ABC Repair System for Datalog-like Theories

Xue Li, Alan Bundy and Alan Smaill

CISA, School of Informatics, University of Edinburgh, 10 Crichton St, Edinburgh, U.K.

Keywords:

Theory Repair, Automated Reasoning, Knowledge Representation, Abduction, Belief Revision, Conceptual

Change, Datalog.

Abstract:

This paper aims to develop a domain-independent system for repairing faulty Datalog-like theories by combi-

ning three existing techniques: abduction, belief revision and conceptual change. Accordingly, the proposed

system is named the ABC repair system. Given an observed assertion and a current theory, abduction adds

axioms which represent the simplest and most likely explanation. Belief revision incorporates a new piece of

information which conﬂicts with the input theory by deleting axioms. Conceptual change uses the reformation

algorithm for blocking unwanted proofs or unblocking wanted proofs. The former two techniques change

an axiom as a whole, while reformation changes the language in which the theory is written. These three

techniques are complementary: abduction adds new axioms, belief revision deletes conﬂicting axioms, while

reformation changes the language of the theory. But they have not previously been combined into one system.

We are working on aligning these three techniques in the ABC repair system, which is capable of repairing lo-

gical theories with better quality than individual techniques. Datalog is used as the underlying logic of theories

in this paper, but the proposed system has the potential to be adapted to theories in other logics.

PRELIMINARY

In this paper, a Datalog-like theory is represented as

T, and T

is a corresponding repaired theory. α and

β are Datalog-like axioms written in the language of

T. All constants and predicates start with an upper-

case letter, while variables start with a lower-case one.

The unary function N calculates the number of the

elements in its argument. The argument of N is a set,

e.g. N (T) is the number of the axioms in T.

1 INTRODUCTION

In a knowledge base, logical theories need to evolve

to keep up with a changing environment and to correct

faulty representations. The need for evolution could

be formalised as the conﬂicts between what the user

knows and what the logical theories derive. By repai-

ring logical theories, these conﬂicts can be ﬁxed, and

then the logical theories evolve for absorbing users’

knowledge. A user could be an automated agent or a

human.

Datalog is a declarative logic programming lan-

guage in FOL, which has resurged in the database

community in recent years (Bellomarini et al., 2018).

In comparison with ﬁrst-order logic (FOL), Datalog

excludes negations, function symbols, and existen-

tial quantiﬁcation. Assertions and rules in Datalog

are formalised as Horn Clauses, which should satisfy

conditions that each assertion does not contain any va-

riables, and each variable which occurs in the head of

a rule also occurs in the body of the same rule (Ceri

et al., 1989). There are different variants of Datalog.

This paper restricts to the basic Datalog introduced

above. For a uniform representation in this paper, we

use the implication symbol =⇒ rather than the tradi-

tional symbol, :- in a Datalog-like theory.

Deﬁnition 1 (Datalog Language).

Although the restriction of Datalog reduces the

expressive power of the logic, it brings signiﬁcant ad-

vantages including that deduction is decidable (Pfen-

ning, 2006); Prolog uniﬁcation is sufﬁcient, and the

reformation algorithm is greatly simpliﬁed, as will be

introduced in §2.2.1.

An example of a Datalog-like theory is the Swan

Li, X., Bundy, A. and Smaill, A.

ABC Repair System for Datalog-like Theories.

DOI: 10.5220/0006959703350342

In Proceedings of the 10th International Joint Conference on Knowledge Discover y, Knowledge Engineering and Knowledge Management (IC3K 2018) - Volume 2: KEOD, pages 335-342

ISBN: 978-989-758-330-8

335

theory as below. It has four axioms saying that Ger-

man is part of Europe, all European swans are white,

and the swan named Bruce is a German swan. One

theorem is that Bruce is white. Imagine the user ob-

serves the fact that Bruce is black. In this scenario,

the theory is faulty so that it needs to be repaired.

A1. German(x) ⇒ European(x).

A2. European(x) ∧ Swan(x) ⇒ White(x).

A3. German(Bruce).

A4. Swan(Bruce).

