Logical Approach to Theorem Proving with Term Rewriting on KR-logic

Tadayuki Yoshida

, Ekawit Nantajeewarawat

, Masaharu Munetomo

and Kiyoshi Akama

Tokyo Software Development Laboratory, International Business Machines Corporation, Tokyo, Japan

Computer Science Program, Sirindhorn International Institute of Technology Thammasat University,

Pathumthani, Thailand

Information Initiative Center, Hokkaido University, Sapporo, Japan

Keywords: Term Rewriting Rules, Logical Problem Solving Framework, Equivalent Transformation, Correctness.

Abstract:

Term rewriting is often used for proving theorems. To mechanizing such a proof method with computation

correctness guaranteed strictly, we follow LPSF, which is a general framework for generating logical problem

solution methods. In place of the ﬁrst-order logic, we use KR-logic, which has function variables, for correct

formalization. By repeating (1) specialization by a substitution for usual variables, and (2) application of an

already derived rewriting rule, we can generate a term rewriting rule from the resulting equational clause.

The obtained term rewriting rules are proved to be equivalent transformation rules. The correctness of the

computation results is guaranteed. This theory shows that LPSF integrates logical inference and functional

rewriting under the broader concept of equivalent transformation.

1 INTRODUCTION

Term rewriting is often used for solving proof prob-

lems (Bird and Wadler, 1988; Dershowitz and Jouan-

naud, 1990). Typical examples are proofs in group

theory, where the axiom of a group consists of laws,

such as associativity law ∀x, y,z ∈ G : x · (y · z) = (x ·

y) · z. The operator (·) here is a mapping G × G → G.

Assuming the axiom of a group, we want to prove that

((a

−1

·a)·(b·b

−1

))

−1

= b·((a·b)

−1

·a). Usually, term

rewriting is used for proving such an equation. If the

terms on both sides of this equation are rewritten re-

peatedly to reach a single term ﬁnally, the equation is

proved.

Such a proof by term rewriting is, however, not

regarded as logical problem solving. To mechaniz-

ing such a proof method, we need to formalize the

problem as a logical problem with computation cor-

rectness guaranteed strictly. We follow the LPSF

(Akama et al., 2019a), which is a general framework

for generating logical problem solving methods for

Model-Intersection (MI) problems (Akama and Nan-

tajeewarawat, 2016). We apply the LPSF to this proof

problem by reformalizing it as an MI problem.

We also need to construct a rule set. We already

know how to make term rewriting rules from equa-

tional clauses (Akama et al., 2019b). Based on this

work, we try to devise a new method of generating

new term rewriting rules from Cs.

Assume that Cs is a background knowledge con-

sisting of equational clauses. We apply to Cs repeat-

edly (1) specialization by a substitution for usual vari-

ables, and (2) application of already derived rewriting

rules. After repetition, we make a term rewriting rule

from the resulting equational clause. We prove the

correctness of the obtained term rewriting rule, i.e., it

is an ET rule. Since ET rules are repeatedly applied

to the original problem, the result of computation is

also correct. Such guarantee of correctness is usually

not clearly discussed in the theory of term rewriting

systems.

The rest of this paper is organized as follows: Sec-

tion 2 gives an introductory example used throughout

the paper. Section 3 reviews KR-Logic, a foundation

to formalize the problem (Akama et al., 2019a). Sec-

tion 4 introduces a class of conditional term rewriting

rules and gives a theoretical bases for handling equal-

ity atoms in KR-Logic. Section 5 proves the correct-

ness of the rule generation method for a given clause

set. Section 6 shows how we can build a conditional

term rewriting system for an introductory example by

producing a set of term rewriting rules from a given

clause set. Section 7 explains a solution built on the

new conditional term rewriting system and the com-

putation steps are shown for an example input. Sec-

tion 8 concludes the paper.

282

Yoshida, T., Nantajeewarawat, E., Munetomo, M. and Akama, K.

Logical Approach to Theorem Proving with Term Rewriting on KR-logic.

