We propose an approach based on Answer Set Programming for reasoning about actions with domain descriptions including ontological knowledge, expressed in the lightweight description logic ℰℒ⊥. We consider a temporal action theory, which allows for non-deterministic actions and causal rules to deal with ramifications, and whose extensions are defined by temporal answer sets. We provide conditions under which action consistency can be guaranteed with respect to an ontology, by a polynomial encoding of an action theory extended with an ℰℒ⊥ knowledge base (in normal form) into a temporal action theory.
The integration of description logics and action formalisms has gained a lot of interest in the
past years [
As usual in many formalisms integrating description logics and action languages [
The approach proposed by Baader et al. [
The paper studies extended temporal action theories, combining the temporal action logic mentioned above with an ℰ ℒ⊥ knowledge base. It is shown that, for ℰ ℒ⊥ knowledge bases in normal form, the consistency of the action theory extensions with the ontology can be verified by adding to the action theory a set of causal laws and state constraints, by exploiting a fragment of the materialization calculus by Krötzsch [29]. Furthermore, sufficient conditions on the action theory can be defined to repair the states resulting from action execution and guarantee consistency with TBox. To this purpose, for each inclusion axiom in TBox, a suitable set of causal laws can be added to the action theory. Our approach provides a polynomial encoding of an extended action theory, with an ℰ ℒ⊥ knowledge base in normal form, into the language of the (DLTL) temporal action logic studied in [23]. The proof methods for this temporal action logic, based on bounded model checking, can then be exploited for reasoning about actions in the extended action theory.
A preliminary version of this work, which does not exploit a temporal action language, has been presented in CILC 2016 [20].
We consider a fragment of the logic ℰ ℒ++ [
We let be a set of concept names, a set of role names and a set of individual names. A concept in ℰ ℒ⊥ is defined as follows:
:= | ⊤ | ⊥ | ⊓ | ∃. | {} where ∈ and ∈ . Observe that complement, disjunction and universal restriction are not allowed in ℰ ℒ⊥.
A knowledge base K is a pair ( , ), where is a TBox containing a finite set of concept inclusions 1 ⊑ 2 and is an ABox containing assertions of the form () and (, ), with , 1, 2 concepts, ∈ and , ∈ .
We will assume that the TBox is in normal form [
In the following we will denote with ,K , ,K and ,K the (finite) sets of concept names, role names and individual names occurring in K .
Definition 1 (Interpretations and models). An interpretation in ℰ ℒ⊥ is any structure (∆ , · ) where: ∆ is a domain; · is an interpretation function that maps each concept name to set ⊆ ∆ , each role name to a binary relation ⊆ ∆ × ∆ , and each individual name to an element ∈ ∆ . Furthermore: ⊤ = ∆ , ⊥ = ∅; {} = { }; ( ⊓ ) = ∩ ; (∃.) = { ∈ ∆ | ∃ ∈ : (, ) ∈ }. An interpretation (∆ , · ) satisfies an inclusion ⊑ if ⊆ ; it satisfies an assertion () if ∈ ; it satisfies an assertion (, ) if ( , ) ∈ .
Given a knowledge base K = ( , ), an interpretation (∆ , · ) is a model of if (∆ , · ) satisfies all inclusions in ; (∆ , · ) is a model of K if (∆ , · ) satisfies all inclusions in and all assertions in . is consistent with if there is a model of satisfying all the assertions in .
In this paper we refer to the notion of the temporal action theory in [19], a rule based fragment of which has been studied in [23, 24], which exploits the dynamic extension of LTL introduced by Henriksen and Thiagarajan, called Dynamic Linear Time Temporal Logic (DLTL) [28]. In DLTL the next state modality is indexed by actions and the until operator is indexed by a program which, as in PDL, can be any regular expression built from atomic actions using sequence (;), nondeterministic choice (+) and finite iteration ( * ). The derived modalities ⟨ ⟩ and [ ] can be defined as: ⟨ ⟩ ≡ ⊤ and [ ] ≡ ¬⟨ ⟩¬ . Similarly, ○ (next), ◇ and □ operators of LTL can be defined. We let Σ be a finite non-empty set of (atomic) actions and we refer to [ 19, 23] for the details concerning complex actions.
A domain description Π is a set of laws describing the effects of actions and their executability preconditions. Atomic propositions describing the state of the domain are called fluents . Actions may have direct effects, described by action laws, and indirect effects, described by causal laws capturing the causal dependencies among fluents.
Let ℒ be a first order language which includes a finite number of constants and variables, but no function symbol. Let be the set of predicate symbols, the set of variables and the set of constant symbols. We call fluents atomic literals of the form (1, . . . , ), where, for each , ∈ ∪ . A simple fluent literal (or s-literal) is an atomic literals (1, . . . , ) or its negation ¬(1, . . . , ). We denote by the set of all simple fluent literals. is the set of temporal fluent literals : if ∈ , then [], ○ ∈ , where is an action name (an atomic proposition, possibly containing variables), and [] and ○ are the temporal operators introduced in the previous section. Let = ∪ ∪ {⊥, ⊤}, where ⊥ represents the inconsistency and ⊤ truth. Given a (simple or temporal) fluent literal , represents the default negation of . A (simple or temporal) fluent literal possibly preceded by a default negation, will be called an extended fluent literal .
The laws are formulated as rules of a temporally extended logic programming language. Rules have the form □(0 ← 1, . . . , , +1, . . . , ) (1) where the ’s are either simple fluent literals or temporal fluent literals, with the following constraints: (i) If 0 is a simple literal, then the body cannot contain temporal literals; (ii) If 0 = [], then the temporal literals in the body must have the form []′; (iii) If 0 = ○ , then the temporal literals in the body must have the form ○ ′. As usual in ASP, the rules with variables will be used as a shorthand for the set of their ground instances.
In the following we use of a notion of state: a set of ground fluent literals. A state is said to be consistent if it is not the case that both and ¬ belong to the state, or that ⊥ belongs to the state. A state is said to be complete if, for each fluent , either or ¬ belong to the state. The execution of an action in a state may possibly change the values of fluents in the state through its direct and indirect effects, thus giving rise to a new state.
While a law as (1) can be applied in all states, a law 0 ← 1, . . . , , +1, . . . , (2) which is not prefixed by □, only applies to the initial state.
A domain description can be defined as a pair (Π , ), consisting of a set of laws Π and a set of temporal constraints . The following action laws describe the deterministic effect of the actions shoot and load for the Russian Turkey problem, as well as the nondeterministic effect of action spin, after which the gun may be loaded or not: □([ℎ]¬ ← ) □[] □([] ← []¬) □([]¬ ← [])
The following precondition laws: □([]⊥ ← ) specifies that, if the gun is loaded, is not executable. The program (¬_ℎ?; )* ; _ℎ?; ; ℎ describes the behavior of the hunter who waits for a turkey until it appears and, when it is in sight, loads the gun and shoots. Actions _ℎ? and ¬_ℎ? are test actions (we refer to [23]). If the constraint ⟨(¬_ℎ?; )* ; _ℎ?; ; ℎ⟩⊤ is included in then all the runs of the domain description which do not start with an execution of the given program will be filtered out. For instance, an extension in which in the initial state the turkey is not in sight and the hunter loads the gun and shoots is not allowed.
As we will see later, this temporal language is also well suited to describe causal dependencies
among fluents as static and dynamic causal laws similar to the ones in the action languages
[
The semantics of a domain description has been defined based on temporal answer sets [23],
which extend the notion of answer set [
We define the satisfiability of a simple, temporal or extended literal in a partial temporal interpretation (, ) in the state 1 . . . , (written (, ), 1 . . . |= ) as: (, ), 1 . . . |= ⊤, (, ), 1 . . . ̸|= ⊥ (, ), 1 . . . |= iff [1; . . . ; ] ∈ , for s-literal (, ), 1 . . . |= [] iff [1; . . . ; ; ] ∈ or
1 . . . , is not a prefix of (, ), 1 . . . |= ○ iff [1; . . . ; ; ] ∈ ,
where 1 . . . is a prefix of (, ), 1 . . . |= iff (, ), 1 . . . ̸|= The satisfiability of rule bodies in a temporal interpretation is defined as usual. A rule □( ← ) is satisfied in a temporal interpretation (, ) if, for all action sequences 1 . . . (including the empty action sequence ), (, ), 1 . . . |= implies (, ), 1 . . . |= . A rule ← is satisfied in a partial temporal interpretation (, ) if, (, ), |= implies (, ), |= .
Let Π be a set of rules over an action alphabet Σ , not containing default negation, and let ∈ Σ .
Definition 2. A temporal interpretation (, ) is a temporal answer set of Π if is minimal (with respect to set inclusion) among the ′ such that (, ′) is a partial interpretation satisfying the rules in Π .
To define answer sets of a program Π containing negation, given a temporal interpretation
(, ) over ∈ Σ , the reduct, Π (, ), of Π relative to (, ) is defined, by extending Gelfond
and Lifschitz’ transform [
Definition 3. The reduct, Π (,1,...),ℎ , of Π relative to (, ) and to the prefix 1, . . . , ℎ of , is the set of all the rules [1; . . . ; ℎ]( ← 1, . . . , ) such that ← 1, . . . , , +1, . . . , is in Π and (, ), 1, . . . , ℎ ̸|= , for all = + 1, . . . , .
The reduct Π (, ) of Π relative to (, ) is the union of all reducts Π (,1,...),ℎ for all prefixes 1, . . . , ℎ of .
In definition above, we say that rule [1; . . . ; ℎ]( ← ) is satisfied in a temporal interpretation (, ) if (, ), 1 . . . |= implies (, ), 1 . . . |= .
Definition 4 ([23]). A temporal interpretation (, ) is an answer set of Π if (, ) is an answer set of the reduct Π (, ).
Observe that a partial interpretation (, ) provides, for each prefix 1 . . . , a partial evaluation of fluents in the state corresponding to that prefix. The (partial) state (,1...) obtained by the execution of the actions 1 . . . in the sequence can be defined as: (,1...) = { : [1; . . . ; ] ∈ }.
Although the answer sets of a domain description Π are partial interpretations, in some cases, e.g., when the initial state is complete and all fluents are inertial, it is possible to guarantee that the temporal answer sets of Π are total. The case of total temporal answer sets is of special interest as a total temporal answer set (, ) can be regarded as a temporal model.
In this section we define a notion of extended temporal action theory, consisting of a temporal
action theory plus an ℰ ℒ⊥ knowledge base. Our approach, following most of the proposals for
reasoning about actions in DLs [
We want to regard DL assertions as fluents that may occur in our action laws as well as in the states of the action theory. Given a (normalized) ℰ ℒ⊥ knowledge base K = ( , ), we require that: (a) for each (possibly complex) concept occurring in K there is a unary predicate ∈ ; (b) for each role name ∈ ,K there is a binary predicate ∈ ; (c) the set of constants includes all the individual names occurring in the K , i.e. ,K ⊆ .
Observe that if a complex concept such as ∃. occurs in K , there exists a predicate name
∃. ∈ and, for each ∈ ,KB , the fluent literals (∃.)() and ¬(∃.)() belong to the
set (we will still call such literals assertions). Although classical negation is not allowed in
ℰ ℒ⊥, we use explicit negation [
A simple literal in is said to be a simple assertion if it has the form () or (, ) or ¬() or ¬(, ), where ∈ K is a base concept in K , ∈ ,K and , ∈ ,K . Observe that {}() and ¬{}() are simple assertions, while (∃.)() and ¬(∃.)() are non-simple assertions.
In order to deal with existential restrictions, in addition to the individual names ,KB occurring in the KB we introduce a finite set of auxiliary individual names, as proposed in [29] to encode ℰ ℒ⊥ inference in Datalog, where contains a new individual name ⊑∃. , for each inclusion ⊑ ∃. occurring in the KB . We further require that ⊆ . Definition 5 (Extended action theory). An extended action theory is a tuple (K , Π , ), where: K = ( , ) is an ℰ ℒ⊥ knowledge base; Π is a set of laws: action, causal, executability and initial state laws (see below); is a set of temporal constraints.
