Side-Change Transformation

Kiyoshi Akama

, Ekawit Nantajeewarawat

and Taketo Akama

Information Initiative Center, Hokkaido University, Sapporo, Japan

Computer Science, Sirindhorn International Institute of Technology, Thammasat University, Pathumthani, Thailand

Modeleet Labs, Sapporo, Japan

Keywords:

Model-Intersection Problem, Extended Clause, Equivalent Transformation, Side-Change Transformation,

Computation Control.

Abstract:

Many logical problems, such as proof problems and query-answering problems, can be mapped into model-

intersection (MI) problems, which constitute one of the largest and most fundamentally important classes of

logical problems. To solve MI problems, many equivalent transformation rules have been employed. In this

paper, we introduce a new transformation, called side-change transformation, and propose unfolding/side-

change computation control, i.e., when neither unfolding nor deﬁnite-clause removal is applicable, an attempt

is made to transform a given problem using side-change transformation so as to derive an equivalent problem

to which unfolding is applicable. The correctness of side-change transformation is shown. While a resolution-

based proof method increases problem size monotonically no matter what control is taken, a reduction of

problem size can often be achieved by using the unfolding/side-change control.

1 INTRODUCTION

Proof problems and query-answering (QA) problems

are two most important and largest classes of logi-

cal problems, and many solutions to their subclasses

have been intensively studied. When we try to solve

some of these problems on FOL

, i.e., the set of all

ﬁrst-order formulas with built-in constraint atoms, we

bump up against the fact that none of these problem

classes have been solved entirely so far. The conven-

tional Skolemization (Chang and Lee, 1973; Fitting,

1996), by which neither the satisﬁability nor the logi-

cal meaning of a formula in FOL

can be preserved

in general (Akama and Nantajeewarawat, 2015b), has

been a major hindrance to solve problems in these two

classes correctly.

To overcome the difﬁculty, new Skolemization,

called meaning-preserving Skolemization (MPS), an

extended clause space, called ECLS

, and model-

intersection (MI) problems were introduced (Akama

and Nantajeewarawat, 2011; Akama and Nantajee-

warawat, 2015a). All proof problems and all QA

problems on FOL

have been mapped using MPS,

with the answers to them being preserved, into MI

problems on ECLS

(Akama and Nantajeewarawat,

2015a; Akama and Nantajeewarawat, 2016a), thereby

guaranteeing the correctness for solving a large class

of logical problems. MI problems on ECLS

consti-

tute one of the largest classes of logical problems and

are of fundamental importance.

To solve MI problems, we use many equivalent

transformation (ET) rules. An appropriate strategy

for applying ET rules is essential to solve logical

problems efﬁciently. One strategy is to repeatedly

apply unfolding and deﬁnite-clause removal (Akama

and Nantajeewarawat, 2016b), by which a decrease

of problem size can be expected. However, in the ex-

tended clause space ECLS

, unfolding is not necessa-

rily applicable to an atom in the right-hand side of a

clause, and computation may often reaches a state in

which neither unfolding nor deﬁnite-clause removal

can be applied.

To get out of such a state, we introduce in this

paper a new transformation called side-change trans-

formation on the extended space ECLS

, by which

we try to transform a given problem in order to obtain

an equivalent problem to which unfolding or deﬁnite-

clause removal is applicable. The correctness of side-

change transformation is shown.

Side-change transformation, which moves atoms

to the other side of a clause, is a very simple transfor-

mation. It does not change problem size (the num-

ber of clauses and the number of atoms in the set

of clauses representing a problem). However, side-

Akama, K., Nantajeewarawat, E. and Akama, T.

Side-Change Transformation.

DOI: 10.5220/0006934802370246

In Proceedings of the 10th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2018) - Volume 2: KEOD, pages 237-246

ISBN: 978-989-758-330-8

237

change transformation contributes to efﬁcient compu-

tation when it is used together with extended unfol-

ding transformation in the space ECLS

, which is lar-

ger than the clause space taken by the conventional

theory (Akama and Nantajeewarawat, 2016b).

The rest of this paper is organized as follows:

Section 2 introduces side-change transformation and

explains its typical usage. Section 3 illustrates

unfolding/side-change computation control for sol-

ving MI problems. Section 4 begins with a review of

theoretical foundation concerning the extended clause

space ECLS

and the semantics of a set of extended

clauses. It then formalizes side-change transforma-

tion and proves the correctness of this transformation.

Section 5 presents some experiments. Section 6 con-

cludes the paper.

2 SIDE-CHANGE

TRANSFORMATION AND ITS

USAGE

2.1 Side-Change Transformation

Let Cs be a clause set and p a predicate occurring in

Cs. An atom whose predicate is p is referred to as a p-

atom. The clause set Cs can be transformed by side-

change transformation of p-atoms (also called side-

change transformation for p) as follows:

1. Determine a new predicate not p for p.

2. Move each p-atom in each clause to its opposite

side in the same clause (i.e., from the left-hand

side to the right-hand side and vice versa) with its

predicate being changed from p to not p.

For example, a singleton clause set

{(p(1,x),q(x,y) ← p(4,x), p(5,x),r(y))}

is transformed by side-change transformation of p-

atoms into

{(not p(4,x),not p(5,x), q(x, y) ← not p(1, x),r(y))}.

Let the clause set obtained from Cs by side-change

transformation of p-atoms into not p-atoms be deno-

ted by SIDECH(Cs, p,not p).

