Unfounded Sets for Disjunctive Hybrid MKNF Knowledge Bases
aa r X i v : . [ c s . A I] F e b Unfounded Sets for Disjunctive Hybrid MKNFKnowledge Bases
Spencer Killen and Jia-Huai You University of Alberta
Abstract
Combining the closed-world reasoning of answer set programming (ASP) withthe open-world reasoning of ontologies broadens the space of applications of rea-soners. Disjunctive hybrid MKNF knowledge bases succinctly extend ASP andin some cases without increasing the complexity of reasoning tasks. However, inmany cases, solver development is lagging behind. As the result, the only knownmethod of solving disjunctive hybrid MKNF knowledge bases is based on guess-and-verify, as formulated by Motik and Rosati in their original work. A main ob-stacle is understanding how constraint propagation may be performed by a solver,which, in the context of ASP, centers around the computation of unfounded atoms ,the atoms that are false given a partial interpretation. In this work, we build towardsimproving solvers for hybrid MKNF knowledge bases with disjunctive rules: Weformalize a notion of unfounded sets for these knowledge bases, identify lowercomplexity bounds, and demonstrate how we might integrate these developmentsinto a solver. We discuss challenges introduced by ontologies that are not presentin the development of solvers for disjunctive logic programs, which warrant somedeviations from traditional definitions of unfounded sets. We compare our workwith prior definitions of unfounded sets.
Minimal Knowledge and Negation as Failure (MKNF), a modal autoepistemic logic de-fined by Lifschitz [7] which extends first-order logic with two modal operators K and not , provides a uniform framework for nonmonotonic reasoning. It was later built uponby Motik and Rosati [8] to define hybrid MKNF knowledge bases, where rule-basedMKNF formulas along with a description logic (DL) knowledge base intuitively encap-sulate the combined semantics of answer set programs and ontologies. One argumentfor using hybrid MKNF is the existence of a proof theory based on guess-and-verify- one can enumerate partitions (a term that corresponds to interpretation in first-orderlogic) and for each one check whether it is an MKNF model. Such an approach is notefficient enough to be practical in a solver.To address the above issue, Ji et al. [5] give a definition of unfounded sets andan abstract DPLL-based solver [9] for normal hybrid MKNF knowledge bases, where1ules are constrained to a single atom in the head.Disjunctive heads in rules are a powerful extension to answer set programming andincrease the expressive power of programs in the polynomial complexity hierarchy [2].In this work, we extend the work of Ji et al. [5] by defining unfounded sets for dis-junctive hybrid MKNF knowledge bases and investigate its properties. The problemturns out to be substantially more challenging than the normal case. We show the fol-lowing main results. First, we show that the problem of determining whether an atomis unfounded w.r.t. a given (partial) partition is coNP-hard. The result is somewhatsurprising in that the claim holds even for normal rules under the condition that the en-tailment relation in the underlying DL is polynomial. This shows that the polynomialconstruction of the greatest unfounded set as given by Ji et al. [5] for the normal case isonly an approximation. Our proof relies on an encoding that takes care of several con-ditions simultaneously (the hardness in the presence of non-disjunctive rules and theentailment relation under DL is polynomial). Then, we formulate a polynomial opera-tor to approximate the greatest unfounded set of disjunctive hybrid MKNF knowledgebases. Unlike the conventional definition of unfounded sets for disjunctive logic pro-gram [6], greatest unfounded sets under our definition exist unconditionally. We iden-tify the conditions under which our approximation becomes exact for normal as wellas for disjunctive hybrid MKNF knowledge bases. These conditions are also the onesunder which the coNP-hardness reduces to polynomial complexity for the normal anddisjunctive cases respectively, thus these results pinpoint the sources that contribute tothe hardness of computing greatest unfounded sets in general. Finally, based on theseresults, we formulate a DPLL-based solver, where the computation of unfounded setsbecomes a process of constraint propagation for search space pruning.The next section provides preliminaries. Section 3 gives the definition of unfoundedsets and studies its properties. Section 4 shows the main technical results concerningthe challenges of computing unfounded sets, which lead to a formulation of a DPLL-based solver in Section 5. Section 6 is about related work. The paper is closed byconcluding remarks in Section 7. Minimal knowledge and negation as failure (MKNF) extends first-order logic withtwo modal operators, K and not , for minimal knowledge and negation as failure re-spectively. MKNF formulas are constructed from first-order formulas using these twomodal operators for closed-world reasoning. Intuitively, K ψ asks whether ψ is knownw.r.t. a collection of “possible worlds” - the larger the set, the fewer facts are known -while not ψ checks whether ψ is not known, based on negation as failure. An MKNFstructure is a triple ( I , M , N ) where I is a first-order interpretation and M and N aresets of first-order interpretations. Operators shared with first-order logic are defined asusual. The satisfiability under an MKNF structure is defined as:• ( I , M , N ) | = A if A is true in I where A is a ground-atom• ( I , M , N ) | = ¬ F if ( I , M , N ) = F • ( I , M , N ) | = F ∧ G if ( I , M , N ) | = F and ( I , M , N ) | = G ( I , M , N ) | = ∃ x , F if ( I , M , N ) | = F [ α / x ] for some ground atom α (where F [ α / x ] is obtained by replacing every occurrence of the variable x with α )• ( I , M , N ) | = K F if ( J , M , N ) | = F for each J ∈ M • ( I , M , N ) | = not F if ( J , M , N ) = F for some J ∈ N Other symbols such as ∨ , ∀ , and ⊃ are interpreted in MKNF as they are in first-orderlogic. An MKNF interpretation M is a set of first-order interpretations; M satisfies aformula F , written M | = MKNF F , if ( I , M , M ) | = F for each I ∈ M . Definition 2.1. An MKNF model M of a formula F is an MKNF interpretation suchthat M | = MKNF F and there does not exist an MKNF interpretation M ′ ⊃ M such that ( I , M ′ , M ) | = F for each I ∈ M ′ .Following Motik and Rosati [8], a hybrid MKNF knowledge base K = ( O , P ) consists of a decidable description logic (DL) knowledge base O (typically called anontology) which is translatable to first-order logic and a set of MKNF rules P . Wedenote this translation as