Action laws describe the immediate effects of actions. They have the form: □([]0 ← 1, . . . , , +1, . . . , ) (3) where 0 is a simple fluent literal and the ’s are either simple fluent literals or temporal fluent literals of the form []. Its meaning is that executing action in a state in which the conditions 1, . . . , hold and conditions +1, . . . , do not hold causes the effect 0 to hold. As an example, [Assign(c, x )]Teaches(x , c) ← () means that executing the action of assigning a course to has the effect that teaches . Non-deterministic effects of actions can be defined using default negation in the body of action laws, as for the action spin in Section 3.
Causal laws describe indirect effects of actions. They have the form: In Π we allow two kinds of causal laws. Static causal laws have the form: (4) (5) (6) □(0 ← 1, . . . , , +1, . . . , ) where the ’s are simple fluent literals. Their meaning is: if 1, . . . , hold in a state and +1, . . . , do not hold in that state, than 0 is caused to hold in that state. Dynamic causal laws have the form:
□(○ 0 ← 1, . . . , , +1, . . . , )
where 0 is a simple fluent literal and the ’s are either simple fluent literals or temporal fluent
literals of the form ○ . For instance, ○ Teacher (x ) ← ○ Teaches(x , y ) ∧Course(y ). Observe
that, differently from [
Precondition laws describe the executability conditions of actions. They have the form: □([]⊥ ← 1, . . . , , +1, . . . , ) where ∈ Σ and the ’s are simple fluent literals. The meaning is that the execution of an action is not possible in a state in which 1, . . . , hold and +1, . . . , do not hold
Initial state laws are needed to introduce conditions that have to hold in the initial state. They have the form:
0 ← 1, . . . , , +1, . . . , where the ’s are simple fluent literals. Observe that initial state laws, unlike static causal laws, only apply to the initial state as they are not prefixed by the □ modality. As a special case, the initial state can be defined as a set of simple fluent literals. For instance, the initial state {, ¬_ℎ, ¬ ℎ} is defined by the initial state laws: , ¬_ℎ, ¬ ℎ.
Following Lifschitz [30] we call frame fluents the fluents to which the law of inertia applies. Persistence of frame fluents from a state to the next one can be captured by introducing in Π a set of causal laws, said persistence laws for all frame fluents .
□(○ ← , ○ ¬
)
□(○¬
← ¬
, ○ )
meaning that, if holds in a state, then will hold in the next state, unless its negation ¬ is
caused to hold (and similarly for ¬ ). Persistence of a fluent is blocked by the execution of an
action which causes the value of the fluent to change, or by a nondeterministic action which may
cause it to change. The persistence laws above play the role of inertia rules in [26], + [25]
and [
If is non-frame with respect to an action , is not expected to persist and may change its value when an action is executed, either non-deterministically: □(○ ← ○ ¬ ) □(○¬ ← ○ ) or by taking some default value (see [23] for some examples).
In the following we assume that persistence laws and non-frame laws can be applied to simple assertions but not to non-simple ones (such as (∃.)()), whose value in a state (as we will see) is determined from the value of simple assertions. For simple assertions () in a domain description, one has to choose whether the concept is frame or non-frame (so that either persistence laws or non-frame laws can be introduced). In particular, we assume that all the nominals always correspond to frame fluents: if {}() (respectively ¬{}()) belongs to a state, it will persist to the next state unless it is cancelled by the direct or indirect effects of an action.
ABox assertions may incompletely specify the initial state. As we want to reason about states corresponding to ℰ ℒ⊥ interpretations, we assume that the laws: ← ¬ and ¬ ← for completing the initial state are introduced in Π for all simple literals (including assertions with nominals). As shown in [23], the assumption of complete initial states, together with suitable conditions on the laws in Π , gives rise to semantic interpretations (extensions) of the domain description in which all states are complete. In particular, to guarantee that each reachable state in each extension of a domain description is complete for simple fluents, we assume that, either the lfuent is frame, and persistence laws are introduced for it, or it is non-frame, and non-frame laws are introduced. Other literals, such as existential assertions, are not subject to this requirement but, as we will see below, the value of existential assertions in a state will be determined from the value of simple assertions. Under the condition above, starting from an initial state which is complete for simple fluents, all the reachable states are also complete for simple fluents. We call well-defined the domain descriptions satisfying this condition (and sometimes we will simply say that Π is well-defined).
The third component of a domain description (, Π , ) is the set of temporal constraints in DLTL, which allow general temporal conditions to be imposed on the executions of the domain description. Their effect is that of restricting the space of the possible executions (or extensions). For a domain description = (Π , ), as introduced in [23], extensions are defined as follows. Definition 6. An extension of a well-defined domain domain description = (Π , ) over Σ is a (total) answer set (, ) of Π which satisfies the constraints in .
In the next section we extend the notion of extension to a domain description (, Π , ) with an ontology.
Given an action theory (K , Π , ), where K = ( , ), we define an extension of (K , Π , ) as an extension (, ) of the action theory (Π , ) satisfying all axioms of the ontology . Informally, each state in the extension is required to correspond to an ℰ ℒ⊥ interpretation and to satisfy all inclusion axioms in TBox . Additionally, the initial state must satisfy all assertions in the ABox .
To define the extensions of an action theory (K , Π , ), we restrict to well-defined domain descriptions (K , Π , ), so that all states in an extension are complete for simple fluents (and for simple assertions). We prove that such states represent ℰ ℒ⊥ interpretations on the language of , provided an additional set of laws Π ℒ() is included in the action theory. Next, we add to Π another set Π of constraints, to guarantee that each state satisfies the inclusion axioms in . Finally, we add to Π the set of laws Π , to guarantee that all assertions in are satisfied in the initial state.
Overall, this provides a transformation of the action theory (K , Π , ) into a new action theory (Π ∪ Π , ), by eliminating the ontology while introducing a set of static causal laws and constraints Π = Π ℒ() ∪ Π ∪ Π , intended to exclude those extensions which do not satisfy the axioms in .
The set of domain constraints and causal laws in Π ℒ() is intended to guarantee that any state of an extension respects the semantics of DL concepts occurring in K . The definition of Π ℒ() is based on a fragment of the materialization calculus for ℰ ℒ⊥, which provides a Datalog encoding of an ℰ ℒ⊥ ontology. Here, the idea is that of regarding an assertion () in a state as the evidence that ∈ in the corresponding interpretation. Let Π ℒ() be the following set of causal laws: (1) □(⊥ ← ⊥ ()) (4) □(__() ← (5) □((∃.)() ← (6) □(¬(∃.)() ← __()) __())
(2) □(⊤() ← ) (, ) ∧ ()) (3) □({}() ← ) (7) □(⊥ ← { (8) □(⊥ ← { (9) □(⊥ ← { }() ∧ () ∧ ()), for ̸= }() ∧ () ∧ ()), for ̸= }() ∧ (, ) ∧ (, )), for ̸= for all , ∈ ∆ , ∈ ,K , ∈ K (the base concepts occurring in K ) and ∈ ,K (the roles occurring in K ). Observe that the first constraint has the effect that a state , in which the concept ⊥ has an instance, is made inconsistent. Law (4) makes __() hold in any state in which there is a domain element such that (, ) and () hold (where __ is an additional auxiliary predicate for ∈ K and ∈ ,K ). Laws (5) and (6) guarantee that, for all ∈ ∆ , either (∃.)() or ¬(∃.)() is contained in the state. State constraints (7-9) are needed for the treatment of nominals and are related to materialization calculus rules (27-29) [29].
Let (, ) be an extension of the action theory (Π ∪ Π ℒ(), ). It can be proven that any state of (, ) represents an ℰ ℒ⊥ interpretation. Given a state , let + be the set of ℰ ℒ⊥ assertions () ((, )), such that () ∈ (resp., (, ) ∈ ). Let − be the set of ℰ ℒ⊥ assertions () ((, )), such that ¬() ∈ (resp., ¬(, ) ∈ ).
Proposition 1. Let (, ) be an extension of the action theory (Π ∪ Π ℒ(), ) and let be a state of (, ). Then there is an interpretation (∆ , · ) that satisfies all the assertions in + and falsifies all assertions in − (and we say that (∆ , · ) agrees with state ).
