MEANING-PRESERVING SKOLEMIZATION

Kiyoshi Akama

and Ekawit Nantajeewarawat

Information Initiative Center, Hokkaido University, Hokkaido, Japan

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

Keywords:

Skolemization, Equivalent transformation, Conjunctive normal form, Question-answering problems.

Abstract:

Skolemization is a well-known method for removing existential quantiﬁers from a logical formula. Although

it always yields a satisﬁability-preserving transformation step, classical Skolemization in general does not

preserve the logical meaning of a source formula. We develop in this paper a theory for extending a space of

logical formulas by incorporation of function variables and show how meaning-preserving Skolemization can

be achieved in an obtained extended space. A procedure for converting a logical formula into an equivalent

one in an extended conjunctive normal form on the extended space is described. This work lays a theoretical

foundation for solving logical problems involving existential quantiﬁcations based on meaning-preserving

formula transformation.

1 INTRODUCTION

Conversion of a given formula into a conjunction of

clauses, called a conjunctive normal form (CNF) or a

clausal normal form, is a normalization process com-

monly used in automated reasoning. Such conver-

sion involves removal of existential quantiﬁcations by

Skolemization (named after Thoralf Albert Skolem),

i.e., by replacement of an existentially quantiﬁed vari-

able with a Skolem term, which is usually determined

by a relevant part of a formula prenex.

Conversion into CNFs is a basic preparation step

for automated proof by resolution and factoring. Most

theories in logic programming are based on clausal

forms. Recently, question-answering problems (QA

problems) have gain wide attention. A problem in this

class is concerned with ﬁnding the set of all ground

instances of a given query atom that are logical conse-

quences of a given formula. Most research works on

solving QA problems are also based on Skolemiza-

tion, including those in systems involving integration

between formal ontological background knowledge

and instance-level rule-oriented components, e.g., in-

teraction between Description Logics and Horn rules

(Donini et al., 1998; Horrocks et al., 2005; Levy and

Rousset, 1998; Motik et al., 2005) in the Semantic

Web’s ontology-based rule layer.

Skolemization, however, does not preserve the

logical meaning of a formula; the formula resulting

from Skolemization is not necessarily equivalent to

the original one. Only the satisﬁability property of a

formula is preserved—the resulting formula is equi-

satisﬁable with the original formula (Chang and Lee,

1973), i.e., it is satisﬁable iff the original formula is.

Equivalent Transformation (ET) of formulas is es-

sential and very useful for solving many kinds of log-

ical problems (Akama and Nantajeewarawat, 2006),

including QA problems. In ET-based problem solv-

ing, a logical formula representing a given problem

is successively transformed into a simpler but logi-

cally equivalent formula. Correctness of computation

is readily guaranteed by any combination of equiva-

lent transformations, which yields many kinds of cor-

rect algorithms for solving logical problems. Since

classical Skolemization does not result in meaning-

preserving transformation, it cannot be used in an ET-

based problem-solving process.

Our primary objective here is to develop a theory

for extending a space of logical formulas by introduc-

tion of function variables and a specialization opera-

tion on them in such a way that “meaning-preserving”

Skolemization can be achieved in an obtained ex-

tended space. Fig. 1 gives a pictorial view of our goal.

Assume that α is a given ﬁrst-order formula with oc-

currences of existential quantiﬁcations. As illustrated

in the ﬁgure, suppose that α is converted into a CNF

β by a sequence of transformation steps based on the

usual normalization procedure on the space, say L

of ﬁrst-order logic. When classical Skolemization is

used in this conversion, α and β are not necessar-

ily logically equivalent and, thus, β does not always

serve as an intermediate equivalent formula for fur-

322

Akama K. and Nantajeewarawat E..

MEANING-PRESERVING SKOLEMIZATION.

DOI: 10.5220/0003692003220327

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2011), pages 322-327

ISBN: 978-989-8425-80-5

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Clauses

on L

¯γ

Figure 1: ET-based problem solving with meaning-pre-

serving Skolemization.

ther transformation preserving the logical meaning of

α. By contrast, in the expected extended logical struc-

ture, referred to as L

, by using meaning-preserving

Skolemization, α is converted into an extended CNF,

say

β, that is logically equivalent to it. Consequently,

α can be further equivalently transformed in the ex-

tended space, for example, by using the meaning-

preserving transformation path from

β to

γ in the ﬁg-

ure. It is expected that our meaning-preserving Skol-

emization framework will provide an important theo-

retical basis for a large class of automated reasoning

tasks.

