773
IEEE TRANSACTIONS ON COMPUTERS, VOL. C-25, NO. 8, AUGUST 1976
Problems and Experiments for and with Automated
Theorem-Proving Prograrms
JOHN D. McCHAREN, ROSS A. OVERBEEK, AND LAWRENCE A. WOS
Abstract-The two objectives of this paper are 1) to give a large
that the omission of one of the two 4-clauses, for example,
and varied problem set, complete clause sets for use in testing automated theorem-proving programs and 2) the presentation of a
number of experiments with an existing program under a variety
corresponding to the usual formulation of associativity of
1) In group theory x3 = e implies the cummutator ((x,y),y) = e.
The proof is rather longer than that of the usually cited test prob-
Although some automated theorem-proving procedures
additi
on In ring theory can materially impede the perforof conditions.
mance of a theorem-proving program. It is, however, the
The problems range from quite trivial to quite difficult, including other-side of this issue which is of concern here. Experisome for which a refutation was not obtained. The statements of ments conducted with abbreviated or with minimal clause
a problem without the full list of corresponding clauses would be sets yield little information about the value of a new
of only cursory value when viewed from either of the objectives of strategy or a new inference rule and, unfortunately, are in
this paper. The problems are taken from the usually studied algebraic systems, Tarskian geometry, Henkin models, and from Wang's fact often misleading. Because of these considerations a
suggested test cases, among others. Among those included herein, large and varied problem set has been assembled, many of
the following are of added interest because of their difficulty, which are presented herein, in addition to the usual
especially from the viewpoint of automated theorem proving.
mathematical formulation, in a clause formulation.
lems for theorem-proving programs. The problem is even somewhat
challenging for the individual unaided by the computer.
2) Using the Tarski axioms for plane geometry, the relation of
betweenness can be shown to be symmetric. This axiom set is quite
are tailored for a specific field of mathematics, much effort
is devoted to the development of procedures which have
value for a wide variety of mathematical problems. In this
regard problems are given, among others, from various
nonintuitive. The problems taken from this field provide an inter- fields of algebra, Tarskian geometry, category theory, and
esting test for programs because, among other considerations, they
use six place functions in the corresponding clause sets.
program verification. But perhaps the most important
The experiments compare the inference rules of UR-resolution, dimension of the problem set given herein is its breadth
hyperresolution, paramodulation, taken separately, and in com- of difficulty. Too often in the literature one encounters
bination. Presented herein are examples employing the set of solutions to quite trivial problems as evidence for the worth
support strategy, various complexity schemes (C-sets), inclusion of a new idea, and even more so, seldom does one find
on whic the citedor fied Included
of lemmas, various demodulation approaches, and various clause t erein eam
representations.
therem examples on which the cited effort failed. Included
The program was developed at Northern Illinois University and among the examples are the often-cited easy problems,
the experiments are from its use at the Argonne National Labora- some problems which the authors consider quite difficult
tory on an IBM 370/195. Time for the experiments, number of but solvable by the program described here, and some
clauses generated, number of clauses retained, percentage of suc- quite difficult which were not provable with the justcessful unifications, and some of the parameters governing each mented pre
experiment are given together with the actual set of clauses em- mentioned program.
ployed therein.
Many experiments, including some that failed, are given
here. In addition to the clauses characterizing a problem,
the values for some of the input parameters governing the
resolution.
approach taken and various statistics resulting from the
INTRODUCTION
corresponding experiments are presented. Although one
E XPERIMENTATION with and evaluation of an au- of the four standard approaches, described in a later sectomated theorem-proving program require a varied tion, was chosen for most of the experiments, some, but not
problem set. A member of such a problem set cannot be all, of the harder theorems were proven with a nonstandard
specified by simply "stating the problem" under consid- approach. Such examples definitely have an ad hoc flavor.
eration. The actual clauses corresponding to the axioms It is not known at this time whether the ad hoc designation
of the underlying theory together with those for the denial is correct or whether a standard approach can be developed
of the theorem in question must be given. It is well known to deal with such examples. Nor is it yet clear how the
.choice of the standard approach is to be made.
roam.
Manuscript received September 22, 1975; revised February 27,
The experiments are offered not only as evidence of the
ptnilpwro
h program
rga
u
lofrteproeo
but
also
for the purpose of
The authors are with the Division of Computer Science, Depaxtment potential power of the
of Mathematical Sciences, Northern Illinois University, DeKalb, IL.
comparative evaluation. Although it might be somewhat
1976.
774
IEEE TRANSACTIONS ON COMPUTERS, AUGUST 1976
difficult to directly use the time taken to obtain a proof cent of success is much less).
3) The use of demodulation appears to be necessary to
because of the difference in computer speeds, the values
associated with clauses generated, and with clauses re- avoid an excessive number of essentially (semantically)
identical clauses. Most previous theorem proving programs
tained do provide a basis for comparison.
BRIEF DESCRIPTION OF THE PROGRAM
The program referenced in this paper has a variety of
features which enhance its performance. Although a detailed description of these features is available elsewhere
[3], a summary will be presented here. Because most of
these features were necessary to obtain proofs of the more
difficult problems, and because many reported implementations lack one or more of these features, it is hoped
that this summary will prove useful to those -designing new
programs, as well as those with existing programs.
1) Only one copy of a literal or term (other than a variable) is maintained. All clauses containing a given literal
will contain references to the same copy of that literal.
Each literal will reference all clauses that contain it, and
each argument of a literal/term will refer back to the containing literal/term. The advantages which accrue to such
a representation are substantial, One of the most significant is that once a unification of two literals has been established, an entire set of resolvents has been calculated.
In the case of paramodulation or demodulation a similar
situation exists when two terms have been unified.
2) A set of lists, referred to as FPA lists, is maintained
such that each list identifies all literals which a) have a
designated sign, b) have a designated predicate symbol,
and c) have a designated symbol occurring as the initial
symbol in a specified argument.
For example, a list will be kept for all negative literals
that have P as the predicate symbol and have g as the first
symbol in the second argument. A reserved symbol will be
used to indicate the presence of any variable (rather than
a specific variable symbol). To illustrate the use of these
lists, suppose that the program must determine which literals of opposite sign might unify with Q(a,g(f(x,c))).
First, the union of the following two FPA lists is
formed:a) the FPA list of all negative literals with Q as the
predicate symbol and a as the first symbol in the first
argument;
b) the FPA list of all negative literals with Q as the
predicate symbol and any variable symbol as the first
symbol in the first argument.
Next, a similar union is formed for the second argument.
The intersection of the two resulting lists gives a set of literals which can be tested with the complete unification
algorithm. Since it is critical that FPA lists can be referenced rapidly, a rather natural key is associated with
each list, and a hash table of pointers to the list headers is
maintained. By using the approach of FPA lists to prescreen literals before attempting complete unifications, the
program normally is successful on at least 50 percent of
attempted unifications (although on some problems in
which the level of nested functions is quite high, the per-
that utilize demodulation would simply read as input a
fixed set of demodulators along with the statement of the
problem. This may be referred to as static demodulation.
However, a significant advantage can be gained by adding
demodulators as positive unit equality clauses are produced. Thus, if
f(a,b) = c
is deduced at some point in the run, the demodulator
f(a,b)=
c
might beneficially be added to the set of demodulators.
This ability to add demodulators as the corresponding
equalities are detected will be called dynamic demodulation. For many of the problems in the set presented herein,
the use of dynamic demodulation was essential (in the
sense that the program would not have obtained a proof,
otherwise). The implementation of dynamic demodulation
required solutions to several problems:
a) Cycles of the form
* tl
t
t1 =}t2=*
must be avoided. To prevent this a total ordering of all
terms is defined and a term can only be demodulated to
"Csimpler" terms.
b) Since the number of active demodulators can become quite large, some means of rapidly selecting relevant
demodulators to use against a given term had to be developed. The selected approach is described in detail
elsewhere [1].
c) When a new demodulator is added, clauses already
existing should be demodulated. To efficiently accomplish
this, FPA lists corresponding to arguments of terms were
introduced (these lists are useful in implementing paramodulation, as well). Once a set of terms which can be
demodulated has been determined, the program can rapidly trace back to the affected clauses.
d) Once the equality
tl
=
t2
has been generated, some means must exist for determining whether or not t, > t2 should be added as a demodulator. In the current program the demodulator will
be added if the difference in complexity of t1 and t2 exceeds a specified input parameter.
4) Subsumption was used on all problems. Any generated clause which was subsumed by an existing clause was
not added to the clause space. Further, if any previously
existing clause was subsumed by a generated clause, it was
removed from the clause space. While proofs of the simple
problems could probably have been obtained without these
features, on the more demanding problems the use of
subsumption was essential. For techniques of implementing a relatively efficient subsumption algorithm the
MC CHAREN et al.: AUTOMATED THEOREM-PROVING PROGRAMS
reader is referred to [3]. Restricted subsumption tests, such
as unit subsumption, have been widely used. However, for
problems that do not formulate naturally as Horn sets, the
general test appears warranted and is, in fact, not prohibitively slow.
5) Previous work has illustrated the advantages of establishing a measure of the "complexity" of terms and
setting a bound on the complexity of allowable terms. In
some cases the straightforward approach of limiting the
number of symbols in a term proved inadequate. Therefore, the current program contains a mechanism for
specifying more precise measures. Not only were clauses
containing terms above a certain complexity rejected, the
theorem prover was biased to process clauses in an order
related to the complexity of the terms contained in each
clause. For a more detailed discussion see [1], [3].
6) For a theorem prover to be able to process a clause
space composed of thousands of clauses, the cost of the
basic operations must not be a linear (or worse) function
of the number of clauses in the space. One such basic operation is to determine whether a designated literal, term,
or clause already exists. To deal with this problem efficiently a hash table is maintained. The key of an entry is
a literal or a term, and each entry contains a pointer to the
corresponding entity (the key itself need not be kept in the
table entry, since the pointer to the literal or term is included). Thus, the time to determine whether a given literal or term already exists or not will remain relatively
constant.
7) The theorem prover contains a variety of parameters
which affect its performance. Although a comprehensive
list is well beyond the scope of this paper, the following are
particularly worth noting:
a) A variety of parameters are used to evaluate the
complexity of a term. For example, ground terms may or
may not be emphasized. Further, a constant which represents a function of other symbols may either be treated as
a more complex term than those of which it is a function,
or it may be assigned an identical complexity. For example,
if
P(a,b,c)
identifies c as the product of a and b, then c may be either
considered as more complex or of equal (or even less)
complexity (depending on input parameters). Of the wide
variety of complexities which could be specified to the
program, one of the following four were used on most of the
problems in the included set:
i) complexity of a term = (number of function and
variable symbols that occur in the term)
ii) complexity of a term = (# of function symbols)
+ 2* (number of variable symbols)
(Let each fVnction symbol be assigned an "intrinsic"
complexity as discussed above.)
iii) complexity of a term = (the sum of the complexities associated with the function symbols) + 1*(number
of variable symbols)
iv) complexity of a term = (the sum of the complex-
775
ities associated with the function symbols) + 2* (number
of variable symbols)
As mentioned in the Introduction, work is in progress
to identify whether or not the more specialized complexity
measures that were required on some of the harder problems can be generalized.
b) A variety of options govern the performance of
subsumption, factoring, and "qualification" (see Winkers
results on qualified hyperresolution [4]). In all of the
problems standard settings were employed for these parameters. The authors intend to investigate the effects of
altering these parameters.
c) A flag can be set to cause the generation of
tl = t2
to automatically result in immediate generation of t2 = t1.
* 8) The program currently supports binary resolution,
unit resolution, hyperresolution, and UR resolution (a
UR-resolvent is a unit clause derived by matching unit
clauses against literals in a nucleus; for a more precise
definition see [1]), paramodulation, and factoring. In this
set of experiments only hyperresolution, UR resolution,
paramodulation, and factoring were employed.
NOTES ON THE PROBLEMS
1) Problems in group theory. These problems are
standard and have appeared in the literature.
2) Proving the isomorphism of two groups. These two
problems are included because they illustrate the difficulties introduced by case analysis.
3) Problems in ring theory. Again these are standard.
Note that the lemma "Boolean rings have characteristic
zero" is not included in the third problem.
4) Problems in Boolean algebras. This axiomization and
these problems were taken from Chapter 2 of Boolean
Algebra and Its Applications by J. Eldon Whitesitt.
5) Problems in Henkin models. These problems were
supplied in private correspondence with D. Luckham.
6) Problems in category theory. This axiomization was
taken from Chapter 1 of Theory of Categories by B.
Mitchell.
7) Problems in elementary geometry. These axioms for
elementary geometry were given in "What is elementary
geometry" by A. Tarski (Symposium of the Axiomatic
Method). Note that the weakened form of the continuity
axiom is used.
8) Problems in elementary set theory. These problems
are interesting because, although they are simple problems
for a mathematician, they prove to be difflcult for the
program. Some progress on these problems is reported in
"An Evaluation of an Implementation of Qualified
Hyper-Resolution" by S. K. Winker (presented at the
IEEE Workshop on Automated Theorem-Proving, June
1975).
9) Wang's Problems. Solutions to EXQ2 and EXQ3
have still not been successfully obtained.