2.2 Sufﬁcient Conditions for Enabling

Unfolding Transformation

Let Cs be a set of clauses. To apply unfolding or

deﬁnite-clause removal to Cs, we need to determine

a set D of deﬁnite clauses and a predicate p such that

(i) D is a subset of Cs, (ii) the head of each deﬁnite

clause in D is a p-atom, and (iii) no p-atom appears

in the left-hand side of any clause in Cs − D.

When unfolding is not applicable, side-change

transformation is useful for deriving from Cs a new

set of clauses to which unfolding or deﬁnite-clause

removal can be applied. Two types of its usage that

enable unfolding are described below.

2.2.1 Type 1

Let p be a predicate and not p a new predicate deter-

mined for side-change transformation for p. Unfol-

ding with respect to not p-atoms can be applied after

side-change transformation for p if each clause C in

Cs satisﬁes the following two conditions:

1. There is at most one p-atom in the right-hand side

of C.

2. If there is exactly one p-atom in the right-hand

side of C, then all user-deﬁned atoms in the left-

hand side of C are p-atoms.

Example 1. Consider the following clauses:

: q(x), p(5,x) ← r(x)

: ← p(x,y),r(y, w), s(y,w)

: p(x,z) ← p(x,y),r(y,z)

: p(x,w), p(x,2) ← p(x,y),t(w)

Each of these clauses satisﬁes the two conditions

above. By side-change transformation of p, these four

clauses are transformed into:

: q(x) ← r(x),not p(5, x)

: not p(x,y) ← r(y,w),s(y, w)

: not p(x, y) ← not p(x,z),r(y,z)

: not p(x,y) ← not p(x, w), not p(x,2),t(w)

We obtain the set {C

} of deﬁnite clauses for

the predicate not p, which can be used for unfolding

with respect to not p(5,x) in the right-hand side of the

ﬁrst clause C

Assume that D

not p

is the set of all deﬁnite clau-

ses for not p in Cs. If there is no not p-atom in the

right-hand side of any clause in Cs −D

not p

, then D

not p

can be removed by deﬁnite-clause removal transfor-

mation.

2.2.2 Type 2

Assume that p and q are distinct predicates. Unfol-

ding with respect to q-atoms can be applied after side-

change transformation for p if each clause C in Cs

satisﬁes the following two conditions:

1. There is at most one q-atom in the left-hand side

of C.

KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development

238

2. If there is exactly one q-atom in the left-hand side

of C, then (i) all other user-deﬁned atoms in the

left-hand side of C are p-atoms and (ii) no p-atom

appears in the right-hand side of C.

Example 2. Consider the following four clauses:

: s(x,y) ← q(x, y)

: q(x, y) ← r(y)

: p(x,y),q(x,y) ← r(x),s(y)

: p(x,y), p(x,z),q(x,y) ← t(y,z)

Each of them satisﬁes the two conditions for Type 2.

By side-change transformation of p, these clauses are

changed into:

: s(x,y) ← q(x, y)

: q(x, y) ← r(y)

: q(x,y) ← not p(x,y),r(x), s(y)

: q(x,y) ← not p(x,y),not p(x,z),t(y,z)

We obtain the set {C

} of deﬁnite clauses for

the predicate q, which can be used for unfolding with

respect to q(x,y) in the right-hand side of C

Assume that D

is the set of all deﬁnite clauses for

q in Cs. If there is no q-atom in the right-hand side of

any clause in Cs − D

, then D

can be removed by

deﬁnite-clause removal transformation.

2.3 Reduction of Problem Size

The resolution proof procedure is one of the most fa-

mous sound and complete proof procedures (Fitting,

1996). It uses two inference rules, i.e., the resolution

inference rule (Robinson, 1965) and the factoring in-

ference rule. Given a set Cs of clauses, the resolution

proof procedure starts with Cs and repeatedly make

inference by resolution or factoring to construct a se-

quence [Cs

,Cs

,...], where Cs

= Cs. When it

reaches a clause set Cs

that contains the empty clause

(←), the proof is completed. Each inference step in-

creases the number of clauses by one.

The solution method for MI problems considered

in this paper uses unfolding/side-change control, i.e.,

it attempts to apply transformation in the preference

order of (i) deﬁnite-clause removal, (ii) unfolding,

and (iii) side-change transformation. By deﬁnite-

clause removal, the number of clauses decreases. Let

p be a predicate and number(p) denote the number

of p-atoms in the right-hand sides of all clauses un-

der consideration. By repeated application of unfol-

ding with respect to p-atoms, we expect number(p)

to become zero and deﬁnite-clause removal can then

be applied for removing the deﬁnition of p. By side-

change transformation, we expect to obtain new clau-

ses to which further unfolding is applicable. Appro-

: FM(x) ← FP(x) C

: FP(john) ←

: FP(mary) ← C

: teach(john,ai) ←

: St(paul) ← C

: AC(ai) ←

: Tp(kr) ← C

: Tp(lp) ←

: curr(x,z) ← exam(x,y),subject(y, z), St(x),

Co(y),Tp(z)

: mdt(x,y) ← curr(x, z), expert(y,z), St(x), Tp(z),

FP(y),AC(w),teach(y,w)

: mdt(x,y) ← St(x), NFP(y)

: exam(paul,ai) ← C

: subject(ai,kr) ←

: subject(ai, lp) ← C

: expert(john,kr) ←

: expert(mary,lp) ←

: AC(x) ← teach(mary,x)

: ← AC(x),BC(x)

: AC(x),BC(x) ← Co(x)

: Co(x) ← AC(x)

: Co(x) ← BC(x)

: FP(x) ← NFP(x)

: ← NFP(x), teach(x,y),Co(y)

: teach(x,y),NFP(x) ← FP(x),func( f

,x,y)

: Co(y), NFP(x) ← FP(x), func( f

,x,y)

Figure 1: Background knowledge for the mdt problem on

ECLS

priate combinations of these three types of transfor-

mation are often useful to reduce problem size.

