The open world assumption makes OWL principally suitable to handle incomplete knowledge in Semantic Web scenarios, however, some scenarios desire closed world reasoning. Autoepistemic description logics allow to realise closed world reasoning in open world settings through epistemic operators. An extension of OWL by epistemic operators therefore allows for non-monotonic features known from closed world systems, such as default rules, integrity constraints or epistemic querying. These features can be beneficially applied in Semantic Web scenarios, where OWL lacks expressiveness.
An important goal in the design of the Ontology Web Language (OWL) [
The open world assumption seems to correctly model much of day-to-day reasoning. However, the framework of first-order logic may be unsuitable for certain situations which require complete knowledge about the world. Consider a table of train departure times. If the table does not explicitly state that a train leaves at 12:47, then we usually conjecture that there is no such train. In other words, for train time-tables we typically use the closed world assumption (CWA), assuming that our knowledge about that part of the world is complete. Under CWA, we conclude that there is no train at 12:47 unless we can prove the contrary. Such inference is non-monotonic, meaning that additional knowledge can invalidate previous conclusions. For example, learning that there is a train at 12:47 invalidates our earlier conjecture.
Many knowledge modelling constructs are related to CWA and cannot be expressed in first-order logic. Default rules allow for modelling exceptions. For example, we may make a common conjecture that people eat meat, unless we know them to be vegetarians. This rule relieves us from the burden of explicitly asking each person whether he is a vegetarian or not.
Constraints also depend on closed world reasoning. For example, we could easily embarrass ourselves by inviting a vegetarian to dinner and then preparing Beef Stroganoff just because we did not require the guest to specify whether he is a vegetarian. To prevent such situations, we might introduce a constraint stating that for each guest his views on eating meat should be known.
Choosing between OWA or CWA is often an all-or-nothing game, thus
posing problems for applications which need to deal with both kinds of information
at once. For example, in an application dealing with travelling vegetarians,
assuming that a person eats meat just because we do not know that he is not a
vegetarian may be wrong; however, assuming that one might get to the
International Vegetarianism Convention by a train which is not listed in the time table,
seems wrong as well. In other words, we believe that many applications require
OWA and CWA in parallel, allowing for local closed world (LCW) reasoning [
A common objection to extending DLs with non-monotonic constructs is that completeness of knowledge can be stated in a purely first-order setting. For example, using nominals one can restrict an interpretation of a concept to exactly the specified set of individuals. However, this solves the problem only partially, since there is no equivalent nominal construct for roles. Moreover, such a solution does not provide introspection — reasoning about the state of the knowledge base. Introspection is not definable in first-order logic, but is necessary for formalising defaults or constraints. Similarly, a common objection to introducing defaults and constraints is that they should be realised outside the logic, for example, by checking for missing information in a preprocessing step. However, it is unclear how to define the semantics of such a step. If the semantics were defined in an ad-hoc manner, we would soon experience the same problems observed in the early frame representation systems, which eventually lead to formal reconstruction of their semantics.