10) SAM's Lemma.
IEEE TRANSACTIONS ON COMPUTERS, AUGUST
776
11) Program verification problems [1]. These problems
were contributed by E. McCharen.
12) Program Verification Problems [11]. These seven
problems were contributed by G. Ernst in a private correspondence.
13) Limit Theorems. These three problems were contributed by W. W. Bledsoe in a private correspondence.
Proofs of the first two theorems have been obtained using
qualified hyperresolution, along with qualified binary
resolution. The details of these experiments, however, are
not presented in this paper.
A. AXIOMS FOR A GROUP (I)
CLOSURE PROPERTY
P(X,Y,F(X,Y))
EXISTENCE OF AN IDENTITY
P(E,X,X)
P(X,E,X)
EXISTENCE OF AN INVERSE
P(G(X),X,E)
P(X,G(X),E)
-P(X,Y,U) -P(Y,Z,V) -P(X,V,W) P(U,Z,W) ASSOCIATIVE PROPERTY
-P(X,Y,U) -PlY,Z,V) -P(U,Z,W) P(X,V,W)
THE OPERATION IS WELL DEFINED
-P(X,Y,Z) -P(X,Y,W) EQUAL(Z,W)
EQUALITY AXIOMS
EQUAL(X,X)
-EQUAL(X,Y) EQUAL(Y,X)
-EQUAL(X,Y) -EQUAL(Y,Z) EQUAL(X,Z)
-EQUAL(X,Y) EQUALIF(X,W),F(Y,W)) EQUALITY SUBSTITUTION AXIOMS
-EQUAL(X,Y) EQUAL(F(W,X),F(W,Y))
-EQUAL(X,Y) EQUAL(G(X),G(Y))
-EQUAL(X,Y) -P(X,W,Z) P(Y,W,Z)
CL -EQUAL(X,Y) -P(W,X,Z) P(W,Y,Z)
CL -EQUAL(X,Y) -P(W,Z,X) P(W,Z,Y)
THEOREM GI. IF X**2 * X FOR ALL X IN A GROUP G, THEN G IS ABEL IAN.
X HAS ORDER 2
CL P(X,X,E)
GENIAL OF THE THEOREM
CL P(A,B,C)
CL -P(B,A,C)
THEOREM G2. THE INVERSE OF EACH ELEMENT IN A GROUJP IS UNIQUE.
DENIAL OF THE THEOREM
CL P(A,B,C)
CL P(B,A,E)
CL P(A,C,E)
CL P(C,A,E)
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
-)INX) O(G(X))
-U(X) -0(Y) -P(X,Y,Z) O(Z)
-F.QUALIX,Y) -O(X) O(Y)
-EQUAL(X,Y) EQUAL(IIW,X),I(W,Y))
-FQUAL(X,Y) EQUAL(I(X,W),I(Y,W))
CL nIX) OMY) O(I(X,Y))
CL (0(X) O(Y) P(X,I(X,Y),Y)
CL n(B)
CL PFB,G(A),C)
CL PIA,C,D)
CL
CL
CL
CL
CL
FOR A GROUP (II)
ELIGIXG,X),XG)
PFXG,X,G(XG,X),E(XG))
P(XG,G(XG,XI,X,E(XG))
-P(XG,Y,Z,YZ) -P(XG,XY,Z,XYZ) ASSOCIATIVE PROPERTY
-P(XG,X,Y,XY)
S,XG S XY, XYZ)
P(XG,XY,Z,XYZ)
-PIXG,X,Y,XY) -P(XG,Y,Z,YZ) -P(XG,X,YZ,XYZ) AXIOMS
EQUALITY
EQlUAL(X,X)
-F4UAL(X,Y) EQUAL(Y,X)
CL
CL
CL
Cl.
CL
CL
CL
CL
CL
CL
CL
DEFINED BELOW ARE
ISOMORPHII.
EQUAL(X,B) EQtIAL(X,C)
EQUAL(X,C2) EQtIAL(X,C3)
P(Gl,C,C,B)
P(G2,Cl,C2,C2)
P(G2,C2,Cl,C2)
P(G2,Cl,C3,C3)
PIG2,C3,Cl,C3)
P(G2,C2,C2,C3)
P(G2,C2,C3,Cl)
P(G2,C3,C2,C1)
P(G2,C3,C3,C2)
EQUAL(K(A),C1)
EQUAL(K(B),C2)
DEFINITION OF THE FUNCTION K
EQUAL(K(C),C3)
EL(Dl, C)
THE DENIAL THAT K IS AN ISOMORPHISM
EL(D2,G1)
P(Gl,Dl,D2,D3)
EL(D3,Gl)
-P(Gl,K(DX),K(D2),K(D3))
**
**
*****
******
**
**
********
****
*
*****
C. AXIOMS FOR A RING
EXISTENCE OF AN ADDITIVE IDENTITY
S(E,X,X)
S(X,E,X)
CLOSURE PROPERTY
S(X,Y,JIX,Y))
P(X,Y,F(X,Y))
EXISTENCE OF INVERSES
S(GSX),X,E)
S(X,G(X),E)
-S(Y,Z,V) -S(X,V,W) SIVX,Z,W) ASSOCIATIVE PROPERTY
-I;(X,Y,VO) -S(Y,Z,V)
-S(VO,Z,W) S(X,V,W)
-!;(X,Y,VO)
-P(X,Y,VO) -P(Y,Z,V) -P(X,V,W) P(VO,Z,W)
-V(IX,Y,VO) -P(Y,Z,V) - F(IO,Z,W) P(X,V,W)
COMMUTATIVE PROPERTY
-I;(X,Y,Z) S(Y,X,Z)
DISTRIBUTIVE PROPERTY
-SIY,Z,V3) -P(X,V3,V4)
S(V1,V2,V4)
-P(X,Y,VX) -P(X,Z,V2)
-S(Vl,V2,V4) P(X,V3,V4)
-P(X,Y,VX) -P(X,Z,V2) -S(Y,Z,V3) -P(V3,X,V4I)
SIVX,V2,V4I)
-P(Y,X,VX) -P(Z,X,V2) -S(Y,Z,V3) -S(Vl,V2,V4)
P(V3,X,V4)
-P(Y,X,Sl) -PIZ,X,V2) -S(Y,Z,V3)
EQUALITY AXIOMS
FQUALIX,X)
-EQUAL(X,Y) EQUAL(Y,X)
-EQUAL(X,Y) -EQUAL(Y,Z) EQUAL(X,Z)
EQUALITY SUBSTITUTION AXIOMS
-EQUALIX,Y) -S(X,W,Z) S(Y,W,Z)
-EQUAL(X,Y) -S(W,X,Z) S(W,Y,Z)
-EQUAL(X,Y) -S(WI,Z,X) S(W,Z,Y)
-DQUALIX,Y) -P(X,W,Z) PIY,W,Z)
-EQUALIX,Y) -P(W,X,Z) P(W,Y,Z)
-EQUAL(X,Y) -P(W,Z,X) P(W,Z,Y)
-EQUAL(X,Y) EQUAL(G(X),G(Y))
EQIUALIF(X,W),F(Y,W))
-EQ(UAL(X,Y)
-FQUAL(X,Y) EQUAL(F(W,X),F(W,Y))
-EQUAL(X,Y) EQUAL(J(X,W),JIY,W))
EQUALIJIW,X),J(W,Y))
-FQJUALIX,Y)-S(X,Y,Z)
EQUAL(W,Z)
-T;IX,Y,W)
*
THEOREM R3. IF X
CL PIX,X,X)
CL P(A,B,C)
CL -P(B,A,C)
*0
0
*
THE
OPERATIONS ARE WELL DEFINED
CANCELLATION LAWS
DENIAL OF THE THEOREM
-
X
*
Y
DENIAL OF THE THEOREM
X * X FOR ALL X, THEN THE RING IS
COMMUTATIVE.
DENIAL OF THE THEOREM
0. AXIOMS FOR A BOOLEAN ALGEBRA
CL
CL
SOMORPHIGSI
P(G2,Cl,Cl,Cl)
-FI A,E,E)
THEOREM R2. (-X) * I-Y)
CL F (A,B,C)
CL P(G(A),G(B),D)
CL -FQUAL(C,D)
DEFINED
Al
P(Gl,BRB,C)
PIGl,B,C,A)
PFGX,C,B,A)
THEOREM RI. X
FOLLOWING ARE THE USUAL AXIOMS OF A GROUP WITH AN
ADDITIONAL ARGUMENT IN THE USUAL PREDICATES AND FUNCTIONS
TO IDlENTIFY THE GROUP.
CLOSURE PROPERTY
CL -DL(X,XG) -EL(Y,XG) P(XG,X,Y,F(XG,X,Y))
CL
CL
CL
CL
CL
CL
ORDER 3
DEFINITION OF A FUNCTIONl K
DENIAL THAT K IS
P(Gl,A,C,C)
P(Gl,C,A,C)
CL
TIlE
CL -F.L(X,XG)
.*
EL(A,
-P(X,Y,W) -P(X,Y,Z) EQUAL(W,Z)
SHED LEMMAS
ESTAFIL
CL -S;tX,Y,Z) -S(X,W,Z) EQUALIY,W)
CL -SIX,Y,Z) -S(W,Y,Z) EQUAL(X,W)
-11(D)
-ELIX,XG) -EL(Y,XG) EL(F(XG,X,Y),XG)
THE OPERATION IS WELL
-P(XGGX,Y,Z) -P(XG,X,Y,W) EQUAL(W,Z)EXISTENCE
OF AN IDENTITY
ELIE(XG),XG)
P(XG,X,E(XG),X)
F(XG,E(XG),X,X)
EXISTENCE OF INVERSES
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
THE FOLLOWING IDENTITIES MAY BE ESTABLISHED,
CL EQUAL(F(X,E),X)
CL DQUAL(F(E,X),X)
CL FQUAL(F(G(X),X),E)
CL FQUALIF(X,G(X)),E)
CL F.QUAL(GIG(X)),X)
CL FQUAL(G(E),E)
THEOREM G7. SUBGROUPS OF INDEX 2 ARE NORMAL.
DENIAL OF THE THEOREM
CL 11(E)
CL
CL
CL
CL
CL
E.QUAL(X,C1) EQUAL(X,C2)
CL -DL(X,G2S
CL PFGI,A,A,A)
CL P(Gl,A,B,B)
CL P(Gl,B,A,B)
CL P(GX,N,B,A)
CL P(G2,C1,C1,Cl)
CL P(G2, Cl,C2,C2)
CL P G2, C2,Cl,C2)
CL P(G2,C2,C2,Cl)
CL EQUAL(K(A),CS)
CL F.QUALIK)B),C2)
CL EL(Dl,Gl)
CL EL(D2,Gl)
CL EL(D3,Gl)
CL P(Gl,Dl,D2,D3)
CL -P(G2, KIDS), K(D2),K(D3))
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
PIJ,GIH),K)
B. AXIOMS
EQUALDE(XG),E(YG))
-EQUAL(XG,YD)
GS. THE GROUPS OF
S)
ELIB, Q)
CL ELIC, aE)
CL ELICl,G2)
CL NLIC2,G2)
CL EL(C3,G2)
CL -ELIX, El) EQUAL(X,A)
CL -EL(X,G2) EQUALIX,Cl)
CL I'(Gl,A,A,A)
CL PFGl,A,B,B)
CL P(GX,B,A,B)
-P(K,GIB),E)
EQUALITY SUBSTITUTION AXIOMS
PIXG,W,Y,Z)
-FQUAL(X,Y) -P(XG,W,Z,X) P(XG,W,Z,Y)
-F.QUAL(XG,YG) EQUAL(F(XG,X,Y),F(YG,X, Y))
-EDQUAL(X,Y) EQUAL(F(XG,X,Z), F(XG,Y,Z)
-EQUAL(X,Y) EQUAL(FIXG,Z,X),F(XG,Z,Y)
-EQUALIXG,YG) EQUAL(G(XG,X),G(YG,X))
-EQUAL(X,Y) EQUAL(GSXG,X),G(XG,Y))
-EQUAL(XG,YG) -EL(X,XG) EL(X,YG)
-EQUAL(X,Y) -EL(X,XG) EL(Y,XG)
-EQUAL(X,Y) -P(XG,W,X,Z)
THEOREM GO. THE GROUPS OF ORDER 2 DEFINED BELOW ARE ISOMORPHIC.
DEFINITION OF THE TWO GROUPS
CL EL(AGOL)
CL EL(B,Gl)
CL EL(Cl,G2)
CL EL(C2,G2)
CL -EL(X,Gl) EQUAL(X,A) EQUALIX,B)
CL
CL
THEOREM G4. THE AXIOMS MAY BE WEAKENED BY ASSUMING THAT E IS JUST A
LEFT IDENTITY AND THAT LEFT INVERSES ESIST. EACH ELEMENT
HAS A RIGHT INVERSE. (THE AXIOMS P(X,E,X) AND P(X,G(X),E)
SHOULD BE DELETED. )
DENIAL OF THE THEOREM
CL -P(A,Y,E)
THEOREM GS. THE AXIOMS MAY BE WEAKENED BY ASSUM1ING THAT E IS JUST A
LEFT IDENTITY AND THAT LEFT INVERSES EXIST. EACHI ELEMENT
HAS A RIGHT IDENTITY. (TIHE AXIOMS P(X,E,X) AND P(X,G(X),E)
SHOULD t%BE DELETED.)