2 ABC REPAIR SYSTEM

The ABC system repairs faulty Datalog-like theories

automatically or partially automatically. Figure 1 gi-

ves the main components of the system. The inputs

given by the user are a Datalog-like theory (T) and

a preferred structure (PS) which will be deﬁned in

§2.1.1. The output is a set of repaired theories. Be-

cause each original repair technique could have multi-

ple repairs to one fault, it is also normal that the com-

bination repair mechanism generates more than one

repair solution and then outputs a set of repaired the-

ories rather than single repaired theory.

Figure 1: The components (C1-C4) of the ABC repair sy-

stem: T is a Datalog-like theory; PS is a preferred structure

deﬁned in §2.1.1; The output is a set of repaired theories.

C1 calculates a minimal set of axioms by pruning

redundancy, which reduces the search space of both

fault detection and repair generation. C2 and C3 are

central parts, which will be discussed in details in §2.1

and §2.2. Generally speaking, when a fault is detected

by C2, its proof/proofs will be provided to C3, ba-

sed on which, C3 generates repairs by combining ab-

duction, belief revision and reformation. This process

will repeat until there are no faults left. In the end, all

repaired theories will be ranked by C4. Ideally, best-

repaired theories could be highlighted for users. As

a part of future work, C4 is denoted in the imaginary

line in Figure 1.

2.1 Fault to be Repaired

This section discusses what faults are to be repaired

and how to detect them based on automated reaso-

ning.

2.1.1 Fault Deﬁnition

Common faults of a logical theory are inconsistency

and incompleteness. Although a Datalog-like theory

is guaranteed to be consistent by never proving a ne-

gative statement, consistency is an underlying requi-

rement of a repaired theory in our system. However,

completeness, which requires a theory to prove all the

possible sentences or their negations in the signature,

may be too strong a requirement. Instead of incom-

pleteness, we are going to focus on the propositions

which the user insists an object theory to prove or not

to prove. The preferred structure is deﬁned as below.

Deﬁnition 2 (Preferred Structure). A preferred

structure (PS) is a structure over the language of a

logical theory (T); PS describes the user’s intention

that a faithful T should follow by giving preferred

base sets:

True Set (T (PS)): Set of the ground propositions

which should be provable by T. These proposi-

tions are called the preferred propositions.

False Set (F (PS)): Set of the ground propositions

which should not be proved by T. These propo-

sitions are called the violative propositions

Obscure Set (O(PS)): Set of the ground propositi-

ons which are neither in T (PS) nor F (PS).

Contradictory Set (C (PS)): Set of the ground pro-

positions which are both in T (PS) and F (PS).

PS is a guideline for determining the faithfulness of a

theory T according to user’s view. As shown in Figure

1, PS is a required input for the ABC system.

Based on the deﬁnition, the last two sets are deci-

ded by the ﬁrst two. Therefore, in the implementation,

only T (PS) and F (PS) need to be given explicitly. In

the swan example, the PS for formalising the user’s

knowledge that Bruce is black so that it cannot also

be white is:

The true set: {Black(Bruce)}

The false set: {White(Bruce)}

An object theory should follow its PS by proving

the preferred propositions, while not proving the vio-

lative propositions. This makes the theory correct and

useful to the user.

∀α, α ∈ T (PS), T ` α (1)

∀β, β ∈ F (PS), T 0 β (2)

Notice that C(PS) should be empty. Otherwise, the

preferred structure is self-contradictory, to which no

consistent theory could be faithful. If there is a pro-

position in C (PS), a warning would be given and the

process of our system has to stop. In this case, the

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

336

user has to empty C (PS) before giving the PS as the

input of the ABC system.

Given a preferred structure with empty C (PS), the

possible faults of a logical theory are incompatibility

and insufﬁciency.

Deﬁnition 3 (Incompatible and Insufﬁcient).

• An incompatible theory with respect to the prefer-

red structure is one which has a theorem that is in

the false set: ∃β, β ∈ F (PS), T ` β.

• An insufﬁcient theory with respect to the preferred

structure is one in which a proposition that is in

the true set, but is not a theorem of the theory:

∃α, α ∈ T (PS), T 0 α.

