COREDIAG: Eliminating Redundancy in Constraint Sets
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C ORE D IAG: Eliminating Redundancy in Constraint Sets
Alexander Felfernig 1 , Christoph Zehentner 1 , Paul Blazek 2
1
Graz University of Technology, Inffeldgasse 16b, 8010 Graz, Austria
alexander.felfernig@ist.tugraz.at
arXiv:2102.12151v1 [cs.AI] 24 Feb 2021
christoph.zehentner@ist.tugraz.at
2
cyLEDGE Media GmbH, Schottenfeldgasse 60, 1070 Vienna, Austria
p.blazek@cyledge.com
ABSTRACT formally, if C={c1 , c2 , ..., cn } is the initial set of con-
straints defined for the knowledge base and one con-
Constraint-based environments such as config- straint ci is redundant, then (C − {ci }) ∪ C is incon-
uration systems, recommender systems, and
scheduling systems support users in different de- sistent (C is the negation of C).
cision making scenarios. These environments Redundancy elimination in knowledge bases is a
exploit a knowledge base for determining solu- topic extensively investigated by AI research. The
tions of interest for the user. The development identification of redundant constraints plays a ma-
and maintenance of such knowledge bases is an jor role, for example, in the development and main-
extremely time-consuming and error-prone task. tenance of configuration knowledge bases (see, e.g.,
Users often specify constraints which do not re- (Sabin and Freuder, 1999)). The authors introduce
flect the real-world. For example, redundant con- concepts for the detection of redundant constraints
straints are specified which often increase both, in conditional constraint satisfaction problems (CC-
the effort for calculating a solution and efforts re- SPs). The approach is based on the idea of analyz-
lated to knowledge base development and main- ing the solution space of the given problem (on the
tenance. In this paper we present a new al- level of individual solutions) in order to detect differ-
gorithm (C ORE D IAG) which can be exploited ent types of redundant constraints. (Piette, 2008) pro-
for the determination of minimal cores (minimal vide an in-depth discussion of the role of redundancy
non-redundant constraint sets). The algorithm is elimination in SAT solving. They introduce an (in-
especially useful for distributed knowledge en- complete) algorithm for the elimination of redundant
gineering scenarios where the degree of redun- clauses and show its applicability on the basis of an
dancy can become high. In order to show the ap- empirical study. The role of redundancies in ontology
plicability of our approach, we present an empir- development is analyzed by (Fahad and Qadir, 2008).
ical study conducted with commercial configura- The authors point out the importance of redun-
tion knowledge bases. dancy elimination and discuss typical modeling errors
that occur during ontology development and mainte-
Keywords: redundant constraints, minimal cores. nance. (Grimm and Wissmann, 2011) introduce algo-
rithms for redundancy elimination in OWL ontologies.
The authors propose an algorithm that computes re-
1 INTRODUCTION dundant axioms by exploiting prior knowledge of the
The central element of a constraint-based ap- dispensibility of axioms. (Levy and Sagiv, 1992) ana-
plication is a knowledge base (constraint set). lyze two types of redundancies in Datalog programs.
When developing and maintaining constraint sets, First, they interpret redundancy in terms of reachabil-
users are often defining faulty constraints (the ity, i.e., rules and predicates are eliminated that are not
system calculates solutions which are not al- part of any derivation tree. Second, redundancy is de-
lowed or – in the worst case – no solution can fined on the basis of the concepts of minimal deriva-
be found) (Bakker et al., 1993; Felfernig et al., 2004) tion trees which do not include any pair of identical
or redundant constraints which are not needed atoms where one is the predecessor of the other one.
to express the domain knowledge in a com- All the mentioned approaches focus on the identifi-
plete fashion (Sabin and Freuder, 1999; Piette, 2008; cation of redundant constraints in centralized scenarios
Fahad and Qadir, 2008; Levy and Sagiv, 1992). In this where a knowledge engineer is interested in identify-
paper we focus on situations where users are defin- ing redundant constraints in the given knowledge base.