DENIAL OF THE THEnREMl
CL -P(K(Y),Y,K(Y))
THEOREM G6. IF X**3 * E FOR ALL X IN A GROUP G, THEN I(X,Y),Y) * E.
X CUBED 1S EQUAL TO X
CL -P(X,X,Y) P(X,Y,E)
CL -P(X,X,Y) P(Y,X,E)
DENIAL OF THE THEORElI
CL P(A,B,C)
CL P(C,G(A),D)
CL P(D,0(B),H)
CL P(H,B,J)
CL
-rQUAL(X,Y) -EQUALIY,Z) EQUAL(X,Z)
-FIQUAL(XG,YT,) -PIXG,X,Y,Z) PIYG,X,Y,Z)
-EQUALGX,Y) -PIXG,X,Z,W) P(XG,Y,Z,W)
THEOREM
CL -EQUAL(B,C)
THEOREM G3. THE AXIOMS MAY BE WEAKENED BY ASSUMING THAT E IS JUST A
LEFT IDENTITY AND THAT LEFT INVERSES EXIST. E IS A RIGHT
IDENTITY. (THE AXIOMS P(X,E,X) AND P(X,G(X),E) SHOULD BE
DELETED.)
DENIAL OF THE THEOREM
CL -P(A,E,A)
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
SUM(X,Y,F1(X,Y))
PROD(X,Y,F2(X,Y))
CLOSURE PROPERTY
1976
777
MC CHAREN et al.: AUTOMATED THEOREM-PROVING PROGRAMS
CL -SUM(X,Y,V3) SUM(Y,X,V3)
CL -PROD(X,Y,V3) PROD(Y,X,V3)
CL SUM(X,C1,X)
CL SUM(C1,X,X)
CL PROD(X,C2,X)
CL PROD(C2,X,X)
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
COMMUTATIVE PROPERTY
EXISTENCE OF IDENTITIES
DISTRIBUTIVE PROPERTY
-PROD(X,Y,VI) -PROD(X,Z,V2) -SUMIY,Z,V3) -PROD(X,V3,V4I) S(IM(Vl,V2,V4)
-PROD(X,Y,Vl) -PROD(X,Z,V2) -SUM(Y,Z,V3) -SUM(Vl,V2,V4) PRODOX,V3,V4)
-PROD(Y,Z,VI) -PROD(Z,X,V2) -SUV:(Y,Z,V3) -PROE(V3,X,V4) SUM4(V1,V2,V4)
-PROD(Y,X,V1) -PRODZ,OX,V2) -SUMIIY,Z,V3) -SUM(Vl,V2,VI) PRtD(V3,X,V4L)
-!;UM(X,Y,V1) -SUM(X,Z,V2) -PROD(Y,Z,V3) -SUtI(X,V3,V4) PROD)(Vl,V2,V4)
-SUM(X,Y,V1) -SUMIX, Z,V2) -PROD(Y,Z,V3) -PRODIVlS,V2,V4) SUtI(X,V3,V4)
-SUM(Y,X,Vl) -SUM(Z,X,V2) -PRODIY,Z,V3) -SUM(V3,X,V4) PROD(V1,V2,V4)
-SUM(Y,X,Vl) -SUMI(Z,X,V2 ) -PRODIY,Z,V3) -PROD(VS,V2,V4) SIJMIV3, X,V4)
EXISTENCE nF COMPLEMENTS
SUM(X,F3(X),C2)
SUM(F3(X),X,C2)
PROD(X,F3(X),C1)
PROD(F3(X),X,Cl)
-SUM(X,Y,U) -SUM(X,Y,V) EQUAL(U,V) THE OPERATIONS ARE WELL DEFINIED
-PROD(X,Y,IJ) -PROD(X,Y,V) EQUAL(U,V)
EQUALITY AXIOMS
EQUAL(X,X)
-EQUAL(X,Y) EQUAL(Y,X)
-EQUAL(X,Yl -EQUAL(Y,Z) EQUAL(X,Z)
-P.QUAL(X,Y) -SUM(X,V1,V2) SUMl(Y,V1,V2) EQUALITY SUBSTITIJTION AXIOt'S
-EQUAL(X,Y) -SUM(Vl,X,V2) SUMIVl,Y,V2)
-EQUAL(X,YV -SUM(Vl,V2,X) SUM(V1,V2,Y)
-FQUAL(X,Y) -PROD(X,V1,V2) PROE(Y,V1,V2)
-EEQUAL(X,Y) -PROD(Vl,X,V2) PROD(Vl,Y,V2)
-EQUAL(X,Y) -PROD(Vl,V2,X) PROD(IV,V2,Y)
-EQUAL(X,Y) EQUAL(FliX,V1),Fl(Y,Vl))
-EQUAL(X,Y) EQUAL(Fl(Vl,X),Fl(Vl,Y))
-EQUAL(X,Y) EQUAL(F2(X,V1),F2(Y,V1))
-EQUAL(X,Y) EQUAL(F2(Vl,X),F2(Vl,YII
-EQUAL(X,VY EQUAL(F3(X),PF3Y))
THEOREM B3.
X + (Y + Z)
CL
SUMIC4,C5,C7)
CL 9UM(C3,C7,C8)
CL
OiUM(C3,C9,C9)
CL SUM(C9,C5,C10)
CL -FQUAL(CB,Cl0)
THEOREM 82. X + X
CL -iIUMIC3,C3,C3)
*
X
-
(X
+
*
X
X
AND
Y)
-PROD(C3,C3,)C3
+ Z
X.
C2 * C2 AND X * Cl
THEOREM 83. X
CL -SUM(C3,C2,C2) -PROD(C3,C1,C1)
THEOREM B4. X + (X * Y)
CL -SUM(C3,F2(C3,C4),C3)
THEOREM B5. X
C2
CL -iUM(C3,C2,C2)
THEO:EM BS.
CL
+
X *
X AND X
*
-
DENIAL
OF
THE THEOREM
DENIAL OF THE THEOREM
Cl.
DENIAL OF THE THEOREM
*
(X + Y)
*
X.
FROD(C3,Fl(C3,C4),C4)
DENIAL OF THE THEOREM
C2.
Cl
-IIROD(C3,Cl,Cl)
DENIAL
Cl.
OF
THE THEOREM
DENIAL OF THE THEOREM
THEOr.EtI B7. THE COMPLEMENT OF ZEPO IS THE IDENTITY.
CL -EQUAL(F3(Cl),C2)
DEtItAL OF THE THEOREM
THEOFREM B0. -(-X) - X
CL -EQUAL(F3(F3(A)),A)
(-X IS THE COMPLEMENT OF X.)
DENIAL OF THE THEOREM
THE REPMAINING THEOREMS IN BOOLEAN ALGEBRAS WERE OBTAINED IN A
NESTED MIANNER I.E., EACH THEOREtM IS ASSUMED AS A LEtMM1A IN ALL
OF TIlE SUCCEEDING THEOREMS.
CL
ESTABLISHED LEMMAS
Uti(X,X,X)
CL
ROD(X,X,X)
jUIIIO,C2,C2)
CL
CL
CDl,CC1)
!UDox,
CL -IROD(X,Y,Z) SUM(X,Z,OX
CL -SUtl(X,Y,Z)
PROD(X,Z,X)
CL !iUtl(X,F2(X,Y,X)
CL
IROD(X,Fl(X,Y),S)
CL -IUtI(IX,IY,XY) -SUM(Y,,YZ) -SUI(X,YZ,XYZI) SUMI(XY,Z,XYZ)
CL -IUIiIOX,Y,XY) -SUM(Y,Z,YZ) -SUS(XY,Z,XYZ) SUM(X,YZ, XYZ)
CL -IIlOD(X,Y,XY) -PROD(Y,Z,YZ) -PROD(X,YZ,XYZ) PROD(XY,Z,XYZ)
CL -PRODSX,Y,XY) -PROD(Y,Z,YZ) -PRODVXY,Z,XYZ) PROD(X,YZ,XYZ)
DEtIIAL OF THE THEOREM.
CL -EQUAL(F3(F3IC3)),C3)
TIiEOiEV 89. FOR ALL X, THE COMPLEMIENT OF X IS UNIQUE.
CL
ADDITIOtIAL LEtMM1A
EQUAL(F3(F3(X)),X)
DENIAL OF THE THEOREM
CL LUtl(C3,C4,C2)
CL
CL
CL
CL
SUI'I(CD,C5,C2)
PROD(C3,C4,,Cl)
CL
CL
CL
CL
-
(-X)
+
(-Y).
-SUII(X,Y,C2) -SUli(X,Z,C2) -PRODtX,Y,CI) ADDITIONAL LEMMA
[IROD(X,Z,C1) EQUAL(Y,Z)
DENIAL OF THE THEOREM
SUMI(C3,C4,C5)
!tiODIF3(C3),F3IC4),C6)
-EQUAL(F3(CS),C6)
IN1
THE
P(X,Y,Z)
DEFINITION OF LESS THAN OR EQUAL
-LE(X,Y) PtX,Y,E)
-PItX,Y,E) LE(X,Y)
X/Y <C X
-I'(XEY,Z) LEtZ,X)
-PI(X,Y,V1) -PtY,Z,V2) -P(X,Z,V3)
(X/Y)/(Y/Z) <. (X/Y)/Z
-P1V3,V2,V4) -P(VD,Z,V5) LE(V4,V5)
<C X
LE(E,X)
X<-Y AND Y<-X ---) X a Y
-I.E(X,Y) -LE(Y, X) EQUAL(X,Y)
X <- 1
L.E(X,D)
P(X,Y,F(X,Y))
-PtX,Y,Z) -P(X,Y,W)
EQUAL(X,X)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
EQUAL(Z,W)
EQUAL(Y,X)
CLOSURE PROPERTY
THE OPERATION IS WELL DEFINED
EQUALITY AXIOMS
-EQUALSY,Z) EQUAL(X,Z)
EQUALITY SUBSTITUTION AXIOMS
-P(X,W,Z) P(Y,W,Z)
-P(W,X,Z) P(W,Y,Z)
-P(W,Z,X)
P(W,Z,Y)
LE(Y,Z)
LE(Z,Y)
-EQUAL(X,Y) -LE(X,Z)
-EQUAL(X,Y) -LESZ,X)
-EQUAL(X,Y) EQUAL(F(X,W),F(Y,W))
-EQUAL(X,Y) EQUAL(F(W,X),F(W,Y))
THE FOLLOWING SET OF THEOREMS WERE OBTAINED IN A NESTED MANNER
I.E., EACH THEOREM ASSUMES THE PREVIOUS THEOREMS AS LEMMAS.
EACH PROBLEM INCLUDES THE STATEMENT OF THE THEOREM WHICH IMMEDIATELY
PRECEDED
X.
THEOREM H3. X/X
CL P(E,X,E)
CL -P(A,A,E)
THEOREM H4. X/0
CL P(X,X,E)
CL -P(A,E,A)
-
-
IT, AND THIS WAS ASSUMED IN THE SUCCEEDING THEOREMS.
DENIAL OF THE THEOREMI
ADDITIONAL LEMMA
DEtJIAL OF THE THEOREM
0.
ADDITIONAL LEMVMA
DENIAL OF THE THEOREM
X.
ADDITIONAL LEMIMA
DE IIAL OF THE THIEOREI'
THEOREM H5. THE RELATION LESS THAN OR EQJAL IS TRANSITIVE.
ADDITIONAL LEtIMA
CL P(X,E,X)
CL I.E(A,A1)
DENIAL OF THE THEOREM
CL LE(Al,A2)
CL -LE(A,A2)
H6. XSY <-Z ----> X/Z <--Y
CL -LE(X,Y) -LE(Y,Z) LE(X,C)
CL P(A,B,Cl)
CL LE(C1,A1)
CL P(A,A1,C21
CL -LEtC2,B)
THEOREM
ADDITIONAL LEMMA
DENIAL OF THE THEOREM
THEOREM H7. X <- Y ----> X/Y <- Z/X.
ADDITIONAL LEMMA
CL -PtX,Y,W1) -LE(Wl,Z) -P(X,Z,W2) LE(W2,Y)
DENIAL OF THE THEOREM
CL l.E(A,B)
CL P(Cl,8,Bl)
CL P(C1,A,A1)
CL -I.Et 1, Al)
H8. X < YV----> X/Z <. V/Z.
ADDITIONAL LEMMA
CL -I.E(X,Y) -PtZ,Y,W1) -P(Z,X,W2) LE(W1,W2)
DENIAL OF THE THEOREM
CL I.E(A,B)
CL PtA,C1,A1)
CL P(B,C1,B1)
CL -I.E(Al, B1)
THEOREM
X'''.
THEOREM H9. FOR ANY X, LET X' BE DEFINED AS 1/SX. THEN X'
ADDITIONAL LEMMA
CL -I.E(X,Y) -P(X,Z,Wl) -P(Y,Z,W2) LE(W1,W2)
DENIAL OF THE THEOREM
CL I (D,A,B)
CL P(D,B,B1)
CL P(D,8B1,8B2)
CL -EQUAL( N, B2)
THEOREM HMO. X' - X/(l/XS').
