A General Schema for Solving Model-Intersection Problems on a

Specialization System by Equivalent Transformation

Kiyoshi Akama

and Ekawit Nantajeewarawat

Information Initiative Center, Hokkaido University, Hokkaido, Japan

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

Keywords:

Model-Intersection Problem, Query-Answering Problem, Equivalent Transformation, Problem Solving.

Abstract:

A model-intersection problem (MI problem) is a pair of a set of clauses and an exit mapping. We deﬁne MI

problems on specialization systems, which include many useful classes of logical problems, such as proof

problems on ﬁrst-order logic and query-answering (QA) problems in pure Prolog and deductive databases.

The theory presented in this paper makes clear the central and fundamental structure of representation and

computation for many classes of logical problems by (i) axiomatization and (ii) equivalent transformation.

Clauses in this theory are constructed based on abstract atoms and abstract operation on them, which can be

used for representation of many speciﬁc subclasses of problems with concrete syntax. Various computation

can be realized by repeated application of many equivalent transformation rules, allowing many possible

computation procedures, for instance, computation procedures based on resolution and unfolding. This theory

can also be useful for inventing solutions for new classes of logical problems.

1 INTRODUCTION

This paper introduces a model-intersection problem

(MI problem), which is a pair hCs,ϕi, where Cs is a

set of clauses and ϕ is a mapping, called an exit map-

ping, used for constructing the output answer from the

intersection of all models of Cs. More formally, the

answer to a MI problem hCs, ϕi is ϕ(

Models(Cs)),

where Models(Cs) is the set of all models of Cs. The

set of all MI problems constitutes a very large class of

problems and is of great importance.

A QA problem is a pair hCs,ai, where Cs is a set

of clauses and a is a user-deﬁned query atom. The

answer to such a QA problem hCs,ai is deﬁned as

the set of all ground instances of a that are logical

consequences of Cs. A QA problem hCs,ai is a MI

problem hCs,ϕ

i, where for any set G of ground user-

deﬁned atoms, ϕ

(G) is the intersection of G and the

set of all ground instances of a. Characteristically, a

QA problem is an “all-answers ﬁnding” problem, i.e.,

all ground instances of a given query atom satisfying

the requirement above are to be found. Many logic

programming languages, including Datalog, Prolog,

and other extensions of Prolog, deal with speciﬁc sub-

classes of QA problems.

The class of proof problems is also a subclass of

MI problems. In contrast to a QA problem, a proof

problem, is a “yes/no” problem; it is concerned with

checking whether or not one given logical formula is a

logical consequence of another given logical formula.

Formally, a proof problem is a pair hE

i, where E

and E

are ﬁrst-order formulas, and the answer to this

problem is deﬁned to be “yes” if E

is a logical con-

sequence of E

, and it is deﬁned to be “no” otherwise.

Historically, proof problems were ﬁrst solved

(Robinson, 1965). Then QA problems on pure Prolog

were solved based on the resolution principle, which

is a solution for proof problems. This approach is

proof-centered. It has been believed that computa-

tion of Prolog is an inference process. The theory of

SLD resolution was used for the correctness of Pro-

log computation. Many solutions proposed so far for

some other classes of logical problems are also basi-

cally proof-centered.

In contrast, it was shown in (Akama and Nantajee-

warawat, 2013) that the set of all proof problems can

be embedded into the set of all QA problems. This

result supports a QA-centered approach to solving

proof problems, i.e., ﬁrst, develop a general solution

for QA problems, and then, apply it as a solution for

proof problems (Akama and Nantajeewarawat, 2012).

Since a QA problem is a MI problem (as will be seen

in Theorem 3), we have

PROOF ⊂ QA ⊂ MI,

Akama, K. and Nantajeewarawat, E..

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation.

In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2015) - Volume 2: KEOD, pages 38-49

ISBN: 978-989-758-158-8

where PROOF, QA, and MI denote the class of all

proof problems, the class of all QA problems, and the

class of all MI problems, respectively. The class of

all MI problems is larger than that of all QA prob-

lems, and it is a more natural class to be solved by the

method presented in this paper. A general solution

method for MI problems can be applied to any arbi-

trary QA problem and any arbitrary proof problem.

MI problems are axiomatically constructed on an

abstract structure, called a specialization system. It

consists of abstract atoms and abstract operations

(extensions of variable-substitution operations) on

atoms, called specializations. These abstract com-

ponents can be any arbitrary mathematical objects as

long as they satisfy given axioms. Abstract clauses

can be built on abstract atoms. This is a sharp contrast

to most of the conventional theories in logic program-

ming, where concrete syntax is usually used. In Pro-

log, for example, usual ﬁrst-order atoms and substi-

tutions with concrete syntax are used, and there is no

way to give a foundation for other forms of extended

atoms and for various specialization operations other

than the usual variable-substitution operation.

An axiomatic theory enables us to develop a very

general theory. By instantiating a specialization sys-

tem to a speciﬁc domain and by imposing certain re-

strictions on clauses, our theory can be applied to

many subclasses of MI problems.

We proposed a general schema of solving MI

problems by equivalent transformation (ET), where

problems are solved by repeated simpliﬁcation. We

introduced the concept of target mapping and pro-

posed three target mappings. Since transformation

preserving a target mapping is ET, target mappings

provide a strong foundation for inventing many ET

rules for solving MI problems on clauses.

An ET-based solution consists of the following

steps: (i) formalize an initial MI problem on some

specialization system, (ii) prepare ET rules, (iii) con-

struct an ET sequence, (iv) compute a set of models

using a target mapping, (v) apply the set-intersection

operation to the resulting set of models, and (vi) apply

an exit mapping to the intersection result to obtain a

solution.

To begin with, Section 2 recalls the concept of

specialization system and formalizes MI problems on

a specialization system. Section 3 deﬁnes the notions

of a target mapping and a representative mapping, and

introduces a schema for solving MI problems based

on equivalent transformation (ET) preserving target

mappings. The correctness of this solution schema

is shown. Section 4 applies the general ET-based

schema in Section 3 to the domain of clause sets with

built-in constraint atoms. A target mapping, MM, is

introduced for associating with each clause set a col-

lection of its speciﬁc models computed in a bottom-up

manner. Section 5 shows an example of solution of a

MI problem. Section 6 concludes the paper.

