=Paper=
{{Paper
|id=Vol-1916/paper4
|storemode=property
|title=A Distribution Semantics for non-DL-Safe Probabilistic Hybrid Knowledge Bases
|pdfUrl=https://ceur-ws.org/Vol-1916/paper4.pdf
|volume=Vol-1916
|authors=Marco Alberti,Evelina Lamma,Fabrizio Riguzzi,Riccardo Zese
|dblpUrl=https://dblp.org/rec/conf/ilp/0001LRZ17
}}
==A Distribution Semantics for non-DL-Safe Probabilistic Hybrid Knowledge Bases==
https://ceur-ws.org/Vol-1916/paper4.pdf
A Distribution Semantics for non-DL-Safe
Probabilistic Hybrid Knowledge Bases
Marco Alberti1 , Evelina Lamma2 , Fabrizio Riguzzi1 , and Riccardo Zese2
1
Dipartimento di Matematica e Informatica – University of Ferrara
Via Saragat 1, I-44122, Ferrara, Italy
2
Dipartimento di Ingegneria – University of Ferrara
Via Saragat 1, I-44122, Ferrara, Italy
{marco.alberti,evelina.lamma,fabrizio.riguzzi,riccardo.zese}@unife.it
Abstract. Logic Programming languages and Description Logics are
based on different domain closure assumptions, closed and the open
world assumption, respectively. Since many domains require both these
assumptions, the combination of LP and DL have become of foremost im-
portance. An especially successful approach is based on Minimal Knowl-
edge with Negation as Failure (MKNF), whose semantics is used to define
Hybrid KBs, composed of logic programming rules and description logic
axioms. Following such idea, we have proposed an approach for defining
DL-safe Probabilistic Hybrid Knowledge Bases, where each disjunct in
the head of LP clauses and each DL axiom is annotated with a probabil-
ity value, following the well known distribution semantics. In this paper,
we show that this semantics can be unintuitive for non-DL-safe PHKBs,
and we propose a new semantics that coincides with the previous one if
the PHKB is DL-safe.
Keywords: Hybrid Knowledge Bases, MKNF, Distribution Semantics
1 Introduction
Usually, complex domains are modeled using either Logic Programming (LP)
languages or Description Logic (DL) languages. These languages share many
similarities because they are both based on first order logic. On the other hand,
the main and remarkable difference is the domain closure assumption: closed-
world assumption for LP and open-world assumption for DLs. However, the
management of many domains, such as in legal reasoning [1], requires different
closure assumptions.
The combination of LP and DL have been proposed by several authors and
one of the most effective approaches is called Minimal Knowledge with Negation
as Failure (MKNF) [15]. MKNF was then applied to define hybrid knowledge
bases (HKBs) [18], which are defined as the combination of a logic program and
a DL KB.
40
A large number of works in LP show how many domains, especially those de-
rived from the real world, are often characterized by uncertain information, and
present approaches and semantics for allowing probabilistic reasoning, leading to
the dawn of the Probabilistic Logic Programming (PLP) field. One of the most
widespread approach is the distribution semantics [24], where a program de-
fines a probability distribution over normal Logic Programs, called worlds, from
which the probability of a query is obtained. The distribution semantics under-
lies many languages such as Probabilistic Logic Programs [10], Logic Programs
with Annotated Disjunctions (LPADs) [27], CP-logic [25] and ProbLog [11].
Similarly, DLs also need to manage uncertainty. The combination with prob-
ability theory have been proposed by several works exploiting graphical models:
[9] and [8] exploit Bayesian networks while [14] combines DLs with Markov net-
works. Other approaches exploit Nilsson’s probabilistic logic [19]: [13,16,17,7]
reason with intervals of probability values. Others make use of databases tech-
niques to store and recover information such as [12].
In [5] we defined DISPONTE (for “DIstribution Semantics for Probabilis-
tic ONTologiEs”), which applies the distribution semantics to DLs, allowing to
associate probability values to axioms of a KB. The probability of queries is
computed as for PLP languages.
In [2] we proposed an approach for defining DL-safe Probabilistic Hybrid
KBs (PHKBs) under the distribution semantics combining LPADs with DLs
under DISPONTE semantics. In a PHKB, if the logic program is stratified, each
program has a unique model, thus query’s probability is the sum of probabilities
of each program that implies the query.