A resolution proof procedure is a refutation met-

hod, i.e., it aims to show that a given clause set has no

model. QA problems, on the other hand, is basically

concerned with ﬁnding all models of a given clause

set. The unfolding/side-change control can be used

not only for proof problems but also for QA problems

and MI problems.

3 COMPUTATION CONTROL

WITH SIDE-CHANGE

TRANSFORMATION

Control of rule application with side-change transfor-

mation is demonstrated by using a puzzle. When ne-

cessary, the reader is referred to the concept deﬁniti-

ons in Section 4.

3.1 Problem Description

Consider the clause set Cs consisting of C

–C

Fig. 1, which provides the background knowledge on

ECLS

for the mayDoThesis problem (for short, the

mdt problem), modiﬁed from the original problem gi-

ven in (Donini et al., 1998). All atoms appearing in

Fig. 1 are user-deﬁned atoms. The clauses C

–C

Side-Change Transformation

239

: teach(john,ai) ←

: AC(ai) ←

: AC(x) ← teach(mary,x)

: ← AC(x),BC(x)

: AC(x),BC(x) ← Co(x)

: Co(x) ← AC(x)

: Co(x) ← BC(x)

: ← NFP(x),teach(x,y), Co(y)

: mdt(paul,mary) ← AC(x),teach(mary,x),

Co(ai)

: mdt(paul,john) ← AC(x),teach(john,x),Co(ai)

: mdt(paul,x) ← NFP(x)

: teach(mary,x), NFP(mary) ← func( f

,mary,x)

: teach(john,x), NFP(john) ← func( f

,john,x)

: Co(x),NFP(mary) ← func( f

,mary,x)

: Co(x),NFP(john) ← func( f

,john,x)

Figure 2: Clauses obtained by transformation of C

–C

Fig. 1.

together provide the conditions for a student to do

his/her thesis with a professor, where for any student

s and any professor p, mdt(s, p) is intended to mean

“s may do his/her thesis with p.”

Suppose that we want to ﬁnd all professors with

whom paul may do his thesis. This problem is formu-

lated as a MI problem hCs,ϕi, where ϕ is a mapping

used for constructing the output answer from the in-

tersection of all models of Cs, deﬁned by

ϕ(G) = {x | mdt(paul,x) ∈ G}

for any set G of ground user-deﬁned atoms. (The

formal deﬁnition of a MI problem is deferred to

Section 4.4.)

3.2 Inapplicability of Unfolding

Consider the clauses C

–C

in Fig. 2, which are

obtained by equivalently transforming the clauses C

–

in Fig. 1. Unfolding is not applicable to them due

to the following reasons:

• Unfolding of the predicate AC is blocked since

contains an AC-atom and a BC-atom in its left-

hand side.

• Unfolding of the predicate Co is blocked since C

and C

contain Co-atoms and NFP-atoms in their

left-hand sides.

• Unfolding of the predicate teach is blocked since

and C

contain teach-atoms and NFP-atoms

in their left-hand sides.

• Unfolding of the predicate mdt is blocked since

the answer to the problem is inﬂuenced by mdt-

atoms.

As will be subsequently illustrated, unfolding with re-

spect to Co-atoms and with respect to teach-atoms

will be made applicable by side-change transforma-

tion for the predicate NFP.

3.3 Rule Application Control with

Side-Change Transformation

With side-change transformation, the clause set Cs

consisting of C

–C

in Fig. 1 is transformed as fol-

lows, where basic reduction rules given in (Akama

and Nantajeewarawat, 2014) (such as removal of valid

clauses, removal of subsumed clauses, and removal of

duplicate atoms) are also used:

1. By (i) unfolding using the deﬁnitions of the predi-

cates FP, Tp, curr, subject, expert, St, and exam,

(ii) removing these deﬁnitions along with the de-

ﬁnition of FM using deﬁnite-clause removal, and

(iii) application of basic reduction rules, the clau-

ses C

–C

are transformed into the clauses C

–

in Fig. 2.

2. Side-change transformation for NFP enables (i)

unfolding using the deﬁnitions of Co and teach,

(ii) elimination of these deﬁnitions using deﬁnite-

clause removal, and (iii) application of basic re-

duction rules.

3. Side-change transformation for BC enables (i) un-

folding using the deﬁnition of AC, (ii) deﬁnite-

clause removal, and (iii) application of basic re-

duction rules.

4. The resolution rule together with basic reduction

rules are then applied.

5. Assume that notNFP is the predicate determined

for side-change transformation for NFP at Step 2.

By (i) side-change transformation for notNFP, (ii)

unfolding of NFP, (iii) removal of independent

func-atoms, and (iv) application of basic reduction

rules, we obtain the clauses C

and C

given as

follows:

: mdt(paul,mary) ←

: mdt(paul,john) ←

As a result, the MI problem hCs,ϕi in Section 3.1

is transformed equivalently into the MI problem

h{C

},ϕi. Hence the answer to the original MI

problem hCs,ϕi coincides with the answer to the sim-

pler MI problem h{C

},ϕi, which can be readily

determined as the set

ϕ(

Models({C

}))

= ϕ({mdt(paul,mary),mdt(paul,john)})

= {mary,john},

where Models({C

}) denotes the set of all mo-

dels of {C

KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development

240

4 CORRECTNESS THEOREM

FOR SIDE-CHANGE

TRANSFORMATION

To provide a theoretical basis for the correctness of

side-change transformation, we ﬁrst review the exten-

ded clause space ECLS

, the semantics of an exten-

ded clause set, and the formal deﬁnition of a MI pro-

blem. We then show that side-change transformation

preserves the answers to MI problems.