CL -P(D,X,Yl) -P(D,Y1,Y2) -P(D,Y2,Y3)
CL FP(D,A,B)
CL P(D,B,B1)
CL P(B,B1,B2)
CL -EQUAL1B,N32)
EQUAL(Yl,Y3)
ADDITIONAL LEMMA
DENIAL OF THE THEOREM
THEOREM Hll. DEFINE THE OPERATION & ON THE SET OF Z', WHERE
Z'
X'/(l/YV). THE OPERATION
1/Z, BY X' & Y'
IS COMMUTATIVE.
ADDITIONAL LEMMA
CL -P(D,X,V1) -P(FI,Yl,Y2) -P(YI,Y2,Y3) EOUAL(Yl,Y3)
DENIAL OF THE THEOREM
CL
(ID,A,A1A)
CL P(D,8,B1)
CL l'(D,B1,132)
CL PIAl,R2,C1)
CL P(D,A1,A2)
CL P(B1,A2,C2)
CL -EQUAL(CD,C2)
F. CATEGORY THEORY AXIOMS
CL -I)EF(X,Y)
CL -P(X,Y,Z)
CL -P(X,Y,XY)
CL -P(X,Y,XY)
CL -*I(X,Y,XY)
CL -PI(Y,Z,YZ)
CL
CL
CL
CL
CL
CL
CL
P(X,Y,FtX,Y))
DEF(X,Y)
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
COMPOSITION IS CLOSED WIIEN DEFINED
ASSOCIATIVITY PROPERTY
-DEF(XY,Z) DEF(Y,Z)
-P(Y,Z,YZ) -DEF(XY,Z) DEF(X,YZ)
-P(XY,Z,XYZ) -P(Y,Z,YZ) P(X,YZ,XYZ)
-DEF(X,YZ) DEF(X,Y)
-P(Y,Z,YZ) -P(X,Y,XY) -DEFIX,YZ) DEF(XY,Z)
-P(Y,Z,YZ) -P(X,YZ,XYZ) -P(X,Y,XY) P(XY,Z,XYZ)
-IDEF(X,Y) -nEF(Y,Z) -IDENT(Y) DEF(X,Z) SPECIAL AXIOM
IDENT(DOMIX))
FROPERTIES OF DOM(S) AND RGE(S)
IEENT(RGEMX)
I)EF(X,OOM(X))
DIEF(RGE(X),X)
PIX,DOM(X),X)
P(RGE(X),X,X)
CL -I)EFIX,Y) -IDENT(X)
E. HENSIN MODELS
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
THEOREM H2. S/X
CL P(X,D,E)
CL -P(E,A,E)
CL
-EQUAL(C4,C5)
THE FOLLOWING AXIOtiS CHARACTERIZE HENIKII tiODELS.
FOLLOWIIING E IS THE ZERO, AND D IS THE IDEtNTITY.
HEANR XS/Y
Z, AND LE tIEANS LESS THAN OR EQUAL.
CL
CL
CL
CL
0.
CL
!'[IOD(C3,C5,Cl)
THEOREMI B1. -(X + Y)
THEOREM HI. X/1
CL -P(A,D,E)
P(X,Y,Y)
DEFINITION OF THE IDENTITY PREnICATE
-IDEF(X,Y) -iENT(Y) P(X,Y,X)
EQUAL(X,X)
EQUALITY AXIOMS
-EQUAL(X,Y) EQUAL(Y,X)
-EQUAL(X,Y) -EQUALIY,Z) EQUALCX,Z)
-EQUAL(X,Y) -P(X,Z,W) P(Y,Z,W)
EQUALITY SUBSTITUTInN AXIOMS
-EQUAL(X,Y) -P(Z,X,W)
-EQUAL(X,Y) -P(Z,W,X)
P(Z,Y,W)
PtZ,W,Y)
-EQUALtX,Y) EQUAL(DOMX),IDOM(Y))
-FQUAL(X,YY) EQUAL(RGE(X),RGE(Y))
-EQUAL(X,Y) -IDENT(X) IDENTtY)
-EQUAL(X,Y) -DEF(X,Z) DEF(Y,Z)
-EQUAL(X,Y) -DEF(Z,X) DEF(Z,Y)
-EQUAL(X,Y) EQUAL(F(Z,X),F(Z,Y))
-EQUAL(X,Y) EQUAL(F(X,Z),F(Y,Z))
-P(X,Y,Z) -P(X,Y,W) EQUAL(Z,W)
COMPOSITION IS WELL DEFINED
THEOREM Cl. kF XY IS A MIONOMORPHISM, THEN Y IS A MONOMORPHISM.
CL P(A,B,C)
DENIAL OF THE THEOREM
CL -PtC,X,W) -P(C,Y,W)
EQUAL(X,Y)
CL P(B,A2,D)
CL
P(B,A1,D)
CL -EQUAL(A2,Al)
THEOREM C2. IF X AND Y ARE MONOMORPHISMS AND XY IS DEFINED, THEN XY IS
A MONOMORPHISM.
CL -P(A,X,W) -P(A,Y,W) EQUAL(X,Y)
DENIAL OF THE THEOREM
CL -P(B,X,W) -P(B,Y,W) EQUALtX,Y)
CL P(A,B,C)
CL P(C,A2,D)
CL P(C,A1,D)
CL -EQUAL(A2,A1)
THEOREM C3. IF XY IS AN EPIMORPHISM, THEN X IS AN EPIMORPHISM.
CL P(A,B,C)
DENIAL OF THE THEOREM
CL -P(X,C,W) -P(Y,C,W) EQUAL(X,Y)
CL P(A2,A,D)
CL P(A1,A,D)
CL -EQUAL(A2,A1)
THEOREM C4. IF X AND Y ARE EPIMORPHISIIS AND XY IS DEFINED, THEN XY IS
AN EPIMORPHISM.
CL -P(X,A,W) -P(Y,A,W) EQUAL(X,Y)
DENIAL OF THE THEOREM
CL -P(X,B,W)
CL
P(A,B,C)
-P(Y,B,W)
EQUAL(X,Y)
778
CL
CL
CL
IEEE TRANSACTIONS ON COMPUTERS, AUGUST 1976
PIA2,C,D)
THEOREM
CL
CL
CL
CL
CL
DOMED)
IS THE
C6. RGE(S)
IS THE
IDENT(J)
-EQUAL(RGE(A),J)
C7.
EQUALEDOMEA),RGEtB))
RGE(Y),
THEN
C8. IF XY IS DEFINED
DEFtA,B)
-EQUALtDOMtA),R EtB))
TtIlE
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
-L(X,V,I,Z)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
EQUAL(F5(X,V1,V2,V3,V4,V5),F5(Y,V1,V2,V3,V4,V5))
EQUALEF5(Vl,X,V2,V3,V4,V5),FS(Vl,Y,V2,V3,V4,V5))
EQUAL(F5(Vl,V2,X,V3,V4,V5),F5(Vl,V2,Y,V3,V4,V5))
EQUAL(F5(Vl,V2,V3,X,V4,V5),F5(Vl,V2,V3,Y,V4,V5))
EQUAL(F5(Vl,V2,V3,V4,X,V5),F5(Vl,V2,V3, V4,Y,V5))
EQUAL(F5(NV,V2,V3,V4,V5,X),FS(Vl,V2,V3,V4,V5,Y))
Tl. BETWEENNESS
R(A,B,C)
FOR
CL
CL
T4.
EVERY
TO.
LINE
THE
Y, Y
SEGMENT
T6.
X
ANY
AND
Z,
-EQUALEA,C)
-EQUALEC,D)
-EQUAL(C,D)
R(A,C,D)
RIC,A,D)
BETWEEN X ANiD Y.
DENIAL OF THE
THEOREM
BETWEEN
AND Y.
DENIAL OF THE
fHAS
DENIAL
SEGMENT
EXISTS
THERE
AS ITS
AN
OF
THE
ISOCELES
BASE.
OF
THE
3
Y
AND
THEN
Z
Z.
IS
POINTS
DENIAL
OF
THE
CL
X
AXIOMS OF UN;ON
-ELL(X,XS) EL(X,UNI(XS,YS))
-ELEX,YS) EL(X,UNI(XS,YS))
-EL(X,IJN XS,YS) ELDX,XS) ELEX,YS)
-EL(X,XS) -EL(X,YS) EL(X,INT(XS,YS) AXIOMS OF INTERSECTION
-EL(X,INTEXS,YS)) EL(X,XS)
-NL(X,INTEXS,YS)) ELEX,YX)
SET EQUALITY AXIOMS
-EQUXS,YX) INCL(XS,YS)
-EQ(XS,YS) INCL(YS,XS)
-INCL(XS,YS) -INCLEYS,XS) EQ(XS,YS)
EQUALITY AXIOMS
EQ(XS,XS)
-EQEXS,YS) EQ(YS,XS)
-EQ(XS,YS) -EQ(YS,ZS) EQ(XS,ZS)
EQE(X,X)
-EQE(X,Y) EQE(Y,X)
-EQE(X,Y) -EQEEY,Z) EQE(X,Z)
EQUALITY SUBSTITIITION AXIOMS
-EQE(X,Y) -EL(X,XS) EL(Y,XS)
-EQ(XS,YS) -EL(X,XS) EL(X,YS)
-EQ(XS,XP) -INCL(XS,YS) INCL(XP,YS)
-EQ(YS,YP) -INCLEXS,YS) INCL(XS,YP)
-EQEXS,XP) EQE(DIFEL(XS,YS),DIFEL(XP,YS))
-EQ(YS,YP) EQE(DEFEL(XS,YS),DIFEL(XS,YP))
-EQSXS,XP) EQ(COMP(XS),COMP(XP))
-EQEXS,XP) EQ(UNI(XS,YS),UNI(XP,YS))
-EQ(YS,YP) EQ(UNI(XS,YS),UNI(XS,YP))
-EQEXS,XP) EQEINT(XSSYS),INT(XP,YS))
-EQ(YS,YP) E(QINT(XS,YS),INT(XS,YP))
Y
POINTS
AND TO THE
Y
OR
X
IS
AND
LEFT
Z
ARE
OF
Y,
2
BETWEEN W AND
DENIAL
OF
Y.
THE
IS
BETWEEN
IS NOT
CL
CL
CL
CL
THEOREM
W
AtIND X
EITHER
W
IS
'CL
CL
CL
CL
THEOREM
CL
CL
-N(E,D,A)
CL
CL
Y
X AND Z AND IIF
POINT X2 IS BETWEEN Y AND W, THEN X2 IS BETWEEN X AND Z.
DENIAL OF THE THEOREM
H(A,C,E)
CL
CL
1(A,D,E)
THE
POINTS
AND
W
ARE
BETWEEN
UNION
I.
CL
IF
OF SETS
-EQ(UNI(AS,BS,UNI(BS,AS))
AXIOMS
COMPLEMENTATION
OF
OF
SETS
IS
COMMUTATIVE.
OF THE
DENIAL
IS
COMMUlTATIVE.
DENIAL
IN
C
THEN
DENIAL
OF
THE
THE
OF
THEOREM
THEOREM
UJNION
THE
OF
A
AND
B
THEOREM
THE
CL
CL
CL
CL
WANG'S
PROBLEMS
WANG
-RNM,B)
-REB, K)
-R(K,M)
REY,M) -P(Y,M) -R(F(Y),M)
R(Y,M) -P(Y,M) -R(F(Y),Y)
NEY,M)
-P(Y,M) PEY,F(Y))
R(Y,M) -P(Y,M) P(F(Y),Y)
R(Y,M) P(Y,M) R(EV,M) R(EV,Y) -P(Y,VS) -P(EV,
R(Y,B) P(Y,B) -REG(Y),B)
fN(Y,B) PEY,B) -R(G(Y),Y)
R(Y,B) P(Y,B) P(Y,G(Y))
REY,B) PEY,B) P(GEY),Y)
fNtY,B) -PEY,B) REV,B) REV,Y) -PEV,V) -P(V,Y)
REY,K) -R(Y,M)
P(Y,K)
R(Y,K) -R(Y,B) P(Y,K)
R(Y,K) R(Y,M) REY,B) -P(Y,K)
EQUALITY AXIOMS
RSX,X)
-R(X,Y) R(Y,X)
-R(X,Y) -R(Y,Z) R(X,Z)
-E(X,Y) -P(X,Z) P(Y,Z)
-R(X,Y) -P(Z,X) P(Z,Y)
-R(X,Y) R(F(X),F(Y))
-R(X,Y) R(G(X),G(Y))
THIS
SOD,A,C)
TB.
OF
EMPTY
INCLUSION
S1.