A similar approach is the one of [20] where a sigma-algebra over complex
relational models is used to allow existentials in ontologies and to define prob-
ability values on such information. The proposals makes use of semantic trees
to define its semantics, such trees model the sequence of random variables and
specify the trace of a generative process with its associated probabilities. The
semantics so defined can be applied to different languages, allowing the integra-
tion of existence, identity, roles and ontologies into a clean semantic framework.
We believe our approach leads to the definition of the same sigma-algebra and
semantics, we leave for future work a detailed comparison with this work.
In this paper, we show that the semantics proposed in [2] can be unintu-
itive for non-DL-safe PHKBs, and we propose a new semantics that behaves
as, arguably, expected. We also show that the new semantics coincides with the
previous one if the PHKB is DL-safe.
The paper is structured as follows. In Section 2, we provide some background
notions and define MKNF HKBs. In Section 3, we introduce our probabilistic
extension to hybrid MKNF knowledge bases, and in Section 4 we define their
semantics. Section 5 concludes the paper.
41
2 Background
This section introduces the necessary background to understand PHKBs. In
particular, Section 2.1 introduces HKBs as the combination of DLs with LP.
Then, Sections 2.3 and 2.2 present probabilistic DLs and probabilistic LPs, which
are combined in PHKBs.
2.1 MKNF Hybrid Knowledge Bases
The logic of Minimal Knowledge with Negation as Failure (MKNF) was in-
troduced in [15]. An MKNF Hybrid Knowledge Base (HKB) [18] is a pair
K = hO, Pi where O is a DL knowledge base and P is a set of LP rules of
the form h ← a1 , . . . , an , ∼b1 , . . . , ∼bm , where ai and bi are atoms; ∼ represents
default negation; a negative literal is a default negated atom.
DLs are a fragment of First Order Logic (FOL) used to model ontologies [4],
thus they can be directly translated into FOL by exploiting a function π, which
maps axioms to first order formulas. A DL knowledge base (KB) is defined using
concepts, roles and individuals. It is a tuple containing a TBox T containing
concept inclusion axioms C v D, where C and D are concepts possibly built us-
ing other concepts and roles, an ABox A containing concept membership axioms
a : C and role membership axioms (a, b) : R, where C is a concept, R is a role
and a, b are individuals, and possibly a RBox R containing transitivity axioms
T rans(R) and role inclusion axioms R v S, where R, S are roles. A DL KB
is usually assigned a semantics in terms of interpretations I = (∆I , ·I ), where
∆I is a non-empty domain and ·I is the interpretation function. This function
assigns an element in ∆I to each a ∈ I, a subset of ∆I to each C ∈ C and a
subset of ∆I × ∆I to each R ∈ R.
A HKB is positive if no negative literals occur in it. Note that we simplify
the definition in [18] by disallowing disjunctions, which we do not need, in LP
rule heads.
Given a HKB K = hO, Pi, an atom in P is a DL-atom if its predicate occurs
in O, a non-DL-atom otherwise. An LP rule is DL-safe if each of its variables
occurs in at least one positive non-DL-atom in the body; a HKB is DL-safe if all
its LP-rules are DL-safe. If there exists at least one LP rule that is not DL-safe,
we say that the HKB is non-DL-safe.