4.1 User-Deﬁned Atoms, Constraint

Atoms, and func-Atoms

We consider an extended formula space that contains

three kinds of atoms, i.e., user-deﬁned atoms, built-

in constraint atoms, and func-atoms. A user-deﬁned

atom takes the form p(t

,...,t

), where p is a user-

deﬁned predicate and the t

are usual terms. A built-in

constraint atom, also simply called a constraint atom

or a built-in atom, takes the form c(t

,...,t

), where c

is a predeﬁned constraint predicate and the t

are usual

terms. Let A

denote the set of all user-deﬁned atoms,

the set of all ground user-deﬁned atoms, A

the set

of all constraint atoms, and G

the set of all ground

constraint atoms.

A func-atom (Akama and Nantajeewarawat, 2011)

is an expression of the form func( f ,t

,...,t

n+1

where f is either an n-ary function constant or an n-

ary function variable, and the t

are usual terms. It is

a ground func-atom if f is a function constant and the

are ground usual terms.

4.2 Extended Clauses and an Extended

Clause Space

An extended clause C is a formula of the form

,...,a

← b

,...,b

,...,f

), where {a

,...,

,...,b

} ⊆ A

∪ A

and f

,...,f

are func-

atoms. All usual variables occurring in C are im-

plicitly universally quantiﬁed and their scope is re-

stricted to the extended clause C itself. The sets

,...,a

} and {b

,...,b

,...,f

} 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), respectively. When there is only one user-

deﬁned atom in lhs(C), C is called an extended deﬁ-

nite clause. When no confusion is caused, an exten-

ded clause is simply called a clause. The set of all

extended clauses is denoted by ECLS

. The extended

clause space in this paper is the powerset of ECLS

Let Cs be a set of extended clauses. Implicit ex-

istential quantiﬁcations of function variables and im-

plicit clause conjunction are assumed in Cs. Function

variables in Cs are all existentially quantiﬁed and

their scope covers all clauses in Cs. With occurren-

ces 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.

4.3 Interpretations and Models

An interpretation is a subset of G

. A ground user-

deﬁned atom g is true under an interpretation I iff g

belongs to I. Unlike ground user-deﬁned atoms, the

truth values of ground constraint atoms are predeter-

mined independently of interpretations. Let TCON

denote the set of all true ground constraint atoms,

i.e., a ground constraint atom g is true iff g ∈ TCON.

A ground func-atom func( f ,t

,...,t

n+1

) is true iff

f (t

,...,t

) = t

n+1

Let C = (a

,...,a

← b

,...,b

,...,f

) be a

ground clause, where {a

,...,a

,...,b

} ⊆ G

∪

and f

,...,f

are ground func-atoms. C is true un-

der an interpretation I (in other words, I satisﬁes C)

iff at least one of the following conditions is satisﬁed:

1. There exists i ∈ {1,...,m} such that a

∈ I ∪

TCON.

2. There exists i ∈ {1,...,n} such that b

/∈ I ∪

TCON.

3. There exists i ∈ {1, . . . , p} such that f

is false.

An interpretation I is a model of a clause set Cs ⊆

ECLS

iff there exists a substitution σ for function

variables that satisﬁes the following conditions:

1. All function variables occurring in Cs are instan-

tiated by σ into function constants.

2. For any clause C ∈ Cs and any substitution θ for

usual variables, if Cσθ is a ground clause, then

Cσθ is true under I.

For any Cs ⊆ ECLS

, let Models(Cs) denote the set

of all models of Cs.

4.4 MI Problems on the Extended

Clause Space

A model-intersection problem (MI problem) on

ECLS

is a pair hCs,ϕi, where Cs ⊆ ECLS

and ϕ is a

mapping from pow(G

) to some set W . The mapping

ϕ is called an extraction mapping (previously called

an exit mapping). The answer to this problem, deno-

ted by ans

(Cs,ϕ), is deﬁned as ϕ(

Models(Cs)),

where

Models(Cs) is the intersection of all models

of Cs. Note that when Models(Cs) is the empty set,

Models(Cs) = G

Side-Change Transformation

241

The notion of an independent extraction mapping

is next introduced; it is used for giving a sufﬁcient

condition for the correctness of side-change transfor-

mation. Given a set P of predicates, let GAtoms(P)

denote the set of all ground atoms in G

the predicates

of which belong to P. An extraction mapping ϕ is said

to be independent of a set P of predicates iff for any

⊆ G

, if (G

−G

)∪(G

−G

) ⊆ GAtoms(P),

then ϕ(G

) = ϕ(G

4.5 Correctness Theorem for

Side-Change Transformation

We show below that side-change transformation is

answer-preserving.

Theorem 1. Let hCs, ϕi be a MI problem on ECLS

Let p be a predicate occurring in Cs and not p a

new predicate determined for side-change transfor-

mation for p. If ϕ is independent of {p, not p}, then

ans

(Cs,ϕ) = ans

(SIDECH(Cs, p,not p),ϕ).

