Function-variable Elimination and Its Limitations
Kiyoshi Akama
1
and Ekawit Nantajeewarawat
2
1
Information Initiative Center, Hokkaido University, Hokkaido, Japan
2
Computer Science, Sirindhorn International Institute of Technology, Thammasat University, Pathumthani, Thailand
Keywords:
Query-answering Problem, Equivalent Transformation, Meaning-preserving Skolemization, Problem Solving.
Abstract:
The famous proof method by the conventional Skolemization and resolution has a serious limitation. It does
not guarantee the correctness of proving theorems in the presence of built-in constraints. In order to understand
this difficulty, we use meaning-preserving Skolemization (MPS) and equivalent transformation (ET), which
together provide a general framework for solving query-answering (QA) problems on first-order logic. We
introduce a rule for function variable elimination (FVE), by which we regard the conventional Skolemization
as a kind of the composition of MPS and FVE. We prove that the FVE rule preserves the answers to a class
of QA problems consisting of only user-defined atoms, while we cannot prove it in the presence of built-in
constraints. By avoiding the application of the FVE rule in MPS & ET computation, we obtain a more general
solution for proof problems, which guarantees the correctness of computation even in the presence of built-in
constraints.
1 INTRODUCTION
One of the most important methods for proving theo-
rems is based on Skolemization and resolution. This
method, however, has a serious limitation in that it
may give an incorrect result in the presence of built-
in predicates. Consider, for example, a simple proof
problem below, which is a modification of the tax-cut
problem given in (Motik et al., 2005). Assume that
• noteq is a predicate for built-in constraint atoms
and for any ground terms t
1
and t
2
, noteq(t
1
,t
2
) is
true iff t
1
6= t
2
, and
• F
1
, F
2
, and F
3
are the first-order formulas given
by:
F
1
: ∀x,∀y,∀z :
[(hasChild(x, y) ∧ hasChild(x,z)∧ noteq(y,z))
→ TaxCut(x)]
F
2
: hasChild(Peter, Paul)
F
3
: ∃x : hasChild(Peter, x)
The problem is to determine whether E
1
logically
entails E
2
, where E
1
= F
1
∧ F
2
∧ F
3
and E
2
=
TaxCut(Peter). Since F
3
is already implied by F
2
,
we know only one person who is a child of Peter.
Hence E
1
6|= E
2
, i.e., E
1
does not logically entail
TaxCut(Peter). The correct answer to this proof prob-
lem is thus “no”.
If we take conventional proof method, however,
this problem is solved incorrectly. By applying Sko-
lemization to E
1
∧ ¬E
2
, a 0-ary function symbol, say
f
h
, is introduced and E
1
∧ ¬E
2
is converted into the
following four clauses:
TaxCut(x) ← hasChild(x,y),hasChild(x,z),
noteq(y,z)
hasChild(Peter, Paul) ←
hasChild(Peter, f
h
) ←
← TaxCut(Peter)
Application of the resolution rule three times yields
a negative clause (← noteq(Paul, f
h
). Since Paul and
f
h
are not equal, we derivean empty clause (←). Thus
a proof is obtained by Skolemization and resolution,
and the answer is “yes”, which contradicts the intu-
itive and correct answer explained earlier.
From this example, the following questions natu-
rally arise:
1. Where does this inconvenience come from?
2. How to develop a theory to deeply understand the
use of Skolemization and resolution and its limi-
tation?
3. Can we invent a new solution method to resolve
this difficulty?
We give an answer to each of these questions by re-
placing the conventional Skolemization and resolu-
212
Akama, K. and Nantajeewarawat, E..
Function-variable Elimination and Its Limitations.
In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2015) - Volume 2: KEOD, pages 212-222
ISBN: 978-989-758-158-8
Copyright
c
 2015 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

