Combining Rules and Ontologies Embedding Non-Ground Logic Programs into Autoepistemic Logic Jos de Bruijn Digital Enterprise Research Institute (DERI) University of Innsbruck, Austria jos.debruijn@deri.org Joint work with: Thomas Eiter (TU Wien), Axel Polleres (Univ. Rey Juan Carlos, Madrid), Hans Tompits (TU Wien) October 28, 2006 1/22
Outline Combinations of Rules and Ontologies First-Order Autoepistemic Logic Embedding Non-Ground Logic Programs 2/22
The Problem ◮ Combination of Ontologies and Rule bases ◮ Ontologies are FOL theories ◮ Classical first-order logic with equality ◮ No restrictions on interpretations (no unique/standard names) ◮ Description Logics are subsets of FOL ◮ Rule bases are non-ground nonmonotonic logic programs ◮ Stable Model Semantics (SMS) for normal/disjunctive programs ◮ For querying, Well-Founded Semantics subset of SMS ◮ Ontology and Rule-based are complementary descriptions of the same domain ◮ No distinction between ontology-predicates and rule-predicates ◮ There are different reasonable semantics for such combinations [de Bruijn et al., 06] ◮ Disclaimer: we do not address decidability/reasoning yet 4/22
The Combination ◮ Combination through embedding in unified formalism ◮ Ontology and Rule base as complementary descriptions of the same domain ◮ Given first-order theory Φ and logic program P ◮ Goal is combined theory ι (Φ , P ) in unified formalism ◮ Faithful embeddings σ (Φ) , τ ( P ) of Φ, P ◮ Faithful combination of Φ and P ◮ σ (Φ) = ι (Φ , ∅ ) ◮ τ ( P ) = ι ( ∅ , P ) ◮ Trivial combination: ι (Φ , P ) = σ (Φ) ∪ τ ( P ) 5/22
Motivation ◮ How to combine OWL with rules? ◮ What is the “Logic framework”? 6/22
Unified Formalism ◮ First-order autoepistemic logic (FO-AEL) [Konolige, 91] as unified formalism ◮ FO-AEL generalizes FOL: trivial embedding σ (Φ) = Φ possible ◮ No unique/standard names assumption ◮ Allows quantifying-in (free variables in the context of modal operator) ◮ Necessary for embedding non-ground logic programs ◮ FO-AEL allows different embeddings of logic programs ◮ Different embeddings for ground logic programs in the standard autoepistemic logic in the literature ◮ Can be extended to non-ground case 8/22
First-Order Autoepistemic Logic ◮ Given a first-order language L with signature Σ L = �F , P� , ◮ modal language L L is obtained by allowing modal operator L in front of formulas ◮ e.g. ∃ x (L p ( x )) , L( p ∨ q ) , ∀ x , y ( r ( x , y ) ⊃ p ( y ) ∧ L( ∃ z (L q ( y , z ) ∨ p ( c )))) ◮ L stands for “knows/believes” ◮ ∀ x ( bird ( x ) ∧ ¬ L ¬ flies ( x ) ⊃ flies ( x )) ◮ Autoepistemic interpretation � w , T � : first-order interpretation w = � U , · I � and belief set T ⊆ L L ◮ If φ is nonmodal, then satisfaction is as in FOL: w | = T φ iff w | = φ 9/22
Associated variable substitution Variable substitution A variable substitution β is a set { x 1 / t 1 , ..., x k / t k } , where x 1 , ..., x k are distinct variables and t 1 , ..., t k are names. Associated variable substitution Given variable assignment B and substitution β , if β = { x / t | x ∈ V , t w = x B , for some name t } , then β is associated with B . Example Consider L with constants F = { a , b , c } an interpretation w = � U , · I � with U = { k , l , m } a w = k , b w = l , and c w = l variable assignment B : x B = k , y B = l , and z B = m . B has two associated variable substitutions, β 1 = { x / a , y / b } and β 2 = { x / a , y / c } , which are not total. 10/22
Satisfaction of modal atoms Satisfaction of modal atoms w , B | = T L φ iff, for some variable substitution(s) β , associated with B , φβ is closed and φβ ∈ T . Extension to arbitrary formulas is as usual Example Consider some interpretation w and the belief set T = { p ( a ) } , then w | = T L p ( a ) and w | = T ∃ x L p ( x ) 11/22
Stable Expansions ◮ Stable expansion: the sets of beliefs of an ideally introspective agent, given some base set Stable expansion A belief set T ⊆ L L is a stable expansion of a base set A ⊆ L L iff T = { φ | A | = T φ } . Example: A = {¬ L p ⊃ q , ¬ L q ⊃ p , p ⊃ r , q ⊃ r } has two stable expansions: T 1 = { p , L p , r , ¬ L q , . . . } , T 1 = { q , L q , r , ¬ L p , . . . } Autoepistemic consequence A formula φ is an autoepistemic consequence of A if φ is included in every stable expansion of A . Example: r is an autoepistemic consequence of A 12/22