The notation that follows holds thereafter. Given

a set A, pow(A) denotes the power set of A and

partialMap(A) the set of all partial mappings on A

(i.e., from A to A). For any partial mapping f from a

set A to a set B, dom( f) denotes the domain of f, i.e.,

dom( f) = {a | (a ∈ A) & ( f(a) is deﬁned)}.

2 CLAUSES AND

MODEL-INTERSECTION

PROBLEMS

2.1 Specialization Systems

A substitution {X/ f(a),Y/g(z)} changes an atom

p(X,5,Y) in the term domain into p( f(a),5,g(z)).

Generally, a substitution in ﬁrst-order logic deﬁnes

a total mapping on the set of all atoms in the term

domain. Composition of such mappings is also re-

alized by some substitution. There is a substitution

that does not change any atom (i.e., the empty sub-

stitution). A ground atom in the term domain is a

variable-free atom.

Likewise, in the string domain, substitutions for

strings are used. A substitution {X/“aYbc”,Y/“xyz”}

changes an atom p(“X5Y”) into p(“aYbc5xyz”).

Such a substitution for strings deﬁnes a total mapping

on the set of all atoms that may include string vari-

ables. Composition of such mappings is also realized

by some string substitution. There is a string substi-

tution that does not change any atom (i.e., the empty

substitution). A ground atom in the string domain is a

variable-free atom.

A similar operation can be considered in the

class-variable domain. Consider, for example, an

atom p(X : animal,Y : dog,Z: cat) in this domain,

where X : animal, Y : dog, and Z: cat represent an

animal object, a dog object, and a cat object, re-

spectively. When we obtain additional information

that X is a dog, we can restrict X : animal into

X : dog and the atom p(X : animal,Y : dog,Z: cat)

into p(X : dog,Y : dog,Z: cat). By contrast, with new

information that Z is a dog, we cannot restrict Z : cat

and the above atom since Z cannot be a dog and a cat

at the same time. More generally, such a restriction

operation may not be applicable to some atoms, i.e.,

it deﬁnes a partial mapping on the set of all atoms.

Composition of such partial mappings is also a par-

tial mapping, and we can determine some composi-

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation

tion operation corresponding to it. An empty substi-

tution that does not change any atom can be intro-

duced. A ground atom in the class-variable domain is

a variable-free atom.

In order to capture the common properties of such

operations on atoms, the notion of a specialization

system was introduced around 1990.

Deﬁnition 1. A specialization system Γ is a quadru-

ple hA,G,S, µi of three sets A, G, and S, and a map-

ping µ from S to partialMap(A) that satisﬁes the fol-

lowing conditions:

1. (∀s

′

′′

∈ S)(∃s ∈ S ) : µ(s) = µ(s

′

) ◦ µ(s

′′

2. (∃s ∈ S)(∀a ∈ A) : µ(s)(a) = a.

3. G ⊆ A.

Elements of A, G, and S are called atoms, ground

atoms, and specializations, respectively. The map-

ping µ is called the specialization operator of Γ. A

specialization s ∈ S is said to be applicable to a ∈ A

iff a ∈ dom(µ(s)).

Assume that a specialization system Γ = hA,G,S,

µi is given. A specialization in S will often be denoted

by a Greek letter such as θ. A specialization θ ∈ S

will be identiﬁed with the partial mapping µ(θ) and

used as a postﬁx unary (partial) operator on A (e.g.,

µ(θ)(a) = aθ), provided that no confusion is caused.

Let ε denote the identity specialization in S , i.e., aε =

a for any a ∈ A. For any θ, σ ∈ S, let θ ◦ σ denote a

specialization ρ ∈ S such that µ(ρ) = µ(σ) ◦ µ(θ), i.e.,

a(θ◦ σ) = (aθ)σ for any a ∈ A.

2.2 User-deﬁned Atoms, Constraint

Atoms, and Clauses

Let Γ

= hA

,µ

i and Γ

= hA

,µ

i be

specialization systems such that S

= S

. Elements

of A

are called user-deﬁned atoms and those of G

are called ground user-deﬁned atoms. Elements of

are called constraint atoms and those of G

are

called ground constraint atoms. Hereinafter, assume

that S = S

= S

. Elements of S are called special-

izations. Let TCON denote the set of all true ground

constraint atoms.

A clause on hΓ

,Γ

i is an expression of the form

,.. .,a

← b

,.. .,b

where m ≥ 0, n ≥ 0, and each of a

,.. .,a

,.. .,b

belongs to A

∪ A

. It is a ground clause on hΓ

,Γ

iff each of a

,.. .,a

,.. .,b

belongs to G

∪ G

Let CLS denote the set of all clauses on hΓ

,Γ

2.3 Interpretations and Models

An interpretation is a subset of G

. Unlike ground

user-deﬁned atoms, the truth values of ground con-

straint atoms are predetermined by TCON (cf. Sec-

tion 2.2) independently of interpretations. A ground

constraint atom g is true iff g ∈ TCON. It is false oth-

erwise.

A ground clause C = (a

,.. .,a

← b

,.. .,b

) is

true with respect to an interpretation G ⊆ G

(in other

words, G satisﬁes C) iff at least one of the following

conditions is satisﬁed:

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

∈ G ∪

TCON.

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

/∈ G ∪

TCON.

A clause C is true with respect to an interpretation

G ⊆ G

(in other words, G satisﬁes C) iff for any spe-

cialization θ such that Cθ is ground, Cθ is true with

respect to G. A model of a clause set Cs ⊆ CLS is an

interpretation that satisﬁes every clause in Cs.

Note that the standard semantics is taken in this

paper, i.e., all models of a formula are considered

instead of speciﬁc ones, such as those considered in

the minimal model semantics (Clark, 1978; Lloyd,

1987) (i.e., the semantics underlying logic program-

ming) and those considered in stable model semantics

(Gelfond and Lifschitz, 1988; Gelfond and Lifschitz,

1991) (i.e., the semantics underlying answer set pro-

gramming).

2.4 Model-Intersection (MI) Problems

Let Models be a mapping that associates with each

clause set the set of all of its models, i.e., Models(Cs)

is the set of all models of Cs for any Cs ⊆ CLS.

