Interval-based Temporal Reasoning with General TBoxes

Until now, interval-based temporal Description Logics (DLs) did—if at all—only admit TBoxes of a very restricted form, namely acyclic macro deﬁni-tions. In this paper, we present a temporal DL that overcomes this deﬁcieny and combines interval-based temporal reasoning with general TBoxes. We argue that this combination is very interesting for many application domains. An automata-based decision procedure is devised and a tight E XP T IME - complexity bound is obtained. Since the presented logic can be viewed as being equipped with a concrete domain, our results can be seen from a different perspective: we show that there exist interesting concrete domains for which reasoning with general TBoxes is decidable.


Motivation
Description Logics (DLs) are a family of formalisms wellsuited for the representation of and reasoning about conceptual knowledge.Whereas most Description Logics represent only static aspects of the application domain, recent research resulted in the exploration of various Description Logics that allow to, additionally, represent temporal information, see, e.g., [Artale and Franconi,2000] for an overview.One approach for temporal reasoning with DLs is to use so-called concrete domains.Concrete domains have been proposed as an extension of Description Logics that allows reasoning about "concrete qualities" of entities of the application domain such as sizes, weights or temperatures [Baader and Hanschke,1991].As was first described in [Lutz et al.,1997], if a "temporal" concrete domain is employed, then Description Logics with concrete domains are a very useful tool for temporal reasoning.Ontologically, temporal reasoning with concrete domains is usually interval-based but may also be point-based or even both.
In this paper, we define a temporal Description Logic based on concrete domains which uses points as its basic temporal entity, but which may also be used as a full-fledged intervalbased temporal DL.More precisely, the presented logic Ì Ä extends the basic Description Logic Ä with a concrete domain that is based on the rationals and predicates and .The well-known Allen relations can be defined in terms of their endpoints [Allen,1983] thus allowing for (qualitative) interval-based temporal reasoning.Since it is an important feature of DLs that reasoning should be decidable, we prove decidability of the standard reasoning tasks by using an automata-theoretic approach which also yields a tight EXP-TIME complexity bound.
Most DLs allow for some kind of TBox formalism that is used to represent terminological knowledge as well as background knowledge about the application domain.However, there exist various flavours of TBoxes with vast differences in expressivity.To the best of our knowledge, all intervalbased DLs and all DLs with concrete domains defined in the literature admit only a very restricted form of TBox, i.e., sets of acyclic macro definitions.Compared to existing Description Logics that are interval-based or include concrete domains, the distinguishing feature of our logic is that it is equipped with a very general form of TBoxes that allows arbitrary equations over concepts.Thus, the presented work overcomes a major limitation of both families of Description Logics.
Our results can be viewed from the perspective of intervalbased temporal reasoning and from the perspective of concrete domains.For the temporal perspective, we claim that the combination of general TBoxes and interval-based temporal reasoning is important for many application areas.In this paper, we present process engineering as an example.From the concrete domain perspective, our results can be viewed as follows: in [Lutz,2001], it is shown that, even for very simple concrete domains, reasoning with general TBoxes is undecidable.It was an open question whether there exist interesting concrete domains for which reasoning with general TBoxes is decidable.In this paper, we answer this question to the affirmative.This paper is accompanied by a technical report containing full proofs [Lutz,2000].

Syntax and Semantics
In this section, we introduce syntax and semantics of the Description Logic Ì Ä.
Definition 1.Let AE , AE Ê , and AE be mutually disjoint and countably infinite sets of concept names, roles, and concrete features.Furthermore, let AE be a countably infinite subset of AE Ê .The elements of AE are called abstract features.
Ò and one concrete feature .The set of Ì Ä- concepts is the smallest set such that 1. every concept name is a concept 2. if and are concepts, Ê is a role, is a concrete feature, Ù ½ Ù ¾ are paths, and È ¾ , then the following expressions are also concepts: , Ù , Ø , Ê , Ê , Ù ½ Ù ¾ È , and .