Embeddings of Logic Programs ◮ Three embeddings for ground normal and two for disjunctive programs have been defined in the literature ◮ We generalize these embeddings to the non-ground case ◮ Different embeddings lead to different semantics of combination ◮ This presentation limited to normal programs UNA axioms By UNA Σ we denote the set of axioms ¬ L( t 1 = t 2 ) ⊃ t 1 � = t 2 , for all distinct names t 1 , t 2 . 14/22
Embedding Normal Programs Normal Logic Program A normal logic program P consists of rules of the form h ← b 1 , . . . , b m , not c 1 , . . . , not c n , (1) where h , b 1 , . . . , b m , c 1 , . . . , c n are (equality-free) atoms. Embedding Let r be a rule of, then: τ HP ( r ) = ∀ � i b i ∧ � j ¬ L c j ⊃ h ; τ EB ( r ) = ∀ � i ( b i ∧ L b i ) ∧ � j ¬ L c j ⊃ h ; τ EH ( r ) = ∀ � i ( b i ∧ L b i ) ∧ � j ¬ L c j ⊃ h ∧ L h . For a normal logic program P , we define: τ x ( P ) = { τ x ( r ) | r ∈ P } ∪ UNA Σ P , x ∈ { HP , EB , EH } . 15/22
Embedding Normal Programs (cont’d) Example P = { p ← q } τ HP ( P ) = { p ⊃ q } has a stable expansion which includes ¬ q ⊃ ¬ p τ EB ( P ) = { p ∧ L p ⊃ q } has a stable expansion which includes ¬ q ⊃ ¬ L p ∨ ¬ p , but not ¬ q ⊃ ¬ p Theorem A Herbrand interpretation M of a normal logic program P is a stable model of P iff there is a consistent stable expansion T of τ x ( P ) such that M and T agree on ground atoms. 16/22
Comparing Embeddings Combination Given a program P and an FO theory Φ then their combination is defined as ι x (Φ , P ) = Φ ∪ τ x ( P ) ⊆ L L , Σ L L = Σ Φ ∪ Σ P . Comparing Embeddings Let A 1 and A 2 be FO-AEL theories. We write ◮ A 1 ≡ A 2 iff A 1 and A 2 have the same stable expansions, ◮ A 1 ≡ g A 2 iff A 1 and A 2 correspond with respect to ground objective formulas, and ◮ A 1 ≡ ga A 2 iff A 1 and A 2 correspond with respect to ground objective atoms. 17/22
Comparing Stable Expansions Φ \ P P rg S afe G rnd T hr ι EB ≡ ι EH U ni ι EB ≡ g ι EH g H orn ι HP ≡ ga ι EB H orn ι HP ≡ ga ι EB ι HP ≡ ga ι EB ≡ ga {∅} ι EH ≡ ga ι x is short for ι x (Φ , P ) ◮ P rg , S afe , and G rnd are the classes of arbitrary, safe, and ground logic programs, respectively ◮ T hr , U ni , g H orn , H orn , and {∅} are the classes of arbitrary, universal, generalized Horn, (Horn with existentials), Horn, and empty FO theories. 18/22
Comparing Stable Expansions (cont’d) ◮ All embeddings correspond wrt. ground atoms in case Φ = ∅ , because all are proper embeddings of SMS ◮ Two sources of difference between ι HP and ι EB : ◮ Combination with disjunctive knowledge (correspondence if Φ is Horn) ◮ Unnamed individuals (do not play a role if P is ground; Φ may be gHorn) ◮ Note: still difference in non-atomic formulas; recall ι HP includes contrapositive of rules ◮ In case unnamed individuals do not play a role (e.g. P is safe, Φ is universal), ι EB and ι EH correspond. ◮ If P is not safe, ι EB and ι EH differ, even if Φ = ∅ Consider P = { p ( x ); q ( x ) ← p ( x ) } τ EH ( P ) = {∀ x ( p ( x ) ∧ L p ( x )) , ∀ x ( p ( x ) ∧ L p ( x ) ⊃ q ( x ) ∧ L q ( x )) } has one consistent stable expansion which includes ∀ x ( q ( x )) τ EB ( P ) = {∀ x ( p ( x )) , ∀ x ( p ( x ) ∧ L p ( x ) ⊃ q ( x )) } has one consistent stable expansion which does not include ∀ x ( q ( x )), because ∀ x (L p ( x )) is not necessarily true when ∀ x ( p ( x )) is true 19/22
Conclusion and Future Work ◮ Ontology and Logic Program as complementary descriptions of same domain ◮ Embedding both into unified formalism (FO-AEL) to obtain semantics for combination ◮ Different embeddings τ x ( · ) lead to different semantics for combination ◮ Comparison of embeddings gives first idea of which embedding to use in a particular setting ◮ Consider nontrivial embedding for FO theory ◮ Consider relationship with other approaches (e.g. MKNF); similarity between ◮ τ HP embedding and DL + log [Rosati, 06] ◮ τ EB /τ EH embedding and dl-programs [Eiter et al., 04] ◮ Consider decidability of and reasoning with (fragments of) FO-AEL 20/22
. . . and I also care about F-Logic ◮ Extension of FO-AEL with F-Logic features ◮ Integration between F-Logic rules and DL ontologies [de Bruijn and Heymans, 2006] 21/22
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