The deﬁned faults represent the conﬂicts between the

input theory and the user’s intention. One of the ap-

plications is when the user veriﬁes a theory by ex-

periments. For example, a professor of medicine is

studying how a medicine affects a human. At ﬁrst, he

investigated relevant materials, so he had built a set

of axioms in his theory. Then he ran a series of ex-

periments to evaluate his theory. In this scenario, the

result of his experiments could be formalised as a pre-

ferred structure, based on which the conﬂicts between

his theory and his experimental results correspond to

incompatibility and insufﬁciency as deﬁned above.

In the swan example, the original theory proves

that Bruce is white, which is an incompatibility, and

not prove that Bruce is black, which is an insufﬁ-

ciency. The deﬁnition of the preferred structure and

the faults deﬁned based on it can reﬂect the conﬂicts

between a theory and empirical evidence correctly.

In conclusion, the faults of an object theory to be

repaired in this paper are insufﬁciency and incompa-

tibility, deﬁned based on a preferred structure repre-

senting the user’s opinion.

2.1.2 Fault Detection

This section shows how to detect the faults of a

Datalog-like theory by employing selected literal (SL)

resolution.

Detecting insufﬁciency or incompatibility is to

check the provability of a target proposition from a

Datalog-like theory. The inference rule, SL resolution

(Kowalski and Kuehner, 1971) can do this detection,

which follows the principles:

• Select one of the most recently introduced literals

to be resolved upon.

• Resolve the selected literal with an input clause.

SL resolution is not only sound and complete (Gal-

lier, 2003), but also decidable for Datalog-like the-

ories (Pfenning, 2006) so that proofs can always be

detected if there are any.

For reducing the search space, proof discovery

process starts with resolving a goal, with an input

clause whose head is complementary to the goal. As

an axiom in Datalog-like theory is a Horn clause,

which has at most one positive literal, the most re-

cently introduced literals of a nonempty resolvent

could only be a disjunction of negative literals, cal-

led sub-goals here. Then we continually resolve the

ﬁrst sub-goal, with an input clause. Again, either an

empty clause or a clause of a disjunction of negative

literals would be the result. Repeat this process until

no resolution step is available. If it ends up with an

empty clause, refutation occurs, which means that the

input theory proves the goal clause.

For detecting incompatibility, the negation of a vi-

olative proposition in F (PS) is set as the initial goal

for refutation based on the input theory. Every viola-

tive proposition, in turn, is checked in the same way.

Incompatibility is detected if the input theory proves

a violative proposition. ⊥ will represent false.

T is incompatible wrt β, iff T

` ⊥,

where T

= T ∪ {¬β}, β ∈ F (PS). (3)

For example, Figure 2 shows how the Swan the-

ory proves the violative proposition of the PS,

White(Bruce). There are four proof steps in total. In

proof step 1, the initial goal is the negation of the vio-

lative proposition, White(Bruce)⇒, which is resolved

with the head of axiom A2, White(x), by substituting

Bruce for x. In the same way, European and German

are resolved in proof step 2 and 3 respectively. In

proof step 4, the empty clause is derived, which me-

ans that the input Swan theory proves White(Bruce),

so the Swan theory is incompatible.

As for a sufﬁcient theory, all the preferred propo-

sitions should be logical consequences of the object

theory. Similar to detecting incompatibility, preferred

propositions need to be checked one by one. By nega-

ting a preferred proposition into a goal, the theory can

be concluded to be sufﬁcient concerning the preferred

proposition if and only if resolving the goal with the

input theory results in an empty clause. Otherwise,

the object theory is insufﬁcient. Repeat the process

until all the preferred propositions are checked.

T is insufﬁcient wrt α, iff T

0 ⊥,

where T

= T ∪ {¬α}, α ∈ T (PS) (4)

At each resolution in the proof, there is always

one parent being an input clause. When generating

repairs, we can directly change the parent which is an

input clause with no traceback needed.

ABC Repair System for Datalog-like Theories

337

Swan Theory: A1. German(x) ⇒ European(x). A3. German(Bruce).

A2. European(x) ∧ Swan(x) ⇒ White(x). A4. Swan(Bruce).