π ( O ) . Rules in P are of the form: K a , . . . , K a k ← K a k + , . . . , K a m , not a m + , . . . , not a n (1)In the above, a , . . . , a n are function-free first-order atoms of the form p ( t , . . . , t i ) where p is a predicate and t , . . . , t i are either constants or variables, with k ≥ m , n , i ≥ r in P is DL-safe if for every variable present in r , there is an occurence ofthat variable in the rule’s positive body inside a predicate that does not occur in K ’sdescription logic.A hybrid MKNF knowledge base K is DL-safe if every rule in P is DL-safe. Aknowledge base that is not DL-safe may not be decidable [8]. This constraint restrictsall variables in P to names explictly referenced in P . Throughout this work and with-out lose of generality we assume that P is ground, i.e. it does not contain variables.Let π ( P ) denote rule set P ’s corresponding MKNF formula: π ( P ) = ^ r ∈ P π ( r ) π ( r ) = ∀ ~ x k _ i = K a i ⊂ m ^ i = k + K a i ∧ n ^ i = m + not a i ! where ~ x is the vector of free variables found in r .The semantics of a hybrid MKNF knowledge base K is obtained by applyingboth transformations to O and P and wrapping O in a K operator, i.e. π ( K ) = π ( P ) ∧ K π ( O ) . We use P , O , and K in place of π ( P ) , π ( O ) , and π ( K ) re-spectively when it is clear from context that the translated variant is intended. Werefer to formulas of the form K a and not a , where a is a first-order atoms, as K -atomsand not -atoms respectively, and we refer to them collectively as modal-atoms. Hy-brid MKNF knowledge bases rely on the standard name assumption which requires3KNF interpretations to be Herbrand interpretations with a countably infinite num-ber of additional constants. In the rest of paper, we may refer to disjunctive hy-brid MKNF knowledge bases simply as knowledge bases for abbreviation, or normalknowledge bases if each rule in the knowledge base has exactly one atom in the head.We outline some definitions and conventions: For a hybrid MKNF knowledge base K = ( O , P ) , we denote the set of all K -atoms in P with KA ( K ) = KA ( P ) where KA ( P ) = { K a | either K a or not a occurs in the head or body of a rule in P } (2)We use K ( body − ( r )) to denote the set of K -atoms converted from not -atoms froman MKNF rule r ’s negative body, i.e. K ( body − ( r )) = { K a | not a ∈ body − ( r ) } anduse body + ( r ) to denote the K -atoms from the positive body of the rule r . The objec-tive knowledge of a hybrid MKNF knowledge base K w.r.t. to a set of K -atoms S ⊆ KA ( K ) , denoted as OB O , S , is the set of first-order formulas { π ( O ) } ∪ { a | K a ∈ S } .A (partial) partition of KA ( K ) is a nonoverlapping pair ( T , F ) , i.e., T ∩ F = /0,where T and F are subsets of KA ( K ) . A partition is total if T ∪ F = KA ( K ) . Adependable partition is a partial partition ( T , F ) with the additional restriction that OB O , T ∪ {¬ b } is consistent for each K b ∈ F or OB O , T is consistent if F is empty. Weadd this restriction for convenience and note that a partial partition that is not depend-able may not be extended to an MKNF model. In practice, a solver includes direct con-sequences of OB O , T in T and it only operates on dependable partitions. We denote thepartition induced by the body of a rule r with body ( r ) = ( body + ( r ) , K ( body − ( r ))) . Arule body is applicable w.r.t. a partition ( T , F ) if body ( r ) ⊑ ( T , F ) , i.e., if body + ( r ) ⊆ T and K ( body − ( r )) ⊆ F . We say that an MKNF interpretation M of K induces a parti-tion ( T , F ) if ^ K a ∈ T M | = MKNF K a ∧ ^ K a ∈ F M | = MKNF ¬ K a (3)If M is an MKNF model of K and M induces the partition ( T ∗ , F ∗ ) , then we saya partition ( T , F ) ⊑ ( T ∗ , F ∗ ) can be extended to an MKNF model. Every partitioninduced by a model is dependable. Note that for any dependable partition ( T ′ , F ′ ) ,every partial partition ( T , F ) ⊑ ( T ′ , F ′ ) is dependable. First defined for normal logic programs by van Gelder et al. [11], unfounded setsencapsulate atoms that must be false w.r.t. a partial interpretation. Critically, given apartition ( T , F ) of KA ( K ) that assigns atoms truth or falsity, an unfounded set of aknowledge base K w.r.t. ( T , F ) is a set of atoms that must be false if ( T , F ) can beextended to an MKNF model.A head-cut R ⊆ P × KA ( K ) is a set of rule atom pairs such that a rule r ∈ P occurs in at most one pair in R and for every pair ( r , h ) ∈ R we have h ∈ head ( r ) . Weuse head ( R ) to denote the set { h | ( r , h ) ∈ R } where R is a head-cut. Definition 3.1.
Let K = ( O , P ) be a disjunctive MKNF knowledge base and ( T , F ) apartial partition of KA ( K ) . A set X of K -atoms is an unfounded set of K w.r.t. ( T , F ) if for each K -atom K a ∈ X and each head-cut R such that:4. head ( R ) ∪ π ( O ) | = a (with O , R can derive K a ), and2. head ( R ) ∪ OB O , T ∪ {¬ b } is consistent for each K b ∈ F and head ( R ) ∪ OB O , T isconsistent if F is empty (the partition ( T ∪ head ( R ) , F ) is dependable),there is a pair ( r , h ) ∈ R such that at least one of the following conditions hold: i. body + ( r ) ∩ ( F ∪ X ) = /0 ( r positively depends on false or unfounded atoms), ii. K ( body − ( r )) ∩ T = /0 ( r negatively depends on true atoms), or iii. head ( r ) ∩ T = /0 (rule head is already satisfied)A K -atom in an unfounded set is called an unfounded atom .We illustrate some general characteristics of this definition with the following ex-ample. Example 1.
Let K = ( O , P ) where O = { ( a ⊃ a ′ ) ∧ ( b ⊃ b ′ ) ∧ ¬ f } P = { K f ← b ; K a ← not b ; K a , K b , K c ← ; K a ′ ← K a ′ ; K b ′ ← K b ′ } Let ( T , F ) be the dependable partition ( { K b } , /0 ) . The K -atom K f is an unfoundedatom w.r.t. ( T , F ) because K f creates an inconsistency in O . K a is an unfoundedatom because the only way of deriving K a relies on ¬ K b which contradicts T . The K -atom K a ′ is unfounded because K a is unfounded and K b ′ is not unfounded because K b is in T . Lastly, K c is an unfounded atom because the only rule that can derive K c has another head-atom ( K b ) in T .An unfounded set X w.r.t. a dependable partition ( T , F ) is a set of K -atoms thatmust be false should ( T , F ) be extended to an MKNF model. A head-cut R is a set ofrules that may be used in conjunction with ( T , F ) to derive a K -atom. A K -atom isunfounded only if every head-cut that can derive it has a pair in it that meets one of theconditions i through iii . Note that if ( T , F ) is dependable, then it is impossible to derivea K -atom in F without violating condition 2 because an empty head-cut can be used toderive any K -atom in T . We demonstrate this property in the following example. Example 2.