Figure 1: A conventional proof diagram.
Figure 2: An ET-based proof diagram.
tion (Robinson, 1965) with the meaning-preserving
Skolemization (Akama and Nantajeewarawat, 2011a)
and equivalent transformation (ET).
Let L
1
be the set of all first-order formulas. Let
CS be the conventional algorithm for transforming a
first-order formula into its clausal form using the con-
ventional Skolemization. Let C L
1
be the powerset
of the set of all usual clauses. Let rf denote resolu-
tion and factoring. Then a typical computation path
by the conventional proof method can be depicted by
Fig. 1. A first-order formula is converted by CS into a
set of usual clauses possibly with new function sym-
bols. For instance, an existentially quantified formula
∃x : p(x) is transformed into a clause set {(p(h) ←)},
where h is a 0-ary function symbol.
In conventional clauses, all variables are univer-
sally quantified and existential quantification can-
not be expressed. Conventional clauses are there-
fore not sufficiently expressive for representing first-
order formulas. To extend clauses with the expres-
sive power of existential quantification, variables of
a new type, called function variables, were intro-
duced (Akama and Nantajeewarawat, 2011a). A
function variable may appear in an atom of a spe-
cial kind, called func-atom, which is generally of
the form func( f,t
1
,...,t
n
,t
n+1
), where f is an n-ary
function variable or an n-ary function constant, and
t
1
,...,t
n
,t
n+1
are usual terms.
To understand the computation path in Fig. 1, we
consider a new path given in Fig. 2, where MPS
is the algorithm for transforming a first-order for-
mula into its extended clausal form using meaning-
preserving Skolemization (Akama and Nantajee-
warawat, 2011a). For instance, an existentially quan-
tified formula ∃x : p(x) is transformed by MPS into
a clause set {(p(x) ← func( f,x))}, where func( f,x)
is a func-atom and f is a 0-ary function variable,
which is not included in L
1
. An existentially quanti-
fied formula cannot be equivalently transformed into
a clausal form in the usual first-order formula space
L
1
. We extended L
1
into a new space, which includes
function variables. In Fig. 2, C L
2
is the powerset of
the set of all extended clauses, which may possibly in-
clude function variables. MPS and extended clauses
will be formally defined in Section 3, where the set of
all extended clauses is referred to as ECLS
F
.
Figure 3: Connecting the conventional proof diagram and
the ET-based proof diagram using FVE.
To connect the diagrams in Fig. 1 and Fig. 2, we
introduce in this paper a partial mapping FVE from
CL
2
to C L
1
as outlined in Fig. 3. For instance, the
clause set {(p(x) ← func( f,x))} is mapped by FVE
to {(p(h) ←)}. The conventional Skolemization, CS,
is identified as the composition of MPS and FVE in
the sense that
{CS(L) | L ∈ L
1
} = {FVE(MPS(L)) | L ∈ L
1
}.
We provein this paper that FVE preserves the answers
to proof problems in a certain restricted class. Since
resolution and factoring are ET rules in the space of
CL
1
, the conventional solution can also be regarded
as ET computation, i.e., the conventional diagram
supports a restricted form of computation compared
with the ET-based proof diagram.
The theory in this paper enables us to compare
the conventional solution and the ET-based solution
for proof problems in the common MPS & ET frame-
work. The limitation of FVE can be precisely inves-
tigated. The difficulty shown by the example at the
beginning of this section can be overcome in the MPS
& ET framework by using ET computation paths that
do not include application of the FVE rule.
The MPS & ET theory has been developed mainly
for solving query-answering (QA) problems. A QA
problem is a pair hK,ai, where K is a first-order for-
mula and a is a user-defined atom, and the answer to
this problem is the set of all ground instances of a that
are logically entailed by K.
While the answer to a proof problem is either
“yes” or “no”, which does not contain any(first-order)
term, the answer to a QA problem is a set of ground
atoms that may contain terms. MPS is necessary for
solving QA problems. Since a new term introduced
by the conventional Skolemization may affect ground
atoms in a model of a given first-order formula, the
conventional Skolemization is inappropriate for de-
veloping a solution for QA problems. So we take
MPS over CS. Since ET includes resolution and fac-
toring, we take the MPS & ET framework over the CS
& rf framework.
It was shown in (Akama and Nantajeewarawat,
2013) that proof problems constitute a specific sub-
class of QA problems. So it is natural to apply the
MPS & ET framework to solve proof problems. The
theory presented in this paper is developed as a the-
Function-variable Elimination and Its Limitations
213

ory for solving QA problems based on the MPS & ET
framework.
The rest of the paper is organized as follows: Sec-
tion 2 formalizes QA problems and proof problems.
Section 3 describes a procedure for converting first-
order formulas using MPS into equivalent formulas in
extended existentially quantified conjunctive normal
forms, and formulates QA problems in clausal forms.
Section 4 defines a target mapping, called MM, which
provides a basis for ET. Section 5 presents the main
theoretical results of this work and the FVE rule. Sec-
tion 6 explains the limitations of the FVE rule. Sec-
tion 7 answers the three questions identified in the
third paragraph of Section 1. Section 8 provides con-
clusions. The proofs of all theorems presented in this
paper can be found in (Akama and Nantajeewarawat,
2015).
The following notation is used. Given a set A,
pow(A) denotes the power set of A and fpow(A) de-
notes the set of all finite subsets of A.
2 QA PROBLEMS AND PROOF
PROBLEMS
2.1 Interpretations and Models
In this paper, an atom occurring in a first-order for-
mula can be either a user-defined atom or a constraint
atom. The semantics of first-order formulas based
on a logical structure given in (Akama and Nantajee-
warawat, 2012) is used. The set of all ground user-
defined atoms, denoted by G, is taken as the interpre-
tation domain. An interpretation is a subset of G. A
ground user-defined atom g is true with respect to an
interpretation I iff g belongs to I. Unlike ground user-
defined atoms, the truth values of ground constraint
atoms are predetermined independently of interpreta-
tions. A model of a first-order formula E is an in-
terpretation that satisfies E. The set of all models of a
first-order formula E is denoted by Models(E). Given
first-order formulas E
1
and E
2
, E
2
is a logical conse-
quence of E
1
, denoted by E
1
|= E
2
, iff every model of
E
1
is a model of E
2
.
2.2 QA Problems
A query-answering problem (QA problem) is a pair
hK,ai, where K is a first-order formula, representing
background knowledge, and a is a user-defined atom,
representing a query. The answer to a QA problem
hK,ai, denoted by ans
qa
(K,a), is defined as the set of
all ground instances of a that are logical consequences
of K. Using Models(K), the answer to a QA problem
hK,ai can be equivalently defined as
ans
qa
(K,a) = (
\
Models(K)) ∩ rep(a),
where rep(a) denotes the set of all ground instances
of a (Akama and Nantajeewarawat, 2013).
2.3 Proof Problems
A proof problem is a pair hE
1
,E
2
i, where E
1
and E
2
are first-order formulas, and the answer to this prob-
lem, denoted by ans
pr
(E
1
,E
2
), is defined by
ans
pr
(E
1
,E
2
) =

“yes” if E
1
|= E
2
,
“no” otherwise.
It is well known that a proof problem hE
1
,E
2
i can be
converted into the problem of determining whether
E
1
∧ ¬E
2
is unsatisfiable (Chang and Lee, 1973),
i.e., whether E
1
∧ ¬E
2
has no model. As a result,
ans
pr
(E
1
,E
2
) can be equivalently defined by
ans
pr
(E
1
,E
2
) =