Section 2 formalizes a class of QA problems and

outlines an ET-based method for solving them. Sec-

tion 3 explains the necessity of meaning-preserving

Skolemization and an extension of a logical space.

After introducing function constants and function

variables, Section 4 formulates an extended logical

space and deﬁnes the meanings of extended formu-

las. Section 5 presents an extended conjunctive nor-

mal form, called existentially quantiﬁed conjunctive

normal form (ECNF), along with an algorithm for

meaning-preserving conversion of a formula into an

ECNF on the extended logical space. Section 6 con-

cludes the paper.

2 QUESTION-ANSWERING

PROBLEMS AND ET-BASED

SOLUTIONS

To begin with, a question-answering problem is de-

ﬁned. It is followed by a general ET-based solution

scheme.

2.1 Question-Answering (QA) Problems

A question-answering problem (QA problem) is a pair

hK,qi, where K is a logical formula and q is an atomic

formula (atom). The answer to a QA problem hK, qi,

denoted by ans(K,q), is deﬁned by

ans(K,q) = {q

′

| (q

′

is a ground instance of q) &

(K |= q

′

)},

i.e., the set of all ground instances of q that follows

logically from K. When K consists of only deﬁnite

clauses, problems in this class are problems that have

been discussed in logic programming (Lloyd, 1987).

When K is a conjunction of axioms and assertions in

Description Logics (Baader et al., 2007), QA prob-

lems are usually called query-answering problems.

2.2 Solving QA Problems by ET

Using the set of all models of K, denoted by

Model(K), the answer to a QA problem hK,qi, can

be equivalently represented as

ans(K,q) = (

Model(K)) ∩ rep(q),

where

Model(K) is the intersection of all models of

K and rep(q) is the set of all ground instances of q.

Calculating

Model(K) directly may require high

computation cost. To reduce the cost, K is trans-

formed into a simpliﬁed formula K

′

such that all mod-

els of K is preservedand

Model(K

′

)∩rep(q) can be

determined at a low cost. Obviously, if Model(K) =

Model(K

′

), then ans(K,q) = ans(K

′

,q).

3 NEED FOR

MEANING-PRESERVING

SKOLEMIZATION

3.1 Use of Conjunctive Normal Forms

A conjunctive normal form (CNF) is a set of clauses,

interpreted as a conjunction. Most important meth-

ods for theorem proving deal with logical formulas

in CNFs, using basic operations such as uniﬁcation,

resolution, unfolding, and factoring. Based on CNFs,

a transformation scheme for solving a QA problem

hK,qi typically consists of two steps:

1. K is converted into a CNF K

′

2. hK

′

,qi is transformed equivalently into hK

′′

,qi,

where K

′

is also a CNF.

From hK

′′

,qi, the answer to the problem hK,qi is de-

termined by

ans(K,q) = (

Model(K

′′

)) ∩ rep(q).

MEANING-PRESERVING SKOLEMIZATION

323

3.2 Traditional Skolemization

For traditional transformation of ﬁrst order formulas

into CNFs, all transformations are basically equiv-

alent transformation. They include, for example,

the implication law (p → q ≡ ¬p ∨ q), the De mor-

gan’s laws (¬(p ∧ q) ≡ ¬p ∨ ¬q), etc. It is well-

known, however, that traditional Skolemization is not

meaning-preserving. For example, the formula

∀x,∃y : p(x, y) (1)

is Skolemized to ∀x : p(x, f(x)), where f is a new

function constant, called a Skolem function. It is ob-

vious that ∀x,∃y : p(x, y) and ∀x : p(x, f(x)) have dif-

ferent meanings. Given any arbitrary ground term t

the former formula states the existence of a ground

term t

such that p(t

) is true, while the latter for-

mula states not only the existence of such a ground

term t

but also that one such t

is f(t

). The set

{p(t, 3) | t is a ground term}, for example, is a model

of the former formula but is not a model of the second

one.