TRIANGLE
II(E,A,C)
-1E(D,E,A)
THEOREM
SET
AXIOMS OF
-INCLE(ONI
THEOREMI
POINTS
THEN
DEFINITION
THE COMPLEMENT OF THE COMPLEMENT OF A SET IS THAT SET
THEOREM
DENIAL OF THE THEOREM
CL -EQ(COMP(COMP(AS)),AS)
CL
YZ
THIE
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
AND
ELEMENTARY SET THEORY
-EL(X,EMPTY)
-INCLCXS,YS) -EL(X,XS) EL(X,YS)
INCL(XS,YS) EL(DIFEL(XS,YS,XS)
INCL(XS,YS) -EL(DIFEL(XS,YS,YS)
EL(X,XS) EL(X,COMPEXS))
-EL(X,XS) -El.(X,COMPEXS))
CL
DISTINCT
-CI(Al,A2,A3)
CL
CL
CL
CL
CL
CL
CL
LINE
C(D,E,A3)
THEOREM
B(A,D,C)
THE
COLINEAR
l(D,E,Al)
CL
CL
T7. GIVEN 2
AND
IS
W
THEOREM
(A, D, E )
(C(A,D,AU)
-(:(A,E,AO) -C(D,E,AO)
THEOREM 54. IF A AND B ARE CONTAINED
CL
INCL(AS,CS)
CL
INCL(BS,CS)
CL
(AS,BS),CS)
THEOREM
BETWEEN
NOT
AND Z,
GIVEN TWO DISTINCT POINTS X AND Y AND THREE POINTS ZC, Z2,
AND Z3 WIIICH ARE COLINEAR WITH X AND Y, THEN ZS, Z2, AND
ARE COLINEAR.
DENIAL OF THE THEOREM
CL -EQIUAL(D,E)
CL
CL
C(D,E,A2)
CL
Y
Y,X, ANn Z,
X ANn Y, ANID X
IF
THE
Y,
THEN
Z3
CL
DISTINCT
-IKQUAL(A,D)
-EQUAL(A,E)
-EQUAL(D,E)
OF
THEOREM
THEOREIi T1O.
THEOREM
MIDPOINT.
A
DENIAL
AXIOM
OF THE
THE
W1ITH
THEOREM, S3. THIE
X
IS
B(A,X,D)
CL -EQUAL(A,C)
CL
X IS
Y,
SEGMENT
THAT
BETWEEN
CL
OF
C2, AND C3
GIVEN 3 DISTINCT AND COLINEAR POINTS X,
A FOURTH POINT W COLINEAR
X AND Y,
WJITH X AND Z AND COLRNEAR W1ITH Y AND Z.
THEOREM S2. THE INTERSECTION
CL -EQ(INT(AS,BS),INT(BS,AS))
RELATION.
DENIAL
FOR
ON
CL
AND
EACH
FOR
-EQUAL(A,D)
-L.(A,X,D,X)
THEOREM
CL
SYMMETRIC
A
DENIAL
AND
X
ALL
BETWEEN
CL
IS
-EQUAL(A,D)
-N(A,X,D) -LEA,X,D,X)
THEOREM
CL
X
ALL
-REC,C,D)
WITH
CL
L(Y,V,W,Z)
LEV,Y,W,Z)
-REC,B,A)
THEOREM
CL
SUBSTITUTION AXIOMS
BEW,Z,Y)
EQUAL(F4(Vl,V2,V3,X),F4(Vl,V2,V3,Y))
-EQUALEX,Y)
Cl,
DENIAL OF
Hl.
EQUALIF4AX,V1,V2,V3),F4(Y,V1,V2,V3Ii
EQUAL1PIVVl,X,V2,V3),F4(Vl,Y,V2,V3))
EQUAL(FN(Vl,V2,X,V3),F4(Vl,V2,Y,V3))
-EQUAL(X,Y)
THE THREE POINTS POINTS
SET ARE NOT COLINEAR.
C(Cl,C2,C3)
FOLLOWING EQEXS,YS) REPRESENTS EQ.UALITY OF SETS, EQE(X,Y)
REPRESENTS EQUALITY OF ELEMENTS. DIFEL(XS,YX) REPRESENTS ARl
ELEMENT IN XS BUT NOT IN YS.
EQUAL(F2(VY,V2,V3,X,V4),F2(VD,V2,V3,Y,V4))
EQUALF2(Vl,V2,V3,V4,X),F2(V1,V2,V3,V4,Y))
EQUALEF3(X,V1,V2,V3,V4),F3EY,V1,V2,V3,V4)))
EQUALEF3tVl,X,V2,V3,V4 ),F3(VD,Y,V2,V3,V4))
EQUAL(F3(Vl,V2,X,V3,V4),F3(VD,V2,Y,V3,V4))
EQUAL(F3(Vl,V2,V3,X,V4),F3(VD,V2,V3,Y,V4))
EQUAL(F3(Vl,V2,V3,V4AX),F3(VY,V2,V3,V4,Y))
-EQUAL(X,Y)
-FQUAL(X, Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
AND Z IMPLIES
IN ANY ORDER.
THEOREM
IN THIE
EQUALtF2tVl,X,V2,V3,V4),F2tVl,Y,V2,V3,V4))
EQUALCF2(EV,V2,X,V3,V4),F2(Vl,V2,Y,V3,V4))
-EQUAL(X,SY)
-EQUAL(X,Y)
X, Y,
DENIAL OF THE
IA,D,E)
-C(A,E,D-) -C(D,A,E) -C(D,E,A) -C(E,A,D) -C(E,D,A)
C
THEOREll T12.
CL
CL
CL
CL
CL
CL
CLAUSES
C(X,Y,Z)
THE COLINEARITY OF 3 POINTS
THEIR COLINEARITY WHEN TAKEN
DIMlENSION AXIOM
EQUALITY
4
C(X,Y,Z)
C(X,Y,Z)
AXIOM
EIIUALEF2(X,V1,V2,V3,V4),F2tY,V1,V2,V3,V4))
-EQUAL(X,Y)
-EQUALDX,Y)
-EQUAL(X,Y)
THEOREM
CL
IS BETWEEN X AND Z, AND
X IS THE SAME AS THAT
-EQUAL(X,Y) -L(V,W,X,Z) L(V,WI,Y,Z)
-EQUAL(X,Y) -L(V,W,Z,X) LIV,W,Y,Z)
-EQUAL(X,Y) EQUAL(FI(X,V1,V2,V3,V4),FP(Y,V1,V2,V3,V4))
-EQUALEX,Y) EQUAL(FI(Vl,X,V2,V3,V4),FI(Vl,Y,V2,V3,V4))
-EQSUAL(X,Y) EQUAL(F)(Vl,V2,X,V3, V4),FltVl,V2,Y, 3,V4))
-F-QUAL(X,Y) EQUAL(FlEVl,V2,V3,X,V4),FltVl,V2,V3,Y,V4))
-EQUAL(X,Y) EQLSALIFE(VS,V2,V3,V4,X),FE(Vl,V2,V3,V4,Y))
T3. FOR
-N(X,D,D)
CL
B(Y,W,Z)
B8W,Y,Z)
-BEW,Z,X)
THEOREM
CL
CL
Y
W TO
LOWER
-EQUAL(X,Y) -L(V,X,IJ,Z)
THEOREM T2.
CL
CL
CL
-I.(X,II,X,V) -L(Y,W,Y,V) -L(Z,W,Z,V) UJPPER DIMENSION AXIOM
EQUAL(W,V) B(X,Y,Z) B(Y,Z,X) B(Z,X,Y)
-I.(V,X,V,Xl) -L(V,Z,V,Zl) -B(V,X,Z) CONTINUITY AXIOM
-N(X,Y,Z) L(V,Y,V,FS(X,Y,Z,Xl,Zl,V))
-.(YV.X,V,Xl) -L(V,Z,V,ZlC -B(V,X,Z)
-OE(X,Y,Z) B(XE,F5(X,Y,Z,XO,Zl,V),Zl)
EQUAL(X,X)
EQ.UALITY AXIOMS
THEOREM
CL
POINT
FROM
-II(C2,C3,Cl)
-I,(C3,Cl,C2)
EQUALtY,X)
-I(X,Y,Z)
-E,(Y,X,Z)
-I;(X,Z,Y)
CL
SEGMENT CONSTRUCTION
-B(X,W,Z)
-REW,X,Z)
N)(A,D,E)
-fI(A,C,D)
-I:(A,D,C)
THE01EMl T10.
FOR GEOMETRY
IE(X,Y,F4(X,Y,W,V))
I.(Y,F4(X,Y,W,V),W,V)
-UE(C1,C2,CO)
-EtQUALtX,Y)
IF THE POINTS Y AND W ARE BETWEEN X AND Z, THEN EITHER
W OR W IS BETWEEN X AND Y.
DENIAL OF THE THEOREM
Y IS BETWEEN X AND
N(A,C,E)
DEFINITION OF COLINEARITY IS DEFINED BY THE FOLLOWING
CL -C(X,Y,Z)
B(X,Y,Z) B(Y,X,Z) B(X,Z,Y)
CL
IDENTITY AXIOM FOR BETWEENNESS
-O,(X,Y,X) EQUAL(X,Y)
TRANSITIVITY AXIOM FOR BETWEENNESS
-H(X,Y,V) -B(Y,Z,V) B(X,Y,Z)
CONNECTIVITY AXIOM FOR BETWEENNESS
-I:(X,Y,Z) -B(X,Y,V) EQUALEX,Y)
fS(X,Z,V) B(X,V,Z)
L.(X,Y,Y,X)
REFLEXIVITY AXIOM FOR EQUIDISTANCE
IDENTITY AXIOM FOR EQUIDISTANCE
-I.(X,Y,Z,Z) EQUAL(X,Y)
-I.(X,Y,Z,V) -L(X,Y,V2,W) L(Z,V,V2,W) TRANSITIVITY AXIOM FOR EQUIDISTANCE
-,0(X,14,V) -B(Y,V,Z) B(X,F(EW,X,Y,Z,V),Y)
PASCH'S AXIOM
-fS(X,W,V) -B(Y,V,Z) B(Z,W,F1(W,X,Y,Z,V))
EUCLID'S AXIOM
-S(X,V,W) -B(Y,V,Z) EQUAL(X,V) B(X,Z,F2(W,X,Y,Z,V))
-II(X,V,W) -S(Y,V,Z) EQUAL(X,V) B(X,Y,F3(W,X,Y,C,V))
-I;(X,V,W) -B(Y,V,Z) EQUAL(X,V) B(F2(W,X,Y,Z,V),W,F3(W,X,Y,Z,V))
-I.(X,Y,X1,Yl) -L(Y,Z,Yl,Zl) -L(X,V,X1,V1) FIVE SEGMENT AXIOM
-I.(Y,V,Y1,Vl) -B(X,Y,Z) -B(XE,Yl,ZS) EQUAL(X,Y) L(Z,V,CZ,Vl)
-FQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
-EQUAL(X,Y)
CL
CL
CL
CL
THEOREM 11.
MEANS THE
THE DISTANCE
THAN
-B(C,AO,D)
-II(A,AO,E)
THEREM T9.
CL
IS DEFINED.
DENIAL OF THE THEOREM
THEN DOMED) a RGEtY).
DENIAL OF THE THEOREM
B(X,Y,Z)
FOLLOWING
L(W,X,Y,Z) MEANS
FROM Y TO Z.
CL
XY
-rDEF(A,B)
G. TARSKI'S AXIOMS
IN
THEOREM
UNIQUE IDEtITITY J SIJCH THAT JX IS DEFINED.
DENIAL OF THE THEOREM
THEOREM
CL
CL
THE
CL
DOMED)
IF
SUCH THAT XI IS nEFINED.
UNIQUE IDENTITY
DENIAL OF
DIEFEJ,A)
THEOREM
CL
CL
CS.
CL
I)EFtA,Q)
IDENTEQ)
-FQUAL(DOM(A),Q)
THEOREM
CL
CL
P(Al,C,D)
-F:QUAL(A2,Al)
PROBLEM
IS
EXQ1
PROPOSED
BY
Y)
AXIOMS
IS ALSO.