Proof: Assume that (i) ϕ is independent of

{p,not p}, (ii) Cs

= SIDECH(Cs, p,not p), and (iii)

is a mapping that changes p-atoms into not p-

atoms. Let m be an arbitrary model of Cs. Let a set

⊆ G

be constructed from m by

= (m − GAtoms({p})) ∪

{π

(x) | x ∈ (GAtoms({p}) − m)}.

Obviously, m

is a model of Cs

. This con-

struction determines a mapping ψ from Models(Cs)

to Models(Cs

). Obviously, ψ

−1

is a mapping from

Models(Cs

) to Models(Cs) and ψ ◦ ψ

−1

is an iden-

tity mapping. Hence ψ is a bijective mapping from

Models(Cs) to Models(Cs

). As a result,

•

Models(Cs) −

Models(Cs

) ⊆ GAtoms({p,

not p}), and

•

Models(Cs

) −

Models(Cs) ⊆ GAtoms({p,

not p}).

Since ϕ is independent of {p,not p}, it follows that

ϕ(

Models(Cs)) is equal to ϕ(

Models(Cs

)).

5 EXPERIMENTS WITH

SIDE-CHANGE

TRANSFORMATION

5.1 The Agatha QA Problem

Consider the “Dreadsbury Mansion Mystery” pro-

blem (Pelletier, 1986), which is described as follows:

Someone who lives in Dreadsbury Mansion killed

: ∃x : (live(x, D) ∧ kill(x,A))

: ∀x : (live(x, D) ↔ (eq(x,A) ∨ eq(x,B) ∨

eq(x,C)))

: ∀x∀y : (kill(x,y) → hate(x, y))

: ∀x∀y : (kill(x,y) → ¬richer(x,y))

: ¬(∃x : (hate(C, x) ∧ hate(A,x) ∧ live(x,D)))

: ∀x : ((neq(x,B) ∧ live(x,D)) → hate(A,x))

: ∀x : ((¬richer(x, A) ∧ live(x, D)) → hate(B, x))

: ∀x : ((hate(A,x) ∧live(x,D)) → hate(B,x))

: ¬(∃x : (live(x,D) ∧

∀y : (live(y,D) → hate(x,y))))

: ∀x : (kill(x,A) → killer(x))

Figure 3: Background knowledge represented by ﬁrst-order

formulas.

Aunt Agatha. Agatha, the butler, and Charles live

in Dreadsbury Mansion, and are the only people who

live therein. A killer always hates his victim, and is

never richer than his victim. Charles hates no one that

Aunt Agatha hates. Agatha hates everyone except the

butler. The butler hates everyone not richer than Aunt

Agatha. The butler hates everyone Agatha hates. No

one hates everyone. The problem is to ﬁnd who is the

killer.

5.2 Formalization by First-Order

Formulas

Assume that eq and neq are predeﬁned binary con-

straint predicates and for any ground usual terms t

and t

, (i) eq(t

) is true iff t

= t

, and (ii) neq(t

)

is true iff t

6= t

. The background knowledge of this

mystery is formalized as the conjunction of the ﬁrst-

order formulas in Fig. 3, where (i) the constants A,

B, C, and D denote “Agatha,” “the butler,” “Charles,”

and “Dreadsbury Mansion,” respectively, and (ii) for

any terms t

and t

, live(t

), kill(t

), hate(t

and richer(t

) are intended to mean “t

lives in t

,”

“t

killed t

,” “t

hates t

,” “t

is richer than t

,” re-

spectively.

5.3 Meaning-Preserving Skolemization

Meaning-preserving Skolemization transforms a ﬁrst-

order formula equivalently into a set of extended

clauses (Akama and Nantajeewarawat, 2011). Let

MPS(E) denote the set of extended clauses resulting

from applying meaning-preserving Skolemization to

a given ﬁrst-order formula E in FOL

. MPS(E) is

obtained from E by repeated subformula transforma-

tion and conversion into a clausal form.

Consider the formulas F

–F

in Fig. 3. Refer-

ring to the clauses C

–C

in Fig. 4, MPS(F

) =

}, MPS(F

) = {C

}, MPS(F

) =

KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development

242

: live(x, D) ← func( f

,x)

: kill(x,A) ← func( f

,x)

: ← live(x,D),neq(x,A),neq(x,B),neq(x,C)

: live(A, D) ←

: live(B, D) ←

: live(C, D) ←

: hate(x,y) ← kill(x,y)

: ← kill(x,y),richer(x, y)

: ← hate(A,x), hate(C, x), live(x,D)

: hate(A,x) ← neq(x, B), live(x,D)

: richer(x,A),hate(B, x) ← live(x,D)

: hate(B,x) ← hate(A,x), live(x,D)

: ← hate(x,y),func( f

,x,y),live(x,D)

: live(y, D) ← live(x,D), func( f

,x,y)

: killer(x) ← kill(x,A)

Figure 4: The initial state with extended clauses.