DOI: 10.5220/0008165902820289

In Proceedings of the 11th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2019), pages 282-289

ISBN: 978-989-758-382-7

2 AN INTRODUCTORY

EXAMPLE

We take a proof problem in the group theory and show

intuitive solution by term rewriting. We will try to

formalize this solution in later sections.

2.1 A Proof Problem of a Group

Consider the following deﬁnition of a group:

hG,(·), e,(

−1

))i is a group iff

1. G is a set,

2. (·) is a mapping from G × G to G,

3. e ∈ G, and

4. (

−1

) is a mapping from G to G,

that satisfy the following conditions:

A1 (associativity) : ∀x,y, z ∈ G : x · (y · z) = (x · y) · z

A2 (identity) : ∀x ∈ G : x ·e = e · x = x

A3 (inverse) : ∀x ∈ G,∃x

−1

∈ G :

x · x

−1

= x

−1

· x = e

Using these three axioms, we want to prove that

((a

−1

· a) · (b · b

−1

))

−1

= b · ((a · b)

−1

· a).

2.2 Informal Proof with Equalities

We formalize the proof problem in 2.1 by introducing

two function-like notations. Assume that

1. (·) is a mapping f from G × G to G, and

2. (

−1

) is a mapping i from G to G.

Using these mappings, we have the following ﬁve

equations corresponding to three axioms in 2.1.

e1: f (x, f (y,z)) = f ( f (x , y),z)

e2: f (x,e) = x

e3: f (e,x) = x

e4: f (x,i(x)) = e

e5: f (i(x),x) = e

Let a,b be a term in G. Let’s take an couple

of terms i( f ( f (i(a),a), f (b,i(b)))) as ((a

−1

· a) · (b ·

−1

))

−1

and f (b, f (i( f (a,b)), a) as b · ((a · b)

−1

· a).

The original proof problem is then considered to de-

termine the equality of these two terms.

In addition to the set of equalities above, we have

four additional equalities derived from the original

equalities.

e6: i(e) = e

e7: f (x, f (y,i( f (x,y)))) = e

e8: f (y,i( f (x,y))) = i(x)

e9: i( f (y,x)) = f (i(x),i(y))

For example, a generation process of e7 is shown

as follows:

f (x,i(x)) = e (p1)

⇒ f ( f (x, y),i( f (x, y))) = e (p2)

⇒ f (x, f (y,i( f (x, y)))) = e (e7)

Starting with p1, a substitution {x/ f (x , y)} is ap-

plied to p1 then p2 is obtained. Applying e1 to p2

results in e7. In similar way, e6 is obtained. Using e7,

e8 and e9 are generated.

By replacement of the terms using these nine

equalities, each term reaches e. This obviously means

that these two are equal;

i( f ( f (i(a),a), f (b,i(b)))) = f (b, f (i( f (a,b)),a).

2.3 How to Give Correctness to the

Informal Method

The informal method is only procedural, i.e.,

1. each term rewriting rule is constructed without

proving it to be semantically correct.

2. each successive transformation process is ob-

tained by application of rewriting rules without

correctness of computation result.

Our theory is correctness-based, i.e.,

1. each term rewriting rule is proved to be an ET

rule.

2. each successive transformation process is ob-

tained with guarantee of correctness of computa-

tion result.

KR-logic is sufﬁcient for correctness-based the-

ory, while ﬁrst order logic is not.

3 REPRESENTATION IN

KR-LOGIC

We use KR-Logic in order to formalize the original

problem in 2.1 as a proof problem on a logical struc-

ture. KR-Logic is considered as an extension of usual

ﬁrst order logic by introduction of existential quan-

tiﬁcation of function variables, which is essential for

representation of proof problems with equality con-

straints.

3.1 KR-logic

We take ECLS

as a logical structure L, where

ECLS

is the space of all clauses (including non-

ﬂat ones) on KR-logic. It is an extension of ECLS

(Akama and Nantajeewarawat, 2018) that contain

Logical Approach to Theorem Proving with Term Rewriting on KR-logic

283

only ﬂat clauses, which is also an extension of CLS

consisting of usual clauses.