Let K = ( O , P ) where O = { a ⊃ b } , and P = { K a ← not b ; K b ← not a } The dependable partition ( { K b } , { K a } ) is the only total dependable partition inducedby an MKNF model of K . Suppose we have the dependable partition ( T , F ) =( { K a } , /0 ) . Note that ( T , F ) cannot be extended to an MKNF model. Neither K a nor5 b is an unfounded atoms w.r.t. ( T , F ) : when R = /0 we have head ( R ) ∪ OB O , T | = a and head ( R ) ∪ OB O , T | = b . Let ( T , F ) = ( /0 , { K a } ) be a different dependable partition.The K -atom K a is an unfounded atom w.r.t. ( T , F ) . The only head-cut that can derive K a is the set R = { ( K a ← not b ) } , however, head ( R ) ∪ OB O , T ∪ {¬ a } can be rewrittenas { a } ∪ OB O , T ∪ {¬ a } which is inconsistent.Under Definition 3.1, atoms in T cannot be unfounded. We formally establish thatno K -atom in T can be an element in an unfounded set in the following lemma. Lemma 3.1 ( T is disjoint from any unfounded set) . Let U be an unfounded set of adisjunctive knowledge base K w.r.t. a dependable partition ( T , F ) of KA ( K ) . Wehave T ∩ U = /0. Proof.
Assume for the sake of contradiction that U ∩ T = /0, and let K a ∈ U ∩ T .Because U is an unfounded set w.r.t. ( T , F ) we have for every head-cut R such that head ( R ) ∪ OB O , T | = a , OB O , T ∪ {¬ b } is consistent for each K -atom K b ∈ F , and OB O , T is consistent, that there is a pair ( r , h ) ∈ R such one of the conditions i , ii ,or iii is satisfied. Let R = /0. We have head ( R ) ∪ OB O , T | = a because K a ∈ T . Be-cause ( T , F ) is dependable, OB O , T ∪ {¬ b } is consistent for each K -atom K b ∈ F , and head ( R ) ∪ OB O , T is consistent. However, there does not exist a pair ( r , h ) ∈ R because R is empty, a contradiction.The property demonstrated in Lemma 3.1 is inherited from the definition of un-founded sets for normal hybrid MKNF knowledge bases [5]. This is quite differentfrom the definition of unfounded sets for disjunctive logic programs: Leone et al. [6]refer to (partial) partitions (called interpretations in their context) where no atom in T is unfounded (under their own definition of unfounded sets) as unfounded-free . Insome respects, unfounded sets under Leone et al. [6] can doubt the truth of K -atoms in T . Since unfounded atoms are assumed to be false, an unfounded set w.r.t. ( T , F ) thatshares K -atoms with T is proof that ( T , F ) cannot be extended to a model. As shownin Lemma 3.1, Definition 3.1 lacks this property. We illustrate this difference in thefollowing example. Example 3.
Let K = ( /0 , P ) where P = { K a , K b ←} and construct the dependablepartition ( T , F ) = ( { K a , K b } , /0 ) . Under Leone et al.’s definition, both { K a } and { K b } are unfounded sets w.r.t. ( T , F ) , however, the set { K a , K b } is not an unfounded setw.r.t. ( T , F ) . Under Definition 3.1, none of the three aforementioned sets are unfoundedsets w.r.t. ( T , F ) due to Lemma 3.1.Leone et al. show that partial partitions that have the unfounded-free property andsatisfy every rule in P are precisely the partial partitions that can be extended to stablemodels [6]. In the example above, the dependable partition ( T , F ) = ( { K a , K b } , /0 ) cannot be extended to an MKNF model and neither K a nor K b is an unfounded atomw.r.t. ( T , F ) . This indicates that unfounded sets under Definition 3.1 cannot be usedto determine whether a partition can be extended to an MKNF model in the same wayas Leone et al. We demonstrate that this is the case even for a normal knowledge basewith an empty ontology. 6 xample 4. Let K = ( /0 , P ) where P = { K a ← not a } . Note that K does nothave an MKNF model. The two possible total partitions are ( T , F ) = ( /0 , { K a } ) and ( T , F ) = ( { K a } , /0 ) . Under both Definition 3.1 and Leone et al.’s definition of un-founded sets, the only unfounded set w.r.t. ( T , F ) is /0. Like Leone et al., we candetermine that ( T , F ) is not an MKNF model of K because there is a rule r ∈ P such that body ( r ) ⊑ ( T , F ) and head ( r ) ∩ T = /0. Under Leone et al.’s definition, theset { K a } is an unfounded set of P w.r.t. ( T , F ) , however, { K a } is not an unfoundedset of K w.r.t. ( T , F ) under Definition 3.1. We cannot use Definition 3.1 to concludethat there is not MKNF model that induces ( T , F ) .The above example demonstrates a limitation that prevents unfounded sets frombeing used as a mechanism for MKNF model checking. This limitation is also presentin the unfounded sets defined by Ji et al. [5], however, it does not inhibit unfoundedsets from being useful in a solver. Following Ji et al [5] and Leone et al. [6], weshow that unfounded sets in Definition 3.1 are closed under union. The property thatall dependable partitions are unfounded-free (Lemma 3.1) removes the need for anadditional restriction on partitions as is needed for disjunctive logic programs [6].The unfoundedness of some K -atoms is dependant on the unfoundedness of other K -atoms (condition i of Definition 3.1), thus new unfounded sets can be constructedby adding certain K -atoms to smaller unfounded sets. Condition iii of Definition 3.1( head ( r ) ∩ T = /0) does not depend on the unfounded set X like it does in Leone et al.’sdefinition (in this context, ( head ( r ) \ X ) ∩ T = /0). Applying Lemma 3.1, ( head ( r ) \ X ) ∩ T = /0 can be rewritten as head ( r ) ∩ T = /0. This results in unfounded sets beingclosed under union in general. We demonstrate this property formally in the followingproposition. Proposition 3.1 (Existence of a greatest unfounded set) . For a disjunctive hybridMKNF knowledge base K = ( O , P ) and a partial partition ( T , F ) of KA ( K ) , thereexists a greatest unfounded set U K ( T , F ) such that X ⊆ U K ( T , F ) for every unfoundedset X of K w.r.t. ( T , F ) . Proof.