To summarise, we believe that OWL should be extended with non-monotonic
constructs. In this paper we sketch a possible solution based on autoepistemic
description logics (ADL) [
Autoepistemic logic is a formalism concerned with the notions of ‘knowledge’ and
‘assumption’ and allows for introspection of knowledge bases, i.e. to ask what
a knowledge base knows or assumes. (See e.g. [
In [
C, D r −→ −→
A | ⊤ | ⊥ | C ⊓ D | C ⊔ D | ¬C | ∀ r.C | ∃ r.C | KC | AC p | Kp | Ap
An epistemic interpretation is a triple ( I, WK, WA ) where I = (ΔI , ·I ) is a first-order interpretation with interpretation domain ΔI and interpretation function ·I , and WK, WA are sets of first-order interpretations, seen as possible worlds for the two modalities K and A in the sense of modal logics. The following equations define how the elements of ALCKN F are epistemically interpreted. ⊤ I,WK,WA = ΔI , ⊥ I,WK,WA = ∅
A I,WK,WA = AI ⊆ ΔI , p I,WK,WA = pI ⊆ ΔI × ΔI (C ⊓ D) I,WK,WA = C I,WK,WA ∩ D I,WK,WA (C ⊔ D) I,WK,WA = C I,WK,WA ∪ D I,WK,WA
(¬C) I,WK,WA = ΔI \ C I,WK,WA
(∀ r.C) I,WK,WA = {a ∈ ΔI | ∀b.(a, b) ∈ r I,WK,WA → b ∈ C I,WK,WA }
(∃ r.C) I,WK,WA = {a ∈ ΔI | ∃b.(a, b) ∈ r I,WK,WA ∧ b ∈ C I,WK,WA }
(KC) I,WK,WA = TJ ∈WK C J ,WK,WA , (AC) I,WK,WA = TJ ∈WA C J ,WK,WA
(Kp) I,WK,WA = TJ ∈WK p J ,WK,WA , (Ap) I,WK,WA = TJ ∈WA p J ,WK,WA
Primitive concepts are interpreted as subsets of ΔI , and primitive roles as
subsets of ΔI× ΔI . The boolean connectives and existential and universal role
quantification are interpreted in terms of set operations on ΔI , as in ALC [
An epistemic interpretation satisfies an inclusion axiom C ⊑ D if C I,WK,WA ⊆ D I,WK,WA , and it satisfies an assertion axiom C(a) or r(a, b) if aI ∈ C I,WK,WA or (aI , bI) ∈ r I,WK,WA , respectively. An epistemic model for an ALCKN F knowledge base KB is a non-empty set M of first-order interpretations such that, for each I ∈ M, the epistemic interpretation ( I, M, M ) satisfies all axioms in KB and there is no set M′ of first order interpretations such that M ⊂ M′ and the epistemic interpretation ( I, M′, M ) also satisfies all axioms in KB . An ALCKN F knowledge base KB is satisfiable if it has an epistemic model. It entails an axiom α, denoted by KB |= α, if α is satisfied in all its epistemic models.
As a special case, a non-epistemic knowledge base KB always has a unique
epistemic model M(KB ), which is just the set of all its first-order models [
The semantics of both epistemic operators, K and A , is defined as an intersection of concept/role extensions over sets of first-order interpretations WK, WA, seen as possible worlds. Therefore they both ensure statements to constantly hold in all possible worlds in these sets. The difference between K and A lies in the restrictions about which worlds belong to WK and WA , respectively.
To see this difference, consider the knowledge bases KB = {∃ r.C(a)}, KB K = {∃ r.KC(a)} and KBA = {∃ r.AC(a)}. The set of all first-order models of KB , denoted by M(KB ), can be verified to be the unique epistemic model for KB . However, M(KB ) is not an epistemic model for KB K, since it contains first-order interpretations in which the r-successors of a do not constantly belong to C over all J ∈ M(KB ). The use of the K-operator in KB K requires the existence of an r-successor for a which belongs to C in all possible worlds, i.e. which is known to be in the extension of C. The set Mx ⊂ M(KB ), defined by {I : I |= r(a, x)∧C(x)} for some x ∈ ΔI , fulfils this condition. It is an epistemic model for KB K, since the epistemic interpretation ( I, Mx, Mx ) satisfies the axiom in KB K whereas ( I, Mx ∪ {I′}, Mx ) does not, for any I′ ∈ M(KB )\Mx1. In this sense K can be paraphrased as “known”.