}, MPS(F

) = {C

}, MPS(F

) = {C

}, MPS(F

)

= {C

}, MPS(F

) = {C

}, MPS(F

) = {C

MPS(F

) = {C

}, and MPS(F

) = {C

}. Let

K = F

∧F

∧···∧F

. Then MPS(K ∧F

) = {C

...,C

5.4 Simpliﬁcation

We simplify the formalization in Fig. 3 for ease of

computation and presentation. F

is simpliﬁed using

into F

as follows:

≡ ∃x : (live(x,D) ∧ kill(x, A))

≡ ∃x : ((eq(x,A) ∨ eq(x,B) ∨ eq(x,C)) ∧

kill(x, A))

≡ ∃x : ((eq(x,A) ∧ kill(x,A)) ∨

(eq(x,B) ∧ kill(x,A)) ∨

(eq(x,C) ∧ kill(x,A)))

≡ ∃x : ((eq(x,A) ∧ kill(A,A)) ∨

(eq(x,B) ∧ kill(B,A)) ∨

(eq(x,C) ∧ kill(C,A)))

≡ kill(A,A) ∧ kill(B,A) ∧ kill(C,A)

≡ F

is simpliﬁed using F

into F

as follows:

≡ ¬(∃x : (live(x,D) ∧

∀y : (live(y,D) → hate(x,y))))

≡ ¬(∃x : (live(x,D) ∧

(hate(x,A) ∧ hate(x,B) ∧ hate(x,C))))

≡ F

Assume that

• K = F

∧ F

, and

• K

= F

∧ F

The Agatha QA problem is hK ∧ F

,killer(x)i,

and the simpliﬁed Agatha QA problem is hK

∧

,killer(x)i. The set of clauses obtained from

∧F

is shown in Fig. 5, where MPS(F

) = {C

: kill(A,A),kill(B,A),kill(C,A) ←

: ← live(x,D),neq(x,A),neq(x,B),neq(x,C)

: live(A, D) ←

: live(B, D) ←

: live(C, D) ←

: hate(x,y) ← kill(x,y)

: ← kill(x,y),richer(x, y)

: ← hate(A,x), hate(C, x), live(x,D)

: hate(A,A) ←

: hate(A,C) ←

: richer(x,A),hate(B, x) ← live(x,D)

: hate(B,x) ← hate(A,x),live(x, D)

: ← hate(x,A), hate(x,B),hate(x,C),live(x,D)

: killer(x) ← kill(x, A)

Figure 5: The simpliﬁed initial state.

MPS(F

) = {C

}, MPS(F

) = {C

MPS(F

) = {C

}, MPS(F

) = {C

}, MPS(F

) =

}, MPS(F

) = {C

}, MPS(F

) = {C

MPS(F

) = {C

}, and MPS(F

) = {C

5.5 Computation Control

We use priority ordering to control selection and ap-

plication of ET rules. For example, assume that

• (subsumed) is an ET rule for elimination of sub-

sumed clauses,

• (ud) is an ET rule for unfolding by a set of deﬁnite

clauses or deletion of a set of deﬁnite clauses, and

• (rr n), where n is a nonnegative integer, is the reso-

lution rule with a resolvent consisting of less than

or equal to n atoms.

To represent a preference order among ET rules, a bi-

nary relation > is used. ET rules together with a bi-

nary relation > are referred to as prioritized ET rules

or a prioritized ET rule set. The prioritized ET rules

(subsumed) > (ud) > (rr 4)

indicate that (subsumed) has priority over (ud) and

(ud) has priority over (rr 4). The relation > speciﬁes

computation control for rule selection at each compu-

tation step as follows:

• If (subsumed) is applicable, then apply it.

• If (subsumed) is not applicable and (ud) is appli-

cable, then apply (ud).

• If (subsumed) and (ud) are not applicable and

(rr 4) is applicable, then apply (rr 4).

For more detailed explanation of computation control

and ET rules, the reader is referred to (Akama et al.,

2018).

Side-Change Transformation

243

: hate(B,C) ←

: hate(B,A) ←

: killer(x) ← kill(x, A)

: hate(x,y) ← kill(x,y)

: hate(A,C) ←

: hate(A,A) ←

: ← kill(x,y),richer(x, y)

: kill(A,A),kill(B,A),kill(C,A) ←

: richer(B, A), hate(B,B) ←

: ← kill(C,A)

Figure 6: The computation state to which unfolding cannot

be applied.

5.6 The State to Which Unfolding Is Not

Further Applicable

Consider the simpliﬁed Agatha QA problem hK

∧

,killer(x)i. Assume that

• (eq) is an ET rule for evaluating eq-atoms,

• (neq) is an ET rule for evaluating ground neq-

atoms,

• (erase) is an ET rule for removing body atoms that

are erasable in a clause, and

• (specAtom) is an ET rule for specializing a clause

into a number of clauses.

By using the prioritized ET rules

(eq) > (neq) > (subsumed) > (erase) >

(specAtom) > (ud),

computation reaches the state in Fig. 6, where the

clause (richer(B, A), hate(B,B) ←) prevents unfol-

ding with respect to richer-atoms and hate-atoms.

Notice the two clauses in Fig. 6 that contain richer-

atoms:

← kill(x, y), richer(x, y)

richer(B,A),hate(B, B) ←

By side-change transformation with respect to richer,

we obtain the following two clauses:

notricher(x, y) ← kill(x,y)

hate(B, B) ← notricher(B,A)

They make unfolding with respect to richer and/or

hate applicable. Unfolding with respect to notricher

and hate yields the following clauses:

← kill(B,A)

kill(A, A), kill(B,A),kill(C,A) ←

killer(x) ← kill(x,A)

← kill(C, A)

Resolution then resolves this state to enable further

application of unfolding. All atoms except killer-

atoms are removed by unfolding. Finally, we obtain

: hate(B,C) ←

: hate(B,A) ←

: kill(B,A),kill(C,A) ←

: hate(x,y) ← kill(x,y)

: hate(A,C) ←

: hate(A,A) ←

: ← kill(x,y),richer(x, y)

: ← hate(B,B)

: richer(B, A), hate(B,B) ←

: ← kill(C,A)

Figure 7: The First Stop for Proof.