Let K denote the conjunction of the following for-

mulas:

: ∀x∀y : (g(x) ∧ g(y) → g($ f (x, y)))

: ∀x : (g(x) → g($i(x)))

: g($e)

: ∀x∀y∀z : ((g(x) ∧ g(y) ∧ g(z))

→ eq($ f (x,$ f (y,z)), $ f ($ f (x , y),z)))

: ∀x : (g(x) → eq($ f (x, $e),x))

: ∀x : (g(x) → eq($ f ($e,x), x))

: ∀x : (g(x) → eq($ f (x, $i(x)),$e))

: ∀x : (g(x) → eq($ f ($i(x),x),$e))

These formulas give deﬁnitions of $ f , $i, and $e

which are corresponding to conditions listed in 2.1.

3.2 Formalization in Clausal Form in

KR-logic

Let term t

and t

= $i($ f ($ f ($i(x),x), $ f (y,$i(y)))) and

= $ f (y,$ f ($i($ f (x, y)),x),

respectively. In order to ensure the equality of these

two terms, we want to prove

K → (∀x∀y : (g(x)∧g(y)) → eq(t

)).

Its negation is

K ∧ ¬(∀x∀y : (g(x) ∧ g(y)) → eq(t

)),

which is equivalent to

K ∧ (∃x∃y : (g(x)∧ g(y) ∧ ¬eq(t

))).

By meaning-preserving Skolemization (MPS),

this formula is transformed into

E : ∃$ f ∃$i∃$e∃$a∃$b :

K ∧ g($a) ∧ g($b) ∧ ¬eq(t

where $a and $b are new function variables.

Each formula in K is transformed into a set of

clausal forms using MPS as follows: MPS(F

) =

, MPS(F

) = C

, MPS(F

) = C

, MPS(F

) = C

MPS(F

) = C

, MPS(F

) = C

, MPS(F

) = C

, and

MPS(F

) = C

The formula E is eventually represented in a

clausal form by:

: ← eq($i($ f ($ f ($i($a),$a),$ f ($b, $i($b)))),

$ f ($b,$ f ($i($ f ($a, $b)),$a)))

: g($ f (x,y)) ← g(x), g(y)

: g($i(x)) ← g(x)

: g($e) ←

: eq($ f (x,$ f (y,z)), $ f ($ f (x, y),z))

← g(x),g(y), g(z)

: eq($ f (x,$e), x) ← g(x)

: eq($ f ($e,x),x) ← g(x)

: eq($ f (x,$i(x)), $e) ← g(x)

: eq($ f ($i(x),x), $e) ← g(x)

: g($a) ←

: g($b) ←

Let Cs be a set of clauses in ECLS

and equal

to {C

,. ..,C

}. Then we solve the given proof

problem by proving that Models({C

} ∪ Cs) =

4 TERM REWRITING RULES

We introduce a class of conditional term rewriting

rules based on the equality atoms in a given clause

set.

4.1 Overview

We introduce a logical structure ECLS

, in which

an equational clause is deﬁned. A conditional term

rewriting rule is deﬁned. We obtain a conditional term

rewriting rule from an equational clause. Rewriting

relation determined by a conditional term rewriting

rule is deﬁned. We propose a sufﬁcient condition of

the correctness of a conditional term rewriting rule

that is obtained from an equational clause.

4.2 Alphabet and Terms of KR-logic

An alphabet hF,V,FC,FV,Pred,Pred

i is assumed,

where F is a set of constructors, V is a set of usual

variables, FC is a set of function constants, FV is

a set of function variables, Pred is a set of user-

deﬁned predicate symbols, and Pred

is a set of built-

in constraint predicate symbols. Each element in

F ∪ FC ∪ FV is associated with a non-negative inte-

ger, called its arity.

Deﬁnition 1. A term on hF,V, FC,FVi, which is

also simply called a term, is inductively deﬁned as

follows:

1. A 0-ary element in F ∪ FC is a term.

2. If v ∈ V ∪ FV, then v is a term.

3. If f ∈ F ∪ FC ∪ FV, the arity of f is n > 0, and

,. ..,t

are terms, then f (t

,. ..,t

) is a term.

The set of all terms on hF,V,FC,FVi is denoted

by T(F,V,FC,FV). Let T(F) = T(F,

0). Let

T(F, FC) = T(F,

0,FC,

0). A term in T(F,FC) is

called a ground term.

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4.3 Term Contexts, Atom Contexts, and

Clause Contexts

Assume that  is a function symbol with arity 0 that

does not belong to F ∪ V ∪ FC ∪ FV. A subterm is a

part of a term, and a subterm is extended by a term-

context into a term. Let t be a term and tc a term

context. t B tc is the term tc{/t}. A term is a part

of an atom, and a term is extended by an atom context

into an atom. Let t be a term and ac an atom context.

t B ac is the atom ac{/t}. An atom is a part of a

clause, and an atom is extended by a clause context

into a clause. Let a be an atom and con a clause con-

text. a B con is the clause con{/a}. The sets of all

term contexts, all atom contexts, and all clause con-

texts are denoted, respectively, by Con

, Con

, and

Con

4.4 Conditional Term Rewriting Rules

4.4.1 Equational Clauses

An equational clause is a clause in ECLS

of the form

eq(t

) ← conds,

where t

and t

are terms in T(F,V, FC,FV), and

conds is a ﬁnite sequence of atoms. The set of all

equational clauses in ECLS

is denoted by EQC.

4.4.2 Conditional Term Rewriting Rule

A conditional term rewriting rule, “CTRR” for short,

is a formula of the form (t

→ t

: conds). A condi-

tional term rewriting rule r of the formula (t

→ t

conds) is often represented by r : (t

→ t

: conds).

The set of all conditional term rewriting rules is de-

noted by CT R.

4.4.3 Equational Clause to CTRR

An equational clause C = (eq(t

) ← conds) deter-

mines a set of two conditional term rewriting rules,

which is denoted by ctrs(C), i.e.,

ctrs(C) = {(t

→ t

: conds),(t

→ t

: conds)}.

4.5 CTRR as Relation of Clauses

A conditional term rewriting rule r determines a rela-

tion on pow(ECLS

), denoted by rel(r), as follows:

Deﬁnition 2. Let r be a conditional term rewriting

rule. Let S

be the set of all substitutions on V.

rel(r) = { (C

) | r = (t

→ t

: conds)

& (tc ∈ Con

) & (ac ∈ Con

)

& (con ∈ Con

)

& (θ ∈ S

) & (condsθ = true)

& (C

= ((((t

θ) B tc) B ac) B con))

& (C

= ((((t

θ) B tc) B ac) B con)) }.

Figure 1 illustrates how the element of rel(r) is

constructed by a couple of clauses C

and C

. As-

sume that θ is given to fullﬁl the conditions denoted

by condsθ, all occurrences of t

θ are valid under the

term context tc, with which all atom occurrences con-

taining t

θ are valid under the atom context ac, in

which a clause containing the atoms with t

θ is valid

under the clause context con. The same validity is

conﬁrmed for t

θ in C

. Then an element of rel(r) is

obtained as a pair of C

and C

term

context

atom

context

clause

context

rel(r)

= (

conds

)

conds

true

Figure 1: Element of rel(r) visualized.

Deﬁnition 3. C

is transformed into C

by a condi-

tional term rewriting rule r, denoted by C

→ C

, iff

) ∈ rel(r).

4.6 Basic Theorem

Theorem 1. Let Cs be a set of clauses. Let C,

, and C

be clauses in ECLS

. If Models(Cs) =

Models(Cs ∪ {C}), ctrs(C) 3 r, then r is model-

preserving.

Proof. According to Theorem 1 in (Akama et al.,

2019b), obviously Models(Cs∪{C

}) = Models(Cs∪

}).

5 MAKING CORRECT TERM

REWRITING RULE

We introduce a mechanism to generate a new condi-

tional term rewriting rule from a set of clauses. Also,

we prove a theorem for giving the correctness of gen-

erated rules.

5.1 Generation of CTRR

We describe the method of generating a CTRR after

transforming a set of equational clauses. A transfor-

Logical Approach to Theorem Proving with Term Rewriting on KR-logic

285

mation is caused by two ways, i.e., substitution and

rewriting.

Deﬁnition 4. Let Cs be a set of clauses. Let C be

a clause in ECLS

. Let R be a set of term rewriting

rules. Cs

C is deﬁned inductively by:

• If Cs 3 C, then Cs

• If Cs

, θ is a substitution, and C

θ = C, then

• If Cs

, r ∈ R, and C

→ C, then Cs

Using above relation, we can make a rewriting

rule.

Deﬁnition 5. Cs

7→ r iff there is a clause C such that

C and ctrs(C) 3 r.

5.2 Main Theorem for Generation

A logical consequence relation, lcr is deﬁned by

lcr(X) = (Models(Cs) = Models(Cs ∪ {X})), where

X is an equational clause. A transformation by sub-

stitution preserves a logical consequence relation.

Proposition 1. Let Cs be a subset of ECLS

, and

C a clause in ECLS

. Let θ be a substitution. If

Models(Cs) = Models(Cs∪{C}), then Models(Cs) =

Models(Cs ∪ {Cθ}).

Proof. Since (1) G ∈ Models(Cs) iff G ∈

Models(Cs ∪ {C}), and (2) G ∈ Models(Cs ∪ {C}) iff

G ∈ Models(Cs ∪ {Cθ}), we have G ∈ Models(Cs)

iff G ∈ Models(Cs ∪ {Cθ}). Hence Models(Cs) =

Models(Cs ∪ {Cθ}).

A logical consequence relation is preserved by re-

peated application of specialization and rewriting.

Proposition 2. Let Cs be a set of clauses. Let C

be a clause in ECLS

. Assume that R is a set of

model-preserving term rewriting rules. If Cs

then Models(Cs) = Models(Cs ∪ {C}).

Proof. Assume that Cs

• (base case) Assume that Cs 3 C. Then, Cs = Cs ∪

{C}. Hence, Models(Cs) = Models(Cs ∪ {C}).

• (inductive case) There is a clause C

, a substitu-

tion θ, and r ∈ R such that (1) Cs

, and (2)

→ C. By the inductive hypothesis and (1),

Models(Cs) = Models(Cs ∪ {C

}). By proposi-

tion 1, Models(Cs) = Models(Cs ∪ {C

θ}). By

r ∈ R and (2), Models(Cs ∪{C

θ}) = Models(Cs∪

{C}). Hence, Models(Cs) = Models(Cs ∪ {C}).

The method of rule generation gives model-

preserving rules.

Theorem 2. Let Cs be a subset of ECLS

. Let R

be a set of term rewriting rules. Let r be a conditional

term rewriting rule such that Cs

7→ r. If R is model-

preserving, then r is also model-preserving.

Proof. Models(Cs) = Models(Cs ∪ {C

}) and from

Theorem 1, Models(Cs ∪ {C

}) = Models(Cs ∪

}). Then Models(Cs) = Models(Cs ∪ {C

}).

A generated rule is model-preserving since all

transformations in the generation sequence and a rule

generation method preserve a logical consequence re-

lation.

Equational

clauses in

CTRRs

11 11

21 22

8 4

Legend

C’

Rule

application

Rule

generation

Clause

CTRR

Generated

CTRR

r’

Transformed

clause

r’

Figure 2: CTRRs expansion by rule generation.

6 CONSTRUCTING TERM

REWRITING SYSTEM WITH

RULE GENERATION

We introduce a method to construct a term rewriting

system which equips rule generation.

6.1 Term Rewriting System Overview

Once a logical structure containing equational clauses

are constructed, then we can develop a term rewriting

system which contains a set of CTRRs growing by

rule generation.

Figure 2 shows the overview of CTRRs expan-

sion. The basic diagram elements shown in Legend

are:

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• Clause, a circled number represents an equational

clause with its number

• Rule application, a solid line between two circles

represents a transformation using pre-generated

CTRR

• CTRR, a diamond with inner number. A white

diamond represents a simple CTRR, while a dark

diamond a CTRR from a transformed clause, and

• Rule generation, a dotted line between a clause

and a CTRR

CTRR #1 to #10 are generated in straightforward

way as shown in 4.4.3. For example, the top diamond

in “CTRRs” section at the right side means r

, which

is generated from C

. It is denoted as ctrs(C

) 3 r

CTRR #11, #13, #17, and #18 are generated from

transformed clauses by applying one or more pre-

generated CTRRs. For example, the negative dia-

mond of 13 is a generated rule from C

which is a

resulting clause of rule application to C

As a result, a set of CTRRs are growing by gen-

erating a new one from a transformed clause using

existing CTRRs.

6.2 Generation of Term Rewriting

Rules

Figure 2 shows the mechanism where conditional

term rewriting rules are generated from a set of equa-

tional clauses. In “Equational clauses in Cs” section

at the left side of Figure 2, C

, C

, and C

are

listed in this order. For example, the circle of #8 in

“Equational clauses in Cs” is connected to the circle

of #16 with a solid line and the diamond of #4, which

means that by application of r

, C

is transformed into

. Then the dark diamond of #13 is generated from

a transformed clause #16.

6.3 Rule Generation Sequences

In Figure 2, we have 14 generation lines. Each line

consists of a rule generation arrow line and zero or

more rule application arrow lines. When we repre-

sent a rule generation line using , ctrs, and 7→, we

call such formal representation a “rule generation se-

quence.” Here are 14 generation sequences appear in

Figure 2.

#4: ctrs(C

) = {r

}

#6: ctrs(C

) = {r

}

#7: ctrs(C

) = {r

}

#8(a): ctrs(C

) = {r

}

#8(b): Cs C

, ctrs(C

) 3 r

;

#8(c): Cs C

, ctrs(C

) 3 r

;

#5(a): Cs C

, ctrs(C

) 3 r

;

#5(b): Cs C

, ctrs(C

) 3 r

;

#5(c): ctrs(C

) = {r

}

6.4 Generation of r

from Cs

The following steps explain the generation sequence

#5(a) for r

from Cs starting with C

(1) Cs 3 C

(2) C

−→ C

(3) C

−→ C

(4) C

−→ C

(5) C

−→ C

(6) ctrs(C

) 3 r

The 4th step is detailed as follows:

(4-1) Take C

: eq(y,$ f ($i(x), $ f (x, y))) ← g(x),g(y)

(4-2) Specialization by θ

= {y/$ f (y,$i($ f (x,y)))}

: eq($ f (y,$i($ f (x, y))),

$ f ($i(x), $ f (x, $ f (y,$i($ f (x,y))))))

← g(x), g(y)

(4-3) Application of r

−→ C

: ($ f (x, $ f (y,$i($ f (x, y)))) → $e

: {g(x), g(y)})

: eq($ f (y,$i($ f (x, y))),$ f ($i(x),$e))

← g(x), g(y)

6.5 Generated Rules

From all 14 generation sequences, we have condi-

tional term rewriting rules as follows:

: (x → $ f (x,$e) : {g(x)})

: ($ f (x,$e) → x : {g(x)})

: (x → $ f ($e,x) : {g(x)})

: ($ f ($e,x) → x : {g(x)})

: ($e → $ f (x, $i(x)) : {g(x)})

: ($ f (x,$i(x)) → $e : {g(x)})

: ($e → $ f ($i(x), x) : {g(x)})

: ($ f ($i(x),x) → $e : {g(x)})

: ($ f ($ f (x, y),z) → $ f (x , $ f (y,z)) :

{g(x),g(y), g(z)})

: ($ f (x,$ f (y,z)) → $ f ($ f (x,y),z) :

{g(x),g(y), g(z)})

: ($ f (x,$ f (y,$i($ f (x,y)))) → $e) :

{g(x),g(y)})

: ($i($e) → $e : {})

: ($ f (y,$i($ f (x, y))) → $i(x) :

{g(x),g(y)})

: ($i($ f (y,x )) → $ f ($i(x ), $i(y)) :

{g(x),g(y)})

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7 LPSF-BASED SOLUTION WITH

TERM REWRITING

We propose a LPSF-based solution for problems us-

ing term rewriting. First, we give a theoretical foun-

dation of the correctness of the solution, followed by

the solution for the sample problem.

7.1 Clause-rule Interaction Tree

A clause is transformed into a clause by repeated ap-

plication of rules, which is represented by a triple.

C ◦ [r

,..., r

] = C

is deﬁned inductively by:

• C ◦ [] = C.

• C ◦ [r

,..., r

n−1

] = C

, and there is θ such that

→ C

We add a generated rule to a triple to form a

quadruple.

Deﬁnition 6. Let Cs a set of clauses. Clause-

Rule Interaction quadruple wrt Cs is a tuple of the

form (C, [r

,..., r

],C

), where C and C

are clauses,

,..., r

and r

are term rewriting rules, C ∈ Cs,

C ◦ [r

,..., r

] = C

, and ctrs(C

) 3 r

Notations for an element designator in a sequence

and a subsequence are introduced.

Deﬁnition 7. Let X be a ﬁnite sequence.

nth(m,X) is deﬁned as the m-th element of X.

Under(m,X ) is deﬁned as the set of all elements

of X until m-th element, i.e., U nder(m,X) =

{nth(1,X),nth(2,X ), .. .,nth(m,X )}.

Two notations for obtaining clauses and rules from

a set of CRI quadruples.

Deﬁnition 8. Let X be a set of Clause-Rule Interac-

tion quadruples. We deﬁne cl(X ) and trs(X) by:

• cl(X) = {C | (C,[r

,..., r

],C

) ∈ X}.

• trs(X) = {r

| (C,[r

,..., r

],C

) ∈ X}.

If a rule is applied for rule generation, the rule

must be already generated.

Deﬁnition 9. Let Cs a set of clauses. Clause-Rule

Interaction Tree wrt Cs, denoted by CRIT (Cs), is

a ﬁnite sequence of clause-rule interaction quadru-

ples wrt Cs such that if (C, [r

,..., r

],C

) =

nth(m,CRIT (Cs)), then {r

,..., r

} ⊆ trs(Under(m −

1,CRIT (Cs))).

Example:

CRIT(Cs) = ( (C

,[],C

...

,[r

],C

,[r

],C

,[r

],C

,[r

],C

))

7.2 Correctness

Rules generated in a Clause-Rule Interaction Tree are

model-preserving.

Theorem 3. Let Cs be a set of clauses. Let r be a

conditional term rewriting rule. If trs(CRIT (Cs)) 3 r,

then r is model-preserving.

Proof. We prove the theorem by induction of the size

of CRIT (Cs). Assume that trs(CRIT (Cs)) 3 r.

• When size(CRIT (Cs)) = 1. Then

(C,[],C

,r) = nth(1,C RIT (Cs)). Hence C = C

and Models(Cs) = Models(Cs ∪ {C

}). Let

= {}. Since Cs

and ctrs(C

) 3 r, we have

7→ r. By Theorem 2, r is model-preserving.

• When size(CRIT (Cs)) > 1. Assume that the the-

orem holds for U nder(m − 1,CRIT (Cs)). Let

R = trs(U nder(m − 1,CRIT (Cs))). Each r

∈

R is model-preserving. Assume also that

(C,[r

,..., r

],C

,r) = nth(m,CRIT (Cs)). Let

= {r

,..., r

}. Then Cs

. Since CRIT (Cs)

is a clause-rule interaction tree wrt Cs, we

have R

⊆ R. Hence R

is model-preserving.

Since Cs

and Proposition 2, we have

Models(Cs) = Models(Cs∪ {C

}). Since Cs

and ctrs(C

) 3 r, we have Cs

7→ r. By Theorem 2,

r is model-preserving.

7.3 Solution for the Sample Problem

The sample proof problem is proven by the applica-

tion of model-preserving term rewriting rules gener-

ated in this paper. All steps of computation are shown

below. M is used as a short-hand for Models for sav-

ing spaces.

M ({(← eq($i($ f ($ f ($i($a),$a), $ f ($b, $i($b)))),

$ f ($b,$ f ($i($ f ($a, $b)),$a))))} ∪ Cs)

(Apply r

with θ

= {x/$a})

= M ({(← eq($i($ f ($e,$ f ($b,$i($b)))),

$ f ($b,$ f ($i($ f ($a, $b)),$a))))} ∪ Cs)

(Apply r

with θ

= {x/$a,y/$b}

= M ({(← eq($i($ f ($e,$ f ($b,$i($b)))),

$ f ($b,$ f ($ f ($i($b),$i($a)),$a))))} ∪ Cs)

(Apply r

with θ

= {x/$b})

KEOD 2019 - 11th International Conference on Knowledge Engineering and Ontology Development

288

= M ({(← eq($i($ f ($e,$e)),

$ f ($b,$ f ($ f ($i($b),$i($a)),$a))))} ∪ Cs)

(Apply r

with θ

= {x/$i($b),y/$i($a), z/$a}

= M ({(← eq($i($ f ($e,$e)),

$ f ($b,$ f ($i($b), $ f ($i($a), $a)))))} ∪ Cs)

(Apply r

with θ

= {x/$e})

= M ({(← eq($i($e),

$ f ($b,$ f ($i($b), $ f ($i($a), $a)))))} ∪ Cs)

(Apply r

with θ

= {x/$a})

= M ({(← eq($i($e),

$ f ($b,$ f ($i($b), $e))))} ∪ Cs)

(Apply r

with θ

= {})

= M ({(← eq($e,$ f ($b,$ f ($i($b),$e))))}∪ Cs)

(Apply r

with θ

= {x/$i($b)}

= M ({(← eq($e,$ f ($b,$i($b))))}∪Cs)

(Apply r

with θ

= {x/$b})

= M ({(← eq($e,$e))} ∪ Cs)

(Remove the eq($e,$e) atom since it is true.)

= M ({(←)} ∪ Cs)

In the last step of transformation, a constraint

solving rule for equality is used. This rule is not a

term rewriting rule. However, it obviously preserves

models. In our computation model, rules of logical

inference and term rewriting can be used together un-

der the principle of equivalent transformation.

8 CONCLUDING REMARKS

Term rewriting rules are used for computation

when the background knowledge contains equational

clauses. We can generate two term rewriting rules

directly from an equational clause. We apply to

an equational clause repeatedly (1) specialization by

a substitution for usual variables, and (2) applica-

tion of an already derived rewriting rule. We make

term rewriting rules directly from the resulting equa-

tional clauses. We have proved that the obtained term

rewriting rule is an ET rule. Since ET rules are re-

peatedly applied to the original problem, the result of

computation is also correct. Such guarantee of cor-

rectness is usually not clearly discussed in the theory

of term rewriting systems.

The informal method is procedural, i.e., seman-

tical structure is not discussed relating to rewriting

rules. Each term rewriting rule is constructed with-

out proving it to be semantically correct. Moreover,

correctness of computation result is guaranteed based

on the correctness of each rewriting rule.

First order logic is not sufﬁcient for correctness-

based theory since it doesn’t have enough expressive

power, while KR-logic is sufﬁcient due to the exis-

tence of function variables.

The theory in this paper is constructed on LPSF,

where KR-logic is used as a canonical logical struc-

ture, and term rewriting rules are used as ET rules.

The resolution rule and the unfolding rule are typi-

cal instances of rules in the domain of logic, while

term rewriting rules are typical instances of functional

rewriting. Hence, logical inference and functional

rewriting co-exist, both of them being instances of a

broader concept of equivalent transformation.

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