Combining Rules and Ontologies Embedding Non-Ground Logic Programs - - PowerPoint PPT Presentation

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

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Outline

Combinations of Rules and Ontologies First-Order Autoepistemic Logic Embedding Non-Ground Logic Programs

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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

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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”?

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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

• perator)

◮ 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 ΣL = F, P, ◮ modal language LL is obtained by allowing modal operator L

in front of formulas

◮ e.g. ∃x(Lp(x)), L(p ∨ q), ∀x, y(r(x, y) ⊃

p(y) ∧ L(∃z(Lq(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 ⊆ LL

◮ If φ is nonmodal, then satisfaction is as in FOL:

w | =T φ iff w | = φ

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Associated variable substitution

Variable substitution

A variable substitution β is a set {x1/t1, ..., xk/tk}, where x1, ..., xk are distinct variables and t1, ..., tk are names.

Associated variable substitution

Given variable assignment B and substitution β, if β = {x/t | x ∈ V, tw = xB, 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} aw = k, bw = l, and cw = l variable assignment B: xB = k, yB = l, and zB = m. B has two associated variable substitutions, β1 = {x/a, y/b} and β2 = {x/a, y/c}, which are not total.

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Satisfaction of modal atoms

Satisfaction of modal atoms

w, B | =T Lφ 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 Lp(a) and w | =T ∃xLp(x)

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Stable Expansions

◮ Stable expansion: the sets of beliefs of an ideally introspective

agent, given some base set

Stable expansion

A belief set T ⊆ LL is a stable expansion of a base set A ⊆ LL iff T = {φ | A | =T φ}.

Example: A = {¬Lp ⊃ q, ¬Lq ⊃ p, p ⊃ r, q ⊃ r} has two stable expansions: T1 = {p, Lp, r, ¬Lq, . . .}, T1 = {q, Lq, r, ¬Lp, . . .}

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

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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 UNAΣ we denote the set of axioms ¬L(t1 = t2) ⊃ t1 = t2, for all distinct names t1, t2.

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Embedding Normal Programs

Normal Logic Program

A normal logic program P consists of rules of the form h ← b1, . . . , bm, not c1, . . . , not cn, (1) where h, b1, . . . , bm, c1, . . . , cn are (equality-free) atoms.

Embedding

Let r be a rule of, then: τHP(r) = ∀

ibi ∧ j¬Lcj ⊃ h;

τEB(r) = ∀

i(bi ∧ Lbi) ∧ j¬Lcj ⊃ h;

τEH(r) = ∀

i(bi ∧ Lbi) ∧ j¬Lcj ⊃ h ∧ Lh.

For a normal logic program P, we define: τx(P) = {τx(r) | r ∈ P} ∪ UNAΣP, x ∈ {HP, EB, EH}.

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Embedding Normal Programs (cont’d)

Example

P = {p ← q} τHP(P) = {p ⊃ q} has a stable expansion which includes ¬q ⊃ ¬p τEB(P) = {p ∧ Lp ⊃ q} has a stable expansion which includes ¬q ⊃ ¬Lp ∨ ¬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 τx(P) such that M and T agree on ground atoms.

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Comparing Embeddings

Combination

Given a program P and an FO theory Φ then their combination is defined as ιx(Φ, P) = Φ ∪ τx(P) ⊆ LL, ΣLL = ΣΦ ∪ ΣP.

Comparing Embeddings

Let A1 and A2 be FO-AEL theories. We write

◮ A1 ≡ A2 iff A1 and A2 have the same stable expansions, ◮ A1 ≡g A2 iff A1 and A2 correspond with respect to ground

• bjective formulas, and

◮ A1 ≡ga A2 iff A1 and A2 correspond with respect to ground

• bjective atoms.

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Comparing Stable Expansions

Φ\P Prg Safe Grnd T hr ιEB ≡ ιEH Uni ιEB ≡g ιEH gHorn ιHP ≡ga ιEB Horn ιHP ≡ga ιEB {∅} ιHP ≡ga ιEB ≡ga ιEH ≡ga

ιx is short for ιx(Φ, P)

◮ Prg, Safe, and Grnd are the classes of arbitrary, safe, and ground

logic programs, respectively

◮ T hr, Uni, gHorn, Horn, and {∅} are the classes of arbitrary,

universal, generalized Horn, (Horn with existentials), Horn, and empty FO theories.

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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 ιHP and ιEB:

◮ 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 ιHP

includes contrapositive of rules

◮ In case unnamed individuals do not play a role (e.g. P is safe,

Φ is universal), ιEB and ιEH correspond.

◮ If P is not safe, ιEB and ιEH differ, even if Φ = ∅

Consider P = {p(x); q(x) ← p(x)} τEH(P) = {∀x(p(x) ∧ Lp(x)), ∀x(p(x) ∧ Lp(x) ⊃ q(x) ∧ Lq(x))} has one consistent stable expansion which includes ∀x(q(x)) τEB(P) = {∀x(p(x)), ∀x(p(x) ∧ Lp(x) ⊃ q(x))} has one consistent stable expansion which does not include ∀x(q(x)), because ∀x(Lp(x)) is not necessarily true when ∀x(p(x)) is true

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Conclusion and Future Work

◮ Ontology and Logic Program as complementary descriptions

• f same domain

◮ Embedding both into unified formalism (FO-AEL) to obtain

semantics for combination

◮ Different embeddings τx(·) 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

◮ τHP embedding and DL+log [Rosati, 06] ◮ τEB/τEH embedding and dl-programs [Eiter et al., 04]

◮ Consider decidability of and reasoning with (fragments of)

FO-AEL

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. . . 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]

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Bibliography

◮ J. de Bruijn, T. Eiter, A. Polleres, H. Tompits. On

representational issues about combinations of classical theories with nonmonotonic rules. In KSEM 2006.

◮ J. de Bruijn and S. Heymans. Translating ontologies from

predicate-based to frame-based languages. In RuleML-2006.

◮ J. de Bruijn, T. Eiter, A. Polleres, and H. Tompits.

Embedding non-ground logic programs into autoepistemic logic for knowledge-base combination. In IJCAI-07.

◮ T. Eiter, T. Lukasiewicz, R. Schindlauer, and H. Tompits.

Combining answer set programming with description logics for the semantic web. In KR 2004.

◮ R. Rosati. DL+log: Tight integration of description logics and

disjunctive datalog. In KR 2006.

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