“yes” if Models(E
1
∧ ¬E
2
) = ∅,
“no” otherwise.
3 MEANING-PRESERVING
TRANSFORMATION ON AN
EXTENDED FORMULA SPACE
After defining an extended formula space and an exis-
tentially quantified conjunctive normal form (ECNF),
a procedure for converting a first-order formula into
an ECNF using meaning-preserving Skolemization
(MPS) is recalled. The notions of an extended clause
space, a plain clause, and a QA problem in a clausal
form are then introduced.
3.1 An Extended Formula Space
We consider an extended formula space that contains
three kinds of atoms, i.e., user-defined atoms, con-
straint atoms, and func-atoms. A func-atom is an ex-
pression of the form func( f,t
1
,...,t
n
,t
n+1
), where f
is either an n-ary function constant or an n-ary func-
tion variable, and the t
i
are usual terms. There are
two types of variables: usual variables and function
variables. A function variable is instantiated into a
function constant or a function variable, but not into a
usual term. Each n-ary function constant is associated
with a mapping from G
n
t
to G
t
, where G
t
denotes the
set of all ground terms. The extended space contains
both universal quantifiers and existential quantifiers.
An extended disjunctive form is a formula of the
form
KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development
214

∀v
1
,...,∀v
m
: (a
1
∨ ···∨ a
n
∨ ¬b
1
∨ ··· ∨ ¬b
p
∨
¬f
1
∨ ··· ∨ ¬f
q
),
where v
1
,...,v
m
are usual variables, each of a
1
,...,
a
n
,b
1
,...,b
p
is a user-defined atom or a constraint
atom, and f
1
,...,f
q
are func-atoms.
An existentially quantified conjunctive normal
form (ECNF) is a formula of the form
∃v
h
1
,...,∃v
h
m
: (C
1
∧ ··· ∧C
n
),
where v
h
1
,...,v
h
m
are function variables and C
1
,...,
C
n
are extended disjunctive forms.
3.2 Conversion Algorithm
Assume that
• the initial space INI is the set of all first-order for-
mulas, and
• the target space FIN is the set of all ECNFs.
Let a formula α in INI be given as input. It is trans-
formed into a formula in FIN as follows:
1. Convert → and ↔ equivalently into ¬, ∧, and ∨,
using the following logical equivalences:
β → γ ≡ ¬β∨ γ
β ↔ γ ≡ (¬β∨ γ) ∧ (¬γ ∨ β)
2. Move ¬ inwards equivalently until each occur-
rence of ¬ immediately precedes an atom, using
the following logical equivalences:
¬(¬β) ≡ β
¬(β∧ γ) ≡ ¬β∨ ¬γ
¬(β∨ γ) ≡ ¬β∧ ¬γ
¬∀x : α ≡ ∃x : ¬α
¬∃x : α ≡ ∀x : ¬α
3. Repeatedly move ∨ in the current formula through
∃, ∀, and ∧ as far as possible using the following
logical equivalences in the left-to-right direction:
(∃x : β) ∨ γ ≡ ∃x : (β∨ γ)
(∀x : β) ∨ γ ≡ ∀x : (β∨ γ)
(β∧ γ) ∨ δ ≡ (β ∨ δ) ∧ (γ∨ δ)
Each time the first and the second equivalences
above are used, if γ includes x as a free variable,
then rename the quantified variable x in (∃x : β)
and (∀x : β), respectively, by using a new variable
name.
4. Repeatedly move ∧ in the current formula through
∀ as far as possible using the following logical
equivalence in the left-to-right direction:
∀x : (β∧ γ) ≡ (∀x : β) ∧ (∀x : γ)
5. If the current formula includes a subformula of
one of the two forms
• ∀x
1
,...,∀x
n−1
,∀x
n
: β
• ∀x
1
,...,∀x
n−1
,∃x
n
: β
such that x
1
,...,x
n−1
,x
n
are not mutually distinct,
then rename these quantified variables by using
new variable names so that different quantifica-
tions in this subformula refer to different vari-
ables.
6. If the current formula includes ∃, then:
(a) Skolemization: From the current formula, se-
lect a subformula
∀x
1
,...,∀x
n
,∃y : β,
where n ≥ 0, such that there is no further
universal quantification over this subformula.
Transform this subformula into
∃h,∀x
1
,...,∀x
n
,∀y : (β∨¬func(h,x
1
,...,x
n
,y)),
where h is a new n-ary function variable that
has not been used so far.
(b) Repeatedly move the new ∃-subformula (intro-
duced at Step 6a) through ∧ as far as possible
using the following logical equivalence in the
left-to-right direction:
(∃h : β) ∧ γ ≡ ∃h : (β∧ γ)
(c) Go to Step 3.
7. Stop with the current formula as the output for-
mula.
It was shown in (Akama and Nantajeewarawat,
2011b) that this algorithm always terminates and
yields an output ECNF in FIN that has the same logi-
cal meaning as the input first-order formula.
3.3 An Extended Clause Space
An extended clause C is a formula of the form
a
1
,...,a
n
← b
1
,...,b
p
,f
1
,...,f
q
,
where each of a
1
,...,a
n
,b
1
,...,b
p
is a user-defined
atom or a constraint atom, and f
1
,...,f
q
are func-
atoms. All usual variables occurring in C are implic-
itly universally quantified and their scope is restricted
to the extended clause C itself. The sets {a
1
,...,a
n
}
and {b
1
,...,b
p
,f
1
,...,f
q
} are called the left-hand side
and the right-hand side, respectively, of the extended
clause C, and are denoted by lhs(C) and rhs(C), re-
spectively. When n = 0, C is called a negative ex-
tended clause. When n = 1, C is called an extended
definite clause, the only atom in lhs(C) is called the
Function-variable Elimination and Its Limitations
215