Preferred Structure: The false set F (PS): {White(Bruce)}

The true set T (PS): {Black(Bruce)}

Inference (goal):

Proof Step 1:

Proof Step 2:

Proof Step 3:

Proof Step 4 (the empty clause):

White(Bruce) =⇒

European(Bruce) ∧Swan(Bruce) =⇒

European(x) ∧ Swan(x) =⇒ White(x)

German(Bruce) ∧ Swan(Bruce) =⇒

German(x) =⇒ European(x)

Swan(Bruce) =⇒

=⇒ German(Bruce)

=⇒

=⇒ Swan(Bruce)

Figure 2: Proving White(Bruce): proof steps of the incompatibility of the Swan theory.

2.2 Repair Generation

In the ABC repair system, repairs are generated ba-

sed on the proofs of faults: incompatibility and in-

sufﬁciency. The desired repairs for each fault are

shown in Table 1: incompatibility could be repaired

by blocking all unwanted proofs of each violative pro-

position, and insufﬁciency could be repaired by un-

blocking a wanted proof of each preferred proposi-

tion. Blocking a proof could be done by breaking any

proof step of it, while unblocking a proof requires to

build all necessary proof steps. As the decision of

which axiom(s) or predicate(s) to be changed is not

unique, usually, the core task here is to avoid unne-

cessary information loss by making minimal changes

ardenfors, 1992).

Table 1: The target of repairing a faulty theory T.

Incompatibility Insufﬁciency

Fault ∃β, β ∈ F (PS), T `

∃α, α ∈ T (PS), T 0

Target T

0 β T

` α

Core

Task

Blocking all proofs

of β.

Unblocking one

proof of α.

Method Break one proof

step in each proof.

Build all necessary

proof steps.

Abduction, belief revision and reformation are

three technical candidates for generating repairs. Dif-

ferent techniques work in different scenarios.

2.2.1 Repair Techniques in the ABC System

Original Abduction. The underlying reasoning of

abduction is the inference to the best explanation. Gi-

ven an observation, abduction seeks for its explana-

tion (Cox and Pietrzykowski, 1986). If the observa-

tion is not a logical consequence of the original the-

ory, abduction will add the explanation as an axiom,

which could be an assertion or a rule. With no extra

information, the solution of abduction is not unique

because there could be different ways of proving the

observation.

Abduction in the ABC System. Regarding a pre-

ferred proposition in T (PS) as the observation, ab-

duction repairs insufﬁciencies by adding axioms.

Abduction could directly add a preferred proposi-

tion. However, directly adding everything that is true

would result in a clumsy theory with lower quality,

e.g., a theory of gases that just added all observations

of gas pressure and volume is inferior to one that in-

cludes Boyle’s law. Therefore, it is better to add a rule

which proves not only the targeted preferred proposi-

tion but also several other axioms and/or other prefer-

red proposition, while not proving any proposition in

F (PS).

Original Belief Revision. The underlying reasoning

of belief revision is deduction. Belief revision works

on incorporating new information which is inconsis-

tent with the original knowledge base (G

ardenfors,

1992). The inconsistent new information is usually

represented as a new belief, e.g. β. Then revision

function adds the new belief and deletes old ones to

make the revised theory consistent, which is blocking

all proofs of ¬β. From the view of proofs, the task

of a revision function is to break the unwanted proofs

with minimal changes.

Belief Revision in the ABC System. To repair the

incompatibility, the ABC system uses belief revision

to block the proofs of the propositions in F (PS). For

blocking these unwanted proofs, belief revision dele-

tes axioms while making minimal changes.

Original Conceptual Change. Conceptual change

uses the reformation algorithm, which is triggered by

the reasoning failure that T proves an unwanted as-

sertion or fails in proving a wanted one (Bundy and

Mitrovic, 2016). The underlying reasoning of refor-

mation is deduction, and originally in FOL. By chan-

ging the language of a theory, reformation inverts the

outcome of the targeted uniﬁcation, which contributes

to blocking or unblocking proofs. The repair types of

reformation include changing the name or the arity of

a predicate, changing the name of a constant or chan-

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

338

ging a constant into a term containing a variable.