We show that unfounded sets are closed under union and the existence of agreatest unfounded set directly follows. Let X a and X b be unfounded sets of K w.r.t. apartial partition ( T , F ) of KA ( K ) . We show that the set X c = X a ∪ X b is an unfoundedset of K w.r.t. ( T , F ) . If ( T , F ) is not dependable, then every set X ⊆ KA ( K ) isan unfounded set of K w.r.t. ( T , F ) including X c . Assume that ( T , F ) is depend-able and for the sake of contradiction, assume X c is not an unfounded set. For some K -atom K a ∈ X c we have a head-cut R s.t. conditions 1 ( head ( R ) ∪ OB O , T | = a )and 2 ( head ( R ) ∪ OB O , T ∪ {¬ b } is consistent for each K b ∈ F or head ( R ) ∪ OB O , T is consistent if F is empty) hold. In this head-cut, there is a pair ( r , a ) such thatnone of the conditions i ( body + ( r ) ∩ ( X c ∪ F ) = /0), ii ( body − ( r ) ∩ T = /0), or iii ( head ( r ) ∩ T = /0) hold. For simplicity, assume K a ∈ X a (proof is identical if K a ∈ X b ).If body + ( r ) ∩ ( X a ∪ F ) = /0 then we have body + ( r ) ∩ ( X c ∪ F ) = /0 and it follows that X c is an unfounded set.This property is a natural result of Lemma 3.1 and differs from Leone et al.’s un-founded sets are closed under union only if ( T , F ) is unfounded-free.7 solver can use any unfounded set to extend a dependable partition’s false atomswithout affecting the models it finds. We now relate unfounded sets to MKNF models. Proposition 3.2.
Let ( T ∗ , F ∗ ) be the partition induced by an MKNF model of a dis-junctive hybrid MKNF knowledge base K . For any dependable partition ( T , F ) ⊑ ( T ∗ , F ∗ ) , U K ( T , F ) ∩ T ∗ = /0. Proof.
Note that ( T ∗ , F ∗ ) is total and dependable. Let ( T , F ) ⊑ ( T ∗ , F ∗ ) and U be anunfounded set of K w.r.t. ( T , F ) . Let U be an unfounded set of K w.r.t. ( T , F ) . Weshow that U ∩ T ∗ = /0 and it follows that U K ( T , F ) ∩ T ∗ = /0. Assume for the sake ofcontradiction that U ∩ T ∗ = /0; have B = U ∩ T ∗ , then construct an MKNF interpretation M ′ such that M ′ = { I | I | = OB O , T and I | = t for each K t ∈ T ∗ \ B }} (4)The dependable partition induced by M ′ is ( T ∗ \ B , F ∗ ∪ B ) . For each b ∈ B , OB O , T = b , thus M ′ ⊃ M . We derive a contradiction by showing U is not an unfounded setof K w.r.t. ( T , F ) . By construction, ( I , M ′ , M ) | = MKNF π ( O ) for each I ∈ M ′ . If ∀ I ∈ M ′ , ( I , M ′ , M ) | = MKNF π ( P ) , then M is not a model, a contradiction. Using ( T ∗ \ B , F ) to denote the partition used to test each rule r ∈ P , observe that if r is notsatisfied w.r.t. ( T ∗ \ B , F ) it of the form body ( r ) ⊑ ( T ∗ \ B , F ) , head ( r ) ∩ T ∗ = /0,and head ( r ) ∩ ( T ∗ \ B ) = /0. r is a rule whose body is satisfied by ( T ∗ \ B , F ) but alltrue atoms in its head come from B . Let R = { ( r , h ) } where b is some atom from head ( r ) ∩ B . Conditions 1 and 2 of Definition 3.1 are met for R to test if U is anunfounded set of K w.r.t. ( T , F ) . We show that none of the conditions i through iii aremet by R showing that U is not an unfounded set w.r.t. ( T , F ) , a contradiction. First, body + ( r ) ⊆ T ∗ \ B gives us body + ( r ) ∩ ( F ∪ U ) = /0. From K ( body − ( r )) ⊆ F , wederive K ( body − ( r )) ∩ T = /0. Finally, using head ( r ) ∩ T ∗ ⊆ B and B ∩ T = /0 (Lemma3.1), we conclude head ( r ) ∩ T = /0. We have shown U ∩ T ∗ = /0, as desired.We’ve shown that if a dependable partition ( T , F ) can be extended to anMKNF model, no unfounded set of K w.r.t. ( T , F ) may overlap with the true atomsin the model. It follows directly from Proposition 3.2 that the following analogousproperty holds for atoms in F . Corollary 3.1.
Let ( T ∗ , F ∗ ) be the partition induced by an MKNF model M of adisjunctive hybrid MKNF knowledge base K . Then, for any dependable partition ( T , F ) ⊑ ( T ∗ , F ∗ ) , M | = MKNF ¬ K u for all u ∈ U K ( T , F ) .With these properties, we’ve shown that unfounded sets can be used to extend apartition without missing any models, i.e., if ( T , F ) can be extended to an MKNF model M then ( T , F ∪ U ) can be extended to the same model M for any unfounded set U w.r.t. ( T , F ) . Due to the inconsistencies that can arise in connection with O , computing the greatestunfounded set w.r.t. a partial partition is intractable in general.8 xample 5. Let K = ( O , P ) where O = ¬ ( a ∧ b ) and P = { K a ← not b ; K b ← not a ; K c ← K c } Under Definition 3.1, K c is an unfounded atom w.r.t. ( /0 , /0 ) , however, with the V ( /0 , /0 ) K operator defined by Ji et al. [5] we have lfp ( V ( /0 , /0 ) K ) = KA ( K ) which misses K c asan unfounded atom. It’s clear that a similar operator for disjunctive knowledge baseswould have the same limitation.In the following, we first give a formal proof of intractability and then we constructan operator for hybrid MKNF knowledge bases with disjunctive rules that adopts thesame approximation technique used by Ji et al. in their V ( T , F ) K operator [5] for hybridMKNF knowledge bases with normal rules.We now show that deciding whether an atom of a normal hybrid MKNF knowledgebase is unfounded iscoNP-hard by comparing the head-cuts that need to be considered to determineunfoundedness with the SAT assignments that need to be considered to determine thesatisfiability of a 3SAT problem. Proposition 4.1.
Let K = ( O , P ) be a normal hybrid MKNF knowledge base suchthat the entailment relation OB O , S | = a can be checked in polynomial time for anyset S ⊆ KA ( K ) and for any K -atom K a ∈ KA ( K ) . Determining whether a K -atom K a ∈ KA ( K ) is an unfounded atom of K w.r.t. a dependable partition ( T , F ) of KA ( K ) is coNP-hard. Proof.