head of C, denoted by head(C), and the set rhs(C) is
also called the body of C, denoted by body(C). When
n > 1, C is called a multi-head extended clause.
When no confusion is caused, an extended clause,
a negative extended clause, an extended definite
clause, and a multi-head extended clause are also
called a clause, a negative clause, a definite clause,
and a multi-head clause, respectively.
The set of all extended clauses is denoted by
ECLS
F
. The extended clause space in this paper is
the powerset of ECLS
F
.
Semantically, an extended clause corresponds to
an extended disjunctive form, and a set of extended
clauses corresponds to an ECNF. Let Cs be a set of
extended clauses. Implicit existential quantifications
of function variables and implicit clause conjunction
are assumed in Cs. Function variables in Cs are all ex-
istentially quantified and their scope covers all clauses
in Cs. With occurrences of function variables, clauses
in Cs are connected through shared function variables.
After instantiating all function variables in Cs into
function constants, clauses in the instantiated set are
totally separated.
When no confusion is caused, a clause C is also
written as
• H ← B, provided that H = lhs(C) and B = rhs(C),
• H ← B,b
1
,...,b
n
, provided that H = lhs(C) and
B∪ {b
1
,...,b
n
} = rhs(C), where n ≥ 1, and
• h ← B, provided that {h} = lhs(C) and B =
rhs(C).
3.4 Plain Clauses
A plain clause and a plain clause set are defined be-
low.
Definition 1. A clause C is plain iff the following
conditions are satisfied:
1. If a func-atom func( f,t
1
,...,t
n
,t
n+1
) occurs in C,
then t
1
,...,t
n
,t
n+1
are usual variables and are all
disctinct.
2. If func-atoms func( f,t
1
,...,t
n
,t
n+1
) and func( f
′
,
t
′
1
,...,t
′
m
,t
′
m+1
) both occur inC, thent
n+1
and t
′
m+1
are different.
A set of clauses is plain iff it consists of plain clauses
solely.
Theorem 1. If a first-order formula is converted by
the algorithm in Section 3.2 into a set Cs of clauses in
ECLS
F
, then Cs is a plain clause set.
3.5 QA Problems in Clausal Forms
Let hK,ai be a QA problem, where K is a first-order
formula. Suppose that K is converted into a set Cs
of extended clauses by the procedure in Section 3.2.
Then the QA problem hK,ai is transformed into the
QA problem hCs, ai, which is said to be in a clausal
form.
Assume that a proof problem hE
1
,E
2
i is given,
where E
1
and E
2
are first-order formulas. This proof
problem can be solved by first constructing a QA
problem hE,yesi, where
• E = E
1
∧ ¬E
2
, and
• yes is a ground atom that does not appear in E.
Since hE,yesi is a QA problem, a solution method for
QA problems can be used for solving it. That is, the
QA problem hE,yesi is transformed into a QA prob-
lem in a clausal form hCs,yesi, where Cs is a set of
extended clauses obtained from E by the conversion
procedure in Section 3.2. The answer to the QA prob-
lem hCs,yesi, i.e., ans
qa
(Cs,yes), is either the single-
ton {yes} or the empty set. As shown in (Akama
and Nantajeewarawat, 2013), the answer to the proof
problem hE
1
,E
2
i, i.e., ans
pr
(E
1
,E
2
), can be obtained
through ans
qa
(Cs,yes) as follows:
ans
pr
(E
1
,E
2
) =

“yes” if ans
qa
(Cs,yes) = {yes},
“no” if ans
qa
(Cs,yes) = ∅.
4 A TARGET MAPPING MM
A target mapping is a key concept for generating so-
lutions for QA problems. We first prove in this sec-
tion that a mapping MM on pow(ECLS
F
), which is
defined in Section 4.2, is a target mapping, i.e., it sat-
isfies Theorems 2 and 4. Transformations that pre-
serve a target mapping always preserve the answers
to QA problems. For elimination of function variables
(Theorem 8), Theorem 5 will play an important role
in Section 5.
4.1 Preliminary Notation for Defining
MM
The notation below is used in Section 4.2 for defining
MM.
1. Assumed that (i) for any constraint atom c, not(c)
is a constraint atom, (ii) for any constraint atom
c and any substitution θ, not(c)θ = not(cθ), and
(iii) for any ground constraint atom c, c is true iff
KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development
216