Conceptual Change in the ABC System. Because

the ABC system is based on Datalog, reformation is

much simpler than that for FOL. For example, diffe-

ring arity of a predicate overloads the predicate, and

changing a constant into a term containing a varia-

ble would introduce a function. These two kinds of

repairs are illegal in the circumstance of Datalog so

that they are abandoned in the ABC system. In this

paper, we use the left-hand side term of each reso-

lution to represent a goal/sub-goal, while the right is

the input clause side. Table 2 gives reformation re-

pairs on different types of resolution problems in the

ABC system, in which the changes are underlined.

R1 replaces P

(

−→

) by P

(

−→

) for making the origi-

nally failed proof step between P

(

−→

) and P

(

−→

)

successful. The remaining repairs are about making

the originally successful proof steps fail: R2 renames

a predicate on the input clause side, R3 changes one

argument of the predicate from the input clause side,

and R4 increases arity by 1, and then adds distinguis-

hed constants as arguments to predicates on both si-

des. Because R4 changes the arity of a predicate, it

needs to be propagated to all instances of the modi-

ﬁed predicate. When the resolution is between a con-

stant goal and a variable from input clause side, it can

always succeed unless the variable is instantiated by

another constant as in R5.

Considering the deﬁnition of the preferred struc-

ture, reformation is capable of repairing both insufﬁ-

Table 2: Repairs of reformation for inverting the outcome

of proof step in the ABC system. P is a predicate; t and u

are either constants or a variables; n ≥ 0, m ≥ 0; 1 ≤ i ≤

n; P refers to a constant when its arity is 0; ’=’ represents

unifying, of which goal/sub-goal is on the left-hand side,

and input clause on the right.

Resolution Pro-

blem

Reformation Repair

(

−→

) 6=

(

−→

)

R1. Replace input side by the

other: P

(

−→

) = P

(

−→

) = P(

−→

)

R2. Rename input side P:

−→

) 6= P

(

−→

R3. Rename one argument in

−→

e.g., let t

6= u

: P(

−→

) 6=

−→

0 n

R4. Add distinguished arguments

C, C

: P(

−→

, C) 6= P(

−→

, C

C = x R5. Instantiate x to C

: C 6= C

ciency (R1) and incompatibility (R2-R5) by changing

the language in which the theory is written.

All Repair Candidates in the ABC System. Apart

from changing language by reformation, or adding or

deleting an axiom as a whole by abduction and belief

revision respectively, a rule can be changed by ad-

ding or deleting a precondition (McNeill and Bundy,

2007). A precondition needs to be deleted when a pre-

ferred proposition should be derived based on a rule,

but it is not because of the improperly strict precondi-

tion. In this case, the repair of deleting the precondi-

tion could be classiﬁed as a variant abduction, which

unblocks a wanted proof. Similarly, the precondition

which can block a rule from proving a violative pro-

position could be added to the rule, which could be

classiﬁed as a variant of belief revision that blocks

unwanted proofs. In conclusion, all repair candidates

in the ABC system are summarised in Table 3.

Table 3: Technique candidates for repairing faults.

Incompatibility Insufﬁciency

Tech.

Belief revision:

deletes axioms;

adds unprovable

preconditions to a

rule.

Abduction: adds

an assertion/ rule;

deletes unprovable

preconditions from

a rule.

Reformation: changes the language of T.

2.2.2 Combination of Techniques

The candidate techniques discussed in the last section

are complementary. Combining them allows us to ﬁx

faults with new repairs. Due to the minimal change

principle, we do not combine techniques in repairing

a single proof step, because each original technique

alone is sufﬁcient already. Now, we can list all pos-

sible repairs for one proof step when tackling incom-

patibility in Table 4, and Table 5 gives all possible

repairs for insufﬁciency.

It is fairly common that several faults are involved

in one theory (McNeill and Bundy, 2007). The ABC

system combines different techniques when there are

multiple problematic proof steps. These problema-

tic proof steps are detected in turn as errors: after

repairing one error, a new one is detected. Here er-

rors could be the faults: insufﬁciency and incompa-

tibility based on a preferred structure; or the issues

against the restriction of Datalog, or some heuristics,

e.g., prefer not to change the equality predicate. Ex-

cept for the original existing errors, there could be two

kinds of introduced errors during a repair process:

1. New introduced errors. A repair may introduce a

error which did not exist previously. For example, a

ABC Repair System for Datalog-like Theories

339

Table 4: Incompatibility repairs: block a proof by breaking

one proof step of it in the ABC system. All symbols mean

the same as in Table 2.