779
MC CHAREN et al.: AUTOMATED THEOREM-PROVING PROGRAMS
CL LE (1,A)
CL -I.Ell,X) -Lf(X,N) D(X)
CL -[2(X) LE(1,X)
CL -NI)X) LE(X,N)
CL -I.QUAL(X,Y) EQUAL(S(X),S(Y))
CL -I.E(X,Y) LE(S(X),S(Y))
CL -I.E(S(X),X)
CL ['( 1)
CL 'i(N)
CL I.E(X,S(X))
CL -UIW(iJ1,X,Y,Z) -LE(W.Jl,'W2) UE)WZ2, X, Y, Z)
CL -tU(W,X,Y,CX) -EQUAL(S(Zl),Z2) -D(Z2) -LFE((X,Z2),II)
CL I.E(X,X)
CL -I.E(X,Y) -LE(Y,X) EIUALIX,Y)
CL -I.E(X,Y) -LE(Y,Z) LE(X,Z)
CL I.E(X,Y) -EQIJAL(X,Y)
CL -FQUALtX,Y) -LE)X,Z) LE)Y,Z)
CL -EQUAL(X.Y) -LE(Z,Y) LE(Z,Y)
CL I:QUAL(X,X)
CL -I{QUALI(XY) EtlUAL(Y,X)
CL -EQUALIX,Y) -EQUALIY,Z) EQUALtX,Z)
THIS PROBLEM IS EXQ2 PROPOSED BY WANG
CL -R(M,B)
CL It( B, K) R(M,K)
CL RtY,J) -R(Y,K) P(Y,J)
CL R(Y,J) RtY,K) -P(Y,J)
CL R(Y,M) -P(Y,M) -R(F(Y),M)
CL R(YM) -P(Y,M) -R(FIY),Y)
CL E(Y,M) -PI(Y,RM) P(Y,F(Y))
CL RIY,M) -PIY,M) P(F(Y),Y)
CL
I(Y,M) PFY,I) R(V1,M) R(VI,Y) -PtY,Vl) -P(YI,Y)
CL
A(Y,B) P(Y,B) -R(G(Y),B)
CL RI(Y,B) P(Y,B) -R(G(Y),Y)
CL [X(Y,B) P(Y,B) P(Y,G(Y))
CL
lO(Y, B) P(Y,B) P(G(Y),Y)
I EY,B)
CL
-P(Y,E) R(V,B) RIV,Y) -P(Y,V) -P(V,Y)
CL R(Y,K) -R(Y,li) P(Y,K)
CL
E(Y,K) -R(Y,B) P(Y,K)
CL lI(Y,K) R(Y,Vl) R(Y,B) -P(Y,K)
EQUALITY AXIOMS
CL ttX,X)
CL -FI(X, Y) R(Y,X)
CL -55(X,Y) -R(Y,Z) RIX,Z)
EQUALITY SUBSTITUTION AXIOMS
CL -R(X,Y) -P(X,Z) P(Y,Z)
CL -DI(X,Y) -PIZ,X) P(Z,Y)
CL -1IIX,Y) RNF(X),F(Y))
CL -RIX,Y) RtGIX),G(Y))
THiEOREIl PV2.
CL QlIJ,T,EX)
CL LE(ltl)
CL -fr5S(1l),VT,EX) -EQUAL(E(EX,1),VT)
THIS PROBLEtl IS EXS3 PROPOSED BY HANG
CL -EI4, B)
CL RIY,J) -R(Y,K) P(Y,J)
CL R(Y,J) R(Y,K) -PlY,J)
CL
(Y,Mf) -P(Y,M) -RtFtY),M)
CL RIY,M) -PIY,MI) -RIFIY),Y)
CL RtY,M) -P(Y,M) P(Y,F(Y))
CL It(Y, M) -P(Y, M) PIF(Y),Y)
CL II(Y,M) P(Y,M) R(Vl,M) R(Vl,Y) -P(Y,Vl) -P(Vl,Y)
CL RYI,B) PIY,B) -R(G(Y),B)
CL
N(Y,B) P(Y,B) -R(G(Y),Y)
CL ltIY,B) PtY,B) P(Y,G(Y))
CL R(Y,B) P(Y,B) P(G(Y),Y)
CL RlIY,B) -P(Y,B) RtV,B) R(V,Y) -P(Y,V) -P(V,Y)
CL ;I(Y,K) -R(Y,M) PtY,K)
CL IY, K) -RIY,B) P(Y,K)
CL RIY,K) R(Y,M) RIY,B) -P(Y,K)
EQUALITY AXIOMS
CL
(NX,Y)
CL -R(X,Y) REY,X)
CL -REX,Y) -RtY,Z) R(X,Z)
EQUALITY SUBSTITUTION AXIOMS
CL -Il(X,Y) -P(X,Z) PtY,Z)
CL -IIX,Y) -P(Z,X) P(Z,Y)
CL -R(X,Y) R(F(X),F(Y))
R(G(X),G(Y))
CL -ECX,Y)
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
THEOUEM PV4.
CL -DllIVJ,VT,VX) -LE(M,N) Q3(S(M),F(M,VJ,VT,VX),VX) nENIAL OF THE THEOREM
CL -4l(VJ,VT,VX) -LE(M,N) UBF(fiM,VJ,VT,VX),VX,l,Mi)
CL -Ql(VJ,VT,VX) -LE(M,N) LEt1,G(M,VJ,VT,VX))
CL -41(VJ,VT,VX) -LE(M,N) LE(G(tO,VJ,VT,VX),M1)
CL -Ql(VJ,VT,VX) -LE(M,N) EQUAL(E(VX,C(M,VJ,VT,VX)),F(tA,VJ,VT,VX))
CL Ql(J,T,EX)
CL LE(l,M)
CL LECS(M ),N)
-Q3(S(SCM)),VT,EX) -LE(1,VK) -LE(VK,S(P)) -EQUAL(E(EX,VK),VT)
L. PROGRAMl VERIFICATION (II)
MAXtX,X,X)
MAX(CZERO,X,X)
MIN(CZERO,X,CZERO)
MlN(X,X,X)
MIN(CONE,X,X)
MIN(Y,Z,X)
-MAXtX,Y,Z) MAX(Y,X,Z)
-MINtX,Y,Z) MAXtX,Z,X)
MINtX,Z,X)
-MAXtX,Y,Z)
-M4INIX,Y,XY) -MINIY,Z,YZ) -MIN(X,YZ,XYZ) MIN(XY,Z,XYZ)
-MINIX,Y,XY) -MINIY,Z,YZ) -MIN(XY,Z,XYZ) MIN(X,YZ,XYZ)
-MAXIX,Y,XY) -MAX(Y,Z,YX) -MAXtX,YZ,XYZ) MAXIXY,Z,XYZ)
-tlAXIX,Y,XY) -MAXIY,Z,YZ) -MAXIXY,Z,XYZ) MAXIX,YZ,XYZ)
-MINIX,Z,X) -MAX(X,Y,XD) -MIN(Y,Z,Yl) -MlNtZ,Xl,Zl) MAX(X,Yl,Zl)
-MINIX,Z,X) MAX(X,Y,X1) -MIN(Y,Z,Yl) -MAXtX,Yl,Zl) MIN(Z,Xl,ZO)
-MIN(X,Z,Z) -MAXtY,ZJY1) -MIN(X,Y,Xl) -MIN(X,Yl,ZO) MAX(Z,Xl,Zl)
-tIIN(X,Z,Z) -MAX(Y,Z,Yl) -MIN(X,Y,Xl) -MAX(Z,Xl,Zl) MIN(X,Yl,Zl)
-MlN(X,Y,Z)
CL -1IN(BS,E2,A2)
K. PROGlAM VERIFICATION (1)
THE FOLLOWING SETS OF CLAUSES AROSE IN A NATURAL MANNER
FROM WORK DONE IN PROGRAM VERIFICATION.
PV1.
-f11(X,Y,Z) -LEIX,Y) Q2(X,Y,Z)
CL -QD1(X,Y,Z) LE(X,Y) QS(X,Y,Z)
CL -Q2(X,Y,Z) Q4(X,Y,Y)
CL -f13(X,Y,Z)
QI(X,Y,X)
THEOREM
CL
AXIOMS
CL LE(XCX)
CL -I.E(X,Y) -LE(Y,X) EQUAL(X,Y)
CL -I.EtX,Y) -LE(YVZ) LE(X,Z)
CL L.E(X,Y) LE(Y,X)
CL LE(X,Y) -EQUAL(X,Y)
CL -EQUAL(X,Y) -LEIX,Z) LEtY,Z)
CL -FQUAL(X,Y) -LE(Z,X) LEIZ,Y)
CL
CL
CL
(lA, B,C)
DENIAL OF THE THEOREM
-I14(A, ,W) -LE(A,W) -LE(B,W) -LE(W,A)
-Q14(A,B,W) -LE(A,W) -LEIB,W) -LE(W,B)
COMMON.
-I1(VJ,VT,VX)
-TI3(VJ, VT,VX)
-(I3(VJ,VT,VX)
-(I4(VJ,VT,VX)
-I14(VJ,VT,VX)
CL -(I5CVJ,VT,VX)
CL -QE(VJ,VT,VX)
-(12(VJ,VT,VX)
A)
CL
CL
EQUAL(PD(S(X)fl,X)
EQUAL(S(PD(X)),X)
-EQUAL(PD(X),PDtY)) EQUAL(X,Y)
-EQUAL(S<X),ISY)) EQUAL(X,Y)
-EQUALtX,Y) EQUALtPDIX),PD(Y))
-EQUAL(X,Y) EQUAL(S(X),S(Y))
-LQUAL(X,Y) EQUALIAIX),A(Y))
LT(PD(X),X)
LT(X,S(C))
-LT(X,Y) -LT(Y,Z) LT(X,Z)
LTOX,Y) LT(Y,X) EQUAL(X,Y)
CL
CL
CL -LTIX,X)
CL -LTIX,Y) -LTIY,X)
CL -EQUALIX,Y) -LT(X,Z) LTIY,Z)
CL -EQUAL(X,Y) -LT(Z,X) LT(Z,Y)
CL
(A V B)'
EQUAL(X,X)
THEO:EUM El.
DENIAL OF THE THEOREM
CL -LT(N,J)
CL LTIK,J)
CL -LT(K,I)
CL LT(I,N)
CL LT(AIJ),A(K))
CL LTIX,I) -LTIX,J) -LT(AIX),A(K))
CL -LTIONE,I) LT(X,I) LTIN,X) -LT(A(X),APEIlI)))
CL -LTIONE,X) -LT(X,I) -LT(A(X),A(PDIX)))
CL -LTIQ,I)
CL -LTtJ,Q)
CL LT(A(Q),A(J))
THEOI:EM E2.
DENIAL OF THE TIHEOREt'
CL -LTtN,J)
CL LT(K,J)
CL -LT(K,I)
CL LT(I,N)
CL LT(AIJ),A(K))
CL LTtX,I) -LTIX,J) -LT(AtX),ACK))
CL -LT(ONE,I) LT(X,I) LT(N,X) -LT(A(X),A(PDtI)))
CL -LT(ONE,X) -LT(X,I) -LTtAIX),AIPD(X)))
CL LT(J,I)
THEORIEM E3.
CL -LT(N,J)
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
LT(K,J)
-LT(K,I)
Q2(VJ,E(VX,1),VX)
Q3(S(1),VT,V
-LE(VJ,N) Q4(VJ,VT,VX)
LE(VJ,N) Q7(VJ,VT,VX)
-D(VJ) -LEIE(VX,VJ),VT) Q6(VJ,VT,VX)
-D(VJ) LEIE(VX,VJ),VT) Q5(VJ,VT,VX)
-D(VJ) Q6(VJ,E(VX,VJ),VX)
Q3ISIVJ),VT,VX)
CL
CL
CL
CL
CL
CL
CL
CL
1(I )))
-LT(Q,I)
-LT(J,Q)
-LT(A(J),A(I))
LT(A(Q),A(K))
Eli.
DENIAL OF THE THEOREM
-LT)(N, C)
-LT(K,L)
LTIL,H)
LT(ONE,L)
LTI(AIK),AIPDIL)))
LT(X,L) -LT(X,S(I)) -LT(A(X),A(K))
-LT(ONE,L) LT(X,L) LT(E,X) -LT(A(X),AIPDIL)))
-LT(OIJE,X) -LT(X,L) -LT(A(X),A(PD(X)))
THEOi:EII E5.
CL
CL
CL
CL
CL
CL
CL
CL
DENIAL OF THE THEOREM.
LUMI,N)
-LT(A(J),A(K))
LT(A(Q),A(K))
LT(X,I) -LT(X,J) -LT(A(X),A(K))
-LT(ONE,I) LT(X,I) LT(N,X) -LT(A(X),A(
-LT(ONIE,X) -LT(X,I) -LT(A(X),AlPD(X)))
THEORiEM
THE RECIAINING THEOREMS HAVETHE FOLLOWING SET OF AXIOMS IN
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
CL
MAX(CONE,X,CONE)
SAM'S LEMMA. LET L BE A MODULAR LATTICE WITH 0 AND 1.
SUPPOSE THAT A AND B ARE ELEMENTS OF L SUCH
THAT AVB AND A&B BOTH HAVE COMPLEMENTS. THEN
B)) & ((A V B)' V ((A & B)'
((A V B)' V ((A & B)'
CL fIIN(A,B,C)
CL MAX(D), C, CONE)
CL
MIN(8,D,E)
CL MIIN(A,E,CZERO)
CL MAX(B,A2,B2)
CL MAX(A,B2,CONE)
CL MAX(A, B,C2)
CL MINA2, C2, CZERO)
M1IN(D,A,D2)
CL
CL IIAX A2, D2, E2)
CL hIN(D,B,A3)
CL IAX(A2,A3,B3)
DENIAL OF THE THEOREMl
THEOI;EIi PV3.
CL -'fl(VJ,VT,VX) -LE(M,N) Q3(S(ti),F(tA,VJ,VT,VX),VX) DENIAL OF THE THFOREM
CL -fllt VJ,VT,VX) -LE(M,N) UB(Ftt,V.YJ,VT,VX),VX,l,t.)