not(c) is not true. Given a set Cs of clauses in
ECLS
F
, let mvRhs(Cs) be defined by
mvRhs(Cs) = {mvRhs(C) | C ∈ Cs},
where for any clause C, mvRhs(C) is the clause
obtained from C as follows: For each constraint
atom c in lhs(C), remove c from lhs(C) and add
not(c) to rhs(C).
2. Assume that (i) FVar is the set of all function
variables and FCon the set of all function con-
stants, and (ii) Map(FVar,FCon) is the set of
all mappings from FVar to FCon. Given σ ∈
Map(FVar,FCon) and a set Cs of clauses in
ECLS
F
, let Csσ = {Cσ | C ∈ Cs}, i.e., Csσ is
the clause set obtained from Cs by instantiating
all function variables appearing in it into function
constants using σ.
3. Let ECLS
NFV
denote the subset of ECLS
F
that
contains all (extended)clauses with no occurrence
of any function variable.
1
Let GCL be the set
of all clauses that consist only of ground user-
defined atoms. Given a set Cs of clauses in
ECLS
NFV
, let ginst(Cs) be defined as a subset of
GCL as follows:
(a) Let Cs
1
be a ground clause set obtained from
mvRhs(Cs) by
Cs
1
= {Cθ | (C ∈ mvRhs(Cs)) &
(θ is a ground instantiation for all
usual variables occurring in C)}.
(b) Let Cs
2
be a ground clause set obtained from
Cs
1
by removing each clause whose right-hand
side contains at least one false constraint atom
or at least one false func-atoms.
(c) Then let ginst(Cs) be a ground clause set ob-
tained from Cs
2
by removing all true constraint
atoms and all true func-atoms from the right-
hand side of each clause in Cs
2
.
4. Let G denote the set of all ground user-defined
atoms. Assume that ⊥ is a ground atom outside G.
Let SEL be defined as the set of all mappings from
fpow(G) to G ∪ {⊥} such that for any sel ∈ SEL
and any X ∈ fpow(G), the following conditions
are satisfied:
• If X = ∅, then sel(X) = ⊥.
• If X 6= ∅, then sel(X) ∈ X.
Let GDC be the set consisting of every definite
clause whose body contains only ground user-
defined atoms and whose head is either a ground
user-defined atom or ⊥. Given a mapping sel ∈
1
Function constants may occur in clauses in ECLS
NFV
.
SEL and a subset Cs of GCL, let dc(sel,Cs) be
defined as a subset of GDC by
dc(sel,Cs) = {dc(sel,C) | C ∈ Cs},
where for each clauseC ∈ Cs, dc(sel,C) is the def-
inite clause obtained from C as follows:
• head(dc(sel,C)) = sel(lhs(C))
• body(dc(sel,C)) = rhs(C)
5. For any definite-clause set D ⊆ GDC, let the
meaning of D, denoted by M (D), be defined as
follows:
(a) Let a mapping T
D
on pow(G ∪{⊥}) be defined
by: for any G ⊆ G ∪ {⊥},
T
D
(G) = {head(C) | (C ∈ D) & (body(C) ⊆ G)}.
(b) M (D) is then defined as the set
S
∞
n=1
T
n
D
(∅),
where T
1
D
(∅) = T
D
(∅) and for each n > 1,
T
n
D
(∅) = T
D
(T
n−1
D
(∅)).
4.2 A Target Mapping MM
Using the notation provided by Section 4.1, we de-
fine a mapping MM (Definition 2) and prove in Theo-
rem 4 that MM is a target mapping, i.e., for any clause
set Cs,
T
Models(Cs) =
T
MM(Cs).
Definition 2. Let Cs be a set of clauses in ECLS
F
. A
collection MM(Cs) of ground-atom sets is defined as
the set
{M (D) | (σ ∈ Map(FVar,FCon)) &
(sel ∈ SEL) &
(D = dc(sel,ginst(Csσ))) &
(⊥ /∈ M (D))}.
In Theorems 2–5 below, assume that Cs is a set of
clauses in ECLS
F
.
Theorem 2. MM(Cs) ⊆ Models(Cs).
Theorem 3. For any m in Models(Cs), there exists
m
′
∈ MM(Cs) such that m
′
⊆ m.
Theorem 4.
T
Models(Cs) =
T
MM(Cs).
Theorem 5 below is used for proving the correct-
ness of elimination of function variables in Theo-
rem 8.
Theorem 5. Models(Cs) = ∅ iff MM(Cs) = ∅.
Function-variable Elimination and Its Limitations
217

5 ELIMINATION OF FUNCTION
VARIABLES
This section investigatesthe FVE partial mapping, de-
fined in Section 5.1, which maps a plain set of clauses
to a set of usual clauses. By the application of FVE,
all function variables in a given plain clause set are
instantiated. Given an arbitrary first-order formula L,
the formula MPS(L) is a plain clause set and it can
be further converted into a set of usual clauses (with-
out function variables), which can also be obtained
by applying the conversion algorithm CS (cf. Fig. 1
in Section 1) to L. The main objective of this section
is to prove Theorem 9, which states that the answer to
a proof problem is preserved by FVE transformation.
5.1 Satisfiability Preservation
Theorem 6 provides a foundation for satisfiability-
preserving transformation.
Theorem 6. Assume that Cs is a plain set of clauses
with no occurrence of any constraint atom. Then the
following conditions are equivalent:
• (∀sel ∈ SEL) : ⊥ ∈ M (dc(sel, ginst(Csσ
M
)))
• (∀σ ∈ Map(FVar,FCon))(∀sel ∈ SEL) : ⊥ ∈
M (dc(sel,ginst(Csσ))).
Now we consider transformation of Cs into Cs
′
,
where
• Cs is a plain set of clauses with no occurrence of
any constraint atom, and
• Cs
′
is obtained from Cs as follows: For anyC ∈ Cs
and any func-atom func(h,v
1
,...,v
n
,u), denoted
by f, occurring in C,
1. remove the func-atom f from C, and
2. replace every occurrence of u in C with f
h
(v
1
,
...,v
n
), where f
h
= funcSym
M
(σ
M
(h)).
This transformation defines a partial mapping,
which is called FVE.
Theorem 7. MM(Csσ
M
) = MM(Cs
′
).
Theorem 8. Models(Cs) = ∅ iff Models(Cs
′
) = ∅.
5.2 An ET Rule for Elimination of
Function Variables
Referring to the transformation of Cs into Cs
′
de-
scribed in Section 5.1, Theorem 9 yields an ET rule
for elimination of function variables, which is hence-
forth called the FVE rule.
Theorem 9. Assume that yes is a ground atom
that does not appear in Cs. Then ans
qa
(Cs,yes) =
ans
qa
(Cs
′
,yes).
5.3 A Solution Method for Proof
Problems
5.3.1 A Procedure for Solving Proof Problems
A procedure for solving proof problems is given be-
low. Let hE
1
,E
2
i be an input proof problem.
1. Construct a QA problem hE,yesi, where
• E = E
1
∧ ¬E
2
, and
• yes is a ground atom that does not appear in E.
2. Convert the QA problem hE,yesi by MPS, using
the conversion procedure in Section 3.2, into a QA
problem hCs,yesi in a clausal form.
3. Convert hCs,yesi by the FVE rule into a QA prob-
lem hCs
′
,yesi.
4. Using ET rules, transform hCs
′
,yesi into a QA
problem hCs
′′
,yesi
5. Determine the answer to the proof problem hE
1
,
E
2
i by
ans
pr
(E
1
,E
2
)=