Conversely, neither any Mx nor any other set of first order interpretations is an epistemic model for KBA, which is unsatisfiable. To see this, consider any set M of first-order interpretations for which ( I, M, M ) satisfies KBA. To verify M as being maximal, ( I, M′, M ) must not satisfy KBA for any set M′ ⊃ M. However, the choice of M′ does not affect the modality A . The set M could only be an epistemic model if it would already be maximal, such that there is no set M′. In this sense, the use of the A-operator in KBA refers to individuals that are assumed to be in the extension of C already, and A can therefore be paraphrased as “assumed”. If this assumption is not justified by other facts then the knowledge base becomes unsatisfiable. The A-operator is directly related to the operator not for negation as failure: AC maps to ¬ not C , which means that any individual assumed to be in C belongs to the complement of those individuals for which there is no evidence to be in C. 1 Since we did not use ∃ Kr.KC(a) in KB K, there are even epistemic models Mσ = {I : I |= Wx∈σ r(a, x) ∧ Vx∈σ C(x)} for any subset σ ⊂ ΔI
An epistemic concept is a concept that contains epistemic operators. We distinguish cases in which epistemic concepts occur inside a knowledge base from those in which they occur outside only, as e.g. in queries.
Querying a knowledge base KB in general means to ask for those individuals that have certain properties specified by a concept. Therefore a query is often defined as a concept C and querying reduces to checking the entailment of concept assertions C(ι) for all known individuals ι in KB . In this sense, an epistemic query is an epistemic concept2 C that is posed as a query to a non-epistemic knowledge base KB . To validate an entailment KB |= C(ι), the assertion C(ι) has to be satisfied by epistemic interpretations ( I, M, M ) for every epistemic model M of KB . However, since here KB is non-epistemic, it is sufficient to consider the set M(KB ) of its first-order models.
An epistemic axiom, either inclusion or assertion, is an axiom that contains an epistemic concept. If epistemic axioms occur in a knowledge base KB , they determine the epistemic models of KB . As some special cases of epistemic axioms we look at default rules and integrity constraints in Section 3. 3
The OWA has been criticised in various Semantic Web related settings based
on DL, such as natural language interfaces [
In our scenario, the pizza delivery services of Giovanni and Alberto allow to order pizzas via the web. They use the vocabulary from a publicly available pizza ontology OP izza to describe the pizzas they offer with semantic annotations OGiovanni and OAlberto in OWL-DL as follows.
OP izza ⊇ { ∃ topping.⊤ ⊑ P izza, Chili ⊑ ¬M ozarella ⊓ ¬T omato ,
V esuf o ⊑ SpicyDish ⊓ ∀ topping.¬Chili , M argarita ≡ ∃ topping.T omato ⊓ ∃ topping.M ozarella ⊓ ∀ topping.(T omato ⊔ M ozarella) } OGiovanni ⊇ { ∃ topping.Chili(normalChili),
∃ topping.Chili ⊓ ¬SpicyDish(mildChili) }
OAlberto ⊇ { M argarita(margarita), V esuf o(vesuf o) }
The ontology OP izza contains knowledge about the relation between
pizzas and their toppings. It defines particular kinds of pizzas, such as V esuf o,
2 Observe that Outside the knowledge base the two operators K and A show the same
behaviour [
The concept SpicyDish in OP izza is intended to indicate whether a pizza is spicy or not. However, Giovanni and Alberto do not consequently use this concept to classify all their pizzas – only some are explicitly said to be spicy or non-spicy. In general, a Semantic Web agent that interprets such annotations and ontologies should not expect to come across situations in which everything is completely specified. It rather gathers pieces of knowledge from different sources and has to draw a conclusion out of these in situations of incomplete information.
In our concrete scenario, consider a Semantic Web agent that is interested in non-spicy pizzas only. Using the OWL-DL concept ¬SpicyDish to query the annotations of Giovanni and Alberto, this agent would only get the pizza mildChili as a result due to the OWA – for the other pizzas there is no evidence to be nonspicy. Intuitively, we would like the pizza margarita to also be in the result of the query, since as humans we make conjectures such as “the toppings tomato and mozarella typically don’t make a pizza spicy”. This more intuitive result can also not be achieved by just posing a closed world query, asking for all spicy pizzas and then inverting the result by taking all the others. In this case we would, besides the pizzas margarita and mildChili, also get the pizza normalChili, which we would intuitively conjecture to be typically spicy due to its chili topping. Thus, there is no straightforward way to incorporate the conjectures about the spiciness into the querying by means of OWL-DL concepts.