Assume that a person A and a person B are in-

terested in knowing which atoms in G

are true and

which atoms in G

are false. They want to know the

unknown set G of all true ground atoms. Due to short-

age of knowledge, A still cannot determine one unique

true subset of G

. The person A can only limit possi-

ble subsets of true atoms by specifying a subset Gs

of pow(G

). The unknown set G of all true atoms

belongs to Gs. One way for A to inform this knowl-

edge to B compactly is to send to B a clause set Cs

such that Gs ⊆ Models(Cs). Receiving Cs, B knows

that Models(Cs) includes all possible intended sets of

ground atoms, i.e., G ∈ Models(Cs). As such, B can

know that each ground atom outside

Models(Cs) is

false, i.e., for any g ∈ G

, if g /∈

Models(Cs), then

g /∈ G. The person B can also know that each ground

atom in

Models(Cs) is true, i.e., for any g ∈ G

, if

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

g ∈

Models(Cs), then g ∈ G. This shows the impor-

tance of calculating

Models(Cs).

A model-intersection problem (MI problem) is a

pair hCs,ϕi, where Cs ⊆ CLS and ϕ is a mapping from

pow(G

) to some set W. The mapping ϕ is called an

exit mapping. The answer to this problem, denoted by

ans

(Cs,ϕ), is deﬁned by

ans

(Cs,ϕ) = 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

2.5 Query-Answering (QA) Problems

Let Cs ⊆ CLS. For any Cs

′

⊆ CLS, Cs

′

is a logical

consequence of Cs, denoted by Cs |= Cs

′

, iff every

model of Cs is also a model of Cs

′

. For any a ∈ A

a is a logical consequence of Cs, denoted by Cs |= a,

iff Cs |= {(a ←)}.

A query-answering problem (QA problem) in this

paper is a pair hCs,ai, where Cs ⊆ CLS and a is a

user-deﬁned atom in A

. The answer to a QA prob-

lem hCs,ai, denoted by ans

(Cs,a), is deﬁned by

ans

(Cs,a) = {aθ | (θ ∈ S ) & (aθ ∈ G

) &

(Cs |= (aθ ←))}.

Theorem 1. For any Cs ⊆ CLS and a ∈ A

ans

(Cs,a) = rep(a) ∩ (

Models(Cs)),

where rep(a) denotes the set of all ground instances

of a.

Proof: Let Cs ⊆ CLS and a ∈ A

. By the deﬁ-

nition of |=, for any ground atom g ∈ G

, Cs |= g iff

g ∈

Models(Cs). Then

ans

(Cs,a)

= {aθ | (θ ∈ S) & (aθ ∈ G

) & (Cs |= (aθ ←))}

= {g | (θ ∈ S) & (g = aθ) & (g ∈ G

) &

(Cs |= (g ←))}

= {g | (g ∈ rep(a)) & (Cs |= (g ←))}

= {g | (g ∈ rep(a)) & (g ∈ (

Models(Cs)))}

= rep(a) ∩ (

Models(Cs)).

Theorem 1 shows the importance of the intersec-

tion of all models of a clause set. By this theorem, the

answer to a QA problem can be rewritten as follows:

Theorem 2. Let Cs ⊆ CLS and a ∈ A

. Then

ans

(Cs,a) = ans

(Cs,ϕ

), where for any G⊆ G

(G) = rep(a) ∩ G.

Proof: It follows from Theorem 1 and the deﬁni-

tion of ϕ

that ans

(Cs,a) = ϕ

(

Models(Cs)) =

ans

(Cs,ϕ

This is one way to regard a QA problem as a MI

problem, which can be understood as follows: The set

Models(Cs) often contains too many ground atoms.

The set rep(a) speciﬁes a range of interest in the set

. The exit mapping ϕ

focuses attention on the part

rep(a) by making intersection with it.

Theorem 3 below shows another way to formalize

a QA problem as a MI problem.

Theorem 3. Let Cs ⊆ CLS and a ∈ A

. Then

ans

(Cs,a) = ans

(Cs∪ {ans(a) ← a},ϕ

), where

for any G ⊆ G

, ϕ

(G) = {x | ans(x) ∈ G}.

Proof: By Theorem 1 and the deﬁnition of ϕ

ans

(Cs,a)

= rep(a) ∩ (

Models(Cs))

= ϕ

(

Models(Cs∪ {ans(a) ← a}))

= ans

(Cs∪ {ans(a) ← a},ϕ

In logic programming (Lloyd, 1987), a problem

represented by a pair of a set of deﬁnite clauses and a

query atom has been intensively discussed. In the de-

scription logic (DL) community (Baader et al., 2007),

a class of problems formulated as conjunctions of

DL-based axioms and assertions together with query

atoms has been discussed (Tessaris, 2001). These two

problem classes can be formalized as subclasses of

QA problems considered in this paper.

3 SOLVING MI PROBLEMS BY

EQUIVALENT

TRANSFORMATION

A general schema for solving MI problems based on

equivalent transformation is formulated and its cor-

rectness is shown (Theorem 8).

3.1 Preservation of Partial Mappings

and Equivalent Transformation

Terminologies such as preservation of partial map-

pings and equivalent transformation are deﬁned in

general below. They will be used with a speciﬁc class

of partial mappings called target mappings, which

will be introduced in Section 3.2.

The expression ans(a) is not an atom in the usual ﬁrst-

order logic space. One way to understand Theorem 3 in the

context of the conventional ﬁrst-order logic is (i) ans(a) is

interpreted as ans(v

,...,v

), where v

,...,v

are the vari-

ables occurring in a, and then (ii) ϕ

(G) = {x | ans(x) ∈ G}

is interpreted as ϕ

(G) = {a

′

| a

′

= ans(t

,...,t

) ∈ G}.

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation

Assume that X and Y are sets and f is a par-

tial mapping from X to Y. For any x,x

′

∈ dom( f),

transformation of x into x

′

is said to preserve f iff

f(x) = f(x

′

). For any x,x

′

∈ dom( f), transformation

of x into x

′

is called equivalent transformation (ET)

with respect to f iff the transformation preserves f,

i.e., f(x) = f(x

′

Let F be a set of partial mappings from a set X

to a set Y. Given x,x

′

∈ X, transformation of x into

′

is called equivalent transformation (ET) with re-

spect to F iff there exists f ∈ F such that the trans-

formation preserves f. A sequence [x

,.. .,x

] of

elements in X is called an equivalent transformation

sequence (ET sequence) with respect to F iff for any

i ∈ {0, 1,.. .,n − 1}, transformation of x

into x

i+1

ET with respect to F. When emphasis is placed on

the initial element x

and the ﬁnal element x

, this se-

quence is also referred to as an ET sequence from x

to x

3.2 Target Mappings

Given a MI problem hCs,ϕi, since ans

(Cs,ϕ) =

ϕ(

Models(Cs)), the answer to this MI problem is

determined uniquely by Models(Cs) and ϕ. As a re-

sult, we can equivalently consider a new MI prob-

lem with the same answer by switching from Cs to

another clause set Cs

′

if Models(Cs) = Models(Cs

′

According to the general terminologies deﬁned

in Section 3.1, on condition that Models(Cs) =

Models(Cs

′

), transformation from x = Cs into x

′

preserves f = Models and is called ET with re-

spect to f = Models, where (i) x,x

′

∈ pow(CLS) and

(ii) Models(x),Models(x

′

) ∈ pow(pow(G)). We can

also consider an ET sequence [Cs

,Cs

,.. .,Cs

] of

elements in pow(CLS) with respect to a singleton set

{Models}. MI problems can be transformed into sim-

pler forms by ET preserving Models.