We show that the described problem is coNP-hard. The 3SAT problem is wellknown to be NP-complete [10]. Let
SAT be an instance of 3SAT in conjunctive normalform such that
CLAUSE = { c , c , . . . , c n } is the set of clauses in SAT and
VAR = { v , v , . . . , v n } is the set of variables in SAT . Determining whether
SAT is unsatisfiableis coNP-hard. We construct a normal hybrid MKNF knowledge base K = ( O , P ) s.t. O = { v ui ⊕ v fi ⊕ v ti | for each v i ∈ VAR where ⊕ is exclusive-or }∪{ ( ^ v i ∈ VAR ¬ v ui ⇐⇒ total ) , total ⊃ sat }∪{ clause i ∨ ¬ total | for each clause c i ∈ CLAUSE where clause i is a formulaobtained by replacing all occurences of v i and ¬ v i in c i with v ti and v fi respectively } (5)and P = { K sat ← K sat } ∪ [ {{ ( K v ti ← not v fi ) , ( K v fi ← not v ti ) } | v i ∈ VAR } (6) This is because in the least fixed point computations of the V ( T , F ) K operator, a default negation not q istrue if K q is not known to be true, and as such, both K a and K b are derived in the first iteration which leadsto inconsistency with O . K sat ← K sat is only required to ensure that K sat is in KA ( K ) .The time to construct the above knowledge base is linear in the number of clauses andvariables in SAT . The first set of formulas in O require exactly one of v ui , v fi , or v ti to be true. This constraint is analogous to a three-valued assignment for SAT where avariable v i ∈ VAR is unassigned if v ui is true, assigned false if v fi is true, and assignedtrue if v ti is true. The second set in O ensures that the atom total is true if and onlyif no variable is unassigned. Finally, the third set of formulas ensure that π ( O ) isinconsistent if the assignment is total and a clause in SAT is not satisfied. We showthat (1) For any K -atom K a and set of K -atoms S , the entailment relation OB O , S | = a is computable in polynomial time and (2) that K sat is an unfounded atom of K w.r.t. ( /0 , /0 ) if and only if SAT is unsatisfiable.(1) We call a set of K -atoms S total if it contains either K v ti or K v fi for each variable v i ∈ VAR . Note that for a variable v i ∈ VAR , the set KA ( K ) only contains K v ti and K v fi ; It does not contain K v ui . Let S ⊆ KA ( K ) . We show that we can, in polynomialtime, determine whether S ∪ π ( O ) is consistent. We split cases where S is total whereit is not. First, assume S is not total: For some variable v i ∈ VAR , neither K v ti nor K v fi is in S . By fixing v ui to be true in a consistent first-order interpretation I of S ∪ π ( O ) ,we ensure the atom total is false. If the atom total is false, we can determine whether OB O , S is consistent in polynomial time because we only need to consider the first twosets of formulas in O . If S is total, we can, in polynomial time, verify that S ∪ π ( O ) is consistent by checking that only one of v ti or v fi is present in S and that every clause clause i is satisfied. After determining whether S ∪ π ( O ) is consistent, we can quicklycheck the relations OB O , S | = v ti and OB O , S | = v fi for any variable v i ∈ VAR : Assuming S ∪ π ( O ) is consistent, the entailment relation OB O , S | = v ti (resp. OB O , S | = v fi ) holdsif and only if K v ti ∈ S (resp. K v fi ∈ S ). When S ∪ π ( O ) is consistent, the entailmentrelation OB O , S | = total holds if and only if S is total. Finally, we have OB O , S | = sat if and only if K sat ∈ S or OB O , S | = total . If S ∪ π ( O ) is inconsistent, the entailmentrelation OB O , S | = K a holds vacuously where K a ∈ KA ( K ) .(2) When determining whether the K -atom K sat is unfounded w.r.t. ( /0 , /0 ) , wemust consider each way to select a head-cut R . We show that there is a correspondencebetween the head-cuts that can disprove the unfoundedness of K sat w.r.t. ( /0 , /0 ) andtotal sat assignments for SAT . Let X = { K sat } be a set that is possibly unfoundedw.r.t. ( /0 , /0 ) . Observe that a larger unfounded set X ′ ⊃ X w.r.t. ( /0 , /0 ) cannot existunless X is an unfounded set w.r.t. ( /0 , /0 ) . A head-cut R cannot be used to disprove theunfoundedness of K sat if either condition 1 or 2 of Definition 3.1 do not hold. Beforecreating a mapping between head-cuts and sat assignments for SAT , we exclude head-cuts that cannot be used to disprove the unfoundedness of K sat , i.e., conditions 1 and 2of Definition 3.1 are met and i , ii , and iii do not hold. Firstly, we exclude head-cuts thatcontain the pair ( r , sat ) because body + ( r ) ∩ X = /0. We further exclude any head-cut R containing a pair of pairs ( r , v ti ) and ( r , v fi ) because head ( R ) ∪ OB O , /0 is inconsistent.Thirdly, we exclude any head-cuts that do not contain either ( r , v ti ) or ( r , v fi ) foreach variable v i ∈ VAR noting that if such a head-cut R also meets the previous two Due to the uniqueness of the second component in such a pair, there should be no confusion about whichrule the first component refers to head ( R ) ∪ OB O , /0 = sat (See (1) for details). The remaining head-cuts have a one to one correspondence with total assignments for SAT : if a head-cutcontains a pair with v ti (resp. v fi ) the corresponding assignment for SAT assigns v i to betrue (resp. false). We have for every such head-cut R that head ( R ) ∪ OB O , /0 | = sat andthat for every pair in ( r , h ) ∈ R we have head ( r ) ∩ T = /0, body + ( r ) ∩ ( F ∪ X ) = /0, and body − ( r ) ∩ T = /0. If head ( R ) ∪ OB O , /0 is consistent, then every clause is satisfied by thecorresponding sat assignment, otherwise, the inconsistency is caused by an unsatisfiedclause ¬ clause i , thus the assignment does not satisfy SAT . If no such head-cut R exists such that head ( R ) ∪ OB O , /0 is consistent, then K sat is unfounded w.r.t. ( /0 , /0 ) and SAT is unsatisfiable. Conversely, if
SAT is unsatisfiable, a head-cut R such that head ( R ) ∪ OB O , /0 is consistent and head ( R ) ∪ OB O , /0 | = sat does not exist, thus { K sat } is an unfounded set w.r.t. ( /0 , /0 ) . We’ve shown that deciding whether an K -atom isunfounded is coNP-hard.It follows that computing the greatest unfounded set of a disjunctive hybridMKNF knowledge base is coNP-hard. Since we are unlikely to find a way to compute U K ( T , F ) in polynomial time, we are motivated to construct a polynomial operatorthat computes an approximation (a subset) of the greatest unfounded set. We definea family of operators Z ( T , F ) K where each operator induced by a dependable partition ( T , F ) computes an approximation of the greatest unfounded set of K w.r.t. ( T , F ) Z ( T , F ) K : 2 KA ( K ) → KA ( K ) X T ∪ { K a | OB O , X | = a for each K a ∈ KA ( K ) } ∪{ K a | ∃ r ∈ P with K a ∈ head ( r ) s.t. body + ( r ) ⊆ X ∧ body + ( r ) ∩ F = /0 ∧ K ( body − ( r )) ∩ T = /0 ∧ head ( r ) ∩ T = /0 ∧{ a , ¬ b } ∪ OB O , T is consistent for each K b ∈ F } (7)This operator is the direct result of combining the V ( T , F ) K operator for normal hybridMKNF [5] with the Φ operator for disjunctive logic programs [6]. It is easy to seethat the Z ( T , F ) K operator is monotonic, and let us use Atmost K ( T , F ) to denote itsleast fixed point. This operator computes a subset of KA ( K ) \ U K ( T , F ) . Firstly,if Atmost K ( T , F ) ∪ π ( O ) is inconsistent, we have KA ( K ) \ Atmost K ( T , F ) = /0; Acompromise to keep the operator computable in polynomial time.To determine whether an atom is unfounded when there are disjunctive rules, wemust consider an exponential number of head-cuts. The Z ( T , F ) K operator instead con-siders the heads of rules all at once and this can result in Atmost K ( T , F ) missing someunfounded atoms even if Atmost K ( T , F ) ∪ π ( O ) is consistent. Example 6.
Let K = ( O , P ) be a disjunctive hybrid MKNF knowledge base where P = { K a , K b ← ; K c ← K c } and O = ( a ∧ b ) ⊃ c . We have that { c } is an unfounded11et of K w.r.t. ( /0 , /0 ) . However, Atmost K ( T , F ) is { a , b , c } and KA ( K ) \ { a , b , c } 6 = U K ( /0 , /0 ) .We intend to identify the class of knowledge bases for which the Z ( T , F ) K operatordoes not miss unfounded atoms as a result of disjunctive heads. First we define a weakhead-cut to be a set of rule atom pairs R w such that R w ⊆ P × KA ( K ) and h ∈ head ( r ) for each pair ( r , h ) ∈ R . Note that this definition is identical to the definition of head-cutswithout the constraint that a rule can appear in at most one pair in R w ; within a weakhead-cut, there may be two pairs ( r , h ) and ( r , h ) such that h = h . In the following,we define a property that captures a subset of knowledge bases where Atmost K ( T , F ) computes U K ( T , F ) if Atmost K ( T , F ) ∪ π ( O ) is consistent. Definition 4.1.
A hybrid MKNF knowledge base K = ( O , P ) is head-independent w.r.t. a dependable partition ( T , F ) if for every K -atom K a ∈ KA ( K ) and every weakhead-cut R w such that head ( R ) ∪ OB O , T | = a , there exists a head-cut R such that R ⊆ R w and head ( R ) ∪ OB O , T | = a .Head-independence means that we cannot derive atoms that we would not be able toderive using only a single atom from each rule head by using multiple atoms in the headof a rule in conjunction with the ontology. The head-independence property is violatedby the knowledge base in Example 6 and it ensures that Atmost K ( T , F ) = U K ( T , F ) .Were we to alter the knowledge base in Example 6 such that the rule K a , K b ← werechanged to the pair of rules K a ← not b and K b ← not a then K would have head-independence. We show formally that for a head-independent knowledge base K , the Z ( T , F ) K operator computes the greatest unfounded set w.r.t. ( T , F ) if Atmost K ( T , F ) ∪ π ( O ) is consistent. Proposition 4.2. If K is head-independent w.r.t. a dependable partition ( T , F ) and Atmost K ( T , F ) ∪ π ( O ) is consistent, then U K ( T , F ) = KA ( K ) \ Atmost K ( T , F ) . Proof.
First we show (1) that no K -atom computed by Atmost K ( T , F ) is unfoundedw.r.t. ( T , F ) and then we show (2) that every atom that is not unfounded w.r.t. ( T , F ) iscomputed by Atmost K ( T , F ) .(1) We first show no K -atom in Z ( T , F ) K ( /0 ) is unfounded. Let K a ∈ Z ( T , F ) K ( /0 ) . Con-struct a weak head-cut R w that contains a pair ( r , h ) for each head K -atom K h ∈ head ( r ) and rule r ∈ P where body + ( r ) ⊆ /0, K ( body − ( r )) ∩ T = /0, and head ( r ) ∩ T = /0.The weak head-cut R w contains every rule that was applied in the computation of Z ( T , F ) K ( /0 ) . We have head ( R w ) ∪ OB O , T | = a . Applying the head-independence con-dition, we obtain a head-cut R such that R ⊆ R w and head ( R ) ∪ OB O , T | = a . For everypair ( r , h ) ∈ R , body + ( r ) ⊆ T , K ( body − ( r )) ∩ T = /0, and head ( r ) ∩ T . The head-cut R shows that K a is not an unfounded atom w.r.t. ( T , F ) , thus it is not a memberof any unfounded set. We show that no atom computed by a successive applicationof Z ( T , F ) K , e.g. Z ( T , F ) K ( Z ( T , F ) K ( /0 )) , is unfounded w.r.t. ( T , F ) . Let Z i be result of ap-plying the Z ( T , F ) K operator i times where Z = /0. We assume that no atom in Z i isunfounded w.r.t. ( T , F ) and show the same for Z i + . Construct a weak head-cut R w that contains a pair ( r , h ) for each head a K -atom K h ∈ head ( r ) and rule r ∈ P where12 ody + ( r ) ⊆ Z i , body − ( r ) ∩ T = /0, and head ( r ) ∩ T = /0. Let K a ∈ Z ( T , F ) K ( Z i ) . Wehave head ( R w ) ∪ OB O , T | = a . Applying the head-independence condition, we obtain ahead-cut R such that R ⊆ R w and head ( R ) ∪ OB O , T | = a . Now we have for each pair ( r , h ) ∈ R , body + ( r ) ⊆ Z i . Knowing that no K -atom in Z i is a member of an unfoundedset, we conclude that a is not an unfounded atom w.r.t. ( T , F ) .(2) We show that if a K -atom K a is not computed by Atmost K ( T , F ) and it is notan unfounded atom w.r.t. ( T , F ) we can derive a contradiction. Let U = KA ( K ) \ Atmost K ( T , F ) . Let K a ∈ U be an K -atom such that there exists a head-cut R where head ( R ) ∪ OB O , T | = a , head ( R ) ∪ OB O , T is consistent and head ( R ) ∪ OB O , T ∪ {¬ b } isconsistent for each K b ∈ F and for each pair ( r , h ) ∈ R , head ( r ) ∩ T = /0 and body − ( r ) ∩ T = /0. If for each pair ( r , h ) ∈ R we have body + ( r ) Atmost K ( T , F ) then K a ∈ Atmost K ( T , F ) , otherwise U is an unfounded set w.r.t. ( T , F ) . Both cases contradictthe initial assumptions.For normal knowledge bases, i.e., where each rule contains only a single head-atom, the head-independence condition is satisfied automatically. If a knowl-edge base K is not head-independent, the Z ( T , F ) K operator computes a subset of U K ( T , F ) . Therefore, for a normal knowledge base and dependable partition ( T , F ) s.t. Atmost K ( T , F ) ∪ π ( O ) is consistent, we have U K ( T , F ) = KA ( K ) \ Atmost K ( T , F ) .The following corollary follows directly from Proposition 4.2. Corollary 4.1.