The epistemic operators introduced in Section 2 can be used as a means to express such conjectures in the knowledge representation formalism, reducing ‘don’t know’ answers that OWL-DL reasoning potentially produces in situations of incomplete knowledge. In the following, we present three different techniques for realising local closed world reasoning by epistemic operators, namely epistemic querying, default rules and integrity constraints, which we apply in our scenario to yield the intuitively desired result when asking for non-spicy pizzas. 3.1
In epistemic queries, as introduced in Section 2, the K-operator is used to refer to locally closed off parts of the domain model. They provide a means to encode conjectures, like the ones made above, directly into the query.
In our example, the agent could pose an epistemic query, asking for “pizzas that are either non-spicy or not known to be spicy but known to have only nonchili toppings”. Querying the annotations of Giovanni and Alberto, this would yield the intuitively desired result as follows.
Q ≡ ¬SpicyDish ⊔ (¬KSpicyDish ⊓ K∀ topping.¬Chili)
OP izza ∪ OGiovanni ∪ OAlberto |= {Q(mildChili), Q(margarita)} The epistemic query yields the pizzas mildChili, since it is declared as non-spicy, and margarita, since it is known to not have any chili topping.
To verify this entailment, we have to check whether the epistemic interpretation ( I, M(KB ), M(KB ) ) satisfies the above concept assertions for all first-order models I ∈ M(KB ), with KB := OP izza ∪ OGiovanni ∪ OAlberto. The assertion for mildChili is satisfied, since this individual is explicitly asserted to the first disjunct ¬SpicyDish of the query in OGiovanni. In the second disjunct, the epistemic concept KSpicyDish is interpreted as the intersection of individuals in SpicyDish over all first-order models of M(KB ). This eliminates those individuals which do not always belong to the concept extension for these models, in this case normalChili, mildChili and margarita. The negated expression ¬KSpicyDish therefore refers to exactly these individuals, which are not known to be spicy dishes. Analogously, for the epistemic concept K∀ topping.¬Chili those individuals are eliminated which are not excluded from having chili toppings, leaving only vesuf o and margarita. Hence, the assertion for margarita is also satisfied.
In general, epistemic queries can be used to make conjectures on the side of a Semantic Web agent, in settings where the original ontologies involved shall be leaved untouched. In such a setting each agent can then make its own conjectures when querying shared ontologies. 3.2
A default rule, according to [
Default rules provide a means to incorporate conjectures into the domain knowledge. In our example scenario, the designers of the domain ontology OP izza could decide to make the conjecture “pizzas with chili toppings are typically spicy, whereas pizzas without chili toppings are typically non-spicy” part of the domain knowledge for pizzas by means of the following default rules.
DP izza = { Da = K∃ topping.Chili ⊓ ¬A¬SpicyDish ⊑ KSpicyDish ,
Db = K∀ topping.¬Chili ⊓ ¬ASpicyDish ⊑ K¬SpicyDish }
By this they would achieve that pizzas, for which there is no evidence of whether
they are spicy or not in OP izza, are concluded to be spicy or non-spicy in OP izza ∪
DP izza by default, depending on whether they are known to have chili toppings or
non-chili toppings only. In a joint ontology OP izza ∪DP izza ∪OGiovanni ∪OAlberto we
would intuitively like these default rules to be applied on the pizzas normalChili
and margarita, concluding that normalChili is spicy and that margarita is not.
Contrarily, we would not like the default rules to be applied on the pizzas
mildChili and vesuf o, since we already know about their spiciness. In order
to verify the appropriate application of the default rules, we will determine the
epistemic models of the ontologies involved. For sake of simplicity, we will only
consider the knowledge base KB := OP izza ∪ OAlberto together with the default
rule Db .