In order to use more partial mappings for simpli-

ﬁcation of MI problems, we extend our consideration

from the speciﬁc mapping Models to a class of partial

mappings, called GSETMAP, deﬁned below.

Deﬁnition 2. GSETMAP is the set of all partial map-

pings from pow(CLS) to pow(pow(G)).

As deﬁned in Section 2.4, Models(Cs) is the set of

all models of Cs for any Cs ⊆ CLS. Since a model is

a subset of G, Models is regarded as a total mapping

from pow(CLS) to pow(pow(G)). Since a total map-

ping is also a partial mapping, the mapping Models is

a partial mapping from pow(CLS) to pow(pow(G)),

i.e., it is an element of GSETMAP.

A partial mapping M in GSETMAP is of par-

ticular interest if

M(Cs) =

Models(Cs) for any

Cs ∈ dom(M). Such a partial mapping is called a tar-

get mapping.

Deﬁnition 3. A partial mapping M ∈ GSETMAP is a

target mapping iff for any Cs ∈ dom(M),

M(Cs) =

Models(Cs).

It is obvious that:

Theorem 4. The mapping Models is a target map-

ping.

Transformation preserving target mappings and

computation of a target mapping constitute a method

for solving MI problems in this paper.

For more general consideration, we introduce a bi-

nary relation  on GSETMAP as follows:

Deﬁnition 4. Let M

∈ GSETMAP. M

 M

iff

the following conditions are satisﬁed:

1. dom(M

) ⊆ dom(M

2. For any Cs in dom(M

(Cs) =

(Cs).

Obviously,  is reﬂexive and transitive. It is also

obvious that:

Proposition 1. For any M ∈ GSETMAP, M is a target

mapping iff M  Models.

By its deﬁnition, a target mapping M satisﬁes the

following two conditions: (i) the domain of M is a

subset of pow(CLS), and (ii) for any clause set Cs in

the domain of M, the intersection of all ground-atom

sets in M(Cs) is equal to the intersection of all mod-

els of Cs. By the ﬁrst condition, since the domain of

M can be smaller than that of the mapping Models,

we can expect a more efﬁcient program for comput-

ing M(Cs) for Cs in the domain of M. By the second

condition, the correctness of transformation and com-

putation is guaranteed (Theorems 5 and 8).

Let hCs,ϕi be a MI problem. If M is a target map-

ping such that M(Cs) is deﬁned, then M can be used

for computing the answer to hCs,ϕi. More precisely:

Theorem 5. Let hCs, ϕi be a MI problem and M ∈

GSETMAP. If M is a target mapping and Cs ∈

dom(M), then ans

(Cs,ϕ) = ϕ(

M(Cs)).

Proof: Assume that M is a target mapping

and Cs ∈ dom(M). Then

M(Cs) =

Models(Cs).

Consequently, ans

(Cs,ϕ) = ϕ(

Models(Cs)) =

ϕ(

M(Cs)).

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

3.3 Representative Mappings

The relations “smaller than” and “ﬁner than” on

GSETMAP are introduced below.

Deﬁnition 5. Let M

∈ GSETMAP. M

is smaller

than M

iff the following conditions are satisﬁed:

1. dom(M

) ⊆ dom(M

2. For any Cs ∈ dom(M

), M

(Cs) ⊆ M

(Cs).

Deﬁnition 6. Let M

∈ GSETMAP. M

is ﬁner

than M

iff the following conditions are satisﬁed:

1. dom(M

) ⊆ dom(M

2. For any Cs ∈ dom(M

) and any m

∈ M

(Cs),

there exists m

∈ M

(Cs) such that m

⊆ m

A smaller target mapping is basically preferable

in order to reduce the cost of computing an answer.

The concept of representative mapping deﬁned below

is useful for constructing small target mappings.

Deﬁnition 7. Let M

∈ GSETMAP. M

is a rep-

resentative mapping of M

iff the following condi-

tions are satisﬁed:

1. M

is smaller than M

2. M

is ﬁner than M

Theorem 6. Let M

∈ GSETMAP. If M

is a rep-

resentative mapping of M

, then M

 M

Proof: Suppose that M

is a representative map-

ping of M

. Let Cs ∈ dom(M

). Since M

is smaller

than M

, M

(Cs) ⊆ M

(Cs). Thus

(Cs) ⊇

(Cs). We show that

(Cs) ⊆

(Cs)

as follows: Assume that g ∈

(Cs). Let m

∈

(Cs). Since M

is ﬁner than M

, there exists

∈ M

(Cs) such that m

⊆ m

. Since g ∈

(Cs),

g belongs to m

. So g ∈ m

, and thus, g ∈

(Cs).

It follows that

(Cs) =

(Cs). Hence M



3.4 Solving MI Problems by Equivalent

Transformation

Next, a schema for solving MI problems based

on equivalent transformation (ET) preserving target

mappings is formulated. The notions of preservation

of target mappings, ET with respect to target map-

pings, and an ET sequence are obtained by specializ-

ing the general deﬁnitions in Section 3.1.

Let π be a mapping, called state mapping, from a

given set STATE to the set of all MI problems. Ele-

ments of STATE are called states.

Deﬁnition 8. Let hS,S

′

i ∈ STATE× STATE. hS,S

′

i is

an ET step with π iff if π(S) = hCs,ϕi and π(S

′

) =

hCs

′

,ϕ

′

i, then ans

(Cs,ϕ) = ans

(Cs

′

,ϕ

′

Deﬁnition 9. A sequence [S

,.. .,S

] of elements

of STATE is an ET sequence with π iff for any i ∈

{0,1,.. .,n− 1}, hS

i+1

i is an ET step with π.