If a knowledge base K is head-independent w.r.t. a dependable par-tition ( T , F ) and Atmost K ( T , F ) ∪ π ( O ) is consistent, then the greatest unfounded setof K w.r.t. ( T , F ) is computable in polynomial time.We have shown that computing the greatest unfounded set of a normal knowledgebase is coNP-hard (Proposition 4.1). Because Atmost K ( T , F ) can be computed inpolynomial time, we conclude that the greatest unfounded set of a normal knowledgebase K can be computed in polynomial time if Atmost K ( T , F ) ∪ π ( O ) is consistentand the greatest unfounded set of a disjunctive knowledge base K can be computed inpolynomial time if Atmost K ( T , F ) ∪ π ( O ) is consistent and K is head-independent.Observe that for the knowledge base constructed in our proof of Proposition 4.1, Atmost K ( T , F ) ∪ π ( O ) is inconsistent. We formally demonstrate the intractability ofcomputing U K ( T , F ) for a disjunctive knowledge base when Atmost K ( T , F ) ∪ π ( O ) is consistent but the head-independence condition is not met. Proposition 4.3.
Let K = ( O , P ) be a disjunctive hybrid MKNF knowledge basesuch that the entailment relation OB O , S | = a can be checked in polynomial time for anyset S ⊆ KA ( K ) and for any K -atom K a ∈ KA ( K ) . Let ( T , F ) be a dependable parti-tion of KA ( K ) such that Atmost K ( T , F ) ∪ π ( O ) is consistent. Determining whether a K -atom K a ∈ KA ( K ) is an unfounded atom of K w.r.t. ( T , F ) is coNP-hard. Proof.
Let
SAT be an instance of 3SAT in conjunctive normal form such that
CLAUSE = { c , c , ..., c n } is the set of clauses in SAT , and
VAR = { v , v , ..., v n } isthe set of variables in SAT . We construct a disjunctive hybrid MKNF knowledge base13 = ( O , P ) s.t. O = { ( v fi ∨ v ti ) ⊕ v ui | for each v i ∈ VAR where ⊕ is exclusive-or }∪{ ( v fi ∧ v ti ) ⊃ sat | for each v i ∈ VAR }∪{ ( ^ c i ∈ CLAUSE clause i ) ∧ ( ^ v i ∈ VAR ¬ v ui ) ! ⊃ sat | where clause i is a formulaobtained by replacing all occurrences of v i and ¬ v i in c i with v ti and v fi respectively } (8)and P = { K sat ← K sat } ∪ [ {{ ( K v ti , K v fi ← ) } | v i ∈ VAR } (9)Let ( T , F ) = ( /0 , /0 ) and observe that Atmost K ( T , F ) ∪ π ( O ) is consistent( Atmost K ( T , F ) = KA ( K ) \ { K sat } ). We show that (1) For any K -atom K a and setof K -atoms S , the entailment relation OB O , S | = a is computable in polynomial time and(2) that K sat is an unfounded atom of K w.r.t. ( /0 , /0 ) if and only if SAT is unsatisfiable.(1) Observe that KA ( K ) ∪ π ( O ) is consistent, therefore, S ∪ π ( O ) is consistent forany set of K -atoms S ⊆ KA ( K ) . The entailment relation OB O , S | = v ti (resp. OB O , S | = v fi ) holds if and only if v ti ∈ S (resp. v fi ∈ S ). What remains to show is that OB O , S | = sat can be checked in polynomial time when K sat S . We call a set of K -atoms S consistent if it does not contain both K v ti and K v fi for every variable v i ∈ VAR . If S isnot consistent, then we have OB O , S | = sat due to the second set of formulas in O . Weassume that S is consistent. We call a set of K -atoms S total if it contains either K v ti or K v fi for each variable v i ∈ VAR . We consider the cases where S is total and where S is not total. If S is not total, we can construct a consistent first-order interpretation of S ∪ π ( O ) such that v ui is true for some v i ∈ VAR , thus OB O , S = sat if S is consistent andnot total. Now we assume that S is total and it follows that V v i ∈ VAR ¬ v ui is satisfied in thethird set of formulas in O . We refer to a model M of S ∪ π ( O ) as a proper model if forevery v i ∈ VAR we have v fi (resp. v ti ) to be false in M if v fi S (resp. v fi S ). Observethat for all models of S ∪ π ( O ) modulo proper models, sat is true because of the secondset of formulas in O (recall that S is total and consistent). Note that for each propermodel M we have M | = v fi ⊕ v ti (where ⊕ is exclusive-or) because S is consistent. Theonly case where OB O , S = sat is if We have OB O , S | = sat if and only if S satisfies everyformula clause i . This can easily be checked in polynomial time.(2) When determining whether the K -atom K sat is unfounded w.r.t. ( /0 , /0 ) , wemust consider each way to select a head-cut R . This part of the proof carries out almostidentically to part 2 of our proof of Proposition 4.1. We only outline the key differences:Rather than relying on head ( R ) ∪ π ( O ) to be inconsistent if R does not correspond toa satisfying assignment of SAT like in our proof of Proposition 4.1, we rely on therebeing a single model of head ( R ) ∪ π ( O ) where sat is false (See (1) for details on propermodels). This is enough to show that head ( R ) ∪ OB O , /0 = sat . When only consideringproper models of head ( R ) ∪ π ( O ) , we can ignore the second set of formulas in O R containing both ( r , v fi ) and ( r , v ti ) is not a valid head-cut. In order to determine whether a K -atom K a is unfounded w.r.t. ( /0 , /0 ) , we mustexhaustively check head ( R ) ∪ π ( O ) for every head-cut R and can conclude that SAT isunsatisfiable. If we know that
SAT is unsatisfiable, there cannot exist a head-cut R thatproves that K a is not an unfounded atom.Intuitively, head-independence means that using multiple atoms from the head ofa rule in conjunction with O cannot derive atoms that cannot be derived using only asingle atom from the head of each rule. The head-independence property is violated bythe knowledge base in Example 6 and it ensures that Atmost K ( T , F ) = U K ( T , F ) . Ifwe were to alter the knowledge base such that the rule K a , K b ← were changed to thepair of rules K a ← not b and K b ← not a then K has head-independence. We showformally that for a head-independent knowledge base, the Z ( T , F ) K operator computes thegreatest unfounded set w.r.t. ( T , F ) . In this section we formulate a DPLL-based solver. First, we construct a well-foundedoperator W ( T , F ) K using the greatest unfounded set approximator from the previous sec-tion: T ( T , F ) K ( X , Y ) = { K a | where OB O , T ∪ X | = a for some K a ∈ KA ( K ) }∪{ K a | where head ( r ) \ F = { K a } and body ( r ) ⊑ ( T ∪ X , F ∪ Y ) for some r ∈ P } ) W ( T , F ) K ( X , Y ) = ( T ( T , F ) K ( X , Y ) ∪ T , ( KA ( K ) \ Z ( T , F ) K ( X , Y )) ∪ F ) We show that this operator maintains the property shown in Proposition 3.2.
Proposition 5.1.
If a dependable partition ( T , F ) can be extended to an MKNF model M , then the dependable partition lfp W ( T , F ) K ( X , Y ) can also be extended to M . Proof.
It follows from Corollary 3.1 that if ( T , F ) can be extended an MKNF model M ,then ( T , F ∪ Z ( T , F ) K ( T , F )) can be extended to M . What’s left to show is that if ( T , F ) canbe extended to an MKNF model M , then ( T ∪ T ( T , F ) K ( T , F )) , F ) can be extended to M .Suppose that there is some K -atom K a in T ∩ T ( T , F ) K ( T , F )) such that M = MKNF K a .Then we either have that OB O , T | = a , and thus M = MKNF π ( O ) or that M = MKNF K h for each K h ∈ head ( r ) and thus M = MKNF π ( P ) . Either case contradicts theassumption that M is an MKNF model of K .Following Ji et al. [5], we construct an abstract solver in Algorithm 1 that prunes thesearch space for solving by using the W ( T , F ) K operator. The CHECK-MODEL procedurechecks whether the MKNF interpretation { I | where I | = π ( O ) , I | = t for each K t ∈ T , and I | = f for each K f ∈ T } is an MKNF model of K whenever the solver reaches a total dependable partition.This procedure is analogous to the NP-oracle required to check a model of a disjunctive15ogic program [1]. Further developments are required for a more precise definition ofthis procedure. Algorithm 1: solver ( K , ( T , F )) ( T , F ) ← W K ( T , F ) ⊔ ( T , F ) ; if T ∩ F = /0 then return false; else if T ∪ F = KA ( K ) then if CHECK-MODEL(T, F) then return true; else return false; else choose a K -atom K a from KA ( K ) \ ( T ∪ F ) ; if solver ( K , ( T ∪ { K a } , F )) then return true; else return solver ( K , ( T , F ∪ { K a } )) ; Proposition 5.2.
Given a partial partition ( T , F ) of KA ( K ) , the invocation of Algo-rithm 1 solver ( K , ( /0 , /0 )) will return true if ( T , F ) can be extended to an MKNF modelof K . Proof.
It follows from Proposition 5.1 that the extension of ( T , F ) on the first line of thealgorithm, ( T , F ) ← W K ( T , F ) , does not miss any models. No models exist that inducea partition ( T , F ) s.t. T ∩ F = /0. Without the use of the W K ( T , F ) operator, the solveralgorithm will explore every partition ( T , F ) ⊆ KA ( K ) × KA ( K ) where T ∩ F = /0.Thus, the usage of the W K ( T , F ) operator simply prunes the search space.Given Proposition 5.2, it is easy to modify Algorithm 1 to report models instead ofreturning a boolean value.We have identified some fundamental challenges in computing unfounded sets forhybrid MKNF knowledge bases that make the problem intractable. The operator con-structed by Ji et al. [5] computes a subset of the greatest unfounded set and we build onthis approximation with an extension for programs with rules with disjunctive heads. Ji et al. establish a definition of unfounded sets for normal hybrid MKNF knowledgebases and construct well-founded operators that can be directly embedded in a solver[5]. We extend their work by introducing a definition of unfounded sets that handles16isjunctive rules, rules that have multiple K -atoms in their heads. Our extension bor-rows from the unfounded-set techniques outlined by Leone et al. [6] for disjunctivelogic programs but with a few noteworthy differences. Namely, our definition cannotbe used directly for model-checking. If the ontology in K is empty, our definition isequivalent to Leon et al.’s for unfounded-free partitions. Similarly, if K is a normalknowledge base, our definition is equivalent to Ji et al.’s definition.Both Ji et al. and Leone et al. outline abstract solvers for finding models of theirrespective languages. These solvers follow the DPLL paradigm of exploring the searchspace for a model. Both solvers substantially prune their search space using unfoundedsets. Because the complexity of model-checking a disjunctive hybrid MKNF knowl-edge bases is greater than that of normal hybrid MKNF knowledge bases [8], our ab-stract solver in this work consults a model checker after a total interpretation has beenguessed. This differs from the solver described by Ji et al. which does not rely ona model checker [5]. Leone et al.’s solver does not deepen its search on partial in-terpretations that assign unfounded atoms as true (partitions that cannot be extendedto models) [6]. This aggressive pruning strategy requires, at each step of the solver,an invocation of an algorithm with a complexity of ∆ P [ O ( log n )] [6]. Industry-gradesolvers, such as Clingo [3] or HEX [2], recognize the impracticality of enumeratingall unfounded sets many times during the solving process and these solvers introduceapproximations techniques. As a caveat of using approximations of unfounded sets,a solver may deepen its search on partial interpretations that cannot be extended tomodels. Because we rely on approximations of greatest unfounded sets, we think itis reasonable for our solver to employ similar strategies used by practical solvers andinclude some partitions that cannot be extended to models in its search.Both Clingo and HEX have additional support for external atoms, atoms whosetruth is dependant on external sources. Clingo 5 defines T -stable semantics [4] toreason about external atoms via external theories. HEX defines semantics for exter-nal atoms using boolean functions that take a total interpretation as input [2]. Forany hybrid MKNF knowledge base, models of the accompanying ontology must bemonotonic [8]. While it may be possible to encode the semantics of hybrid MKNFknowledge bases using either the HEX or Clingo extensions, neither solution exploitsthe monotonicity of external sources and both support nonmonotonic models of theexternal theories. We’ve provided a definition of unfounded sets for disjunctive hybrid MKNF knowledgebases, studied its properties, and formulated an operator to compute a subset of thegreatest unfounded set of a knowledge base. This leads to a DPLL-based solverwhere after each decision constraint propagation is carried out by computing addi-tional true and false atoms on top of the current partial partition. Our methods can bedirectly embedded into a solver for a drastic increase in efficiency when compared toa guess-and-verify solver, the current state of art for reasoning with disjunctive hybridMKNF knowledge bases. The addition of ontologies to answer set programs bringsnew challenges, namely, there is a complexity increase in computing unfounded sets17ven in the case of normal hybrid MKNF knowledge bases. We leave computing un-founded sets in light of inconsistencies that arise because of O to future work. References [1] Rachel Ben-Eliyahu and Rina Dechter. “Propositional semantics for disjunc-tive logic programs”. In:
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