“yes” if ans
qa
(Cs
′′
,yes) = {yes},
“no” if ans
qa
(Cs
′′
,yes) = ∅.
If the clause set Cs
′′
contains an empty clause (←),
then Models(Cs
′′
) = ∅ and thus
T
Models(Cs
′′
) = G.
In this case, ans
qa
(Cs
′′
,yes) = {yes}. If the clause
set Cs
′′
has a model, i.e., Models(Cs
′′
) 6= ∅, then
(
T
Models(Cs
′′
)) ∩ {yes} = ∅. Hence, in this case,
ans
qa
(Cs
′′
,yes) = ∅.
5.3.2 Example
To illustrate application of the above procedure, let
first-order formulas F
1
, F
2
, and F
3
be given by:
F
1
: ∀x : [barber(x) →
(∀y : ((person(y) ∧ ¬shave(y,y))
→ shave(x,y)))]
F
2
: ∀x : [barber(x) →
(∀y : ((person(y) ∧ shave(y,y))
→ ¬shave(x,y)))]
F
3
: ∀x : (barber(x) → person(x))
In plain words, they represent the following knowl-
edge:
KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development
218

F
1
: Every barber shaves every person who does not
shave himself.
F
2
: Every barber does not shave a person who shaves
himself.
F
3
: A barber is a person.
Let us consider a proof that this knowledge entails the
nonexistence of any barber, i.e., consider the proof
problem hE
1
,E
2
i, where E
1
= F
1
∧ F
2
∧ F
3
and E
2
is
the first-order formula ¬(∃x : barber(x)), which intu-
itively means no barber exists.
Conversion of (E
1
∧ ¬E
2
), which is ((F
1
∧ F
2
∧
F
3
) ∧ ¬E
2
), by MPS yields a set Cs consisting of the
following four extended clauses, where f is a 0-ary
function variable:
C
1
: shave(x,x),shave(y,x) ← barber(y), person(x)
C
2
: ← barber(x), person(y),shave(y,y),shave(x, y)
C
3
: person(x) ← barber(x)
C
4
: barber(x) ← func( f,x)
By the application of the FVE rule, Cs is trans-
formed into Cs
′
= (Cs − {C
4
}) ∪ {C
′
4
}, where C
′
4
is
given by:
C
′
4
: barber( f) ←
The resulting clause set Cs
′
can then be transformed
equivalently using ET rules as follows:
• By successively unfolding at barber-atoms three
times, removal of the definite clause defining the
barber predicate, unfolding at person-atoms two
times, and removal of the definite clause defining
the person predicate, Cs
′
is transformed into Cs
′
1
consisting of the following two clauses:
shave( f, f),shave( f, f) ←
← shave( f, f),shave( f, f)
• By removing duplicate atoms, Cs
′
1
is transformed
into Cs
′
2
consisting of the two clauses:
shave( f, f) ←
← shave( f, f)
• By unfolding at the shave-atom in the second
clause above, Cs
′
2
is transformed into Cs
′
3
consist-
ing of the two clauses:
shave( f, f) ←
(←)
Finally, a clause set Cs
′′
= Cs
′
3
is obtained. Since Cs
′′
contains an empty clause, ans
qa
(Cs
′′
,yes) = {yes}.
Hence ans
pr
(E
1
,E
2
) = “yes”.
6 LIMITATIONS BY EXAMPLES
The FVE rule, introduced in Section 5.2, has the fol-
lowing two limitations:
1. Its applicability does not cover all QA problems.
It is applicable to a specific class of QA problems
that corresponds to a class of proof problems.
2. It is not applicable to QA problems with built-in
constraint atoms.
These limitations are demonstrated by means of ex-
amples below.
6.1 Incorrect Results when Applied to
QA Problems
We show that application of the FVE rule to a QA
problem may result in an incorrect result. In particu-
lar, after illustrating a QA problem on first-order logic
(Section 6.1.1) and its equivalent QA problem in a
clausal form (Section 6.1.2), we show that the corre-
sponding QA problem obtained by applying the FVE
rule gives an undesirable result (Section 6.1.3).
6.1.1 A QA Problem on First-order Logic
Assuming that A and B are constant symbols, consider
a QA problem hK,ai on first-order logic, where K is
the first-order formula
(∃x : kill(x, A)) ∧ (kill(A,A) ∨ kill(B,A))
and a is the atom kill(x,A). Since (∃x : kill(x, A))
follows logically from (kill(A,A) ∨ kill(B,A)), this
QA problem is equivalent to the QA problem hK
′
,ai,
where K
′
= (kill(A,A) ∨ kill(B,A)). Among others,
Models(K
′
) contains the models
• {kill(A,A)},
• {kill(B,A)},
• {kill(A,A),kill(B,A)}.
Hence
T
Models(K
′
) = ∅ . Therefore ans
qa
(K
′
,a) is
the empty set. So is ans
qa
(K,a).
6.1.2 A QA Problem on Clauses with Function
Variables
By MPS, K is converted into a clause set Cs
1
consist-
ing of the two clauses, where f is a 0-ary function
variable:
kill(x,A) ← func( f,x)
kill(A,A),kill(B,A) ←
The conversion yields the QA problem hCs
1
,ai in
a clausal form. The set of all models of Cs
1
, i.e.,
Models(Cs
1
), contains the models
• {kill(A,A)},
• {kill(B,A)},
• {kill(A,A),kill(B,A)}.
Function-variable Elimination and Its Limitations
219