We can construct an ET step by using transfor-

mation preserving a target mapping. ET steps used

for solving MI problems are mainly realized based on

target mappings.

Theorem 7. Let S, S

′

∈ STATE. Assume that π(S) =

hCs,ϕi, π(S

′

) = hCs

′

,ϕi, and M is a target mapping

such that M(Cs) = M(Cs

′

). Then hS,S

′

i is an ET step

with π.

Proof:

ans

(Cs,ϕ)

= ϕ(

Models(Cs))

= (since M is a target mapping)

= ϕ(

M(Cs))

= (since M(Cs) = M(Cs

′

))

= ϕ(

M(Cs

′

))

= (since M is a target mapping)

= ϕ(

Models(Cs

′

))

= ans

(Cs

′

,ϕ)

As shown below, we can solve MI problems by

constructing ET sequences.

Theorem 8. Assume that:

• hCs,ϕi is a MI problem.

• [S

,.. .,S

] is an ET sequence with π.

• π(S

) = hCs,ϕi and π(S

) = hCs

,ϕ

• M is a target mapping such that Cs

∈ dom(M).

Then ans

(Cs,ϕ) = ϕ

(

M(Cs

)).

Proof:

ans

(Cs,ϕ)

= ϕ(

Models(Cs))

= (since [S

,.. .,S

] is an ET sequence)

= ϕ

(

Models(Cs

))

= (since M is a target mapping)

= ϕ

(

M(Cs

)).

4 TARGET MAPPINGS FOR

CLAUSES AND COMPUTATION

Next, three target mappings are introduced, i.e., τ

for sets of positive unit clauses, τ

for sets of deﬁ-

nite clauses, and MM for sets of arbitrary clauses in

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation

CLS. Based on these target mappings, an ET solution

for MI problems on clauses is given according to the

general schema of Section 3.

4.1 A Target Mapping for Sets of

Positive Unit Clauses

A positive unit clause is a clause of the form (a ←),

where a is a user-deﬁned atom. Let PUCL denote the

set of all positive unit clauses. For any user-deﬁned

atom a, let rep(a) denote the set of all ground in-

stances of a.

A partial mapping τ

∈ GSETMAP is deﬁned as

follows:

1. For any F ⊆ PUCL, τ

(F) is the singleton set

{

{rep(a) | (a ←) ∈ F}}.

2. For any Cs ⊆ CLS such that Cs 6⊆ PUCL, τ

(Cs)

is undeﬁned.

Theorem 9. τ

is a representative mapping of Models

and is a target mapping.

Proof: Assume that F ⊆ PUCL. Let m

{rep(a) | (a ←) ∈ F}. Obviously, m

is a model

of F, i.e., m

∈ Models(F). So τ

is smaller than

Models. Now let m ∈ Models(F). For any (a ←) ∈ F

and any g ∈ rep(a), g is true with respect to m, i.e.,

g ∈ m. Then m

⊆ m. So τ

is also ﬁner than Models,

whence τ

is a representative mapping of Models. By

Theorem 6 and Proposition 1, τ

is a target map-

ping.

4.2 A Target Mapping for Sets of

Deﬁnite Clauses

A deﬁnite clause is a clause whose left-hand side con-

tains exactly one user-deﬁned atom and no constraint

atom. Let DCL denote the set of all deﬁnite clauses.

Given a deﬁnite clause C, the atom in the left-hand

side ofC is called the head of C, denoted by head(C),

and the set of all user-deﬁned atoms and constraint

atoms in the right-hand side of C is called the body

of C, denoted by body(C). Assume that D is a set of

deﬁnite clauses in DCL. The meaning of D, denoted

by M (D), is deﬁned as follows:

1. A mapping T

on pow(G) is deﬁned by: for any

set G ⊆ G, T

(G) is the set

{head(Cθ) | (C ∈ D) & (θ ∈ S) &

(each user-deﬁned atom in body(Cθ) is in G) &

(each constraint atom in body(Cθ) is true)}.

2. M (D) is then deﬁned as the set

∞

n=1

(∅),

where T

(∅) = T

(∅) and for each n > 1, T

(∅)

= T

n−1

(∅)).

Then a partial mapping τ

∈ GSETMAP is deﬁned

below.

1. For any D ⊆ DCL, τ

(D) is the singleton set

{M (D)}.

2. For any Cs ⊆ CLS such that Cs 6⊆ DCL, τ

(Cs) is

undeﬁned.

Theorem 10. τ

is a representative mapping of

Models and is a target mapping.

Proof: Let D ⊆ DCL. Since M (D) is a model

of D, {M (D)} ⊆ Models(D). So τ

is smaller than

Models. Let m ∈ Models(D). Since M (D) is the least

model of D, M (D) ⊆ m. So τ

is ﬁner than Models.

Then τ

is a representative mapping of Models, and

thus, by Theorem 6 and Proposition 1, it is a target

mapping.

Theorem 11. For any F ⊆ PUCL, τ

(F) = τ

(F).

Proof: For any F ⊆ PUCL, τ

(F) = {M (F)} =

{

{rep(a) | (a ←) ∈ F}} = τ

(F).

4.3 A Target Mapping for Clause Sets

Given a clauseC, the set of all user-deﬁned atoms and

constraint atoms in the left-hand side of C is denoted

by lhs(C) and the set of all those in the right-hand

side of C is denoted by rhs(C). A clause C is said

to be positive if lhs(C) is not empty; it is said to be

negative otherwise.

It is assumed henceforth that (i) for any constraint

atom c, not(c) is a constraint atom; (ii) for any con-

straint atom c and any specialization θ, not(c)θ =

not(cθ); and (iii) for any ground constraint atom c,

c is true iff not(c) is not true.

The following notation is used for deﬁning a tar-

get mapping MM for arbitrary clauses in CLS (Deﬁ-

nition 10).

1. Let Cs be a set of clauses possibly with con-

straint atoms. MVRHS(Cs) is deﬁned as the set

{MVRHS(C) | C ∈ Cs}, where for any clause C ∈

Cs, MVRHS(C) is the clause obtained from C as

follows: For each constraint atom c in lhs(C), re-

move c from lhs(C) and add not(c) to rhs(C).

2. Let Cs be a set of clauses with no constraint

atom in their left-hand sides. For any G ⊆ G,

GINST(Cs,G) is deﬁned as the set

{RMCON(Cθ) | (C ∈ Cs) & (θ ∈ S) &

(each user-deﬁned atom in Cθ is in G) &

(each constraint atom in rhs(Cθ) is true)},

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

where for any clause C

′

, RMCON(C

′

) is the clause

obtained fromC

′

by removing all constraint atoms

from it.

3. Let Cs be a set of clauses possibly with constraint

atoms. For any G ⊆ G, INST(Cs,G) is deﬁned by

INST(Cs,G) = GINST(MVRHS(Cs), G).

4. Let Cs be a set of ground clauses with no con-

straint atom. We can construct a set of deﬁ-

nite clauses from Cs as follows: For each clause

C ∈ Cs,

• if lhs(C) = ∅, then construct a deﬁnite clause

the head of which is ⊥ and the body of which

is rhs(C), where ⊥ is a special symbol not oc-

curring in Cs;

• if lhs(C) 6= ∅, then (i) select one arbitrary atom

a from lhs(C), and (ii) construct a deﬁnite

clause the head of which is a and the body of

which is rhs(C).

Let DC(Cs) denote the set of all deﬁnite-clause

sets possibly constructed from Cs in the above

way.

Proposition 2. Let Cs ⊆ CLS. For any m ⊆ G, m is a

model of Cs iff m is a model of INST(Cs,G).

Proof: INST(Cs,G) is obtained from Cs by (i)

moving constraint atoms in the left-hand sides of

clauses into their right-hand sides, (ii) instantiation of

variables into ground terms, (iii) removal of clauses

containing false constraint atoms in their right-hand

sides, and (iv) removal of true constraint atoms from

the remaining clauses. Each of the operations (i), (ii),

(iii), and (iv) preserves models.

A mapping MM is deﬁned below.

Deﬁnition 10. A mapping MM ∈ GSETMAP is de-

ﬁned by

MM(Cs) = {M (D) | (D ∈ DC(INST(Cs, G))) &

(⊥ /∈ M (D))}

for any Cs ⊆ CLS.

Theorem 12. MM is a representative mapping of

Models and is a target mapping.

Proof: First, we show that MM is smaller than

Models. Let Cs ⊆ CLS. Suppose that m ∈ MM(Cs).

Let Cs

′

= INST(Cs,G). Then there exists D such that

m = M (D), D ∈ DC(Cs

′

), and ⊥ /∈ M (D). We show

that m is a model of Cs

′

as follows:

• Let C

be a positive clause in Cs

′

. Since D ∈

DC(Cs

′

), there exists C ∈ D such that head(C) ∈

lhs(C

) and body(C) = rhs(C

). Since m satis-

ﬁes C, m also satisﬁes C

. Hence m satisﬁes every

positive clause in Cs

′

• Let C

be a negative clause in Cs

′

. Since D ∈

DC(Cs

′

), there existsC

′

∈ D such that head(C

′

) =

⊥ and body(C

′

) = rhs(C

). Since ⊥ /∈ M (D),

m does not include body(C

′

). So rhs(C

) 6⊆ m,

whence m satisﬁes C

. Hence m satisﬁes every

negative clause in Cs

′

So m is a model of Cs

′

. By Proposition 2, m is a model

of Cs, i.e., m ∈ Models(Cs).

Next, we show that MM is ﬁner than Models. Let

Cs ⊆ CLS. Suppose that m

′

∈ Models(Cs), i.e., m

′

a model of Cs. Let Cs

′

= INST(Cs, G). By Propo-

sition 2, m

′

is also a model of Cs

′

. Let D be a set

of deﬁnite clauses obtained from Cs

′

by constructing

from each positive clause C in Cs

′

a deﬁnite clause C

′

as follows:

1. Select an atom a from lhs(C) as follows:

(a) If rhs(C) ⊆ m

′

, then select an atom a ∈ lhs(C) ∩

′

(b) If rhs(C) 6⊆ m

′

, then select an arbitrary atom a ∈

lhs(C).

2. Construct C

′

as a deﬁnite clause such that

head(C

′

) = a and body(C

′

) = rhs(C).

It is obvious that m

′

is a model of D. Let m

′′

= M (D).

Since m

′′

is the least model of D, m

′′

⊆ m

′

. Since

′

is a model of Cs

′

, m

′

satisﬁes all negative clauses

in Cs

′

. Since m

′′

⊆ m

′

, m

′′

also satisﬁes all negative

clauses in Cs

′

. It followsthat ⊥ /∈ M (D). Hence m

′′

∈

MM(Cs).

So MM is a representative mapping of Models.

By Theorem 6 and Proposition 1, MM is a target map-

ping.

Theorem 13. For any D ⊆ DCL, MM(D) = τ

(D).

Proof: Let D ⊆ DCL. Then DC(INST(D,G)) is

the singleton set {INST(D,G)}. Obviously, M (D) =

M (INST(D,G)) and ⊥ /∈ M (INST(D, G)). It follows

that

MM(D) = {M (D

′

) | (D

′

∈ DC(INST(D,G))) &

(⊥ /∈ M (D

′

))}

= {M (INST(D,G))}

= {M (D)}

= τ

(D).

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation

Figure 1: Target mappings and ET computation paths.

4.4 Computation Cost for Solving MI

Problems

Given a set Cs of clauses, a user-deﬁned atom a, and

an exit mapping ϕ, the answer to the MI problem

hCs,ϕi, i.e., ans

(Cs,ϕ) = ϕ(

Models(Cs)), can be

directly obtained by the computation shown in the

leftmost path in Fig. 1.

By Theorems 4, 9, and 12, each of Models, τ

, and

MM is a target mapping. By Theorem 8, with M = τ

ans

(Cs,ϕ) can be obtained as follows:

1. Construct S

= hCs,ϕi.

2. Construct an ET sequence based on Models and

MM starting with S

and ending with S

hCs

,ϕ

i such that Cs

∈ dom(τ

3. ans

(Cs,ϕ) = ϕ

(

(Cs

)).

For the discussion below, the following notation is

assumed:

• For any S, S

′

∈ STATE, let trans(S,S

′

) denote the

transformation of S into S

′

, and time(trans(S, S

′

))

denote the computation time required for this

transformation step.

• Let π be a state mapping. For any target mapping

τ and S ∈ STATE, let comp(τ,S) denote the com-

putation of ϕ(

τ(Cs)), where π(S) = hCs, ϕi, and

let time(comp(τ,S)) denote the amount of time re-

quired for this computation.