Resolution Problem (Technique Abbr.) Repair

−→

) = P(

−→

)

(R) P(

−→

) 6= P

(

−→

(R) P(

−→

) 6= P(

−→

0 n

(R) P(

−→

, C) 6= P(

−→

, C

(B) Delete the axiom where

+P(

−→

) came from.

(B) Add an unprovable precon-

dition −Q(

−→

) to the axiom at

either side.

C = x (R) Instantiate x to C

Table 5: Insufﬁciency repairs: unblock a proof by building

all necessary proof steps in the ABC system. All symbols

mean the same as in Table 2.

Resolution Problem (Technique Abbr.) Repair

(

−→

) 6= P

(

−→

)

(R) P

(

−→

) = P

(

−→

(A) Delete −P

(

−→

), a

precondition from its axiom.

(A) Add a rule which

proves +P

(

−→

(A) Add the assertion:

(

−→

predicate in an unwanted proof is changed, if it is also

necessary for proving a preferred proposition, then

this repair causes newly introduced insufﬁciency.

2. Recurred errors. A repair may affect a previous

one, especially when these two repairs are provided

by different algorithms, which could be difﬁcult to

avoid. For example, an axiom is added by abduction,

and it could be changed by reformation later on. In

this case, developing a set of heuristics is necessary,

which is a part of future work.

We need to continue repairing until no error re-

mains, and then a ﬁnal solution could be generated

completely. By applying three candidate techniques

in parallel for each error, we can get all possible repair

combinations, as shown in Figure 3. For example, if

belief revision deletes an axiom, after which reforma-

tion changes the language of the theory to tackle anot-

her error, then the ﬁnal repaired theory is generated by

combining belief revision and reformation.

Figure 3: Combining repairs when repairing multiple er-

rors. T

, T

refer to repaired theories.

2.2.3 Repair Postulates

Resembling the idea of the AGM postulates for be-

lief revision (G

ardenfors, 1992), several postulates

are formulated in this section. The ABC repair sy-

stem should generate repairs which satisfy these pos-

tulates. The underlying motivation is that we want

to generate reasonable combination repairs and make

a minimal change. All the information conveyed by

axioms are seen as not gratuitous so that we do not

want any unnecessary informational loss. On the ot-

her hand, we do not want to add extra axioms unless

it is necessary.

1. ∃T

, iff T (PS) ∩ F (PS) =

An input theory can be repaired, if and only if the

preferred structure is not self-contradictory. In ot-

her words, for achieving a faithful theory, the pre-

ferred structure must not be self-contradictory.

2. T

= T, iff

∀α, α ∈ T (PS), T ` α; ∀β, β ∈ F (PS), T 0 β

Do nothing if the input theory already satisﬁes

preferred structure.

3. If T is a Datalog-like theory, then T

is also a

Datalog-like theory.

This postulate guarantees that the repaired theory

also follows the restrictions of Datalog if the input

theory is a Datalog-like theory. We assume that

no matter working in which logic, repairs should

not break the restriction of the logical format con-

vention. Otherwise, it would come up with so-

mething that the system cannot understand, and

cause unexpected problems.

4. N (T

) ≤ {N (T) + N (T (PS))}

N is the function giving the size of a set. This

postulate means that the number of the axioms in

a repaired theory should not be greater than the

total number of the original theory and the true

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

340

set of the preferred structuren (T (PS)).

In table 4, no repairs of incompatibility would in-

crease the size of the theory. In table 5, the ﬁrst

two repairs do not change N (T). As for adding a

rule, N (T

) should not be bigger than N (T). Be-

cause if a rule can only prove the current preferred

proposition, it is more reliable for us to add the

proposition directly as an assertion. We assume a

rule should be able to derive two assumptions at

least; otherwise, it is not effective and should not

be considered. The last repair in table 5, which di-

rectly adds a preferred proposition as an assertion,

increases the size of the theory by one for each

preferred proposition. It is not a wise decision ob-

viously if you add more than one axiom for deri-

ving a preferred proposition, in which case, why

not just add the preferred one?