ing redundant constraints which – when deleted from In such scenarios it is assumed that only a small subset
the constraint set (knowledge base) – do not change of the given constraints is redundant (this assump-
the semantics of the remaining constraint set. More tion is also denoted as low redundancy assumption
122nd International Workshop on Principles of Diagnosis
(Grimm and Wissmann, 2011)). Existing algorithms The following configuration task will be used as
are focusing on such centralized scenarios. In this a working example throughout the paper. The vari-
paper we go one step further and propose an algorithm able type represents the type of the car, pdc is the
which is especially useful in distributed knowledge park distance control feature, fuel represents the av-
engineering scenarios where we can expect a larger erage fuel consumption per 100 kilometers, a skibag
number of redundant constraints due to the fact that allows convenient ski stowage inside the car, and 4-
different contributors add constraints which are related wheel represents the actuation type (4-wheel supported
to the same topic (see, e.g., (Chklovski and Gil, 2005; or not supported). These variables represent the possi-
Richardson and Domingos, 2003)) – we denote the ble combinations of customer requirements. The set
assumption of larger sets of redundant constraints CKB = {c1 , c2 , c3 , c4 , c5 } defines additional re-
the high redundancy assumption. For example, we strictions on the set of possible customer requirements
envision a scenario where a large number of users CR = {c6 , c7 , c8 , c9 , c10 }.
propose constraints to be applied by a constraint- • V = {type, f uel, skibag, 4 − wheel, pdc}
based configuration or recommendation engine
(Felfernig and Burke, 2008) and the task of an un- • D={
derlying diagnosis algorithm is to identify minimal dom(type) = {city, limo, combi, xdrive},
sets of constraints which retain the semantics of the dom(f uel) = {4l, 6l, 10l},
original constraint set – we denote such constraint sets dom(skibag) = {yes, no},
as minimal cores. Note that the following discussions dom(4 − wheel) = {yes, no},
are based on the assumption of consistent constraint dom(pdc) = {yes, no}}
sets. Methods for consistency restoration are dis- • CKB = {
cussed in (Bakker et al., 1993; Felfernig et al., 2004; c1 : 4 − wheel = yes → type = xdrive,
Friedrich and Shchekotykhin, 2005; c2 : skibag = yes → type 6= city,
Felfernig et al., 2011). c3 : f uel = 4l → type = city,
The major contributions of our paper are the fol- c4 : f uel = 6l → type 6= xdrive,
lowing. First, we introduce a new algorithm which c5 : type = city → f uel 6= 10l}
allows for a more efficient determination of redun-
dant constraints especially in the context of distributed • CR = {
(community-based) knowledge engineering scenarios. c6 : 4 − wheel = no,
Second, we present the results of a performance anal- c7 : f uel = 4l,
ysis of our algorithm conducted with real-world con- c8 : type = city,
figuration knowledge bases. c9 : skibag = no,
The remainder of this paper is organized as follows. c10 : pdc = yes}
In Section 2 we introduce a simple example configu- On the basis of this example configuration task we
ration knowledge base from the automotive domain. now give a definition of a corresponding configuration
In Section 3 we introduce a basic algorithm for the (solution).
determination of redundant constraints in centralized
settings (S EQUENTIAL). In Section 4 we introduce the Definition (Configuration) : A configuration (so-
C ORE D IAG algorithm. Thereafter we report the re- lution) for a configuration task is an instantiation
sults of a performance evaluation conducted with real- I={v1 = ins1 , v2 = ins2 , ..., vn = insn } where insk
world configuration knowledge bases (Section 5). The is an element of the domain of vk . A configuration is
paper is concluded with Section 6. consistent if the assignments in I are consistent with
the constraints in C. A complete solution is one in
2 WORKING EXAMPLE which all the variables are instantiated. Finally, a con-
For illustration purposes we use a car configuration figuration is valid if it is both, consistent and complete.
knowledge base throughout this paper. A configura- A configuration for our example configuration task
tion task can be defined as a basic constraint satisfac- would be I = {4 − wheel = no, f uel = 4l, type =
tion problem (CSP) (Tsang, 1993) (see the following city, skibag = no, pdc = yes}.
definition).1
3 DETERMINING REDUNDANT
CONSTRAINTS
Definition (Configuration Task) : A configura-
tion task can be defined as a CSP (V, D, C). V = Let us now consider a simple adaptation of the original
′
{v1 , v2 , ..., vn } is a set of finite domain variables. D set of constraints CKB which we denote with CKB .