Using this notation, the time of the above solution by

the ET sequence [S

,.. .,S

] with τ

above is eval-

uated by

= Σ

i=1

time(trans(S

i−1

)) + time(comp(τ

)).

By the deﬁnition of τ

, time(comp(τ

)) is very

small. Assuming each transformation step in the ET

: FM(x) ← FP(x)

: FP(john) ←

: FP(mary) ←

: teach(john,ai) ←

: St(paul) ←

: AC(ai) ←

: Tp(kr) ←

: Tp(lp) ←

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

Co(y),Tp(z)

: mayDoThesis(x,y) ← curr(x,z),expert(y,z),

St(x),Tp(z),FP(y),

AC(w),teach(y,w)

: mayDoThesis(x,y) ← St(x),NFP(y)

: exam(paul,ai) ←

: subject(ai,kr) ←

: subject(ai,lp) ←

: 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(y, x),NFP(y) ← FP(y),funcf

(y,x)

: Co(x),NFP(y) ← FP(y),funcf

(y,x)

: funcf

(john,ai) ←

: ← funcf

(mary,ai)

Figure 2: Clauses representing the background knowledge

of the modiﬁed mayDoThesis problem.

sequence from S

to S

is also very small, the value

is small enough and the solution by this ET se-

quence with τ

can be efﬁcient. This is a basic strat-

egy to obtain an efﬁcient solution for a MI problem.

In order to use τ

after repeated equivalent

transformation, the clause set Cs

determined by

π(S

), where S

is the ﬁnal state obtained from

the ET sequence [S

,.. .,S

], must be inside

dom(τ

). In other words, the role of the ET sequence

,.. .,S

] is to construct Cs

that enters dom(τ

)

starting from S

5 EXAMPLE

Usual ﬁrst-order atoms are used for illustration be-

low. To apply the proposed theory in this section,

a specialization system hA

,S, µ

i corresponding

to the usual ﬁrst-order space is used, where A

is the

set of all ﬁrst-order atoms, G

is the set of all ground

ﬁrst-order atoms, S is the set of all substitutions on

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

, and µ

provides the specialization operation cor-

responding to the usual application of substitutions in

S to atoms in A

5.1 Problem Description

Let Cs be the set consisting of the clauses C

–

in Fig. 2. These clauses are obtained from

the mayDoThesis problem given in (Donini et al.,

1998) with some modiﬁcation.

All atoms ap-

pearing in Fig. 2 belong to A

. The unary pred-

icates NFP, FP, FM, Co, AC, BC, St, and Tp

denote “non-teaching full professor,” “full profes-

sor,” “faculty member,” “course,” “advanced course,”

“basic course,” “student,” and “topic,” respectively.

The clauses C

–C

together provide the conditions

for a student to do his/her thesis with a profes-

sor, where mayDoThesis(s, p), curr(s,t), expert(p,t),

exam(s,c), and subject(c,t) are intended to mean “s

may do his/her thesis with p,” “s studied t in his/her

curriculum,” “p is an expert in t,” “s passed the exam

of c,” and “c covers t,” respectively, for any student s,

any professor p, any topic t, and any course c.

Let a be the atom mayDoThesis(paul,x). We con-

sider the QA problem hCs,ai, which is to ﬁnd all stu-

dents who may do their theses with paul. Let ϕ be

deﬁned by: for any G ⊆ G

ϕ(G) = {mayDoThesis(paul, x) | ans(x) ∈ G},

where ans is a unary predicate denoting “answer.”

The QA problem hCs,ai above can then be trans-

formed into a MI problem hCs ∪ {C

},ϕi, where C

is the clause given by:

: ans(x) ← mayDoThesis(paul,x)

Using rules for transformation of clauses given

in Sections 5.2–5.4, how to compute the answer to

the MI problem hCs ∪ {C

},ϕi is illustrated in Sec-

tion 5.5.

5.2 Unfolding Operation

Assume that:

• Cs ⊆ CLS.

• D is a set of deﬁnite clauses in CLS.

• occ is an occurrence of an atom b in the right-hand

side of a clause C in Cs.

To represent the original mayDoThesis problem in a

clausal form, extended clauses with function variables are

used. To change atoms with function variables into user-

deﬁned atoms, the funcf

predicate is used in the clauses

–C

By unfolding Cs using D at occ, Cs is transformed

into

(Cs− {C}) ∪ (

[

{resolvent(C,C

′

,b) | C

′

∈ D}),

where for each C

′

∈ D, resolvent(C,C

′

,b) is deﬁned

as follows, assuming that ρ is a renaming substitution

for usual variables such that C and C

′

ρ have no usual

variable in common:

1. If b and head(C

′

ρ) are not uniﬁable, then

resolvent(C,C

′

,b) = ∅.

2. If they are uniﬁable, then

resolvent(C,C

′

,b) = {C

′′

where C

′′

is the clause obtained from C and C

′

as follows, assuming that θ is the most general

uniﬁer of b and head(C

′

ρ):

(a) lhs(C

′′

) = lhs(Cθ)

(b) rhs(C

′′

) = (rhs(Cθ) − {bθ}) ∪ body(C

′

ρθ)

The resulting clause set is denoted by UNFOLD(Cs,

D,occ).

5.3 ET by Unfolding and Deﬁnite-clause

Removal

For any predicate p, let Atoms(p) denote the set of

all atoms having the predicate p. Equivalent trans-

formation (ET) of clauses using unfolding and using

deﬁnite-clause removal are formulated below.

Theorem 14. Let Cs ⊆ CLS and a ∈ A

. Assume that:

1. q is the predicate of the query atom a.

2. p is a predicate such that p 6= q.

3. D is a set of deﬁnite clauses in Cs that satisﬁes the

following conditions:

(a) For any deﬁnite clause C ∈ D,

head(C) ∈ Atoms(p).

(b) For any clause C

′

∈ Cs− D,

lhs(C

′

) ∩ Atoms(p) = ∅.

4. occ is an occurrence of an atom in Atoms(p) in

the right-hand side of a clause in Cs− D.

Then the following two sets are equal:

• (

Models(Cs)) ∩ rep(a)

• (

Models(UNFOLD(Cs,D,occ))) ∩ rep(a).

Theorem 15. Let Cs ⊆ CLS and a ∈ A

. Assume that:

1. q is the predicate of the query atom a.