3.3 Introduction of Meaning-Preserving

Skolemization

The basic idea of meaning-preserving Skolemization

is to use existentially quantiﬁed function variables in-

stead of function constants. For example, Formula (1)

is transformed into

∃h,∀x : p(x,h(x)), (2)

where h is a function variable. Intuitively, h is an

unknown function that associates with any arbitrar-

ily given ground term t

a ground term h(t

) such that

p(t

,h(t

)) is true. An alternative form of (2) is

∃h,∀x,y : (p(x, y) ∨ (h(x) 6= y)), (3)

which is intuitively equivalent to (2).

3.4 Need for an Extended Space

Formulas (2) and (3) above both contain a function

variable and a quantiﬁcation on that function variable,

which are not included in usual ﬁrst-order formulas.

The use of them leads to an extension of the basic

concepts for the ﬁrst-order logic. Let L

be the space

of all conventional ﬁrst-order formulas. We need the

following extensions:

• An extended space L

that includes not only usual

terms, atoms, and formulas in L

, but also func-

tion variables, quantiﬁcations on function vari-

ables, and formulas containing them.

• An extended deﬁnition of the truth value of a for-

mula with quantiﬁcations on function variables,

which determines the semantics of formulas on

the extended space.

This requirement raises a question: “How to deﬁne

the extended space L

and the semantics of formulas

thereon?” Section 4 provides an answer to this ques-

tion.

4 AN EXTENDED SPACE FOR

SKOLEMIZATION

4.1 Function Constants and Function

Variables

A usual function symbol in ﬁrst-order logic denotes

an unevaluated function; it is used for constructing a

syntactically new term from existing terms (possibly

recursively) without evaluating those existing terms.

A different class of functions is used in the extended

space L

. A function in this class is an actual math-

ematical function; it takes ground terms as input, and

associates with them an output ground term. The in-

put ground terms are evaluated for determining the

output. We called a function in this class a function

constant.

In order to clearly separate function constants and

function variables from usual functions and usual

terms, a new built-in predicate func is introduced.

Given any n-ary function constant or n-ary function

variable

func(

f,t

,...,t

n+1

where the t

are usual terms, is considered as an atom

of a new type, called a func-atom. When

f is a func-

tion constant and the t

are all ground, the truth value

of this atom is evaluated as follows: it is true iff

f(t

,...,t

) = t

n+1

Accordingly, function constants and function

variables are syntactically differentiated from usual

terms. Function constants and function variables ap-

pear only as the ﬁrst arguments of func-atoms, while

usual terms appear as other arguments of them. Ar-

guments of usual atoms can only be usual terms. By

such clear-cut separation, we need not consider uniﬁ-

cation of usual terms and function variables/function

constants. This makes a computation process eas-

ier to understand since computation methods simi-

lar to those used in the usual ﬁrst-order logic can be

adopted.

The space L

is then extended into the space L

inclusion of func-atoms and quantiﬁcations on func-

KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development

324

tion variables. There are two disjoint classes of atoms

in L

• func-atoms introduced above;

• usual atoms, constructed in the usual way from

ordinary predicates and usual terms.

From these atoms, formulas in L

are constructed us-

ing logical connectives (i.e., ¬, ∧, ∨, →, and ↔) and

quantiﬁcations in the usual way, except that in ad-

dition to quantiﬁcations on usual variables, function

variables are also quantiﬁed. Like a quantiﬁcation on

a usual variable, a quantiﬁcation on a function vari-

able v

is either a universal quantiﬁcation ∀v

or an

existential quantiﬁcation ∃v

In the following, let Var denote the set of all

usual variables and FVar the set of all function vari-

ables. For any expression E, any v ∈ Var (respec-

tively, any v

∈ FVar), and any usual term t (respec-

tively, any function constant f), let E{v/t} (respec-

tively, E{v

/ f}) denote the expression obtained from

E by replacing each occurrence of v with t (respec-

tively, each occurrence of v

with f).

4.2 Interpretations and Models

Let G be the set of all ground atoms. An interpreta-

tion is a subset of G . Given an interpretation I, the

truth value of a closed formula under I is deﬁned as

follows:

1. For any ground atom g, g is true under I iff g ∈ I.

2. For any closed formula α, ¬α is true under I iff α

is false under I.

3. For any closed formulas α and β, α ∧ β (respec-

tively, α ∨ β, α → β, and α ↔ β) is true under I

iff α and β are true (respectively, at least one of α

and β is true, at least one of ¬α and β is true, and

α and β have the same truth value) under I.

