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blems and presents a solution method based on ET.
Section 5 gives a simple example for showing pro-
blem formalization and a solution, and illustrating the
effects of computation control. Section 6 proposes
prioritized ET rules for computation control. Correct-
ness of computation with prioritized ET rules is also
proved. Section 7 describes a method for computa-
tion control with prioritized ET rules. Section 8 ex-
plains ET rules used in this paper mainly by exam-
ples. Section 9 provides conclusions.
The notation that follows holds thereafter. Given
a set A, pow(A) denotes the power set of A. Given
two sets A and B, Map(A,B) denotes the set of all
mappings from A to B, and for any partial mapping
f from A to B, dom( f ) denotes the domain of f , i.e.,
dom( f ) = {a | (a ∈ A) & ( f (a) is defined)}.
2 INSUFFICIENCY OF THE
CONVENTIONAL THEORY
2.1 Incompleteness of the Usual Clause
Space
Let CLS be the set of all clauses consisting only of
user-defined atoms, and CLS
c
the set of all clauses
consisting of user-defined atoms and built-in atoms.
Corresponding to these, let FOL be the set of all first-
order formulas consisting only of user-defined atoms,
and FOL
c
the set of all first-order formulas consisting
of user-defined atoms and built-in atoms.
Let SKO be a mapping such that each first-order
formula in FOL
c
is transformed into a set of clauses in
CLS
c
by SKO using conventional Skolemization and
other ET rules. It is well-known that SKO transforms
each first-order formula in FOL into a set of clauses
in CLS preserving satisfiability. This enables conven-
tional resolution-based theorem proving, which moti-
vates us to consider SKO and CLS as a foundation for
logical problem solving.
However, we would like to stress that SKO and
CLS have serious limitations:
• SKO does not generally preserve the logical mea-
nings of formulas in FOL and those in FOL
c
.
• Existential quantification cannot be represented
by clauses in CLS nor those in CLS
c
.
• SKO does not generally preserve satisfiability for
FOL
c
.
Thus CLS and CLS
c
are not appropriate for entirely
solving all proof problems, QA problems, and MI
problems on FOL and FOL
c
.
2.2 A New Extended Clause Space
Conventional clauses are not sufficiently expres-
sive for equivalently representing first-order formulas
since all variables in a clause are universally quan-
tified and no existential quantification is allowed.
Instead of the usual clause space, we use an exten-
ded clause space, called the ECLS
F
space, in which
a clause may contain three kinds of atoms: built-in
constraint atoms, user-defined atoms, and func-atoms.
Variables of a new type, called function variables, ap-
pear in func-atoms in the positions of their first argu-
ments, and are existentially quantified at the top le-
vel of a clause set under consideration. Existential
quantification of usual variables in first-order logic is
alternatively represented by existential quantification
of function variables in ECLS
F
.
2.3 Model-Intersection Problems
A proof problem is concerned with checking whether
one given logical formula entails another given logi-
cal formula. The proof problems solved by conven-
tional Skolemization and resolution are on FOL, not
FOL
c
. However, the class of proof problems on FOL
is not sufficient for practical use, since it cannot deal
with most of useful built-in constraint atoms.
A query-answering (QA) problem is concerned
with finding all ground instances of a given query
atom that are logical consequences of a given formula.
The logic programming community and the seman-
tic web community consider QA problems together
with proof problems. However, they deal with only
subclasses of QA problems. No general theory of QA
problems on FOL
c
has been constructed.
The class of model-intersection problems (MI pro-
blems) has been invented for constructing a general
theory of logical problem solving (Akama and Nan-
tajeewarawat, 2016a). MI problems enable us to con-
struct a unified theory of logical problem solving. A
MI problem on extended clauses is a pair of a set of
extended clauses and an extraction mapping. The ans-
wer to a MI problem is the value obtained by applying
its extraction mapping to the intersection of all the
models of the conjunction of its extended clauses. MI
problems constitute a very large class of logical pro-
blems, and include both proof problems and QA pro-
blems (see Section 2.4 and Fig. 1).
2.4 Meaning-Preserving Skolemization
The usual clause space taken by conventional logic
programming is too small to consider all proof pro-
blems on FOL
c
and all QA problems on FOL
c
. These
Computation Control by Prioritized ET Rules
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