′
= {dom(v1 ), dom(v2 ), ..., dom(vn )} is a set of cor- CKB includes an additional constraint ca which has
responding domain definitions where dom(vk ) is the been added by a knowledge engineer.
domain of the variable vk . C = CKB ∪ CR where ′
CKB = {
CKB = {c1 , c2 , ..., cq } is a set of domain-specific ca : skibag 6= no → type = limo ∨
constraints (the configuration knowledge base) and type = combi ∨
CR = {cq+1 , cq+2 , ..., ct } is a set of customer re- type = xdrive,
quirements (as well represented as constraints). c1 : 4 − wheel = yes → type = xdrive,
c2 : skibag = yes → type 6= city,
1
Note that the presented concepts are as well applicable c3 : f uel = 4l → type = city,
to other types of knowledge representations such as SAT or c4 : f uel = 6l → type 6= xdrive,
description logics. c5 : type = city → f uel 6= 10l}
222nd International Workshop on Principles of Diagnosis
It is obvious that ca is redundant since it does not S EQUENTIAL in situations with a large amount of re-
further restrict the solution space defined by the con- dundant constraints in CKB . Large amounts of redun-
straints CKB = {c1 , c2 , c3 , c4 , c5 }. In order to dis- dant constraints typically occur in distributed knowl-
cuss constraint redundancy on a more formal level, we edge engineering scenarios where a large number of
introduce the following definitions. users specify rules that in the following have to be ag-
gregated into one consistent constraint set (see, e.g.,
(Chklovski and Gil, 2005)).
Definition (Redundant Constraint) : Let ca be
a constraint element of the configuration knowledge In the following section we introduce the C ORE -
base CKB . ca is called redundant iff CKB − {ca } |= D IAG algorithm which is a valuable alternative to S E -
ca . If this condition is not fulfilled, ca is said to be QUENTIAL in situations with a large number of redun-
non-redundant. Redundancy can also be analyzed by dant constraints. After having introduced C ORE D IAG
checking CKB − {ca } ∪ CKB for consistency – if we will analyze the performance of both algorithms
consistency is given, ca is non-redundant. (S EQUENTIAL and C ORE D IAG) on the basis of real-
Iterating over each constraint of CKB , executing the world configuration knowledge bases (Section 5).
non-redundancy check CKB − {ca } ∪ CKB , and
deleting redundant constraints from CKB results in a 4 COREDIAG
set of non-redundant constraints (the minimal core).
If the non-redundancy check fails (no solution can The C ORE D IAG algorithm (together with C ORE D) is
be found), the constraint ca is redundant and can be based on the principle of divide-and-conquer: when-
deleted from CKB . Otherwise (the non-redundancy ever a set S which is a subset of CKB is inconsistent
check is successful), ca is non-redundant. with CKB , it is or contains a minimal core, i.e. a set
of constraints which preserve the semantics of CKB .
Definition (Minimal Core) : Let CKB be a config- In our implementation C ORE D is responsible for de-
uration knowledge base. CKB is denoted as minimal termining such minimal cores, C ORE D IAG returns the
core iff ∀ci ∈ CKB : CKB − {ci } ∪ CKB is consis- complement of a minimal core which is a maximal set
tent. Obviously, CKB ∪ CKB |= ⊥. of redundant constraints in CKB . C ORE D is based on
The principle of the following algorithm the principle of QuickXPlain (Junker, 2004) – as a con-
(S EQUENTIAL - Algorithm 1) is often used for de- sequence a minimal core (minimal set of constraints
termining such redundancies (see, e.g., (Piette, 2008; that preserve the semantics of CKB ) can be interpreted
Grimm and Wissmann, 2011)). as a minimal conflict, i.e., a minimal set of constraints
that are inconsistent with CKB .
Algorithm 1 S EQUENTIAL(CKB ): ∆ C ORE D allows the determination of preferred min-
imal cores since the algorithm is based on the as-
{CKB : configuration knowledge base} sumption of a strict lexicographical ordering of the
{CKB : the complement of CKB } constraints in CKB . On an informal level a pre-
{∆: set of redundant constraints} ferred minimal core can be characterized as follows:
CKBt ← CKB ; if we have different options for choosing a mini-
for all ci in CKBt do mal core, we would select the one with the most
if isInconsistent((CKBt − ci ) ∪ CKB ) then agreed-upon constraints. For more details on the role
CKBt ← CKBt − {ci }; of strict lexicographical orderings of constraints we
end if refer the reader to the work of (Junker, 2004) and
end for (Felfernig et al., 2011).