So
T
Models(Cs
1
) = ∅. Hence ans
qa
(Cs
1
,a) is the
empty set, which is the same as ans
qa
(K,a) in Sec-
tion 6.1.1.
6.1.3 A QA Problem Obtained by Applying the
FVE Rule
By application of the FVE rule to Cs
1
, with a function
symbol f
h
corresponding to the function variable f,
we obtain a QA problem hCs
2
,ai, where Cs
2
consists
of the following two clauses:
kill( f
h
,A) ←
kill(A,A),kill(B,A) ←
Models(Cs
2
) contains the models
• {kill( f
h
,A),kill(A,A)},
• {kill( f
h
,A),kill(B,A)},
• {kill( f
h
,A),kill(A,A),kill(B,A)},
along with other models each of which contains
kill( f
h
,A). Then
T
Models(Cs
2
) = {kill( f
h
,A)}. As
a result, ans
qa
(Cs
2
,a) = {kill( f
h
,A)}. This an-
swer differs from ans
qa
(K,a) in Section 6.1.1 and
ans
qa
(Cs
1
,a) in Section 6.1.2, and is incorrect due to
inappropriate use of the FVE rule.
6.2 Incorrect Results in the Presence of
Built-in Constraint Atoms
Next, we show that in the presence of a built-in con-
straint atom, application of the FVE rule even to a
proof problem may yield an incorrect result. For this
purpose, we use the tax-cut proof problem with a con-
straint atom on first-order logic given in Section 1
and its equivalent proof problem on extended clauses
(Section 6.2.1). Then we show that, compared with
the (correct) answers to these proof problems, the cor-
responding proof problem obtained by applying the
FVE rule gives an incorrect result (Section 6.2.2).
6.2.1 A Proof Problem on Clauses with Function
Variables
Consider the proof problem hE
1
,E
2
i in Section 1,
where E
1
= F
1
∧ F
2
∧ F
3
and E
2
= TaxCut(Peter). By
applying MPS to the conjunction E
1
∧ ¬E
2
, a 0-ary
function variable, say f, is introduced and E
1
∧ ¬E
2
is converted into a clause set Cs
3
consisting of the fol-
lowing clauses:
C
1
: TaxCut(x) ← hasChild(x,y),hasChild(x,z),
noteq(y,z)
C
2
: hasChild(Peter, Paul) ←
C
3
: hasChild(Peter, x) ← func( f,x)
C
4
: ← TaxCut(Peter)
Semantically, Cs
3
is equivalent to {C
1
,C
2
,C
4
}, which
does not yield a contradiction. The expected result
“no” shown in Section 1 is thus well supported.
If we simplify Cs
3
by equivalent transformation
without using the FVE rule, Cs
3
is transformed as fol-
lows:
• By unfolding of hasChild, Cs
3
is transformed into
a clause set {C
4
,C
5
,C
6
,C
7
,C
8
}, where C
5
–C
8
are
given by:
C
5
: TaxCut(Peter) ← noteq(Paul,Paul)
C
6
: TaxCut(Peter) ← noteq(Paul,x),func( f,x)
C
7
: TaxCut(Peter) ← noteq(x, Paul),func( f,x)
C
8
: TaxCut(Peter) ← noteq(x,y),func( f,x),
func( f,y)
• Since noteq(Paul, Paul) is false, C
5
is removed.
• Since func( f,x) and func( f,y) give x = y and
noteq(x,x) is false, C
8
is removed.
{C
4
,C
6
,C
7
} is obtained, which includes the function
variable f. When letting f() = Paul, C
6
and C
7
are
removed and the resulting clause set is {C
4
}, which is
satisfiable. So {C
4
,C
6
,C
7
} is satisfiable, and we can-
not obtain a contradiction. This transformation result
shows clearly that the answer to this proof problem is
“no”.
6.2.2 A Proof Problem Obtained by Applying
the FVE Rule
Assume that f
h
is a function symbol that corre-
sponds to the function variable f. By application
of the FVE rule to Cs
3
, we have a clause set Cs
4
=
{C
1
,C
2
,C
′
3
,C
4
}, where C
′
3
is given by:
C
′
3
: hasChild(Peter, f
h
) ←
It can be shown that Cs
4
yields a contradiction by
repeatedly applying equivalent transformation as fol-
lows:
• After application of unfolding three times at
hasChild-atoms, Cs
4
is transformed into a clause
set {C
2
,C
′
3
,C
4
,C
′
5
,C
′
6
}, whereC
′
5
and C
′
6
are given
by:
C
′
5
: TaxCut(Peter) ← noteq(Paul, f
h
)
C
′
6
: TaxCut(Peter) ← noteq( f
h
,Paul)
• Since the constraint atoms in their bodies, i.e.,
noteq(Paul, f
h
) and noteq( f
h
,Paul), are true, C
′
5
and C
′
6
are transformed into:
C
′′
5
: TaxCut(Peter) ←
C
′′
6
: TaxCut(Peter) ←
From the final clause set, which contains C
4
, C
′′
5
, and
C
′′
6
, a contradiction is obtained. This contradiction
KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development
220

misleads us that the answer to this proof problem is
“yes”, which is incorrect compared to the expected
result given in Section 1 and the result obtained in
Section 6.2.1. Such inappropriate use of the FVE rule
yields a wrong answer.
7 DISCUSSION
7.1 Incorrect Results Arising from
Built-in Atoms
Incorrect results caused by the conventional Skolemi-
zation and resolution stems basically from the limita-
tion of Skolemization. In particular:
1. If there is no built-in atom in an input first-order
formula, the composition of MPS and FVE gives
a set of usual clauses, which is also obtained by
the conventional Skolemization.
2. If there is a built-in atom in an input first-order
formula, MPS preserves its logical meaning but
FVE cannot guarantee the preservation of its log-
ical meaning, which means that the conventional
Skolemization may produce an incorrect result.
The tax-cut example shown earlier (cf. Section 1 and
Section 6.2) contains a built-in atom noteq(y,z), and
the fact that noteq(Paul, f) is true for any new func-
tion symbol f adopted by the conversion algorithm
CS (cf. Fig. 1) produces an incorrect result.
More generally, the maximality of the instantia-
tion of function variables according to the usual Sko-
lemization is broken if a ground atom set contains ar-
bitrary built-in atoms. Hence Theorem 8 cannot be
obtained.
7.2 Understanding the Conventional
Skolemization and Resolution
The theory developed in the previous sections pro-
vides a deep understanding of the conventional Sko-
lemization and resolution. We have stated (in Sec-
tion 1) that the conventional Skolemization, CS, is
identified as the composition of MPS and FVE in the
sense that
{CS(L) | L ∈ L
1
} = {FVE(MPS(L)) | L ∈ L
1
}.
This is a simplified explanation for the purpose of
readability. More precisely, we need to consider that
(i) by MPS, some function variables may be newly
introduced, and (ii) by FVE, some function sym-
bols may be associated with function variables. The
conventional Skolemization may also introduce some
new function symbols. Then we have the following
proposition based on the theory in the previous sec-
tions: (i) each set of clauses obtained by sequentially
applying MPS and FVE to a first-order formula, with
any selection of function variables and a mapping to
associate function variables with function symbols,
can also be obtained by application of the conven-
tional Skolemization using some function symbols,
and (ii) vice versa.
Limitations of the conventional Skolemization are
thus identified mainly by the limitations of FVE trans-
formation. FVE preserves the answers to proof prob-
lems (Theorem 9); however, its applicability is rather
limited:
1. FVE does not admit inclusion of built-in atoms in
input extended clauses in ECLS
F
, and
2. FVE can be applied to only a restricted class of
QA problems.
This is a sharp contrast to most of important ET trans-
formations, such as unfolding, resolution, factoring,
subsumption, side changing, and definite-clause re-
moval transformation, which have been invented on
the space of extended clauses (Akama and Nantajee-
warawat, 2014). These ET transformations preserve
the answers to QA problems and can be applied to any
QA problem possibly with built-in constraint atoms.
One limitation of the resolution method is that it
uses only resolution and factoring. Obviously, from
the viewpoint of ET, we can use other ET rules and
we should use them for more efficient computation.
7.3 Inventing a New Proof Method
Since proof problems are formalized as QA problems,
the MPS & ET method can be applied to proof prob-
lems. A new procedure for solving proof problems is
given below.
Let hE
1
,E
2
i be an input proof problem.
1. Construct a QA problem hE,yesi, where
• E = E
1
∧ ¬E
2
, and
• yes is a ground atom that does not appear in E.
2. Convert the QA problem hE,yesi by MPS, using
the conversion procedure in Section 3.2, into a QA
problem hCs,yesi in a clausal form.
3. Using ET rules, transform hCs,yesi into a QA
problem hCs
′
,yesi.
4. Determine the answer to the proof problem hE
1
,
E
2
i by
ans
pr
(E
1
,E
2
)=

“yes” if ans
qa
(Cs
′
,yes) = {yes},
“no” if ans
qa
(Cs
′
,yes) = ∅.
Function-variable Elimination and Its Limitations
221

By ET transformation at Step 3, we may basically
try to simplify Cs using ET rules. When we reach a
set of extended clauses that contains an empty clause
(←), we can stop with the answer “yes”. When we
reach a set of positive extended clauses, we can stop
with the answer “no”. All ET rules, which preserve
the answers to QA problems, can be used at Step 3.
The reader may refer to examples given in (Akama
and Nantajeewarawat, 2012; Akama and Nantajee-
warawat, 2013; Akama and Nantajeewarawat, 2014).
8 CONCLUSIONS
The MPS & ET theory takes MPS in place of the con-
ventional Skolemization and ET in place of inference
rules. The work developed in this paper enables us to
consider the conventional Skolemization and the con-
ventional solution for proof problems in the MPS &
ET theory.
In this paper, the FVE rule has been proposed and
its correctness has been proved (i.e., the FVE rule pre-
serves the answers to a class QA problems). The con-
ventional Skolemization is identified as application of
MPS transformation followed by equivalent transfor-
mation using the FVE rule. Since the resolution and
factoring inference rules are ET rules and proof prob-
lems are a subclass of QA problems, the conventional
solution for proof problems is a special case of the
MPS & ET solution for QA problems.
This paper has also investigated the limitations of
the FVE rule, which are also limitations of the con-
ventional Skolemization and the conventional solu-
tion for proof problems. They are:
1. The conventional Skolemization may fail to pre-
serve satisfiability of a given formula in the pres-
ence of built-in constraints.
2. The conventional solution for proof problems
based on Skolemization and resolution cannot
guarantee the correctness of an obtained answer
when built-in constraints are included in a given
problem representation.
By removing the FVE rule, which is less general
than other important ET rules, we have a new proof
method with correctness of computation being guar-
anteed even in the presence of built-in constraints.
ACKNOWLEDGEMENTS
This research was partially supported by JSPS KAK-
ENHI Grant Numbers 25280078 and 26540110.
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