: ← kill(B,A)

: ← kill(C,A)

: kill(B,A),kill(C,A) ←

Figure 8: The Second Stop for Proof.

a singleton clause set {(killer(A) ←)} and reach the

conclusion that Agatha is the answer.

Consequently, we add the side-change transfor-

mation rule (chSide) with the lowest priority, resulting

in the prioritized ET rules

(eq) > (neq) > (subsumed) > (erase) >

(specAtom) > (ud) > (chSide),

in order to have a complete fully automatic computa-

tion from the initial state to the ﬁnal result. We have

a 62-step solution, in which (ud) is applied 30 times,

(chSide) 1 time, (rr 5) 5 times, (subsumed) 16 times,

(erase) 9 times, and (neq) 1 time.

5.7 Formalization of Proof by

First-Order Formulas

Let C

be the clause (kill(B,A),kill(C,A) ←), and

consider a proof problem hK

i, referred to as the

simpliﬁed Agatha proof problem, which is to prove

that K

|= {C

}, i.e., neither B nor C killed A. Note

that the answer to the simpliﬁed Agatha QA problem

,killer(x)i being {A} does not mean that neither

B nor C killed A in any model. B and/or C may be a

killer in some model.

5.8 Two Stops by Inapplicability of

Unfolding

Consider the simpliﬁed Agatha proof problem

,{C

}i. By application of unfolding, com-

putation reaches the state in Fig. 7, which is

very similar to the case of the simpliﬁed Aga-

tha QA problem (Section 5.6), i.e., the clause

(richer(B,A),hate(B, B) ←) prevents unfolding with

respect to richer-atoms and hate-atoms. By

side-change transformation with respect to richer,

KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development

244

notricher-atoms and hate-atoms can be unfolded. Af-

ter removal of notricher-atoms and hate-atoms by

(ud), we reach the state in Fig. 8. The clause

((kill(B, A), kill(C,A) ←) blocks further unfolding.

By side-change transformation, we have:

notkill(B,A) ←

notkill(C,A) ←

← notkill(B,A),notkill(C,A)

By unfolding with respect to notkill-atoms, we have

an empty clause, i.e., it is proved that ¬kill(B,A) ∧

¬kill(C, A).

By experiments, we have found that there is a

proof consisting of 53 steps, in which (ud) is app-

lied 25 times, (chSide) 2 times, (subsumed) 14 times,

(erase) 11 times, and (neq) 1 time.

5.9 A Solution with Forwarding

Consider the same simpliﬁed Agatha proof problem

,{C

}i. We can also solve this proof problem

using forwarding transformation in place of side-

change transformation, where forwarding is an ET

rule, denoted by (fwd), for removing atoms in the

left-hand side of a clause by using a negative clause

(Akama et al., 2018). Notice the blocker of unfol-

ding in Fig. 7, i.e., (richer(B,A),hate(B,B) ←), is

changed into (richer(B,A) ←) by forwarding with

(← hate(B,B)). Similarly, the blocker of unfolding in

Fig. 8, i.e., (kill(B, A), kill(C, A) ←), is changed into

(kill(B, A) ←) by forwarding with (← kill(C, A)).

By the computation control

(eq) > (neq) > (subsumed) > (erase) >

(specAtom) > (ud) > (fwd),

we have a 55-step solution, in which (ud) is applied

24 times, (subsumed) 18 times, (fwd) 2 times, (erase)

10 times, and (neq) 1 time.

5.10 A Solution with Resolution

Consider again the proof problem hK

,{C

}i. To

solve this problem, resolution can be used in place

of side-change transformation.

In case of Fig. 7, the resolvent of the clau-

ses (richer(B,A),hate(B,B) ←) and (← hate(B, B))

is (richer(B,A) ←). In case of Fig. 8, the resol-

vent of (kill(B,A),kill(C,A) ←) and (← kill(C, A))

is (kill(B, A) ←).

By the computation control

(eq) > (neq) > (subsumed) > (erase) >

(specAtom) > (ud) > (rr 10),

we have a 57-step solution, in which (ud) is applied 24

times, (subsumed) 20 times, (rr 10) 2 times, (erase) 10

times, and (neq) 1 time.

5.11 A Solution using Resolution

Without Unfolding

The proof problem hK

,{C

}i can also be solved wit-

hout unfolding by mainly applying resolution through

the prioritized ET rules

(eq) > (neq) > (subsumed) > (erase) >

(specAtom) > (rr 3) > (rr 8) > (rr 20).

By experiments, we know that there is a proof consis-

ting of 43 steps, in which (rr 8) is applied 6 times, (rr

3) 6 times, (subsumed) 24 times, (erase) 6 times, and

(neq) 1 time.

5.12 Applying Resolution to the Agatha

QA Problem

Consider again the simpliﬁed Agatha QA problem

∧F

,killer(x)i. When we apply resolution to this

QA problem, we have the ﬁnal set of clauses in Fig. 9.

By the clause (killer(A) ←), we know that A is a kil-

ler. However, we do not know whether no one else

is a killer. Compared with the solution by unfolding,

resolution is not appropriate to provide a solution for

QA problems.

Notice the three clauses (kill(A, A) ←), (←

kill(B, A)), and (← kill(C, A)) in Fig. 9, which show

that A is a killer, while B and C are not. The clauses

in Fig. 9 do not explicitly conclude that A is the only

killer, which suggests that resolution alone is helpful

to understand the solution but is not enough to show

the exact answer to the given QA problem.

6 CONCLUSIONS

MI problems on the extended space ECLS

consti-

tute one of the largest classes of logical problems and

are of fundamental importance. Since all proof pro-

blems on FOL

and all QA problems on FOL

can be

mapped into MI problems on ECLS

, improvements

to a solution method for MI problems on ECLS

are

essential to efﬁciently solve logical problems. We in-

troduced side-change transformation on ECLS

and

proved the correctness of this transformation. We pro-

posed unfolding/side-change computation control, by

which we expect a reduction of problem size by de-

creasing the number of predicates in problem clauses.

Such a reduction often contributes to more efﬁcient

computation. This computation strategy is in sharp

contrast to the resolution proof procedure, which mo-

notonically increases the problem size regardless of

whatever control it may take.

Side-Change Transformation

245

: hate(B,x) ← live(x,D),kill(x, A)

: hate(B,x) ← kill(A,x)

: ← kill(x,C), kill(x,A),kill(x,B),live(x,D)

: ← kill(x,B),kill(x,A),hate(x,C),live(x,D)

: ← kill(x,A),hate(x,B), hate(x,C), live(x,D)

: ← kill(C,x), kill(A,x)

: ← kill(A,x),hate(C,x)

: killer(A) ←

: ← richer(A, A)

: ← kill(B,A)

: ← kill(A,B)

: ← hate(A,B)

: ← kill(B,B)

: ← hate(B,B)

: ← kill(C,A)

: ← kill(C,C)

: hate(B,C) ←

: ← hate(C,C)

: hate(B,A) ←

: ← hate(C,A)

: ← live(x,D),neq(x,A),neq(x,B),neq(x,C)

: live(A, D) ←

: live(B, D) ←

: live(C, D) ←

: ← kill(x,y),richer(x, y)

: ← hate(A,x), hate(C, x)

: ← hate(x,A), hate(x,B),hate(x,C),live(x,D)

: hate(A,A) ←

: hate(A,C) ←

: hate(x,y) ← kill(x,y)

: hate(B,x) ← hate(A,x)

: richer(x,A),hate(B, x) ← live(x,D)

: killer(x) ← kill(x, A)

: richer(B, A) ←

: kill(A,A) ←

Figure 9: A ﬁnal state of a solution by resolution.

ACKNOWLEDGEMENTS

This research was partially supported by JSPS KA-

KENHI Grant Numbers 25280078 and 26540110.

REFERENCES

Akama, K. and Nantajeewarawat, E. (2011). Meaning-

Preserving Skolemization. In 3rd International Con-

ference on Knowledge Engineering and Ontology De-

velopment, pages 322–327, Paris, France.

Akama, K. and Nantajeewarawat, E. (2014). Equiva-

lent Transformation in an Extended Space for Sol-

ving Query-Answering Problems. In 6th Asian Con-

ference on Intelligent Information and Database Sy-

stems, LNAI 8397, pages 232–241, Bangkok, Thai-

land.

Akama, K. and Nantajeewarawat, E. (2015a). A General

Schema for Solving Model-Intersection Problems on

a Specialization System by Equivalent Transforma-

tion. In 7th International Joint Conference on Kno-

wledge Discovery, Knowledge Engineering and Kno-

wledge Management, Volume 2: KEOD, pages 38–49,

Lisbon, Portugal.

Akama, K. and Nantajeewarawat, E. (2015b). Function-

variable Elimination and Its Limitations. In 7th In-

ternational Joint Conference on Knowledge Disco-

very, Knowledge Engineering and Knowledge Mana-

gement, Volume 2: KEOD, pages 212–222, Lisbon,

Portugal.

Akama, K. and Nantajeewarawat, E. (2016a). Model-

Intersection Problems with Existentially Quantiﬁed

Function Variables: Formalization and a Solution

Schema. In 8th International Joint Conference on

Knowledge Discovery, Knowledge Engineering and

Knowledge Management, Volume 2: KEOD, pages

52–63, Porto, Portugal.

Akama, K. and Nantajeewarawat, E. (2016b). Unfolding

Existentially Quantiﬁed Sets of Extended Clauses. In

8th International Joint Conference on Knowledge Dis-

covery, Knowledge Engineering and Knowledge Ma-

nagement, Volume 2: KEOD, pages 96–103, Porto,

Portugal.

Akama, K., Nantajeewarawat, E., and Akama, T. (2018).

Computation Control by Prioritized ET Rules. In 10th

International Joint Conference on Knowledge Disco-

very, Knowledge Engineering and Knowledge Mana-

gement, Volume 2: KEOD, Seville, Spain.

Chang, C.-L. and Lee, R. C.-T. (1973). Symbolic Logic and

Mechanical Theorem Proving. Academic Press.

Donini, F. M., Lenzerini, M., Nardi, D., and Schaerf, A.

(1998). AL-log: Integrating Datalog and Description

Logics. Journal of Intelligent Information Systems,

16:227–252.

Fitting, M. (1996). First-Order Logic and Automated The-

orem Proving. Springer-Verlag, second edition.

Pelletier, F. J. (1986). Seventy-Five Problems for Testing

Automatic Theorem Provers. Journal of Automated

Reasoning, 2(2):191–216.

Robinson, J. A. (1965). A Machine-Oriented Logic Ba-

sed on the Resolution Principle. Journal of the ACM,

12:23–41.

KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development

246