4. For any v ∈ Var, a closed formula ∀v : E is true

under I iff for any ground term t, E{v/t} is true

under I.

5. For any v ∈ Var, a closed formula ∃v : E is true

under I iff there exists at least one ground term t

such that E{v/t} is true under I.

6. For any v

∈ FVar, a closed formula ∀v

: E is true

under I iff for any function constant f , E{v

/ f}

is true under I.

7. For any v

∈ FVar, a closed formula ∃v

: E is

true under I iff there exists at least one function

constant f such that E{v

/ f} is true under I.

An interpretation I is a model of a closed formula α

iff α is true under I.

4.3 A Safe Extension into a New Space

The introduction of function variables necessitates the

extension of the original space L

into the extended

one L

. A solution path for meaning-preserving Sko-

lemization is:

1. First, transform a given formula α on L

into the

same formula on L

2. Next, transform α in the space of L

into an ex-

tended conjunctive normal form.

Obviously, L

is a subset of L

. Moreover, L

is a

safe extension of L

, i.e.,

any formula on L

have the same meaning as

the same formula on L

the reason being that all formulas on L

do not in-

clude function variables and function constants, and

the deﬁnitions of the truth values of closed formulas

with quantiﬁed function variables under an interpre-

tation do not affect the truth values of formulas on

. The ﬁrst step in the solution path above is thus an

equivalent transformation step.

The second step raises another question: “What

are extended conjunctive normal forms?” Section 5

provides an answer to this question along with an al-

gorithm for the second step.

5 AN ALGORITHM FOR

MEANING-PRESERVING

SKOLEMIZATION

Based on the notion of a formula tree, an extended

conjunctive normal form, called an existentially quan-

tiﬁed conjunctive normal form (ECNF), is deﬁned.

An algorithm for transforming a formula on L

into

an equivalent ECNF on L

is then presented.

5.1 Formula Trees

Given a formula α on L

, the formula tree of α, de-

noted by FT(α), is a binary tree constructed induc-

tively as follows:

1. If α is an atomic formula, then FT(α) is a one-

vertex binary tree whose root is α.

2. If α = ¬β, then FT(α) is a binary tree such that

• root(FT(α)) = ¬, and

• root(FT(α)) has only one child, with FT(β) be-

ing the subtree of FT(α) rooted at this child.

3. If α = β ∧ γ (respectively, β ∨ γ, β → γ, β ↔ γ),

then FT(α) is a binary tree such that

MEANING-PRESERVING SKOLEMIZATION

325

• root(FT(α)) = ∧ (respectively, ∨, →, ↔), and

• root(FT(α)) has two children, with FT(β) be-

ing the subtree of FT(α) rooted at the left child

and FT(γ) being the subtree of FT(α) rooted at

the right child.

4. If α = ∀v : β (respectively, ∃v : β), where v ∈ Var,

then FT(α) is a binary tree such that

• root(FT(α)) = ∀v (respectively, ∃v), and

• root(FT(α)) has only one child, with FT(β) be-

ing the subtree of FT(α) rooted at this child.

5. If α = ∀v

: β (respectively, ∃v

: β), where v

∈

FVar, then FT(α) is a binary tree such that

• root(FT(α)) = ∀v

(respectively, ∃v

), and

• root(FT(α)) has only one child, with FT(β) be-

ing the subtree of FT(α) rooted at this child.

The following notation is used in subsequent text:

• For any v ∈ Var, a ∀v-vertex and an ∃v-vertex in a

formula tree are also called a ∀Var-vertex and an

∃Var-vertex, respectively.

• For any v

∈ FVar, a ∀v

-vertex and an ∃v

-vertex

in a formula tree are also called a ∀FVar-vertex

and an ∃FVar-vertex, respectively.

5.2 Existentially Quantiﬁed

Conjunctive Normal Forms (ECNF)

A formula α on L

is said to be in an existentially

quantiﬁed conjunctive normal form (ECNF) iff α is a

closed formula and every path from the root to a leaf

of the formula tree of α consists of

1. zero or more ∃FVar-vertices, followed by

2. zero or more ∧-vertices, followed by

3. zero or more ∀Var-vertices, followed by

4. zero or more of ∨-vertices, followed by

5. either (i) a leaf vertex representing a usual atom

or (ii) a ¬-vertex followed by a leaf vertex repre-

senting a usual atom or a func-atom.

