomi formulas . . 8, 9

rule of ase analysis . . . . . . . . . . . . . . . . . . . . .8

Russell, Bertrand . . . . . . . . . . . . . . . . . . . . . . . 6

S

set of support . . . . . . . . . . . . . . . . . . . . . . . . . 11

Skolem fun tions . . . . . . . . . . . . . . . . . 7, 8, 10

Skolem, T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

splitting rule . . . . . . . . . . . . . . . . . . . . . . . . . 8, 9

subproblems . . . . . . . . . . . . . . . . . . . . . . . . . 6, 7

T

Tarski, Alfred . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Turing, Alan . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

U

uni� ation . . . . . . . . . . . . . . . . . . . . . . . . . 10, 11

unit preferen e . . . . . . . . . . . . . . . . . . . . . . . . 11

unit rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

V

Voghera, Neri . . . . . . . . . . . . . . . . . . . . . . . . . . .8

W

Wang, Hao . . . . . . . . . . . . . . . . . . . . . . . . 6, 7, 9

Whitehead, A. N. . . . . . . . . . . . . . . . . . . . . . . .6

Documents

Davis- Prehistory and Early History of Automated Deduction