∆ ← CKB − CKBt ; The C ORE D IAG algorithm generates CKB from
return ∆; CKB . It then activates C ORE D (see Algorithm 3)
which determines a minimal core on the basis of
The approach of S EQUENTIAL is straightforward: a divide-and-conquer strategy that divides the con-
each individual constraint ci is evaluated w.r.t. redun- straints in C into two subsets (C1 and C2 ) with the goal
dancy by checking whether CKBt − ci is still in- to figure out whether one of those subsets already con-
consistent with CKB . If this is the case, ci can be tains a minimal core. If C2 contains a minimal core,
considered as redundant. If CKBt − ci is consistent C1 is not further taken into account. If C contains only
one element (cα ) and B is still consistent, then cα is
with CKB , ci is a non-redundant constraint since its part of the minimal core.
deletion induces consistency with CKB . Applying the
′
algorithm S EQUENTIAL to our example CKB results
in ∆ = {ca } since CKB − {ca } ∪ CKB is inconsistent Algorithm 2 C ORE D IAG (CKB ): ∆
and no further constraint ci can be deleted from CKB {CKB = {c1 , c2 , ..., cn }}
such that CKB − {ca } − {ci } is still inconsistent. {CKB : the complement of CKB }
The problem of checking whether a given constraint {∆: set of redundant constraints}
can be inferred from the remaining part of a constraint
set has shown to be Co-NP-complete in the general CKB ← {¬c1 ∨ ¬c2 ∨ ... ∨ ¬cn };
case (Piette, 2008). The major goal of our work was return(CKB − C ORE D(CKB , CKB , CKB ));
to figure out whether there exist alternative algorithms
that have a better runtime performance compared to
322nd International Workshop on Principles of Diagnosis
Redundancy Rate
KB (|cKB |) Alg. ˜0-10% ˜50% ˜75% ˜87.5%
Bike A (32) S 32.0 / 205.4 / 0 64.0 / 408.6 / 32 128.0 / 1209.0 / 96 256.0 / 4073.2 / 224
Bike A (32) CD 63.0 / 614.4 / 0 88.8 / 863.2 / 32 106.6 / 1352.0 / 96 107.4 / 1737.2 / 224
Bike B (35) S 35.0 / 256.8 / 1 70.0 / 616.4 / 36 140.0 / 1710.0 / 106 280.0 / 4854.0 / 246
Bike B (35) CD 68.6 / 693.4 / 1 94.0 / 960.8 / 36 109.6 / 1365.2 / 106 117.8 / 1893.0 / 246
Bike C (37) S 37.0 / 297.0 / 1 74.0 / 696.6 / 38 148.0 / 1824.8 / 112 296.0 / 5722.8 / 260
Bike C (37) CD 72.4 / 703.6 / 1 101.2 / 1091.2 / 38 114.8 / 1524.2 / 112 122.2 / 2115.4 / 260
Bike D (34) S 34.0 / 280.2 / 1 68.0 / 606.8 / 35 136.0 / 1672.0 / 103 272.0 / 5033.6 / 239
Bike D (34) CD 66.2 / 727.6 / 1 94.8 / 1031.8 / 35 104.8 / 1433.8 / 103 114.8 / 2000.8 / 239
Bike E (35) S 35.0 / 254.2 / 9 70.0 / 601.0 / 44 140.0 / 1628.6 / 114 280.0 / 5124.8 / 254
Bike E (35) CD 60.8 / 663.0 / 9 83.4 / 821.4 / 44 96.0 / 1182.6 / 114 103.6 / 1671.2 / 254
Bike F (33) S 33.0 / 274.0 / 1 66.0 / 601.8 / 34 132.0 / 1573.8 / 100 264.0 / 4525.0 / 232