A Hybrid Knowledge Base is given semantics by transforming it into an
MKNF formula. More precisely, the transform π defined for DLs is extended as
follows to support LP rules (where π(O) is the translation of O by means of π):
– if C is a rule of the form h ← a1 , . . . , an , ∼b1 , . . . , ∼bm and X is the vector of
all variables in C, π(C) = ∀X(K a1 ∧ . . . ∧ K an ∧ not b1 ∧ . . . ∧ . . . not bm ⊃
K h)
V
– π(P) = C∈P π(C)
– π(hO, Pi) = K π(O) ∧ π(P)
42
The syntax of MKNF is the syntax of first order logic augmented with modal
operators K and not . In the following, ∆ is the Herbrand universe of the signa-
ture at hand. An MKNF structure is a triple (I, M, N ) where I as a first-order
interpretation over ∆ and M and N are sets of first order interpretations over
∆. Entailment of a closed formula by an MKNF structure is defined as follows:
(I, M, N ) |= p ⇔p∈I
(I, M, N ) |= ¬ϕ ⇔ (I, M, N ) 6|= φ
(I, M, N ) |= ϕ1 ∧ ϕ2 ⇔ (I, M, N ) |= ϕ1 and (I, M, N ) |= ϕ2
(I, M, N ) |= ∃x : ϕ ⇔ (I, M, N ) |= ϕ[α/x] for some α ∈ ∆
(I, M, N ) |= K ϕ ⇔(J, M, N ) |= ϕ for all J ∈ M
(I, M, N ) |= not ϕ ⇔(J, M, N ) 6|= ϕ for some J ∈ N
An MKNF interpretation is a set M of interpretations over ∆. An interpre-
tation M is an MKNF model of a closed formula ϕ iff
– (I, M, M ) |= ϕ for all I ∈ M
– for all M 0 ⊃ M , for some I 0 ∈ M 0 (I 0 , M 0 , M ) 6|= ϕ
A formula ϕ entails a formula φ, written ϕ |=MKNF φ, iff for all MKNF
models M of ϕ and for all I ∈ M (I, M, M ) |= φ.
2.2 Probabilistic Logic Programs
We consider Logic Programs with Annotated Disjunctions (LPADs) [26], which
consist of a finite set of annotated disjunctive clauses ri of the form hi1 :
Πi1 ; . . . ; hini : Πini ← bi1 , . . . , bimi . Here, bi1 , . . . , bimi are logical literals which
form the body of ri , denoted by body(ri ), while hi1 , . . . hini are logical Pni atoms and
{Πi1 , . . . , Πini } are real numbers in the interval [0, 1] such that P k=1 Πik ≤ 1.
ni
If ni = 1 and Πi1 = 1 the clause is a non-disjunctive clause. If k=1 Πik < 1,
the head of the annotated disjunctive clause implicitly contains an extra atom
nullP that does not appear in the body of any clause and whose annotation is
ni
1 − k=1 Πik . The grounding of an LPAD P is denoted by ground(P).
An atomic choice is a triple (ri , θj , k) where ri ∈ P, θj is a substitution
that grounds ri and k ∈ {1, . . . , ni } identifies a head atom of ri . It corresponds
to an assignment Xij = k, where Xij is a multi-valued random variable which
corresponds to Ci θj .
A set of atomic choices κ is consistent if only one head is selected from a
ground clause. In this case Q it is called a composite choice. The probability P (κ) of
a composite choice κ is (ri ,θj ,k)∈κ Πik . A selection σ is a set of atomic choices
that, for each clause ri θj in ground(P), contains an atomic choice (ri , θj , k). It
identifies a world wσ of P, i.e., a normal logic program defined as wσ = {(hik ←
body(ri ))θj |(ri , θj , k) ∈ σ}.
We consider only sound LPADs, where each possible world has a total well-
founded model, so wσ |= q means that the query q is true in the well-founded
43
model of the program wσ . The probability of a query q given a world wσ is
P (q|wσ ) = 1 if wσ |= q and 0 otherwise. The probability of q is then:
X X X
P (q) = P (q, wσ ) = P (q|wσ )P (wσ ) = P (wσ ) (1)
wσ ∈LT wσ ∈LT wσ ∈LT :wσ |=q
Given an LPAD P, WP is the set of all P’s possible worlds. A composite
choice, or a set of composite choices, determine sets of worlds. In particular,
given a composite choice κ, the set of worlds determined by κ is the set of worlds
identified by total choices that are subsets of κ, i.e., ωκ = {wσ | κ ⊆ σ}. SGiven a
set K of composite choices, the set of worlds determined by K is ωK = κ∈K ωκ ;
two sets K1 and K2 of composite choices are equivalent if ωK1 = ωK2 .
We assign probabilities to sets of worlds, rather than to individual worlds,
as follows. Given an LPAD P, let ΩP be the set of sets of worlds determined
by countable sets of countable composite choices. As shown in [21], ΩP is a σ-
algebra over WP , so a probability measure µ : ΩK → [0, 1] can be defined over
ΩP .
A set of composite choices is pairwise incompatible if any two choices from the
set are incompatible; the probability of a pairwise incompatible set of composite
choices is the sum of the probabilities of its elements.
Given a ground query q, a composite choice κ is an explanation for q if w |= q
for all w ∈ ωκ . A set K of composite choices is covering for q if {w | w |= q} ⊆ ωK .
The author of [21] shows that for each countable set K of countable compos-
ite choices, there exists a pairwise incompatible countable set K 0 of countable
composite choices that is equivalent to K, in the sense that they identify the
same set of worlds.
For sound LPADs each query q has a countable covering set K of countable
infinite explanations [21]; since there exists a pairwise incompatible set K 0 that
is equivalent to K, we can define the probability of q as µ(K 0 ).
2.3 Probabilistic Description Logics
DISPONTE [5] applies the distribution semantics to probabilistic ontologies, al-
lowing the definition of probabilistic knowledge bases O, that are sets of certain
and probabilistic axioms. Certain axioms are regular DL axioms, while proba-
bilistic axioms take the form Π :: a, where Π is a real number in [0, 1] and a is
a DL axiom.
An atomic choice for an axiom a is a pair (a, i), where i is 1 if a is selected and
2 otherwise. Composite choices, set of composite choices and the other concepts
from the previous subsection can be defined similarly. A world, here, is obtained
by including in it all certain axioms and a subset of the uncertain axioms. The
probability of the world is given by the product of the probability Π for the
included axioms and 1 − Π for the excluded ones. The probability of a query is
then the sum of the probabilities of the worlds where the query holds (see Eq. 1).
44
3 Probabilistic Hybrid Knowledge Bases
A Probabilistic Hybrid Knowledge Base (PHKB) is a pair K = hO, Pi where O
is a DISPONTE knowledge base and P is an LPAD without function symbols.
In [2], a PHKB’s semantics is given by first grounding it over all the constants
in the PHKB. A world is the deterministic ground HKB obtained by selecting,
for each clause hi1 : Πi1 ; . . . ; hini : Πini ← bi1 , . . . , bimi , one of the disjuncts in
the head and some of the DL axioms. The world’s probability is the product of
the probabilities of the selected head disjuncts and the selected axioms. In terms
of the definitions given in Section 2.2, that is the probability of the set of worlds
whose only element is the world at hand.
Definition 1. Given a world w, the probability of a query q is defined as P (q|w) =
1 if w |=MKNF K q and 0 otherwise.
The probability of the query is its marginal probability:
X
P (q) = P (w)P (q|w) (2)
w
Example 1. The following KB K models the insurgence of a protest against
animal testing:
P =
(C1 ) protest : 0.6 ←
activist(X), ∼cruelT oAnimals(X).
activist(kevin).
(C2 ) activist(nadia) : 0.3.
O=
∃hasAnimal.pet v ¬cruelT oAnimals
(kevin, fluffy) : hasAnimal
(E1 ) 0.4 :: fluffy : cat
cat v pet
This KB has 16 worlds and the query protest is true in four of them, those con-
taining activist(nadia) and protest ← activist(nadia), ∼cruelT oAnimals(nadia),
plus other two, those in which activist(nadia) is absent and fluffy : cat and
protest ← activist(kevin), ∼cruelT oAnimals(kevin) are present.
So the probability of protest is 0.3 · 0.6 + 0.7 · 0.4 · 0.6 = 0.18 + 0.168 = 0.438.
This semantics is defined regardless of the PHKB’s DL-safety, but it can give
unintuitive results for non-DL-safe PHKBs, essentially because a non-DL-safe
HKB may not have the same MKNF models of its grounding over its constants.
Consider, for example, the following non-DL-safe HKB.
Example 2. Let K = hO, Pi, where
P =person(X) ←∼dog(X).
O =guard u person v soldier
∃commands.soldier v commander
john : ∃commands.guard
45
In a model of K’s, no individual is a dog in all interpretations, so each indi-
vidual is a person. This means that in all interpretations, the guard that john
commands is a person, and due to the first axiom, a soldier; in other words,
in each interpretation john commands a soldier, and is a commander. Thus,
K |= K commander(john).
However, the grounding over the known individuals yields the following clause:
P = person(john) ←∼dog(john).
so the only individual known to be a person is john, and (except in the
worlds where john commands himself) the guard that john commands can-
not be inferred to be a soldier . So the grounding of the HKB does not entail
K soldier (john).
4 Semantics for non-DL-safe PHKBs
In this section, we propose a generalization of the semantics that is given by
grounding the PHKB not only over the constants occurring in it, but also on the
countable supply of constants provided by the standard name assumption [18].
We call ∆ the resulting countable set of constants.
Intuitively, a possible world is obtained by selecting one annotated disjunct
for each ground clause in P, and some of the axioms in O, as in the semantics
proposed in [2], but since worlds are obtained from infinite choice and so their
probability is 0, in order to have a non-zero probability for a query we assign
probabilities to sets of worlds, rather than to individual worlds.
We do so by extending to PHKBs the semantics for LPADs with function
symbols recalled in Section 2.2. In particular, we extend the notion of atomic
choice to axioms: an atomic choice for an axiom a determines whether a is
selected, and is of the form (a, ∅, i), where i is 1 if a is selected and 2 otherwise.
The second element, ∅, of the triple is there so atomic choices for rules and
axioms are syntactically uniform.
A selection σ determines the world wσ , i.e., the HKB composed of:
– one rule for each grounding substitution θ of each rule r in P, where (r, θ, i) ∈
σ, whose head is the i-th disjunct of r/θ and whose body is r/θ’s body;
– the axioms a for which (a, ∅, 1) is in the selection.
Given a PHKB K, WK is the set of all K’s possible worlds. A composite
choice, or a set of composite choices, determine sets of worlds, as for LPADs.
Given a PHKB K, let ΩK be the set of sets of worlds determined by finite or
countable sets of finite or countable composite choices; a probability measure
µ : ΩK → [0, 1] is defined over ΩK .
If a query q has a countable covering set K of countable explanations, then
there exists a pairwise incompatible set K 0 with the same property, and whose
probability µ(K 0 ) is defined; that is defined as q’s probability given K.
46
Definition 2. Let K be a PHKB and K be a (finite or countable) covering set
of (finite or countable) explanations for a query q. Then q’s probability given
K PK (q) is the probability of a pairwise incompatible set K 0 of explanations
equivalent to K, which is guaranteed to exist.
Example 3. Consider a probabilistic version of Example 2: let K = hO, Pi, where
P =person(X) : 0.5 ←∼dog(X).
O =guard u person v soldier
∃commands.soldier v commander
john : ∃commands.guard
In the last axiom there is an (unknown) individual that is a guard and that
john commands. Let us call her u.
K |= K commander(john) is entailed by the worlds where the clause with
substitution X/u for the first disjunct is selected. So {{(C1 , X/u, 1)}} is a (finite)
covering set of (finite) explanations. Its probability is 0.5.
Next, we show that the semantics proposed here generalizes the one presented
in [2] for the PHKBs allowed there, i.e., DL-safe PHKBs without function sym-
bols.
Proposition 1. Given a DL-safe PHKB without function symbols, the proba-
bility of any query is the same under the semantics in Definition 1 and the one
in Definition 2.
Proof. A DL-safe KB is equivalent to its grounding over the constants that occur
in it, and if function symbols are not allowed there are finitely many worlds;
each world that entails the query is identified by a selection. The set of such
selections is a pairwise incompatible covering set of explanations for the query,
and its probability is identical to the one given Definition 1.
5 Conclusions
In this paper, we define a semantics for Probabilistic Hybdrid Knowledge Bases,
which is equivalent to that given in [2], but that is applicable also for non-DL-
safe PHKBs. We also show that in case of a DL-safe PHKB the two semantics
coincide.
For the future we plan to provide a reasoner for such semantics. The idea is
to follow [3], where is defined the SLG(O) procedure for HKBs under the well
founded semantics. SLG(O) integrates a DL reasoner into the SLG procedure
in the form of an oracle in order to manage the DL part of the HKBs. The oracle
returns the LP atoms that would have to be true for the query to succeed.
We intend to follow a similar approach for PHKBs, integrating the TRILL
probabilistic DL reasoner [29,28] with the PITA algorithm [23] for PLP reason-
47
ing. We also plan to develop a web application for using the system, similarly to
what we have done for TRILL3 [6] and PITA4 [22].
Moreover, we plan to perform a detailed comparison with alternative ap-
proaches for existential constructs in probabilistic logics, such as the one of [20].
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