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation

2. p is a predicate such that p 6= q.

3. D is a set of deﬁnite clauses in Cs that satisﬁes the

following conditions:

(a) For any deﬁnite clause C ∈ D,

head(C) ∈ Atoms(p).

(b) For any clause C

′

∈ Cs− D,

lhs(C

′

) ∩ Atoms(p) = ∅.

4. For any clause C

′

∈ Cs− D,

rhs(C

′

) ∩ Atoms(p) = ∅.

Then the following two sets are equal:

• (

Models(Cs)) ∩ rep(a)

• (

Models(Cs− D)) ∩ rep(a)

5.4 Other Transformations

5.4.1 Elimination of Subsumed Clauses and

Elimination of Valid Clauses

A clause C

is said to subsume a clause C

iff there

exists a substitution θ for usual variables such that

lhs(C

)θ ⊆ lhs(C

) and rhs(C

)θ ⊆ rhs(C

). If a

clause set Cs contains clauses C

and C

such that

subsumes C

, then Cs can be transformed into

Cs− {C

A clause is valid iff all of its ground instances are

true. Given a clause C, if some atom in rhs(C) be-

longs to lhs(C), then C is valid. A valid clause can be

removed.

5.4.2 Side-change Transformation

Assume that p is a predicate occurring in a clause

set Cs and p does not appear in a query atom under

consideration. The clause set Cs can be transformed

by changing the clause sides of p-atoms as follows:

First, determine a new predicate notp for p. Next,

move all p-atoms in each clause to their opposite side

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

right-hand side and vice versa) with their predicates

being changed from p to notp. Side-change transfor-

mation is useful for decreasing the number of atoms

in a multi-head clause (i.e., a clause whose left-hand

side contains more than one atom) in Cs when (i) ev-

ery negative clause in Cs has at most one p-atom in

its right-hand side and (ii) every non-negative clause

in Cs has more p-atoms in its left-hand side than those

in its right-hand side.

: 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)

: ans(y) ← NFP(x)

: ans(john) ← AC(x),teach(john,x),Co(ai)

: ans(mary) ← AC(x),teach(mary,x),Co(ai)

: ans(john) ← AC(x),teach(john,x),

NFP(john),Co(ai)

: ans(mary) ← AC(x),teach(mary,x),

NFP(mary),Co(ai)

: teach(john,ai),NFP(john) ←

: Co(ai),NFP(john) ←

Figure 3: Clauses obtained by application of unfolding and

application of basic transformation rules.

: ans(x),notNFP(x) ←

: notNFP(john) ←

: ans(john) ←

: ← BC(ai)

Figure 4: Clauses obtained by further application of trans-

formation rules.

5.5 ET Computation

The clause set Cs∪ {C

}, consisting of C

–C

, given

in Section 5.1 is transformed using ET rules provided

by Sections 5.2–5.4 as follows:

• By (i) unfolding using the deﬁnitions of the predi-

cates mayDoThesis, FP, Tp, curr, subject, expert,

St, exam, funcf

, and FM, (ii) removing these def-

initions using deﬁnite-clause removal, and (iii) re-

moval of valid clauses, the clauses C

–C

are

transformed into the clauses C

–C

in Fig. 3.

• Side-change transformation for NFP enables (i)

unfolding using the deﬁnitions of teach, Co,

and AC, (ii) elimination of these deﬁnitions us-

ing deﬁnite-clause removal, (iii) removal of valid

clauses, and (iv) elimination of subsumed clauses.

By such side-change transformation followed by

transformation of these four types, C

–C

are

transformed into the clauses C

–C

in Fig. 4.

• Side-change transformation for notNFP enables

unfolding using the deﬁnitions of BC and NFP.

By unfolding and deﬁnite-clause removal, C

–

are transformed into C

, i.e., (ans(john) ←).

As a result, the MI problem hCs∪ {C

},ϕi in Sec-

tion 5.1 is transformed equivalently into the MI prob-

lem h{(ans(john) ←)},ϕi. Hence

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

ans

(Cs∪ {C

},ϕ)

= ans

({(ans(john) ←)},ϕ)

= ϕ(

Models({(ans(john) ←)}))

= {mayDoThesis(paul,john)}.

6 CONCLUSIONS

A model-intersection problem (MI problem) is a pair

hCs,ϕi, where Cs is a set of clauses and ϕ is an exit

mapping used for constructing the output answer from

the intersection of all models of Cs. The proposed

ET-based solution for MI problems consists of the fol-

lowing steps: (i) formalize a given problem as an MI

problem on some specialization system, (ii) prepare

ET rules from clauses, (iii) construct an ET sequence,

(iv) compute a set of models using a target mapping,

(v) apply the set-intersection operation to the result-

ing set of models, and (vi) apply an exit mapping to

the intersection result to obtain a solution.

The class of MI problems considered in this pa-

per has many parameters, such as abstract atoms, spe-

cializations, restriction on forms of clauses, etc. By

instantiating these parameters, we can obtain theories

for subclasses of QA and proof problems correspond-

ing to conventionalclause-based theories, such as dat-

alog, Prolog, and many other extensions of Prolog.

We introduced the concept of target mapping and

proposed three target mappings, i.e., τ

for sets of

positive unit clauses, τ

for sets of deﬁnite clauses,

and MM for arbitrary sets of clauses. These target

mappings provide a strong foundation for inventing

many ET rules for solving MI problems on clauses.

Most kinds of ET rules, including the resolution and

factoring ET rules, are realized by transformations

that preserve these target mappings. For instance,

a proof based on the resolution principle can be re-

garded as ET computation using the resolution and

factoring ET rules. By introducing new ET rules, we

can devise a new proof method (Akama and Nantajee-

warawat, 2013). By inventing additional ET rules, we

have been successful in solving a large class of QA

problems (Akama and Nantajeewarawat, 2014).

By instantiation, the class of MI problems on spe-

cialization systems produces, among others, one of

the largest classes of logical problems with ﬁrst-order

atoms and substitutions. The ET solution has been

proved to be very general and fundamental since its

correctness for such a large class of problems has

been shown in this paper. By its generality, the theory

developed in this paper makes clear the fundamental

and central structure of representation and computa-

tion for logical problem solving.

ACKNOWLEDGEMENTS

This research was partially supported by JSPS KAK-

ENHI Grant Numbers 25280078 and 26540110.

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A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation