The Rule Interchange Format (RIF) is an emerging W3C format that allows rules to be exchanged between rule systems. Uncertainty is an intrinsic feature of real world knowledge, hence it is important to take it into account when building logic rule formalisms. However, the set of truth values in the Basic Logic Dialect (RIF-BLD) currently consists of only two values (t and f). In this paper, we first present two techniques of encoding uncertain knowledge and its fuzzy semantics in RIF-BLD presentation syntax. We then propose an extension leading to an Uncertainty Rule Dialect (RIF-URD) to support a direct representation of uncertain knowledge. In addition, rules in Logic Programs (LP) are often used in combination with the other widely-used knowledge representation formalism of the Semantic Web, namely Description Logics (DL), in order to provide greater expressive power. To prepare DL as well as LP extensions, we present a fuzzy extension to Description Logic Programs (DLP), called Fuzzy DLP, and discuss its mapping to RIF. Such a formalism not only combines DL with LP, as in DLP, but also supports uncertain knowledge representation.
modeling lack well-defined boundaries or, precisely defined criteria of relationships with other
concepts. To take care of these knowledge representation needs, different approaches for integrating
uncertain knowledge into traditional rule languages and DL languages have been studied [
The Rule Interchange Format (RIF) is being developed by W3C’s Rule Interchange Format (RIF)
Working Group to support the exchange of rules between rule systems [
According to the final report from the URW3 Incubator group, uncertainty is a term intended to
include different types of uncertain knowledge, including incompleteness, vagueness, ambiguity,
randomness, and inconsistency [
The main contributions of this paper are: (1) two techniques of encoding uncertain information in
RIF as well as an uncertainty extension to RIF; (2) an extension of DLP to Fuzzy DLP and the mapping
of Fuzzy DLP to RIF. Two earlier uncertainty extensions to the combination of DL and LP that we can
expand on are [
The rest of this paper is organized as follows. Section 2 reviews earlier work on the interoperation
between DL and LP in the intersection of these two formalisms (known as DLP) and represents the
DL-LP mappings in RIF. Section 3 addresses the syntax and semantics of fuzzy Logic Programs, and
then presents two techniques of bringing uncertainty into the current version of RIF presentation syntax
(hence its semantics and XML syntax), using encodings as RIF functions and RIF predicates. Section 4
adapts the definition of the set of truth values in RIF-FLD for the purpose of representing uncertain
knowledge directly, and proposes the new Uncertainty Rule Dialect (RIF-URD), extending RIF-BLD.
Section 5 extends DLP to Fuzzy DLP, supporting mappings between fuzzy DL and fuzzy LP, and gives
representations of Fuzzy DLP in RIF and RIF-URD. Finally, Section 6 summarizes our main results
and gives an outlook on future research.
$. Description Logic Programs and Their Representation in RIF
In this section, we summarize the work on Description Logic Programs (DLP) [
The paper [
In RIF presentation syntax, the quantifiers Exists and Forall are made explicit, rules are written with a “:-” infix, variables start with a “?” prefix, and whitespace is used as a separator.
Table 1 summarizes the mappings in [
RIF Forall ?x (D(?x) :- C(?x)) Forall ?x (D(?x) :- C(?x)) Forall ?x (C(?x) :- D(?x))
R(a b) Forall ?x ?y (R(?x ?y) :- P(?x ?y)) Forall ?x ?y (P(?x ?y) :- R(?x ?y)) Forall ?x ?y (R(?x ?y) :- P(?y ?x)) Forall ?x ?y (P(?y ?x) :- R(?x ?y)) Forall ?x ?y ?z (
R(?x ?z) :- And(R(?x ?y) R(?y ?z)))
Forall ?x ?y (P(?x ?y) :- R(?x ?y))
Forall ?x (D(?x) :- And(C1(?x) C2(?x))
Forall ?x ?y (P(?x ?y) :- And(R1(?x ?y) R2(?x ?y))
Fuzzy set theory was introduced in [
In this section, we first present the syntax and semantics for fuzzy Logic Programs based on Fuzzy
Sets and Fuzzy Logic [
H (c B1, !, Bn
(1)
where H , Bi are atoms, n , 0 , and the factor c is a real number in the interval (0,1] [
The fuzzy LP language proposed by [
In the current paper, we follow this work and use the following form to represent a fuzzy rule.
H (x") ( B1 (x"1 ), !, Bn (x"n ) / c (2)
Here H (x"), Bi (x"i ) are atoms, x", x"i are vectors of variables or constants, n , 0 and the confidence factor c (also called certainty degree) is a real number in the interval (0,1]. For the special case of fuzzy facts this becomes H / c . These forms with a “/” symbol have the advantages of avoiding possible confusion with the equality symbol usually used for functions in logics with equality, as well as using a unified and compact format to represent fuzzy rules and fuzzy facts.
The semantics of such fuzzy LP is an extension of classical LP semantics. Let BR stand for the
Herbrand base of a fuzzy knowledge base KBLP . A fuzzy Herbrand interpretation H I for KBLP is
defined as a mapping BR - [
Therefore, a fuzzy knowledge base KBLP is true under H I iff every rule in KBLP is true
under H I . Such a Herbrand interpretation H I is called a Herbrand model of KBLP . Furthermore, a
rule is true under H I iff each variable-free instance of this rule is true under H I . A variable-free
instance of a rule (3) is true under H I iff val(H , H I ) , c . min{val(Bi , H I ) | i +{1,!, n}}
( min{} ! 1 if n ! 0 ). In other words, such an interpretation can be separated into the following two
parts [
(1) The body of the rule consists of n atoms. Our confidence that all these atoms are true is interpreted under Gödel’s semantics for fuzzy logic: val((B1,!, Bn ), H I ) ! min{val(Bi , H I ) | i +{1,!, n}} (2) The implication is interpreted as the product: val(H , H I ) ! c . val((B1,!, Bn ), H I )
For a fuzzy knowledge base KBLP , the reasoning task is a fuzzy entailment problem written as KBLP |! H / c ( H + BR , c + (0,1] ).
Example 3.1. Consider the following fuzzy LP knowledge base: (1) (2) we have that KBLP |! cheapFlight( flight0001,1800) / 0.63 .
Example 3.2. Consider the following fuzzy LP knowledge base: A(x) ( B(x), C(x) / 0.5 (1) C(x) ( D(x) / 0.5 (2) B(d ) / 0.5 (3) D(d ) / 0.8 (4) Applying the semantics we discussed, val(cheapFlight( flight0001,1800), HI ) ! 0.9*0.7 ! 0.63 , so We have that KBLP |! A(d ) / 0.2 . The reasoning steps of example 3.2 are described as follows: val( A(d ), H I ) ! 0.5 . min(val(B(d ), H I ), val(C(d ), H I )) **according to (1) ! 0.5 . min(val(B(d ), H I ), 0.5. val(D(d ), H I )) **according to (2) ! 0.5 . min(0.5, 0.5 . val(D(d ), H I )) **according to (3) ! 0.5 . min(0.5, 0.5 . 0.8) **according to (4) ! 0.5 . 0.4 ! 0.2
One technique to encode uncertainty in logics with equality such as the current RIF-BLD (where
equality in the head is “At Risk”) is mapping all predicates to functions and using equality for letting
them return uncertainty values [
Forall ?x1 … ?xk ( h(t1 … tr)=?ch :- And(b1(t1,1 … t1,s1)=?c1 ' bn(tn,1 … tn,sn)=?cn ?ct =External(numeric-minimum(?c1 ' ?cn)) ?ch=External(numeric-multiply(c ?ct)) )
Each predicate in the fuzzy rule thus becomes a function with a return value between 0 and 1. The
semantics of the fuzzy rules is encoded in RIF-BLD using the built-in functions numeric-multiply from
RIF-DTB[
A fact of the form
H
/ c can be represented in RIF-BLD presentation syntax as ) (* <http://example.org/fuzzy/membershipfunction > *) Document(
Group ( (* "Definition of membership function left _ shoulder(0, 4000,1000, 3000) "[] *) Forall ?y( left&shoulder0k4k1k3k(?y)=1 :- And(External(numeric-less-than-or-equal(0 ?y))
External(numeric-less-than-or-equal(?y 1000)))) Forall ?y( left&shoulder0k4k1k3k(?y)=External(numeric-add(External(numeric-multiply(-0.0005 ?y)) 1.5)) :- And(External(numeric-less-than(1000 ?y))
External(numeric-less-than-or-equal(?y 3000)))) Forall ?y( left&shoulder0k4k1k3k(?y)=0 :- And(External(numeric-less-than(3000 ?y))
External(numeric-less-than-or-equal(?y 4000)))) Note that membership function left _ shoulder(0, 4000,1000, 3000) is encoded as three rules.
Document(
Import (<http://example.org/fuzzy/membershipfunction >) Group ( )
Forall ?x ?y( cheapFlight(?x ?y)=?ch :- And(affordableFlight(?x ?y)=?c1
?ch=External(numeric-multiply(0.4 ?c1)))) Forall ?x ?y(affordableFlight(?x ?y)=left&shoulder0k4k1k3k(?y))
) The Import statement loads the left&shoulder0k4k1k3k function defined at the given “<…>” IRI. Example 3.4 We can rewrite example 3.2 in RIF functions as follows:
Another encoding technique is making all n-ary predicates into (1+n)-ary predicates, each being functional in the first argument which captures the certainty factor of predicate applications. A fuzzy rule in the form of equation (2) from Section 3.1 can then be represented in RIF-BLD as
Forall ?x1 … ?xk ( h(?ch t1 … tr) :- An?dc(tb=1(E?cx1tet1r,n1a…l(ntu1m,s1)eri'c-mbinn(i?mcnutmn,1(?…c1 t'n,sn)?cn))
?ch=External(numeric-multiply(c ?ct)) ) ) ) Likewise, a fact of the form
H
/ c can be represented in RIF-BLD as h(c t1 … tr) Example 3.5 We can rewrite example 3.1 in RIF predicates as follows,
Document(
Import (<http://example.org/fuzzy/membershipfunction >) Group (
Forall ?x ?y( cheapFlight(?ch ?x ?y) :- And(affordableFlight(?c1 ?x ?y)
?ch=External(numeric-multiply(0.4 ?c1))) )
Forall ?x ?y(affordableFlight(?c1 ?x ?y) :- ?c1 =left&shoulder0k4k1k3k(?y)) ) )
In this section, we adapt the definition of the set of truth values from RIF-FLD and its semantic structure. We then propose a RIF extension for directly representing uncertain knowledge.
In previous sections, we showed how to represent the semantics of fuzzy LP with RIF functions and predicates in RIF presentation syntax. We now propose to introduce a new dialect for RIF, RIF Uncertainty Rule Dialect (RIF-URD), so as to directly represent uncertain knowledge and extend the expressive power of RIF.
The set TV of truth values in RIF-BLD consists of just two values, t and f . This set forms a two-element Boolean algebra with t ! 1 and f ! 0 . However, in order to represent uncertain knowledge, all intermediate truth values must be allowed. Therefore, the set TV of truth values is extended to a set with infinitely many truth values ranging between 0 and 1. Our uncertain knowledge representation is specifically based on Fuzzy Logic, thus a member function maps a variable to a truth value in the 0 to 1 range.
Definition 1. (Set of truth values as a specialization of the set in RIF-FLD). In RIF-FLD, /t
denotes the truth order, a binary relation on the set of truth values TV . Instantiating RIF-FLD, which
just requires a partial order, the set of truth values in RIF-URD is equipped with a total order over the 0
to 1 range. In RIF-URD, we specialize /t to / , denoting the numerical truth order. Thus, we
observe that the following statements hold for any element ei , ej or ek in the set of truth values TV
in the 0 to 1 range, justifying to write it as the interval [
(1) The set TV is a complete lattice with respect to / , i.e., the least upper bound (lub) and the greatest lower bound (glb) exist for any subset of / .
(2) Antisymmetry. If ei / e j and ej / ei then ei ! ej . (3) Transitivity. If ei / e j and ej / ek then ei / ek . (4) Totality. Any two elements should satisfy one of these two relations: ei / e j or ej / ei . (5) The set TV has an operator of negation, #: TV - TV , such that a). # ei ! 1 % ei . b). # is self-inverse, i.e., ## ei ! ei .
Let TVal(5 ) denote the truth value of a non-document formula, 5 , in RIF-BLD. TVal(5 ) is a mapping from the set of all non-document formulas to TV , I denotes an interpretation, and c is a real number in the interval (0,1].
Definition 2. (Truth valuation adapted from RIF-FLD). Truth valuation for well-formed formulas in RIF-URD is determined as in RIF-FLD, adapting the following three cases.
(8) Conjunction (glbt becomes min): TValI ( And (B1 ! Bn )) ! min(TVal(B1 )!TVal(Bn )) . (9) Disjunction (lubt becomes max): TValI (Or(B1 ! Bn )) ! max(TVal(B1 )!TVal(Bn )) (11) Rule implication ( t becomes 1, f becomes 0, condition valuation is multiplied with c ): TValI (conclusion : % condition / c) ! 1 if TValI (conclusion) , c . TValI (condition) TValI (conclusion : % condition / c) ! 0 if TValI (conclusion) 2 c . TValI (condition)
Forall ?x1 … ?xk (
h(t1 … tr) :- And(b1(t1,1 … t1,s1) ' bn(tn,1 … tn,sn)) ) / c A fuzzy rule in the form of equation (2) from Section 3.1 can be directly represented in RIF-URD as Document(
Group ( ) Likewise, a fact of the form
H
/ c can be represented in RIF-URD as h(t1 … tr) / c
Such a RIF-URD document of course cannot be executed by an ordinary RIF-compliant reasoner. RIF-URD-compliant reasoners will need to be extended to support the above semantics and the reasoning process shown in Section 3.1.
Example 3.6 We can directly represent example 3.1 in RIF-URD as follows:
Document(
Import (<http://example.org/fuzzy/membershipfunction >) Group (
Forall ?x ?y(
cheapFlight(?x ?y) :- affordableFlight(?x ?y) ) / 0.4
Forall ?x ?y(affordableFlight(?x ?y)) / left&shoulder0k4k1k3k(?y) ) )
In this section, we extend Description Logic Programs (DLP) [
Based on Fuzzy Sets and Fuzzy Logic [
Since DL is a subset of FOL, it can also be seen in terms of that subset of FOL, where individuals are equivalent to FOL constants, concepts and concept descriptions are equivalent to FOL formulas with one free variable, and roles and role descriptions are equivalent to FOL formulas with two free variables.
A concept inclusion axiom of the form C ! D is equivalent to an FOL sentence of the form $x.C(x) - D(x) , i.e. an FOL implication. In uncertainty representation and reasoning, it is important to represent and compute the degree of subsumption between two fuzzy concepts, i.e., the degree of overlap, in addition to crisp subsumption. Therefore, we consider fuzzy axioms of the form C ! D ! c generalizing the crisp C ! D . The above equivalence leads to a straightforward mapping from a fuzzy concept inclusion axiom of the form C ! D ! c ( c + (0,1] ) to an LP rule as follows: D(x) ( C(x) / c .
The intersection of two fuzzy concepts in fuzzy DL is defined as (C1 #$C2 )I (x) ! min(C1I (x), C2I (x)) ; therefore, a fuzzy concept inclusion axiom of the form C1 # C2 ! D ! c including the intersection of C1 and C2 can be transformed to an LP rule D(x) ( C1 (x), C2 (x) / c . Here the certainty degree of (variable-free) instantiations of the atom D(x) is defined by the valuation val(D, H I ) ! c . min{val(Ci , H I ) | i +{1, 2}} . It is easy to see that such a fuzzy concept inclusion axiom can be extended to include the intersection of n concepts ( n 6 2 ).
Similarly, a role inclusion axiom of the form R ! P is equivalent to an FOL sentence consisting of an implication between two roles. Thus we map a fuzzy role inclusion axiom of the form R ! P ! c ( c + (0,1] ) to a fuzzy LP rule as P(x, y) ( R(x, y) / c . Moreover, $ni!1 Ri ! P ! c can be transformed to P(x, y) ( R1 (x, y),!, Rn (x, y) / c .
A concept equivalence axiom of the form C " D can be represented as a symmetrical pair of FOL implications: $x.C(x) - D(x) and $x.D(x) - C(x) . Therefore, we map the ‘fuzzified’ equivalence axiom C " D ! c into C(x) ( D(x) / c and D(x) ( C(x) / c ( c + (0,1] ). As later examples show, such mappings in hybrid knowledge bases are directed from rules to DL expressions, hence if we have two rules of the forms C(x) ( D(x) / c1 and D(x) ( C(x) / c2 ( c1, c2 + (0,1] ), they are mapped to a DL expression as C " D ! c with c ! min(c1, c2 ) . Similarly, we map two rules R(x, y) ( P(x, y) / c1 and P(x, y) ( R(x, y) / c2 into a role equivalence axiom of the form R " P ! min(c1, c2 ) , as well as two rules R(x, y) ( P( y, x) / c1 and P( y, x) ( R(x, y) / c2 into an inverse role equivalence axiom of the form P " R% ! min(c1, c2 ) .
A DL assertion C(a) (respectively, R(a, b) ) is equivalent to an FOL atom of the form C(a) (respectively, R(a, b) ), where a and b are individuals. Therefore, a fuzzy DL concept-individual assertion of the form C(a) ! c corresponds to a ground fuzzy atom C(a) / c in fuzzy LP, while a fuzzy DL role-individual assertion of the form R(a,b) ! c corresponds to a ground fuzzy fact R(a, b) / c .
Table 2 summarizes the mappings in Fuzzy DLP. For simplicity, in Fuzzy DLP as defined in this paper we do not use fuzzy forms for all of DLP, excluding value restrictions and transitive role axiom, and assuming c ! 1 whenever / c is omitted.
Forall ?x(
C(?x)=?ch :- And(D(?x)=c1 ?ch=External(numeric-multiply(c c1))) Forall ?x(
D(?x)=?ch :- And(C(?x)=c1 ?ch=External(numeric-multiply(c’ c1))) Forall ?x(
C(?ch ?x) :- And(D(?c1 ?x) ?ch=External(numeric-multiply(c c1))) Forall ?x(
D(?ch ?x) :- And(C(?c1 ?x) ?ch=External(numeric-multiply(c’ c1))) Forall ?x(C(?x) :- D(?x)) / c Forall ?x(D(?x) :- C(?x)) / c’ R(x, y) ( P(x, y) / c, P(x, y) ( R(x, y) / c' R " P ! min(c,c' ) Forall ?x ?y(
R(?x ?y)=?ch :- And(P(?x ?y)=c1 ?ch=External(numeric-multiply(c c1))) Forall ?x ?y(
P(?x ?y)=?ch :- And( R(?x ?y)=c1 ?ch=External(numeric-multiply(c’ c1))) Forall ?x ?y(
R(?ch ?x ?y) :- And(P(?c1 ?x ?y) ?ch=External(numeric-multiply(c c1))) Forall ?x ?y(
P(?ch ?x ?y) :- And(R(?c1 ?x ?y) ?ch=External(numeric-multiply(c’ c1))) Forall ?x ?y(R(?x ?y) :- P(?x ?y)) / c Forall ?x ?y(P(?x ?y) :- R(?x ?y)) / c’ R(x, y) ( P( y, x) / c, P( y, x) ( R(x, y) / c' P " R% ! min(c,c' ) Forall ?x ?y(
R(?x ?y)=?ch :- And(P(?y ?x)=c1 ?ch=External(numeric-multiply(c c1))) Forall ?x ?y(
P(?y ?x)=?ch :- And( R(?x ?y)=c1 ?ch=External(numeric-multiply(c’ c1))) Forall ?x ?y(
R(?ch ?x ?y) :- And(P(?c1 ?y ?x) ?ch=External(numeric-multiply(c c1))) Forall ?x ?y(
P(?ch ?y ?x) :- And(R(?c1 ?x ?y) ?ch=External(numeric-multiply(c’ c1))) Forall ?x ?y(R(?x ?y) :- P(?y ?x)) / c Forall ?x ?y(P(?y ?x) :- R(?x ?y)) / c’ C(a) / c C(a) ! c C(a)=c C(c a) C(a) /c
R(a,b) / c R(a,b) ! c R(a b)=c R(c a b) R(a b) /c
In summary, Fuzzy DLP is an extension of Description Logic Programs supporting the following concept and role inclusion axioms, range and domain axioms, concept and role assertion axioms to build a knowledge base: $ni!1Ci ! D ! c ,
C " D ! c , # ! $R.C , # ! $R% .C , $ni!1 Ri ! P ! c , P " R ! c , P " R% ! c , R& ! R , C(a) ! c , and R(a, b) ! c , where C, D, C1,!Cn are atomic concepts, P, R are atomic roles, a, b are individuals, c + (0,1] and n , 1 . Notice that the crisp DLP axioms in DLP are special cases of their counterparts in Fuzzy DLP. For example, C ! D is equal to its fuzzy version $ni!1Ci ! D ! c for n ! 1 and c ! 1 .