A formula in an ECNF is similar to a usual con-

junctive normal form in that it contains a conjunction

of clauses, each of which is a disjunction of literals.

There are, however, two main differences:

1. A formula in an ECNF contains existential quan-

tiﬁcations on function variables; it has the form

∃v

,...,∃v

: β,

where the v

are function variables and β has the

same form as a usual conjunctivenormal form ex-

cept that the negations of func-atoms may appear

in β, i.e., β is a conjunction of disjunctions of

(i) usual atoms, (ii) negated usual atoms, and (iii)

negated func-atoms.

2. While usual Skolem functions may appear in

usual atoms, function variables can appear only

in func-atoms.

Given usual atoms a

,...,a

,...,b

and func-

atoms f

,...,f

, a disjunction

∨ ··· ∨ a

∨ ¬b

∨ ··· ∨ ¬b

∨ ¬f

∨ ··· ∨ ¬f

contained in an ECNF is often written as

,...,a

← b

,...b

,...,f

5.3 Conversion Algorithm

Assume that

• the initial space INI is the set of all formulas on

that are also formulas on L

, and

• the target space FIN is the set of all formulas in

ECNFs on L

Let a formula α in INI be given as input and T =

FT(α). To transform α into a formula in FIN, T is

changed successively by the steps described below.

Fig. 2 depicts an outline of the procedure.

1. Preparation:

(a) Convert → and ↔ equivalently into ¬, ∧, and

∨, using the following logical equivalences:

• β → γ ≡ ¬β ∨ γ

• β ↔ γ ≡ (¬β ∨ γ) ∧ (¬γ∨ β)

(b) Rename quantiﬁed variables so that for any

two occurrences of quantiﬁcations Qv and Q

′

where v,w ∈ Var, v 6= w.

2. Move ¬ inwards: Move ¬ inwards equivalently

until each occurrence 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. Move down ∨-vertices: Repeatedly move down

∨-vertices in the current state of T through ∃Var-

vertices, ∀Var-vertices, and ∧-vertices as far as

possible using the following logical equivalences:

(∃x : β) ∨ γ ≡ ∃x : (β ∨ γ)

(∀x : β) ∨ γ ≡ ∀x : (β ∨ γ)

(β∧ γ) ∨ δ ≡ (β∨ δ) ∧ (γ∨ δ)

4. Move up ∧-vertices: Repeatedly move up ∧-

vertices in the current state of T through ∀Var-

vertices as far as possible using the following log-

ical equivalence:

∀x : (β ∧ γ) ≡ (∀x : β) ∧ (∀x : γ)

KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development

326

Figure 2: An overview of the conversion procedure.

5. If T includes an ∃Var-vertex, then:

(a) Skolemization: In T, select a subformula

∀x

,...,∀x

,∃y : β,

where n ≥ 0, such that there is no further uni-

versal quantiﬁcation over this subformula in T.

Transform this subformula into

∃h,∀x

,...,∀x

,∀y :(β∨¬func(h,x

,...,x

,y)),

where h ∈ FVar such that h has not been used

so far.

(b) Move up an ∃FVar-vertex: Repeatedly move up

the new ∃FVar-vertex (introduced at Step 5a)

through ∧-vertices as far as possible using the

following logical equivalence:

(∃FVar : β) ∧ γ ≡ ∃FVar : (β ∧ γ)

6. Stop with the formula represented by the current

state of T as the output formula.

It is shown in (Akama and Nantajeewarawat,

2011) that this algorithm always terminates and yields

an output ECNF in FIN that has the same logical

meaning as the input formula.

6 CONCLUSIONS

ET-based computationoften requires a search in a cer-

tain formula space for a simpliﬁed formula that is log-

ically equivalent to an originally given one. Exten-

sion of logical formulas in general enlarges the search

space both for ﬁnding a suitable equivalent logical

formula and for ﬁnding meaning-preserving formula

transformation sequences, thereby increasing the pos-

sibility of ﬁnding efﬁcient computation paths. The

theory for extending a space of formulas by introduc-

tion of function variables presented herein allows one

to use Skolemization as an equivalent transformation

step. It opens up new possibilities for employing ET-

based computation to solve logical problems with un-

restricted use of existential quantiﬁcations.

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