A TBox axiom is an expression of the form Ú with and are concepts.A finite set of TBox axioms is a TBox.Throughout this paper, we will denote atomic concepts by the letter , (possibly complex) concepts by the letters , roles by the letter Ê, abstract features by the letter , concrete features by the letter , paths by the letter Ù, and elements of the set by the letter È .We will sometimes call the TBox formalism introduced above general TBoxes to distinguish it from other, weaker formalisms such as the ones in [Nebel,1990].As most Description Logics, Ì Ä is equipped with a Tarski-style semantics.
Definition 2. An interpretation Á is a pair ´¡Á ¡ Á µ, where ¡ Á is a set called the domain and ¡ Á is the interpretation function mapping each concept name to a subset Á of ¡ Á , each role name Ê to a subset Ê Á of ¡ Á ¢ ¡ Á , each abstract feature to a partial function Á from ¡ Á to ¡ Á , and each concrete feature to a partial function Á from ¡ Á to the rationals É.For paths Ù ½ ¡ ¡ ¡ Ò , we set ´ ¡ .This property ensures that every concept can be converted into an equivalent one in the so-called negation normal form (NNF).The NNF of concepts, in turn, is used as a starting point for devising satisfiability algorithms.It is not hard to see that is not admissible in this sense.However, as we will see in Section 4, the conversion of Ì Ä-concepts into equivalent ones in NNF is nevertheless possible.

Temporal Reasoning with Ì Ä
Although Ì Ä does only provide the relations " " and " " on time points, it is not hard to see that the remaining relations can be defined by writing, e.g., Ù ¾ Ù ½ Ø Ù ½ Ù ¾ for Ù ½ Ù ¾ .However, we claim that Ì Ä cannot only be used for point-based temporal reasoning but also as a full-fledged interval-based temporal Description Logic.As observed by Allen [1983], there are 13 possible relationships between two intervals such as, for example, the Ñ Ø× relation: two intervals ½ and ¾ are related by Ñ Ø× iff the right endpoint of ½ is identical to the left endpoint of ¾ -see [Allen,1983] for an exact definition of the other relations.As we shall see, these 13 Allen relations (as well as the additional relations from the Allen algebra, see [Allen,1983]) can be defined in Ì Ä.
In the following, we present a framework for mixed intervaland point-based reasoning in Ì Ä and apply this framework in the application area of process engineering The representation framework consists of several conventions and abbreviations.We assume that each entity of the application domain is either temporal or atemporal.If it is temporal, its temporal extension may be either a time point or an interval.Left endpoints of intervals are represented by the concrete feature , right endpoints of intervals are represented by the concrete feature Ö, and time-points not related to intervals are represented by the concrete feature Ø.All this can be expressed by the following TBox Ì £ : Here, is an abbreviation for Ú Ú . Let us now define the Allen relations as abbreviations.For example, ´ ¼ µ ÓÒØ Ò× is an abbreviation for ¼ Ù ¼ Ö Ö where ¼ ¾ ´AE µ £ , i.e., and ¼ are words over the alphabet AE .Note that Similar abbreviations are introduced for the other Allen relations.We use × Ð to denote the empty word.For example, ´ × Ð µ ×Ø ÖØ× is an abbreviation for Ù Ö Ö Intuitively, × Ð refers to the interval associated with the abstract object at which the ´ × Ð µ ×Ø ÖØ× concept is "evaluated".
Since we have intervals and points available, we should also be able to talk about the relationship of points and intervals.More precisely, there exist 5 possible relations between a point and an interval [Vilain,1982], one example being ×Ø ÖØ×Ô that holds between a point Ô and an interval if Ô is identical to the left endpoint of .Hence, we can define ´ Ø ¼ µ ×Ø ÖØ×Ô as an abbreviation for Ø ¼ and similar abbreviations for ÓÖ Ô, ÙÖ Ò Ô, ¬Ò × ×Ô, and Ø ÖÔ.
This finishes the definition of the framework.We claim that the combination of interval-based reasoning and general TBoxes is important for many application areas such as reasoning about action and plans [Artale and Franconi,2000].The examples presented here are from the area of process engineering that was first considered by Sattler in a DL context [Sattler,1998].However, Sattler's approach does not take into account temporal aspects of the application domain.We show how this can be done using Ì Ä thus refining Sattler's proposal.
