Book Reviews Automated Theorem Proving 
Automated Theorem Proving: A Logical Basis 
D.W. Loveland 
North-Holland Publishing Co., New York, 1978, 
432 pp., $43.50, ISBN 0-7204-0499-1. 
To determine whether someone understands a 
text, such as a story, essay, or poem, he is asked 
questions that require him to draw inferences from 
what he has read. Since the text, questions, and 
answers are all in natural language, a theory of natu- 
ral language understanding is not satisfactory if it 
cannot support a model of how questions are an- 
swered. When linguists propose explanations for 
natural language, therefore, they must consider the 
inference procedures that will be needed to extract 
information from the representations in their theo- 
ries. 
The inference process associated with the an- 
swering of questions can be formally characterized 
as theorem proving, the subject of Loveland's book. 
Loveland presents mostly various methods of theo- 
rem proving by resolution, but the most attractive 
method he presents is a non-resolution approach 
that extends the problem reduction method in artifi- 
cial intelligence. In the problem reduction method, 
a question Q is reduced to a set of subquestions P1, 
P2 ..... Pn by application of the assertion 
P1 &P2&...&Pn ~ Q 
which is called an implication. The terms Pi and Q 
are atomic statements or their negations. Loveland 
points out that the problem reduction method is not 
complete, i.e., that it cannot always answer answera- 
ble questions. From the assertions 
P=Q, "-p ~ Q 
for example, the question Q cannot be answered yes 
(shown to be a theorem) even though that is logical- 
ly implied. (The incompleteness comes from the 
fact that negation is a primitive in first order logic. 
See Black \[1\] and Smullyan \[3\] for systems that do 
not have negation as a primitive and for which prob- 
lem reduction is complete.) 
Loveland's extension to the problem reduction 
method, named the MESON format (called a format 
because many design choices are left to the implem- 
enter), adds several rules to the problem reduction 
method which make it complete. These rules do not 
complicate the method very much; the most impor- 
tant new rule, for instance, states that when answer- 
ing a question Q, if one of the resulting subques- 
tions is ~Q, then that subquestion is considered to 
be successfully answered in the affirmative. (This 
rule is essentially proof by contradiction.) The ME- 
SON format is partially described elsewhere 
(Loveland and Stickel \[2\]), but this book is the 
source for a full description and a proof of its com- 
pleteness. 
The book is divided into six chapters. The first 
two chapters review the basic concepts of first order 
logic and explain the basic resolution procedure. 
Chapter 3 presents several refinements of resolution, 
including unit preference, set-of-support, linear re- 
finements, and model elimination. Chapter 4 dis- 
cusses subsumption, a technique that removes re- 
dundant expressions from further consideration. 
Chapter 5 adds paramodulation, the inference rule 
that handles equality in the context of a resolution- 
based theorem proving system. The last chapter is 
devoted to the MESON format. In a sense Chapter 
6 is the climax of the book because the MESON 
format is justified on the basis of theorems about 
resolution in the preceding five chapters. 
This book is a well organized and well written 
reference for mechanical theorem proving methods 
presented at the algorithmic level. More than this 
should not be expected. It assumes that the reader 
has an acquaintance with formal logic. It proves 
rigorously nearly every theorem presented, and there 
are many. Many technical terms are defined 
throughout the book, as is typical of mathematical 
treatments. Although theorem proving consists of 
two parts, a mechanism that defines a search space 
and a control that guides the search in that space, 
the techniques described in the book are only the 
space defining mechanisms. Details of the guiding 
controls are still the subject of research. 
Daniel Chester, University of Texas at Austin 

References 
1. Black, F. A deduction question-answering system, Seman- 
tic Information Processing, Marvin Minsky, ed., MIT 
Press, Cambridge, MA (1968), 354-402. 
2. Loveland, D. W. and Stickel, M. E. A hole in goal trees: 
Some guidance from resolution theory, Proc. Third 1JCAL 
Stanford, Calif. (August 1973), 153-161. Also in IEEE 
Trans. on Computers C-25 (April 1976), 335-341. 
3. Smullyan, R. M. Theory of Formal Systems. Annals of 
Mathematics Studies, No. 47, Princeton University Press, 
Princeton, N.J., 1961. 