Therefore, when repairing a theory based on its

preferred structure, we should not result in a new

theory containing more axioms than the original

theory and the true set of the preferred structure

in total.

2.2.4 Repairs for Swan Theory by ABC System

This section will discuss the current repairs of the afo-

rementioned Swan theory. The given theory and its

preferred structure can be found in Figure 2. The pro-

blem could be caused by the fact that European swan

is ambiguous: it may mean ’the European variety of

swans’ or ’swans resident in Europe’. In the scenario

that all European variety swans are white, but Bruce

is resident in Europe, the target theory is as below,

where the desired repairs are underlined.

TA1. German(x, y) ⇒ European(x, y).

TA2. European(x, Variety) ∧ Swan(x) ⇒ White(x).

TA3. German(Bruce, Resident).

TA4. Swan(Bruce).

TA5. Black(Bruce).

In the rest of this section, we are going to discuss how

the ABC system repairs the Swan theory and whet-

her it can generate the targeted theory as we proposed.

The order of solving incompatibility and insufﬁciency

is immaterial. The ABC system tackles incompatibi-

lity ﬁrstly in multiple ways:

1. Delete A4: Swan(Bruce).

2. Rename European in A1:

German(x) ⇒ Europeandash(x).

3. Add a constant argument to European in A2:

European(x, Dummy1) ∧ Swan(x) ⇒ White(x).

The ﬁrst repair, which deletes A1, blocks proof step 4

in Figure 2, which is proper in the scenario that Bruce

is another kind of a bird rather than a swan. The se-

cond repair blocks proof step 2, which does not make

sense in this scenario, although it solves the incom-

patibility. It is possible that some repairs are difﬁcult

to explain from the perspective of semantics in some

scenarios. The last one is quite close to the desired

repairs. As this repair changes the arity of a predi-

cate, it has to be propagated to all instances of the

predicate. Here European in A2 is seen as the trigger

literal, which is assigned a unique constant Dummy1,

while the other instances are assigned the common

constant DummyDefault or a variable for rules. After

propagation, the partially repaired theory is:

Repairs:

add argument(European).

add argument(German).

Repaired Theory:

A1’. German(x, y) ⇒ European(x, y).

A2’. European(x, Dummy1) ∧ Swan(x) ⇒ White(x).

A3’. German(Bruce, DummyDefault).

A4 . Swan(Bruce).

For aligning with changed European in rule A1’, the

arity of German in the rule is also changed from one

to two by adding the new variable y, because Data-

log requires that each variable in the head of a rule

must also exist in its body. However, the changing of

the arity is not triggered by German, so that the new

arguments of all its instances are either variables for

rules or DummyDefault for facts, e.g., DummyDefault

in A3’.

For the current theory, the new proof step 3 failed

because its sub-goal German(Bruce, Dummy1) can-

not unify the input literal German(Bruce, DummyDe-

faut), neither other input literals. Here Dummy1 is

inherited from European(x, Dummy1) in A2’ during

resolution. Now the incompatibility has been solved.

The ABC system will continue to repair insufﬁciency.

The best repair in our scenario is adding the preferred

proposition as an axiom directly.

Repairs:

add argument(European).

add argument(German).

add axiom: Black(Bruce).

Repaired Theory:

A1’. German(x, y) ⇒ European(x, y).

A2’. European(x, Dummy1) ∧ Swan(x) ⇒ White(x).

A3’. German(Bruce, DummyDefault).

A4 . Swan(Bruce).

A5 . Black(Bruce).

Now the theory is faithful concerning the preferred

structure. The repair solution is generated by com-

bining reformation and abduction, and the solution

satisﬁes the postulates of the ABC system. By in-

terpreting Dummy1 as variety and DummyDefault as

resident, the produced theory is the desired one gi-

ven at the beginning of this section. The current

ABC Repair System for Datalog-like Theories

341

ABC system does not deal with semantic interpreta-

tion, which is something worthwhile to research in the

future.

In this section, it can be seen that a single techni-

que is not enough for generating the best repairs for a

theory in some scenarios. Therefore, the ABC repair

system works better than the individual techniques it

combines.

3 FUTURE WORK

1. Give semantic meaning to dummy term introdu-

ced by reformation, which can make repaired theories