In previous sections, we presented two techniques of encoding uncertainty in RIF and proposed a method based on an extension of RIF for uncertainty representation. Subsequently, we also showed how to represent Fuzzy DLP in RIF-BLD and RIF-URD in Table 2.
Layered on Fuzzy DLP, we can build fuzzy hybrid knowledge bases in order to build fuzzy rules on top of ontologies for the Semantic Web and reason on such KBs.
Definition 3. A fuzzy hybrid knowledge base KBhf is a pair 2 KDL , KLP 6 , where KDL is the finite set of (fuzzy) concept inclusion axioms, role inclusion axioms, and concept and role assertions of a decidable DL defining an ontology. KLP consists of a finite set of (fuzzy) hybrid rules and (fuzzy) facts.
A hybrid rule r in KLP is of the following generalized form (we use the BNF choice bar, |): Here, H ( y"), H (z"), Bi ( y"i ), Qj (z"j ) are atoms, & precedes a DL atom, y", z", y"i , z"j are vectors of " " " variables or constants, where y and each yi have arbitrary lengths, z" and each z j have length 1 or 2, and c + (0,1] . Also, & atoms and / c degrees are optional (if all & atoms and / c degrees are missing from a rule, it becomes a classical rule of Horn Logic).
Such a fuzzy hybrid rule must satisfy the following constraints: (1) H is either a DL predicate or a rule predicate ( H + 7T % 7R ). H is a DL predicate with the form &H , while it is a rule predicate without the & operator.
(2) Each Bi (1 2 i / l ) is a rule predicate ( Bi + 7R ), and Bi ( yi ) is an LP atom. (3) Each Qj ( 1 2 j / n ) is a DL predicate ( Qj + 7T ), and Qj (z j ) is a DL atom. (4, pure DL rule) If a hybrid rule has head &H , then each atom in the body must be of the form &Qj (1 2 j / n ); in other words, there is no Bi ( l ! 0 ). A head &H without a body ( l ! 0 , n ! 0 ) constitutes the special case of a pure DL fact.
Example 5.1. The rule & CheapFlight( x, y) ( AffordableFlight( x, y) / c is not a pure DL rule according to (4), hence not allowed in our hybrid knowledge base, while CheapFlight(x, y) ( & AffordableFlight(x, y) / c is allowed.
A hybrid rule of the form & CheapFlight( x, y) ( & AffordableFlight( x, y) / c can be mapped to a fuzzy DL role subsumption axiom AffordableFlight ! CheapFlight ! c .
Our approach thus allows DL atoms in the head of hybrid rules which satisfy the constraint (4, pure DL rule), supporting the mapping of DL subsumption axioms to rules. We also deal with fuzzy subsumption of fuzzy concepts of the form C ! D ! c as shown in Example 5.1.
An arbitrary hybrid knowledge base cannot be fully embedded into the knowledge representation
formalism of RIF with uncertainty extensions. However, in the proposed Fuzzy DLP subset, DL
components (DL axioms in LP syntax) can be mapped to LP rules and facts in RIF. A RIF-compliant
reasoning engine can be extended to do reasoning on a hybrid knowledge base on top of Fuzzy DLP by
adding a module that first maps atoms in rules to DL atoms, and then derives the reasoning answers
with a DL reasoner, e.g. Racer or Pellet, or with a fuzzy DL reasoner, e.g. fuzzyDL [
Our fuzzy extension directly relates to Lotfi Zadeh’s semantics of fuzzy sets and fuzzy logic. We do not yet cover here other researchers’ semantics, for example, Jan Lukasiewicz’s. Nor do we cover other uncertainty formalisms, based on probability theory, possibilities, or rough sets. Future work will include generalizing our fuzzy extension of hybrid knowledge bases to some of these different kinds of uncertainty, and parameterizing RIF-URD to support different theories of uncertainty in a unified manner.
Complementing the RIF-URD presentation syntax, XML elements and attributes like <degree>, @mapkind, and @kind, following those of Fuzzy RuleML, can be introduced for the RIF-URD XML syntax. Another direction of future work would be the extension of uncertain knowledge to various combination strategies of DL and LP without being limited to DL queries.