Bike F (33) CD 64.6 / 632.8 / 1 88.6 / 931.2 / 34 108.2 / 1345.6 / 100 110.8 / 1822.2 / 232
Bike G (36) S 36.0 / 281.4 / 2 72.0 / 660.6 / 38 144.0 / 1729.8 / 110 288.0 / 5434.4 / 254
Bike G (36) CD 70.6 / 714.6 / 2 96.0 / 939.8 / 38 111.6 / 1409.6 / 110 122.4 / 2081.8 / 254
Bike H (24) S 24.0 / 194.2 / 0 48.0 / 398.8 / 24 96.0 / 1047.0 / 72 192.0 / 3010.0 / 168
Bike H (24) CD 47.0 / 443.4 / 0 63.0 / 587.4 / 24 77.2 / 869.8 / 72 80.0 / 1240.4 / 168
Bike I (35) S 35.0 / 268.4 / 1 70.0 / 647.0 / 36 140.0 / 1696.4 / 106 280.0 / 4976.4 / 246
Bike I (35) CD 68.4 / 708.6 / 1 93.6 / 985.0 / 36 112.2 / 1371.4 / 106 117.0 / 1897.0 / 246
Bike J (46) S 46.0 / 366.8 / 4 92.0 / 867.8 / 50 184.0 / 2309.8 / 142 368.0 / 7234.4 / 326
Bike J (46) CD 88.4 / 896.0 / 4 119.4 / 1258.8 / 50 139.8 / 1886.8 / 142 142.0 / 2413.6 / 326
Bike K (35) S 35.0 / 254.0 / 1 70.0 / 805.4 / 36 140.0 / 1852.8 / 106 280.0 / 5146.8 / 246
Bike K (35) CD 68.8 / 712.4 / 1 95.6 / 1021.6 / 36 108.8 / 1374.2 / 106 117.6 / 1945.4 / 246
Bike L (37) S 37.0 / 290.0 / 2 74.0 / 645.8 / 39 148.0 / 1822.0 / 113 296.0 / 5740.8 / 261
Bike L (37) CD 71.4 / 716.4 / 2 96.6 / 1001.6 / 39 113.2 / 1425.8 / 113 111.0 / 1829.8 / 261
Bike 2 (32) S 32.0 / 883.0 / 3 64.0 / 2386.4 / 35 128.0 / 8218.8 / 99 256.0 / 37784.4 / 227
Bike 2 (32) CD 61.2 / 2165.2 / 3 85.4 / 3749.6 / 35 97.2 / 5693.2 / 99 108.0 / 10276.8 / 227
esvs (21) S 21.0 / 340.0 / 0 42.0 / 870.8 / 21 84.0 / 2771.8 / 63 168.0 / 10231.8 / 147
esvs (21) CD 41.0 / 724.0 / 0 56.0 / 1170.6 / 21 65.6 / 1844.0 / 63 71.0 / 3296.4 / 147
fs (16) S 16.0 / 291.6 / 1 32.0 / 664.0 / 17 64.0 / 1989.2 / 49 128.0 / 7238.0 / 113
fs (16) CD 30.6 / 658.8 / 1 42.0 / 933.4 / 17 49.2 / 1504.2 / 49 52.2 / 2431.8 / 113
hypo (21) S 21.0 / 116.6 / 1 42.0 / 321.0 / 22 84.0 / 975.4 / 64 168.0 / 3297.6 / 148
hypo (21) CD 40.6 / 383.8 / 1 55.2 / 549.0 / 22 62.2 / 802.2 / 64 71.0 / 1293.0 / 148
large2 (185) S 130.0 / 2552.8 / 75 260.0 / 4721.8 / 260 520.0 / 7860.0 / 445 1040.0 / 15025.4 / 630
large2 (185) CD 76.8 / 1868.8 / 75 79.8 / 2085.6 / 260 96.6 / 2834.0 / 445 103.0 / 3870.4 / 630
Table 1: Application of S EQUENTIAL (S) (Algorithm 1) and C ORE D IAG (C) (Algorithm 2) to configuration
knowledge bases of www.itu.dk/research/cla/externals/clib. ”Bikex”: bicycles; ”esvs”: corporate networks; ”fs”:
financial services (insurances); ”hypo”: financial services (investments); ”large2”: electronic circuits. Evaluation
data: (#TP-calls / runtime (ms) / #redundant constraints).
5 EVALUATION ber of constraints contained in the minimal core (the
We now compare the performance of C ORE D IAG with lower the number of constraints in the minimal core,
the S EQUENTIAL algorithm discussed in Section 3. the better the performance of C ORE D IAG).