CL -Qll(VJ,VT,VX) -LE(ttA.I) LE)OG(M4,VJ,VT,VX),ti)
CL -frl(VJ,VT,VX) -LE(Ml,N) EQUAL(E(VX,(M1,VJ,VT,VX)),F(tI,VJ,VT,VX)
CL
)ll)J,T,EX)
CL LEC( 1, b1)
CL l.EtStM),N)
CL -fl3(SISIM)),VT,EX) -UB(VT,EX,l,StM)
J. AXIOMS FOR A MODULAR LATTICE
CL
IB(W, X,Y, Z2)
DENIAL OF THE THEOREM
-LTI(t,It)
LT(I,M)
L1T(I,NI)
-LT(I,UE4E)
LTMAUI ),A(I))
LT(X,I) -LT(X,SOII)) -LT(A(X),A(tI))
-LT(OIE,I) LT(X,)) LT(N,X) -LT(A(X),A(PD(I)))
-LT(O:IE,X) -LTtX,I) -LT(A(X),A(PD(XO))
780
IEEE TRANSACTIONS ON COMPUTERS,
THEOFLEli E6.
DERIAL OF THE THEOREM
CL -I.TI(N, K)
CL -I.T (K,I)
CL LTI,M)
CL -LTI ,1i)
CL LT( I, l)
CL -LT(I,OlJE)
CL -UQUAL(K,Il)
CL LTIA(M) ,A(MK))
CL -LT(OXJE,X) -LT(X,I) -LT(A(X),A(PV(X)))
CL -LTtOIIE,X) LT(I,I) LT(N,X) -LT(A(X),A(PD)I)))
CL LT(X,I) -LT(E,S(PI)) -LT(A(X),A(K))
t-iODELS (EQUAL ITY FOPRUL AT
HENKI N
0.
THE FOLLOWING WLERE OBTAltNED IN A
FORMULATION OF HENIKIN MODELS.
NESTEr MANNER
AS
AUGUST
ION)
IN' TIE PREVIOLUS
CL -LE(X,Y) EQUAL(F(X,Y),E)
CL -EQUAL(F(X,Y),E) LE(X,Y)
CL LE(F(X,Y),X)
CL LE(F(F(X,Z),F(Y,Z) ),F(F(X, Y),Z))
CL LE(E,X)
THEOf.Etl E7.
CL -LE(X,Y) -LE(Y,X) EQUAL(X,Y)
CL LE(X,D)
CL -LT(O,XE,X) -LT(X,Pl) -LT(A(X),AIPDtX)))
THEOREtM HP1. XfI = 0.
CL -EQUAL(F(A,D),E)
DENIAL OF THE
THEOR:EMi HP2. O/X = 0 FOR ALL X.
CL EQUAL(F(X,D),E)
CL -EQUAL(F(E,A),E)
ADDITIONAL LEtMMA
EtEIIAL OF THE THEOREM
CL -LT(N,L)
DENIAL OF TIIE THEOREtI
CL
LTIDOIE,L)
CL LT(AEL),A(PD(L)))
CL -LTMOE,N) LT(X,tl) LT(N,X) -LT(A(X),A(PDEN)))
CL
Hl. LI1IlT THEOREtIS
CL
LQIAL(SUI(X,CO),X)
CL
AXIOM 1
AXIOM 2
CL -I.TtCO,X) LT(CO,FP31(X))
CL -I.TtCO,X) -LT(ABS(SUM(Y,MHtIUS(A))),FP31(X))
CL
CL
CL
CL -I.T(CO,X)
CL
CL
CL
CL
DENIAL OF THE THEOREM
DENIAL OF THE THERREIV
-LT(ABSESUM(Y,O41INUS(A))),FP31(X))
I.T(ABS(SUM(F(Y),IT INUS( LX)) ),X)
-I.T(CO,X) LT(CO,FP32(X))
-I.T(CO,X) -LT(ABS(SUIl(Y,MlINEIS(A) )), FP32(X))
I.T(ABS(SUIE(G(Y),I,INUS(L2))),X)
I.T)C0,1)
-I.T(CO,X) LT(ABS(SUM(FP33(X),I.i;NUS(A))),X)
-I.T(CO,X)
-U.T(ABS(SUIl(SUMF(FfP33(X)),tMiRE4US(LU)),SU
I,MIEUS(L2)))),B)
5);(G(FP33(X)),
CL -EQUAL(X,Y) -LT)X,Z) LT(Y,Z)
EQUALITY
CL -F:QUAL(X,Y) -LT(Z,X) LT(Z,Y)
CL -F:QUAL(X,Y) EQUAL(ABS(X),ABS(Y))
CL -FQUAL(X,Y) EQUAL(SUI.I(X,Z),SUEIl(Y,C))
SUBSTITIITUTIO
AXIOM4S
CL -EQUAL(X,Y) EQUAL(SUM(Z,X),ESUl(Z,Y))
CL -EQUAL(X,Y) EQUAU(MINUS(X),fIlMIIUS(Y))
CL -EQUAL(X,Y)
CL -EQUAL(X,Y)
CL -EQUAL(X,Y)
CL -FQUAL(X,Y)
CL -QEUAL(X,Y)
CL -EQUAL(X,Y)
THEOREM) HP6.
CL
CL
CL
CL
AXIOM
AXIOM
5
6
AX l
CL
-I.T(CO,X) -LT(ABS(SUM(Y,MINUS(A) )),FP31(X))
LT(ABS(SUM( F( Y),MINUSE Ll) ) ) ,X)
LATION
ADDIT
DENIAL
TRArNSITIVE.
IS
OtIAL
OF
LEMMA
THE
THEOREM
<- Z, THEt) X/Z <- Y FOR ALL X, Y, AND Z.
ADDITIOtIAL LEMMtlA
DENIAL OF THE THEOREM
LE(C1,A1)
EQUAL(FMA,Al),C)
-LE(C2,B)
(-
Z/X FOR ALL X, Y, AEJD Z
ADDITIONAL LEMIMA
DEtIIAL OF THIE THEOREtM
THEOREM HP8. IF X <- Y, THEN X/Z <- Y/Z FOR ALL X, Y, ANDE Z.
ADDITIOMAL LEEtlIA
CL -LEEX,Y) LE(F(Z,Y),F(Z,X))
CL LE(A,B)
DENIAL OF THE THEOPEM
CL EQUALM(A,C1),A1)
CL EQUALEF(B,Cl),Bl)
CL -LEA1,1B1)
BE DEFINED AS
CL
-EQUAL(F(D,A),F(D,F(D,F(D,A))))
CL
EQUAL(F(D,X), F(D,F(D,F(D,X))))
HP1O.
FOR
X',
WHERE
EQUAL(FED,B),81)
X'
1/X. THENJ X2
-
X''
ADDITIONIAL LEMlMA
DENIAL OF THE THEOREM
1/X, X'
X'/(l/X').
ADDITIONAL LEMMVA
DENIAL OF THE THEOREtM
EQUAL(F(B,B1),B2)
-EQUAL(B,B2)
MPll.
THE OPERATOR
Z = 1/Z, AS
OP
CL
CL
CL
CL
CL
CL
ANY
EQUAL(F(D,A),B)
THEOREM
Oli 8
ERATI1E).
X'
IS DIFINED OEl THE SET OF
& Y'
X'/EI/Y'). & IS A
Z',
WHERE
CO1IMUTATIVE
EQUAL(F(D,X),F(F(D,X),F(D,F(D,X))))ADDITIOEIAL LEtEiMA
EQUAL(F(D,A),A1)
EQUAL(F(D,B),B1)
EQUAL(F(D,B1),B2)
EQUAL(F(A1,B2),C1)
EQUAL(f(D,A1),A2)
DENIAL OF THE THEOREtM
CL EQUAL(F(B1,A2),C2)
CL -EQUAL(C1,C2)
LT(ABS(SUM(FP33(X),MI1l)JS(A))),X)
EI. GROUP AXIOMS (EQUALITY FORMULATION)
EXISTENCE OF ANi IDENTITY
EQUAL(F(E,X),X)
EQUAL(F(X,E),X)
EQUAL(F(G(X),X),E)
EXISTENC OF INVERSES
EQUAL(F(F(X,Y),Z),F(X,lF(Y,Z)))
ASSOCIATIVE PROPERTY
EQUAL(F(X,G(X)),E)
EQUAL(X,X)
THEOREM GP1. IF X**2 = E FOR ALL X, THEtI THE GROUP IS COtMMUTATIVE.
DENIAL OF THE THEOREMi
CL EQUAL(F(X,X),E)
CL EQUAL(F(A,B),C)
CL -EQUAL(F(B,A),C)
THEOMEM GP2. ALL SUBGROUPS OF INDEX TWO ARE NORtMAL.
ESTABLISHED LEMMAS
CL EQUAL(F(E,X),X)
CL EQUAL(F(X,E),X)
CL EQUAL(F(G(X),X),E)
CL EQUAL(F(X, ((X)),E)
CL EQUALIG(G(X)),X)
CL EQUAL(G(E),E)
CL W(E)
DEtlIAL OF THE THEOREtM
CL -O(X) O(G(X))
CL -O(X) -O(Y) -EQUAL(F(X,Y),Z) O(Z)
CL OX) O(Y) O(I(X,Y))
CL
IF X/Y
-LE(X,Y) -LE(Y,Z) LE(X,Z)
EQUAL(F(A,B),Cl)
-LT(ABS(SUH(SUW(F(FP33(X)),G(FP33(X))),tIIlNUS(SUMELS,L2)))),B)
CL
CL
ADDITIOtNAL LEH1IA
DENIAL OF THE THEOREtM
THEOREM HP7. IF X <( Y, THEE) Z/Y
CU -LEE(FX,Y),Z) LE(F(X,Z),Y)
CL LE(A,B)
CL EQUAL(F(C1,B),81)
CL EQUAL(FMC1,A),Al)
CL -LE(Bl,Al)
CL
LT(ABS(SUEl(G(Y),IIII4US( L2) ) ),X )
CL
ALL X.
CL LE(A1,A2)
CL -LE(A,A2)
CL
E'IUALITY AXIOFIS
CL -LT(CO,X) LT(CO,FP32(X))
CL -LT(CO,X) -LT(ABS(SUHi(Y,IilINUS(A))),FP32tX))
CL
CL
CL
CL
CL
CL
X FOR
THEOREMl
ADDITIONAL LEMflA
DEN AL OF THE TIIEOREM
THEOREM lIP5. THE LESS THAN OR EQUAL It
CL EQUAL(F(X,E),X)
CL LE(A,A1)
CL
CL
THEOKIEM BL3.
CL -I.T(CO,X) LT(CO,FP31(X))
CL LT(CO,B)
CL -LT(CO,X)
CL -LT(CO,X)
EQUAL(F(X,X),E)
THEOr:EM
CL EQUAL(SUMl(SUM(X,Y),Z),SUM(X,SUH1(Y,Z)))
CL EQUAL(SUM(X,Y),SUM(Y,X))
CL -I.T(CO,EXA) LT(CO,H(XA))
CL EQUAL(MIINUS(SUM(X,Y)),SUM(I.tINUS(X,MINUS(Y)))
CL
ALL X.
THEONEM HP9. FOR ANY X, LET X
FOR ALL X.
CL -LE(X,Y) LE(F(X,Z),F(Y,Z))
EQUAL(H(X),H(Y))
EUtJAL(F(X),F(Y))
EQIIALUG(X),G(Y))
EQEJAL(FP31(X),FP31(Y))
EQUAL(FP32(X),FP32(Y))
EQUAL(FP33(X),FP33(Y))
CL -EQUAL(X,Y) EQUAL(Fl(X,Z),F1(Y,Z))
CL -EQUAL(X,Y) EQUAL(F1(Z,X),FS(Z,Y))
CL EQUAL(X,X)
CL -EQUAL(X,Y) EQUAL(Y,X)
CL -EQUALUX,Y) -EQUAL(Y,Z) EQUAL(X,Z)
FOR
CL -EQUAL(F(A,E),A)
LT(ABS(SUM(F(Y),MIIUS(L1))),X)
-I.T(CO,X) LT(CO,FP32(X))
-I.T(CO,X) -LT(ABS(SUM(Y,MINUS(A))),FP32(X))
I.T(ABS(SUM(G(Y), (IINUS(L2))),X)
I.T (CO, B)
-I.T(CO,X) LT(ABS(SUM(FP33(X),FIINUS(A))),X)
-I.T(CO,X)
-I.T(SUM1(ABS(SUM(F(FP33(X) ),,1INUS(Ll)) ),ABS(SO)Il(G(FP33(X) ),MIrUXLS( L2))) ), B)
AXIOMI 4
-I.TtSUIl(ABS(X),AES(Y)),XA) LT(A6S(SUEl(X,Y)),XA)
THEOREEI BL2.
CL -I.T(CO,X) LT(CO,FP31(X))
CL
CL
THEOREM HP4. X/0
AXIOM 3
AXIOM 7
THEOREM BL1.
CL
0
lIP3. X/X
ECUALEFEE,X),E)
CL -EQUAL(F(A,A),E)
-LT(X,Y) -LT(Y,Z) LT(X,Z)
-I.T(CO,X) -LT(CO,Y) LT(CO,Fl(X,Y))
-I.T(CO,X) -LT(CO,Y) LT(Fl(X,Y),X)
-LT(CO,X) -LT(CO,Y) LT(Fl(X,Y),Y)
CL -L.TtX,H(XA)) -LT(Y,H(XA)) LT(SUM(X,Y),XA)
CL -I.TECO,XA) LTECO,H(XA))
CL
CL
EQUAL(X,X)
THEOREM
CL ELQUALtSUMtCO,X), X)
CL -LT(X,X)
CL
CL
CL
CL
1976
0(X) OY) EQUAL(F(X,I(X,Y)),Y)
0(B)
EQUAL(F(B,G(A)),C)
EQUALMFA,C),D)
CL -0(D)
************************************* *
RESULTS
The following statistics report the results of early experiments using the program upon problems presented in
the previous section. Note the following:
1) The number of successful and unsuccessful unifications are presented. These values include all calls to the
unification module. In particular the uses of unification
during subsumption and demodulation tests are included.