3 We exclude prerequisite-free defaults (no presence of α) and cases where α=⊤, see [
If we check whether M2 is also an epistemic model of KB ∪ {Db}, we observe that ( I, M2, M2 ) does also satisfy the axioms in KB ∪ {Db}. However, M2 does not fulfil the maximality condition: if we consider the set M′2 := M2 ∪ {I′}, for some I′ ∈ M1, then ( I, M′2, M2 ) does not contradict the inclusion because margarita is in SpicyDish J ,M′2,M2 for all J ∈M2.
Having determined M1 as the only epistemic model of KB ∪ {Db}, and with a similar reasoning for the pizzas in OGiovanni together with Da , our agent can draw the following additional conclusions.
OP izza ∪ DP izza ∪ OGiovanni ∪ OAlberto |=
{ SpicyDish(normalChili), ¬SpicyDish(margarita) } In particular, this leaves no pizza for which it cannot be concluded whether it is spicy or not. Therefore the agent can now safely use the non-epistemic query concept ¬SpicyDish to retrieve the desired non-spicy pizzas.
In general, default rules can be used for including commonly agreed conjectures in the domain knowledge. This relieves Semantic Web agents from the burden of making such conjectures themselves. 3.3
So far, in our scenario, we derived additional conclusions, based on conjectures, to deal with incomplete knowledge about the spiciness of pizzas in ontologies. An alternative would be to not allow such incomplete information about spiciness, and to force pizza delivery services to explicitly classify all their pizzas accordingly by means of integrity constraints.
An integrity constraint is used to check the state of a knowledge base without
deriving new facts – something that cannot be done in OWL-DL. In [
In our example, the designers of OP izza could include an integrity constraint requiring that any pizza is classified as either spicy or non-spicy, invalidating knowledge bases with non-classified pizzas.
IC P izza = { KP izza ⊑ (ASpicyDish ⊔ A¬SpicyDish) } The integrity constraint in IC P izza says that any individual that is known to be a pizza can either be assumed to be spicy or assumed to be non-spicy. Recall that such an assumption, expressed through the A-operator, requires a justification by other facts in a knowledge base. Both the pizzas normalChili and margarita fail to be determined as either spicy or non-spicy, which results in both OP izza ∪ IC P izza ∪ OGiovanni and OP izza ∪ IC P izza ∪ OAlberto being unsatisfiable.
To exemplarily verify this unsatisfiability for the knowledge base KB := OP izza ∪ IC P izza ∪ OAlberto, consider a set of first-order interpretations M1 = {I ∈ M(KB ) : I |= SpicyDish(margarita)}, in which the pizza margarita is constantly spicy. The epistemic interpretation ( I, M1, M1 ) satisfies KB , since margarita is in P izza J ,M1,M1 and also in SpicyDish J ,M1,M1 for all J ∈ M1. However, for any set M′1 ⊃ M1 the epistemic interpretation ( I, M′1, M1 ) also satisfies KB , since margarita is still in SpicyDish J ,M′1,M1 for all J ∈ M1. A similar reasoning can be done for a set of first-order models in which margarita is constantly non-spicy, and hence KB is unsatisfiable.
In general, integrity constraints can be used in cases where conjectures cannot be safely made on either side and where modelers should be forced to explicate certain information. Observe, that in OWL-DL there is no way to express such an integrity constraint allowing to detect the improper modelling in Giovanni’s and Alberto’s ontologies. 4
In this paper we have presented a case for extending OWL with non-monotonic
features by means of autoepistemic description logics [
Although [
While modelling ontologies in OWL-DL is already a complicated task for
many users, ADLs make things even more complicated for the modeller. A
promising alternative to the free use of epistemic operators would therefore be
to employ direct modelling constructs for default rules or integrity constraints.
In any case, a more expressive knowledge representation formalism requires
additional care to be taken, and new modelling methodologies and patterns need
to be introduced in order to handle e.g. situations with conflicting defaults [