The worst case complexity (and best case complex- Table 1 reflects the results of our analysis conducted
ity) of S EQUENTIAL in terms of the number of needed with the knowledge bases of the configuration bench-
consistency checks is n (the number of constraints mark.2 The tests have been executed on a standard
in CKB ). Worst case and best case complexity are desktop computer (Intel®Core™2 Quad CPU Q9400
identical since S EQUENTIAL checks the redundancy CPU with 2.66GHz and 2GB RAM) using the CLib
of each individual constraint ci with respect to CKB . library. We compared the performance of S EQUEN -
In contrast, the worst case complexity of C ORE D IAG TIAL and C ORE D IAG for the different configuration
depends on the number of redundant constraints in knowledge bases. In order to show the advantages of
CKB . The worst case complexity of C ORE D IAG in C ORE D IAG that come along with an increasing num-
terms of the number of needed consistency checks is ber of redundant constraints, we generated three addi-
2c ∗ log2 ( nc ) + 2c where n is the number of constraints tional versions from the benchmark knowledge bases
in CKB and c is the minimal core size. The best case (see Table 1) that differ in their redundancy rate R (see
complexity in terms of the number of needed consis- Formula 1). The number of iterations per setting was
tency checks can be achieved if all constraints element set to 10; for each iteration we applied a randomized
of the minimal core are positioned in one branch of the constraint ordering. Note that an evaluation of the indi-
C ORE D search tree: log2 ( nc ) + 2c. Consequently, the
2
performance of C ORE D IAG heavily relies on the num- www.itu.dk/research/cla/externals/clib.
422nd International Workshop on Principles of Diagnosis
Redundancy Rate
KB redundancy-free ˜0-10% ˜50% ˜75% ˜87.5%
Bike A 9.9 10.2 14.2 33.7 43.0
Bike B 9.3 9.5 29.0 24.6 41.9
Bike C 8.4 11.0 18.3 32.3 44.0
Bike D 7.8 8.9 25.6 28.0 42.6
Bike E 6.3 13.0 19.0 24.2 41.5
Bike F 7.7 12.8 19.5 22.3 41.4
Bike G 7.3 10.3 16.9 26.0 44.8
Bike H 10.5 11.0 13.6 24.1 37.2
Bike I 8.0 9.0 15.7 31.5 45.4
Bike J 9.1 19.3 24.8 26.6 49.2
Bike K 7.7 11.4 17.3 24.8 43.5
Bike L 9.3 13.4 25.8 26.7 45.5
Bike 2 44.8 48.3 84.1 162.0 323.4
esvs 22.7 26.0 45.1 79.5 157.8
fs 22.6 24.2 44.0 76.5 149.7
hypo 8.3 8.4 19.1 26.0 47.6
large2 15.5 16.6 22.0 25.5 36.4
Table 2: Time in ms needed for calculating a solution for a given configuration knowledge base version.
Algorithm 3 C ORE D(B, D, C = {c1 , c2 , ..., cp }): ∆ redundant constraints included (see Table 2) – for this
evaluation as well the number of iterations per setting
{B: consideration set} was set to 10; for each iteration we applied a random-
{D: constraints added to B} ized constraint ordering.
{C: set of constraints to be checked for redundancy}
if D 6= ∅ and inconsistent(B) then
return ∅; 6 CONCLUSIONS
end if The detection of redundant constraints plays a major
if singleton(C) then role in the context of (configuration) knowledge base
return(C); development and maintenance. In this paper we have
end if proposed two algorithms which can be applied for the
k ← ⌈ r2 ⌉; identification of minimal cores, i.e., minimal sets of
C1 ← {c1 , c2 , ..., ck }; constraints that preserve the semantics of the original
C2 ← {ck+1 , ck+2 , ..., cp }; knowledge base. The S EQUENTIAL algorithm can be
∆1 ← C ORE D(B ∪ C2 , C2 , C1 ); applied in settings where the number of redundant con-
∆2 ← C ORE D(B ∪ ∆1 , ∆1 , C2 ); straints in the knowledge base is low. The second al-
return(∆1 ∪ ∆2 ); gorithm (C ORE D IAG) is more efficient but restricted in
its application to knowledge bases that contain a large
number of redundant constraints.
vidual properties of the used knowledge bases is within
the scope of future work. 7 ACKNOWLEDGEMENTS
The work presented in this paper has been conducted
|redundant constraints in CKB | within the scope of the research project ICONE (Intel-
R(CKB ) = (1)
|constraints in CKB | ligent Assistance for Configuration Knowledge Base
Development and Maintenance) funded by the Aus-
In addition to the original version (redundancy rate trian Research Promotion Agency (827587).
= ˜0-10%) we generated three knowledge bases with
the redundancy rates 50%, 75%, and 87.5%. For ex-
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