2) As mentioned in a previous section four standard
complexity measures have been employed, as well as
nonstandard measures. The four standard types are as
follows:
a) complexity of a term = (number of function and
variable symbols that occur in the term)
b) complexity of a term = (number of function
symbols) + 2*(number of variable symbols)
781
MC CHAREN et al.: AUTOMATED THEOREM-PROVING PROGRAMS
(Let each function symbol be assigned an "intrinsic"
^Owssa;x.1
AIOn.rh
I
a< .i~uu~eu
c) complexity of a term = (the sum of the complexities associated with the function symbols) + 1 (number
of variable symbols)
d) complexity of a term - (the sum of the complexities associated with the function symbols) + 2* (number
of variable symbols)
Most of the runs utilized the first measure. Hence, unless
otherwise specified, that measure is assumed.
CONCLUSION
^
PROBLEM
INFERENCIE
GI
GI
G2
G2
G3
G3
G4
G5
G5
Gf
G7
RI
R2
R3
B1
B2
UR
HYPER
UR
83
B4
85
B6
B7
Hi
Hl
H2
H3
H4
H4
HS
H6
H7
H8
Hg
HIO
Hll
RULE
SET OF
SUPPORT
CONC
CONC
CONC
CONC
EVERYTH ING
EVERYTHING
EVERYTHI NG
EVERYTHI NG
EVERYTHING
EVERYTH NG
- CONC
EVERYTHING
EVERYTHING
EVERYTH NG
EVERYTHING
EVERYTHING
EVERYTHI NG
EVERYTHING
EVERYTHING
EVERYTHI NG
EVERYTHI NG
EVERYTHING
HYPER
UR
HYPER
UR
UR
HYPER
HYPER
HYPER
HYPER
HYPER
HYPER
UR
UR
UR
UR
UR
UR
RE
HYPER
UR
UR
UR
UR
HYPER
HYPER
HYPER
CONC
CONC
CONC
EVERYTHI NG
EVERYTHI NG
EVERYTHING
EVERYTHING
CONC
UR
UR
UR
UR
HYPER
HYPER
COtIC
CLAUSES CLAUSES
KEPT
GENERATED
270
350
44
220
173
371
86
394
370
21522
31071
2299
15600
31436
40263
42416
38833
424 16
4395
4395
1804
24
21
22
1024
660 8
4198
2 996
16 8 58
7351
377
696
960
15579
3858
7156
7369
7042
2 5144
11201
140
348
2429
39
39
19
44
61
85
45
94
84
539
1981
351
890
1178
1692
1141
1622
1141
778
781
699
23
22
23
202
282
85
343
406
1147
195
235
310
862
394
665
706
650
109
84
66
169
835
HYPER
HYPER
HYPER
HYPER
HYPER
HYPER
HYPER
HYPER
CONC
CONC
EVERYTHING
EVERYTHING
EVERYTIHING
EVERYTH NG
EVERYTHING
EVERYTHING
EVERYTH I NG
EVERYTHING
EVERYTHING
EVERYTHING
S3
HYPER
EVERYTHING
2076
735
EXQ1
EXQ2
HYPER
HYPER
HYPER
EVERYTHING
EVERYTHING
EVERYTHI NG
%4608
582
2192
Cl
C2
C3
C4
Ti
T2
T3
Si
S2
SAM'S
L EMMA
PY1
PV2
PV3
PV4
E1
E2
E3
E4
E5
E7
Epl
HP1
HP2
HP3
HP4
HP5
HP6
HP7
HP8
HP9
HP10
Hp11
**U
**U
**U
**U
**U
**U
**U
**U
**U
**U
**U
&
&
&
&
&
&
&
A
&
&
&
H
H
H
H
H
H
H
H
H
H
H
EVERYTHING
EVERYTHING
EVERYTHING
EVERYTHING
CONC
CONC
CONC
CONC
CONC
CONC
CONC
CONC
CONC
EVERYTHING
EVERYTHING
CONC
CONC
CONC
CONC
EVERYTH I NG
CONC
CONC
*NONSTAND MEANS NONSTANDARD
46931
22105
108
134
10071
1169
750
65
342
130
133
483
191
10
11
54
295
47
840
303
525
463
1143
4720
292
28
44
521
175
279
45
148
63
64
181
84
11
12
26
45
26
211
117
166
105
307
1157
CLAUSES
DELETED
0
0
0
0
17
36
11
40
36
343
1 76 7
.219
5 89
918
933
835
842
551
82
0
0
0
8
31
43
96
105
466
32
50
68
256
2 10
532
581
0
62
60
469
1515
0
6
0
56
16
118
0
49
0
0
22
36
0
0
0
0
0
6
6
10
8
25
89
**U & H MEANS UR AND HYPER
nThe problems presented in this
very trivial to the
should
paper range
useful both
to
program
The
proof has
provides specific goals
future. The results from
set
those with existing
and those currently developing
of problems for which
a
from the
quite difficult. This varied problem
yet
been obtained by
at which to aim in the
the experiments cited herein
permit comparison and evaluation of the presented proautomated theorem-proving programs.
gram with other
TIME
SUCCESSFUUL
UPINIFICATICON
1.01 SEC.
709
1.03 SEC.
701
.97 SEC.
643
.92 SEC.
463
1.93 SEC.
715
1.97 SEC.
1077
5.48 SEC.
384
2.15 SEC.
1526
2.44 SEC.
1071
54.28 SEC.
60067
2 MIN. 16.86 SEC. 1 559 36
9.83 SEC.
8564
45.08 SEC.
7769 5
2 MIN. 5.26 SEC. 225902
2 MIN. 57.19 SEC. 288648
1 MIN. 0.59 SEC.
58382
NO PROOF (4 MIN.) 273988
56.97 SEC.
211510
28.53 SEC.
22 8571
27.49 SEC.
28714
16.20 SEC.
1376 2
2.42 SEC.
20
6
2.79 SEC.
6
2.52 SEC.
5.11 SEC.
8260
23.52 SEC.
7872 5
11.91 SEC.
25559
11.97 SEC.
36671
41.39 SEC.
103514
34.17 SEC
99628
5.95 SEC.
4969
5.21 SEC.
6139
6.85 SEC.
9798
39.13 SEC.
100380
22.28 SEC.
332 84
336 15
20.82 SEC.
34185
22.75 SEC.
33220
20.32 SEC.
76309
2 MNIH. 3.17 SEC.
1 MIN. 14.04 SEC. 43402
310
2.7 SEC.
3178
4.26 SEC.
50949
OUT OF 240K OF
CORE AT 43.51 SEC.
40955
OUT OF 240K OF
CORE AT 41 SEC.
1 MIN. 21.27 SEC. 139726
NO PROOF (9 MIN.) 843992
77156
30.00 SEC.
2.5 SEC.
2.09 SEC.
26.13 SEC.
3.77 SEC.
4.95 SEC.
1.48 SEC.
2.73 SEC.
2.01 SEC.
2.88 SEC.
3.12 SEC.
3.22 SEC
1.99 SEC.
1.61 SEC.
2.81 SEC.
2.35 SEC,
1.65 SEC,
3.43 SEC.
3.64 SEC.
2.94 SEC.
3.00 SEC.
5.17 SEC.
57.68 SEC.
***%
a
388
213
36880
2591
4007
228
1371
604
605
2399
863
4
5
197
1053
117
2103
768
1607
1721
3848
17580
PERCENT SUCCESSFUL
UNSUCCESSFUL
UNIFICATION
121
134
42
41
186
337
90
385
312
124888
5766
3764
16336
175738
124 80 7
47457
311628
75392
17566
176 5 7
12940
0
0
0
2 544
38 737
10094
7953
804 18
23053
1087
15 89
2600
54669
11498
116 54
11725
114 74
299711
169214
126
3325
66936
***%
85%
84%
94%
92%
79%
76%
81%
80%
78%
33%
97%
70%
82%
56%
70%
55%
47%
74%
60%
62%
62%
100%
100%
100%
82%
67%
72%
56%
82%
79%
79t
65%
73%
74%
75%
75%
20%
20%
71%
49%
76%
43%
7962
57321
44967
95%
52
49
88%
1
2
45
994
91
2573
526
1610
3348
7004
103290
TYPE 3
TYPE 2
*NONSTAND
*NONSTAND
*NONSTAND
TYPE 3
60%
53325
106 8 7
1162
660
36
244
73
72
363
133
COMPLEXITY
63%
81%
78%
69%
86%
86%
85%
89%
89%
87%
87%
80%
29%
81%
51%
56%
45%
40%
SO%
34%
35%
14%
*NONSTAND
*NON STAND
'NONSTAND
IEEE TRANSACTIONS ON
782
COMPUTERS, VOL. c-25, NO. 8, AUGUST 1976
Northern Illinois University in 1973, where he is presently an Assistant
It appears that the program discussed here is substantially Professor.
more powerful than other such resolution-based programs
cited in the literature.
REFERENCES
Ross A. Overbeek was born on May 16, 1949.
[1] J. McCharen, R. A. Overbeek, and L. Wos, "Complexity and related
ley State College, Grand Rapids, MI in 1970
and the M.S. and Ph.D. degrees from the Pennsylvania State University, University Park, in
[2]
[3]
[4]
[5]
enhancements for automated theorem proving programs," Comput.
Math. Appi., to be published.
J. A. Robinson, "Automated deduction with hyper-resolution," Int.
J. Ass. Comput. Math., Vol. 12, pp. 23-41,1965.
R. A. Overbeek, "An implementation of hyper-resolution," Comput.
Math. Appl., Vol. 1, pp. 201-214, 1975.
S. K., Winker, "An evaluation of an implementation of qualified
hyperresolution," presented at the IEEE Workshop on Automated
Theorem Proving, June 1975.
L. Wos, G. A. Robinson, D. F. Carson, and L. Shalla, "The concept
of demodulation in theorem proving," J. Ass. Comput. Math., Vol.
14, pp. 698-709, 1967.
He received the B.Ph. degree from Grand Val-
1970 and 1971, respectively.
Since 1971 he has been a member of the faculty of the Department of Mathematical Sci-
ences at Northern Illinois University, De Kalb,
where he is now an Associate Professor of computer science. His research interests include
automated theorem proving and program verification. He has coauthored
texts in Cobol and assembler language programming.
John D. McCharen was born on October 8,
* 1941. He received the B.S. degree from Southwestern at Memphis, Memphis, TN, in 1963,
and the Ph.D degree from Louisiana State Unil versity, Baton Rouge, LA, in 1969.
*
He served one year as Assistant Professor at
Louisiana State University before joining the
Department of Mathematical Sciences at
Northern Illinois University, DeKalb, in 1970.
He joined the Computer Science Division of the
* Department of Mathematical Sciences at
Lawrence A. Wos was born on July 13, 1930.
He received the B.A. and M.S. degrees from the
University of Chicago, Chicago, IL, in 1950 and
1954, respectively. He received the Ph.D. degree from the University of Illinois, Urbana, in
1957.
Since 1957 he has been at Argonne National
Laboratories, Argonne, IL, where he is presently an Associate Mathematician. His major publications have been in automated theorem
proving.
Resolution, Refinements, and Search Strategies:
A Comparative Study
GERALD A. WILSON
Abstract-This paper describes a comparative study of six binary
inference systems and two search strategies employed in resolution-based problem solving systems. A total of 152 problems, most
of which were taken from the recent literature, were employed.
Each of these problems was attempted under a standard set of
conditions using each inference system and each search strategy,
for a total of twelve attempts for each problem. Using a variety of
Manuscript received September 9, 1975; revised March 4, 1976. Support, including personnel and some computing time, was provided by
National Science Foundation Grant GJ-43622 to the University of
Maryland. The Computer Science Center provided most of the computer
time required for this study.
G. A. Wilson was with the Departnent ofComputer Science, University
of Maryland, College Park, MD 20742. He is now with the Computer
Science Laboratory, Naval Research Laboratory, Washington, DC 20375.
J. Minker is with the Department of Computer Science, University of
Maryland, College Park, MD 20742.
AND
JACK MINKER
performance measures, a large number of hypotheses were examined in an effort to provide insight into the behavior of each inference system/search strategy combination. Whenever possible,
the authors employed distribution-free statistical tests to minimize
the subjectivity of the comparisons and hypothesis testing. Conclusions are presented concerning the effectiveness of the binary
inference systems and search strategies, some effects of employing
different problem representations, and certain characteristics of
problems found to be significant in the overall system performance.
Suggestions are made as to additional techniques that might enable
theorem provers to solve practcal problems.
Index Terms-Comparative analysis, linear resolution, P1 resolution, problem solving, proof procedure system, Q* algorithm,
resolution inference, search strategies, set of support resolution,
Z* algorithm, SL resolution, theorem proving.