As mentioned, we are interested in the states satisfying the TBox of . Provided Π is welldefined, for each extension (, ) of the action theory (Π ∪ Π ℒ(), ), any state is consistent and complete for all simple literals (and hence, by (5) and (6), for all assertions). We say that satisfies the TBox if for all interpretations (∆ , · ) such that (∆ , · ) agrees with state , (∆ , · ) is a model of .
The requirement that each should satisfy the TBox can be incorporated in the action theory through a set of constraints, that we call Π , by exploiting the fact that the TBox is in normal form. Π contains the following state constraints: □(⊥ ← □(⊥ ← □(⊥ ← □(⊥ ← () ∧ ()), for each ⊑ in ; () ∧ () ∧ ()), for each ⊓ ⊑ in ; () ∧ (∃.)()), for each ⊑ ∃. in ; (∃.)() ∧ ()), for each ∃. ⊑ in ; where , ∈ K , ∈ K ∪ {⊥} and ∈ ,K ∪ . For = ⊥, the condition () is omitted. The following proposition can be proved for a well-defined Π . Proposition 2. Let (, ) be an extension of the action theory (Π ∪ Π ℒ(), ) and let be a state of (, ); satisfies iff satisfies all constraints in Π .
We can then add the constraints in Π to an action theory (Π ∪ Π ℒ(), ) to single out the extensions (, ) whose states all satisfy the TBox . In a similar way, we can restrict to answer (, ) satisfies all assertions in , by defining a set of initial state sets (, ) whose initial state constraints as: Π = {⊥ ← ()) | () ∈ } ∪ {⊥ ← (, )) | (, ) ∈ } We define the extensions of the extended action theory (K , Π , ) as the extensions of the action theory (Π ∪ Π , ), where Π = Π ℒ() ∪ Π ∪ Π .
Given this notion of extension of an action theory (K , Π , ), we can verify, in the temporal action logic, whether a sequence of actions is executable from the initial state (executability problem) or whether any execution of an action sequence makes some property (e.g., assertion) hold in all reachable states (temporal projection problem).
Let us consider the example from the introduction.
Example 1. Let K = ( , ) be a knowledge base such that = {∃Teaches.Course ⊑ Teacher } and = { Person(john), Course(cs1 )}. We assume that all simple assertions, i.e., Person(x ), Teacher (x ), Course(x ), Teaches(x , y ), are frame for all , ∈ ,K ∪ .
Let us consider a state 0 where John does not teach any course and is not a teacher. If an action Assign(cs1 , john) were executed in 0, given a Π containing the action law [Assign(cs1 , john)]Teaches(john, cs1 ), the resulting state would contain ∃Teaches.Course)(john), but would not contain Teacher (john), thus violating the state constraint □(⊥ ←
(∃Teaches.Course)() ∧ Teacher ()), in Π . In this case, there is no extension of the action theory in which action Assign(cs1 , john) can be executed in the initial state.
As observed in [
Can we identify which and how many conditions are needed to guarantee that an action theory is able to repair the state, after executing an action, so to satisfy all inclusions of a (normalized) ℰ ℒ⊥ TBox, when possible? Let us continue Example 1.
Example 2. Consider the case when action retire(john) is executed in a state 1 where Person(john), Course(cs1 ), Teaches(john, cs1 ) and Teacher (john) hold. Suppose that the action law: [retire(john)]¬Teacher (john) is in Π . Then, ¬Teacher (john) will belong to the new state (let us call it 2), but 2 will still contain the literals: Course(cs1 ), Teaches(john, cs1 ), which persist from the previous state (as Course and Teaches are frame fluents). Hence, 2 would violate the TBox . To avoid this, Π should contain some causal law to repair the inconsistency, for instance, □(¬Teaches(x , y ) ← ¬ Teacher (x ) ∧ Course(y )). By this causal law, when John retires he stops teaching all the courses he was teaching before. In particular, he stops teaching 1. On the contrary, □(¬Course(y ) ← ¬ Teacher (x ) ∧ Teaches(x , y )) would be unintended.
As we can see, the causal laws needed to restore consistency when an action is executed can in essence be obtained from the inclusions in the TBox and from their contrapositives, even though not all contrapositives are always wanted.
In general, while defining a domain description, one has to choose which causal relationships are intended and which are not. The choice depends on the domain and should not be done automatically, but some sufficient conditions to restore the consistency of the resulting state (if any) with a TBox (in normal form) can be defined.
The case of a (normalized) inclusion ⊑ , with , ∈ , is relatively simple; the execution of an action with effect () (but not ()), in a state in which none of () and () holds, would lead to a state which violates the constraints in the KB . Similarly for an action with effect ¬(). Deleting () should cause () to be deleted as well, if we want the inclusion ⊑ to be satisfied. Hence, to guarantee that the TBox is satisfied in the new state, for each inclusion ⊑ , two causal laws are needed: □(() ← ()) and □(¬() ← ¬ ()).
For an axiom ⊓ ⊑ ⊥, consider the concrete case ⊓ ⊑ ⊥, representing mutually exclusive states of a claim in a process of dealing with an insurance claim, we expect the following causal laws to be included:
□(¬() ← ()) □(¬() ← ()) even though the second one is only useful if a claim can become pending again after having become (temporarily) approved. For the general case, we have the following.
Let us define a set Π ( ) of causal laws associated with as follows: • For ⊑ in : □(() ← ()) and □(¬() ← ¬ ()); • For ⊓ ⊑ : □(() ← () ∧ ()) and at least one among
□(¬() ← ¬ () ∧ ()) and □(¬() ← ¬ () ∧ ()); • For ⊑ ∃.: □((, ⊑∃.) ← ()), □((⊑∃.) ← ()) and □(¬() ← ¬ (∃.)()); • For ∃. ⊑ in : □(() ← (∃.)(), □(¬(∃.)() ← ¬ ()) and at least one of: □(¬(, ) ← ¬ () ∧ ()) and □(¬() ← ¬ () ∧ (, )).
Proposition 3. Given a well-defined action theory (Π , ) and a TBox , any state of an extension (, ) of (Π ∪ Π ℒ() ∪ Π ( ) ∪ Π , ) satisfies .
Observe that, for the case for ⊑ ∃., from (, ⊑∃.) and (⊑∃.) (∃.)() is caused by laws (4-5) in Π ℒ(). While the causal laws in Π ( ) are sufficient to guarantee the consistency of a resulting state with TBox , one cannot exclude that some action effects are inconsistent with TBox and cannot be repaired (e.g., an action with direct effects () and ¬(), conflicting with an axiom ⊑ in ). In such a case, the action would not be executable.
Notice that the encoding above of ℰ ℒ⊥ TBox into a set of temporal laws only requires a polynomial number of laws to be added to the action theory (in the size of ). Based on this mapping, the proof methods for our temporal action logic, which are based on the ASP encodings of bounded model checking [22, 23, 24], can be exploited for reasoning about actions in an action theory extended with an ℰ ℒ⊥ knowledge base.
In this paper we have proposed an approach for reasoning about actions by combining a temporal
action logic in [23], whose semantics is based on a notion of temporal answer sets, and an ℰ ℒ⊥
ontology. It is shown that, for ℰ ℒ⊥ knowledge bases in normal form, the consistency of the
action theory extensions with the ontology can be verified by adding to the action theory a set
of causal laws and state constraints, by exploiting a fragment of the materialization calculus
by Krötzsch [29]. Starting from the idea by Baader et al. [
Many of the proposals in the literature for combining DLs with action theories focus on
expressive DLs. In their seminal work [
The requirement of acyclic TBoxes is lifted in the work by Liu et al. [32], where an approach to the ramification problem is proposed which does not use causal relationships, but exploits occlusion to provide a specification of the predicates that can change through the execution of actions. The idea is to leave to the designer of an action domain the control of the ramifications.
Similar considerations are at the basis of the approach by Baader et al. [
Ahmetai et al. [
In [
Our semantics for actions, as many of the proposals in the literature, requires that a state
provides a complete description of the world and is intended to represent an interpretation of
the ℰ ℒ⊥ knowledge base. An alternative approach has been adopted in [
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