Assume that our goal is to represent information about an automated chemical production process that is carried out by some complex technical device.The device operates each day for some time depending on the number of orders.It needs a complex startup and shutdown process before resp.after operation.Moreover, some weekly maintenance is needed to keep the device functional.Let us first represent the underlying temporal structure that consists of weeks and days.
The axiom states that each week consists of seven days, where the 'th day is accessible from the corresponding week via the abstract feature Ý .The temporal relationship between the days are as expected: Monday starts the week, Sunday finishes it, and each day temporally meets the succeeding one.Note that this implies that days 2 to 6 are during the corresponding week although this is not explicitly stated.Moreover, each week has a successor week that it temporally meets.We now describe the startup, operation, shutdown, and maintenance phases.
Here ×Ø ÖØ, ÓÔ, × ÙØ, and Ñ ÒØ are abstract features and "AE" is used for better readability (i.e., paths ½ ¡ ¡ ¡ are written as ½ AE ¡ ¡ ¡ AE AE ).The TBox implies that phases are related to the corresponding day as follows: startup via ×Ø ÖØ× or ÙÖ Ò , shutdown via ÙÖ Ò or ¬Ò × ×, and operation via ÙÖ Ò .Until now, we did not say anything about the temporal relationship of maintenance and operation.This may be inadequate, if, for example, maintenance and operation are mutually exclusive.We can take this into account by using additional axioms where ÇÎÄÈ is replaced by ÕÙ Ð, ÓÚ ÖÐ Ô×, ÓÚ ÖÐ ÔÔ -Ý, ÙÖ Ò , ÓÒØ Ò×, ×Ø ÖØ×, ×Ø ÖØ -Ý, ¬Ò × ×, or ¬Ò × -Ý yielding 9 axioms.Until now, we have modelled the very basic properties of our production process.Let us define some more advanced concepts to illustrate reasoning with Ì Ä.For example, we could define a busy week as , each day, the startup process starts at the beginning of the day and the shutdown finishes at the end of the day.Say now that it is risky to do maintenance during startup and shutdown phases and define expressing that, in a risky week, the maintenance phase is not strictly separated from the startup and shutdown phases.
A Ì Ä reasoner could be used to detect that Ù×ÝÏ Ú Ê × ÝÏ , i.e., every busy week is a risky week: in a busy week, the week is partitioned into startup, shutdown, and operation phases.Since maintenance may not OVLP with operation phases (see (£)), it must OVLP with startup and/or shutdown phases which means that it is a risky week.We can further refine this model by using mixed point-based and interval-based reasoning, see [Lutz,2000] for examples.

The Decision Procedure
In this section, we prove satisfiability of Ì Ä-concepts w.r.t.
TBoxes to be decidable and obtain a tight EXPTIME complexity bound.This is done using an automata-theoretic approach: first, we abstract models to so-called Hintikka-trees such that there exists a model for a concept and a TBox Ì iff there exists a Hintikka-tree for and Ì .Then, we build, for each Ì Ä-concept and TBox Ì , a looping automaton ´ Ì µ that accepts exactly the Hintikka-trees for ´ Ì µ.
In particular, this implies that ´ Ì µ accepts the empty lan- guage iff is unsatisfiable w.r.t.Ì .
Definition 5. Let Å be a set and ½.A -ary Å-tree is a mapping Ì ½ £ Å that labels each node « ¾ ½ £ with Ì ´«µ ¾ Å. Intuitively, the node « is the -th child of «.We use ¯to denote the empty word (corresponding to the root of the tree).
A looping automaton ´É Å Á ¡µ for -ary Å-trees is defined by a set É of states, an alphabet Å, a subset Á É of initial states, and a transition relation ¡ É Figure 1: A constraint graph containing no -cycle that is unsatisfiable over AE.
½ £ A looping automaton accepts all those Å-trees for which a run exists, i.e., the language Ä´ µ of Å-trees accepted by is Ä´ µ Ì there is a run of on Ì In [Vardi and Wolper,1986], it is proved that the emptiness problem for looping automata is decidable in polynomial time.
A Hintikka-tree for and Ì corresponds to a canonical model for and Ì .Apart from describing the abstract domain ¡ Á of the corresponding canonical model Á together with the interpretation of concepts and roles, each Hintikkatree induces a directed graph whose edges are labelled with predicates from .These constraint graphs describe the "concrete part" of Á (i.e., concrete successors of domain objects and their relationships).Definition 6.A constraint graph is a pair ´Î µ, where Î is a countable set of nodes and Î ¢Î ¢ a set of edges.We generally assume that constraint graphs are equality closed, i.e., that ´Ú½ Ú ¾ µ ¾ implies ´Ú¾ Ú ½ µ ¾ .
A constraint graph ´Î µ is called satisfiable over Å, where Å is a set equipped with a total ordering , iff there exists a total mapping AE from Î to Å such that AE´Ú ½ µ È AE´Ú ¾ µ for all ´Ú½ Ú ¾ È µ ¾ .In this case, AE is called a solution for .
The following theorem will be crucial for proving that, for every Hintikka-tree, there exists a corresponding canonical model.More precisely, it will be used to ensure that the constraint graph induced by a Hintikka-tree, which describes the concrete part of the corresponding model, is satisfiable.
Theorem 7. A constraint graph is satisfiable over Å with Å ¾ É Ê iff does not contain a -cycle.
Note that Theorem 7 does not hold if satisfiability over AE is considered due to the absence of density: if there exist two nodes Ú ½ and Ú ¾ such that the length of -paths (which are defined in the obvious way) between Ú ½ and Ú ¾ is unbounded, a constraint graph is unsatisfiable over AE even if it contains no -cycle, see Figure 1.
The decidability procedure works on Ì Ä-concepts and TBoxes that are in a certain syntactic form.To define this normal form, we first introduce the well-known negation normal form.

Definition 8 (NNF).
A concept is in negation normal form (NNF) if negation occurs only in front of concept names.Every concept can be transformed into an equivalent one in NNF by eliminating double negation and using de Morgan's law, the duality between and , and the following equivalences: ´ where ¡ denotes the exchange of predicates, i.e., is and is .With ÒÒ ´ µ, we denote the equivalent of in NNF.
A TBox Ì is in NNF iff all concepts in Ì are in NNF.We can now extend NNF to the so-called path normal form.Definition 9 (Path Normal Form).A Ì Ä-concept is in path normal form (PNF) iff it is in NNF, and, for all subconcepts Ù ½ Ù ¾ È of , we have either ( 1 ´ µ ´ Ùµ ´ Ù℄ Ù℄ µ Ù ´Ùµ For every Ì Ä-concept , the corresponding concept ´ µ is obtained by replacing all subconcepts Ù ½ Ù ¾ È of with Ù ½ ℄ Ù ¾ ℄ È Ù ´Ù½ µÙ ´Ù¾ µ and with ℄ .We extend the mapping to TBoxes in the obvious way.Let be a Ì Äconcept and Ì a Ì Ä-TBox.By Definition 8, we may assume both and Ì to be in NNF.It is now easy to check that the translation is polynomial and that is satisfiable w.r.t.Ì iff ´ µ is satisfiable w.r.t.´Ì µ (see [Lutz,2000]). t Hence, it suffices to prove that satisfiability of concepts in PNF w.r.t.TBoxes in PNF is decidable.We often refer to TBoxes Ì in their concept form Ì : Ì Ù Ú ¾Ì ÒÒ ´ Ø µ We now define Hintikka-trees for concepts and TBoxes Ì (both in PNF) and show that there exists Hintikka-tree for and Ì iff there exists a model for and Ì .
Let be a concept and Ì a TBox.With Ð´ Ì µ, we denote the set of subconcepts of and Ì .We assume that existential concepts Ê in Ð´ Ì µ with Ê ¾ AE Ê Ò AE are linearly ordered, and that ´ Ì µ yields the -th ex- istential concept in Ð´ Ì µ.Furthermore, we assume the abstract features used in Ð´ Ì µ to be linearly ordered and use ´ Ì µ to denote the -th abstract feature in Ð´ Ì µ.The set of concrete features used in Ð´ Ì µ is denoted with ´ Ì µ.Hintikka-pairs are used as labels of the nodes in Hintikka-trees.Definition 11 (Hintikka-set, Hintikka-pair).Let be a concept and Ì be a TBox.A set © Ð´ Ì µ is a Hintikkaset for ´ Ì µ iff it satisfies the following conditions: © for all concept names ¾ Ð´ Ì µ, We say that for some ½ ¾ ¾ AE and È ¾ .A Hintikka-pair ´© µ for ´ Ì µ consists of a Hintikka-set © for ´ Ì µ and a set of tuples ´ ½ ¾ È µ With ´ Ì µ , we denote the set of all Hintikka-pairs for Intuitively, each node « of a (yet to be defined) Hintikkatree Ì corresponds to a domain object of the corresponding canonical model Á .The first component © « of the Hintikkapair labelling « is the set of concepts from Ð´ Ì µ satisfied by .The second component « states restrictions on the relationship between concrete successors of .If, for example, ´ ½ ¾ µ ¾ « , then we must have Á ½ ´ µ Á ¾ ´ µ.
Note that the restrictions in « are independent from con- cepts ½ ¾ È ¾ © « .As will become clear when Hintikkatrees are defined, the restrictions in « are used to ensure that the constraint graph induced by the Hintikka-tree Ì , which describes the concrete part of the model Á , does not contain a -cycle, i.e., that it is satisfiable.This induced constraint graph can be thought of as the union of smaller constraint graphs, each one being described by a Hintikka-pair labelling a node in Ì .These pair-graphs are defined next.
Definition 12 (Pair-graph).Let be a concept, Ì a TBox, and Ô ´© µ a Hintikka-pair for ´ Ì µ.The pair-graph ´Ôµ ´Î µ of Ô is a constraint graph defined as follows: 1. Î is the set of paths enforced by Ô 2.
Note that, due to path normal form and the definitions of Hintikka-pairs and pair-graphs, we have ¼ for every edge extension ¼ of a pair-graph ´Î µ.
We briefly comment on the connection of completions and the -component of Hintikka-pairs.Let « and ¬ be nodes in a Hintikka-tree Ì and let and be the corresponding domain objects in the corresponding canonical model Á .Edges in Hintikka-trees represent role-relationships, i.e., if ¬ is successor of « in Ì , then there exists an Ê ¾ AE Ê such that ´ µ ¾ Ê Á .Assume ¬ is successor of « and the edge between « and ¬ represents relationship via the abstract feature , i.e., we have Á ´ µ .The second component ¬ of the Hintikka-pair labelling ¬ fixes the relationships between all concrete successors of that " talks about".For example, if ´ ½ ¾ µ ¾ © « and ´ ¿ ¾ µ ¾ © « , where © « is the first component of the Hintikka-pair labelling «, then " talks about" the concrete ½ -successor and the concrete ¿ -successor of .Hence, ¬ either contains ´ ¿ ½ µ or ´ ½ ¿ È µ for some È ¾ .This is formalized by demanding that the pair-graph ´Ì ´«µµ of the Hintikka-pair labelling « together with all the edges from the -components of the successors of « are a completion of ´Ì ´«µµ.Moreover, this completion has to be satisfiable, which is necessary to ensure that the constraint graph induced by Ì does not contain a -cycle.An appropriate way of thinking about thecomponents is as follows: at «, a completion of ´Ì ´«µµ is "guessed".The additional edges are then "recorded" in the -components of the successor-nodes of «.
Definition 13 (Hintikka-tree).Let be a concept, Ì be a TBox, the number of existential subconcepts in Ð´ Ì µ, and be the number of abstract features in Ð´ Ì µ.
Whereas most properties of Hintikka-trees deal with concepts, roles, and abstract features and are hardly surprising, (H11) ensures that constraint graphs induced by Hintikkatrees contain no -cycle.By "guessing" a completion as explained above, possible -cycles are anticipated and can be detected locally, i.e., it suffices to check that the completions ÔÐ´Ì «µ are satisfiable as demanded by (H11).Indeed, it is crucial that the cycle detection is done by a local condition since we need to define an automaton which accepts exactly Hintikka-trees and automata work locally.It is worth noting that the localization of cycle detection as expressed by (H11) crucially depends on path normal form.
The following two lemmas show that Hintikka-trees are appropriate abstractions of models.Lemma 14.A concept is satisfiable w.r.t. a TBox Ì iff there exists a Hintikka-tree for ´ Ì µ.
To prove decidability, it remains to define a looping automaton ´ Ì µ for each concept and TBox Ì such that ´ Ì µ accepts exactly the Hintikka-trees for ´ Ì µ.
Definition 15.Let be a concept, Ì be a TBox, the number of existential subconcepts in Ð´ Ì µ, and be the number of abstract features in Ð´ Ì µ.The looping automaton ´ Ì µ ´É ´ Ì µ ¡ Áµ is defined as follows: ¯É ´ Ì µ , Á ´© µ ¾ É ¾ © , and Note that every state is an accepting state, and, hence, every run is accepting.The following lemma is easily obtained.
Since the size of Ð´ Ì µ is linear in the size of and Ì , it is straightforward to verify that the size of ´ Ì µ is exponential in the size of and Ì .This, together with Lemmas 10, 14, and 16, and the polynomial decidability of the emptiness problem of looping automata [Vardi and Wolper,1986], implies the upper bound given in the following theorem which states the main result of this paper.The lower bound is an immediate consequence of the fact that Ä with general TBoxes is EXPTIME-hard [Schild,1991].Theorem 17. Satisfiability and subsumption of Ì Ä- concepts w.r.t.TBoxes are EXPTIME-complete.

Conclusion
There are several perspectives for future work of which we highlight three rather interesting ones: firstly, the presented decision procedure is only valid if a dense strict linear order is assumed as the underlying temporal structure.For example, the concept is satisfiable w.r.t. the TBox over the temporal structures É and Ê (with the natural orderings) but not over AE.To see this, note that Ì induces a constraint graph as in Figure 1.Hence, it would be interesting to investigate if and how the presented algorithm can be modified for reasoning with the temporal structure AE.
Secondly, Ì Ä does only allow for qualitative temporal reasoning.It would be interesting to extend the logic to mixed qualitative and quantitative reasoning by additionally admitting unary predicates Õ and Õ for each Õ ¾ É.
Thirdly, we plan to extend Ì Ä to make it suitable for reasoning about entity relationship (ER) diagrams.As demonstrated in, e.g., [Calvanese et al.,1998;Artale and Franconi,1999], Description Logics are well-suited for this task.By using an appropriate extension of Ì Ä, one should be able to capture a new kind of temporal reasoning with ER diagrams, namely reasoning over ER diagrams with "temporal" integrity constraints.For example, a temporal integrity constraint could state that employees birthdays should be before their employment date.An appropriate extension of Ì Ä for this task could be by (unqualified) number restrictions, inverse roles, and a generalized version of the concrete domain constructor Ù ½ Ù ¾ È .An extension of the presented automata-theoretic decision procedure to this more complex logic seems possible.

Definition 4 (Concrete Domain). A concrete domain is a pair
and Ü ¾ É, then we call Ü a concrete successor of in Á .We write for Ø ´¡ ¨ µ, where ¡ is a set called the domain, and ¨ is a set of predicate names.Each predicate name È ¾ ¨ is associated with an arity Ò and an Ò-ary predicate È ¡ Ò .
for some ¾ AE and ½ ¾ ¾ AE .A Ì Ä TBox Ì is in path normal form iff it is in NNF and all concepts appearing in Ì are in path normal form.Lemma 10.Satisfiability of Ì Ä-concepts w.r.t.Ì Ä-TBoxes can be reduced to satisfiability of Ì Ä-concepts in PNF w.r.t.Ì Ä-TBoxes in PNF.
paths Ù in to concepts as follows: