A Temporal Description Logic for Reasoning about Actions and Plans
A class of interval-based temporal languages for uniformly representing and reasoning about actions and plans is presented. Actions are represented by describing what is true while the action itself is occurring, and plans are constructed by temporal…
Authors: A. Artale, E. Franconi
Journal of Articial In telligence Researc h 9 (1998) 463-506 Submitted 4/98; published 12/98 A T emp oral Description Logic for Reasoning ab out Actions and Plans Alessandro Artale ar t ale@irst.itc.it ITC-IRST, Co gnitive and Communic ation T e chnolo gies Division I-38050 Povo TN, Italy Enrico F ranconi franconi@cs.man.a c.uk Dep artment of Computer Scienc e, University of Manchester Manchester M13 9PL, UK Abstract A class of in terv al-based temp oral languages for uniformly represen ting and reasoning ab out actions and plans is presen ted. Actions are represen ted b y describing what is true while the action itself is o ccurring, and plans are constructed b y temp orally relating actions and w orld states. The temp oral languages are mem b ers of the family of Description Logics, whic h are c haracterized b y high expressivit y com bined with go o d computational prop erties. The subsumption problem for a class of temp oral Description Logics is in v estigated and sound and complete decision pro cedures are giv en. The basic language T L - F is considered rst: it is the comp osition of a temp oral logic T L { able to express in terv al temp oral net w orks { together with the non-temp oral logic F { a F eature Description Logic. It is pro v en that subsumption in this language is an NP-complete problem. Then it is sho wn ho w to reason with the more expressiv e languages T LU - F U and T L - ALC F . The former adds disjunction b oth at the temp oral and non-temp oral sides of the language, the latter extends the non-temp oral side with set-v alued features (i.e., roles) and a prop ositionally complete language. 1. In tro duction The represen tation of temp oral kno wledge has receiv ed considerable atten tion in the Ar- ticial In telligence comm unit y in an attempt to extend existing kno wledge represen tation systems to deal with actions and c hange. A t the same time, man y logic-based formalisms w ere dev elop ed and analyzed b y logicians and philosophers for the same purp oses. In this class of logical formalisms, prop erties suc h as expressiv e p o w er and computabilit y ha v e b een studied as regards t ypical problems in v olving ev en ts and actions. This pap er analyzes from a theoretical p oin t of view the logical and computational prop erties of a kno wledge represen tation system that allo ws us to deal with time, actions and plans in a uniform w a y . The most common approac hes to mo del actions are based on the notion of state change { e.g., the formal mo dels based on the original situation c alculus (McCarth y & Ha y es, 1969; Sandew all & Shoham, 1994) or the Strips -lik e planning systems (Fik es & Nilsson, 1971; Lifsc hitz, 1987) { in whic h actions are generally considered instan taneous and dened as functions from one state to another b y means of pre- and p ost-conditions. Here, an explicit notion of time is in tro duced in the mo deling language and actions are dened as o c curring over time intervals , follo wing the Allen prop osal (Allen, c 1998 AI Access F oundation and Morgan Kaufmann Publishers. All righ ts reserv ed. Ar t ale & Franconi 1991). In this formalism an action is represen ted b y describing the time course of the w orld while the action o ccurs. Concurren t or o v erlapping actions are allo w ed: eects of o v erlapping actions can b e dieren t from the sum of their individual eects; eects ma y not directly follo w the action but more complex temp oral relations ma y hold. F or instance, consider the motion of a p oin ter on a screen driv en b y a mouse: the p oin ter mo v es b ecause of the mo v emen t of the device on the pad { there is a cause-eect relation { but the t w o ev en ts are con temp orary , in the common-sense notion of the w ord. A class of in terv al temp oral logics is studied based on Description Logics and inspired b y the w orks of Sc hmiedel (1990) and of W eida and Litman (1992). In this class of formalisms a state describ es a collection of prop erties of the w orld holding at a certain time. A ctions are represen ted through temp oral constrain ts on w orld states, whic h p ertain to the action itself. Plans are built b y temp orally relating actions and states. T o represen t the temp oral dimen- sion classical Description Logics are extended with temp oral constructors; th us a uniform represen tation for states, actions and plans is pro vided. F urthermore, the distinction made b y Description Logics b et w een the terminological and assertional asp ects of the kno wledge allo ws us to describ e actions and plans b oth at an abstract lev el (action/plan t yp es) and at an instance lev el (individual actions and plans). In this en vironmen t, the subsumption calculus is the main inference to ol for managing collections of action and plan t yp es. Action and plan t yp es can b e organized in a subsumption-based taxonom y , whic h pla ys the role of an action/plan library to b e used for the tasks kno wn in the literature as plan retriev al and individual plan recognition (Kautz, 1991). A renemen t of the plan r e c o gnition no- tion is prop osed, b y splitting it in to the dieren t tasks of plan description classic ation { in v olving a plan t yp e { and sp e cic plan r e c o gnition with r esp e ct to a plan description { in v olving an individual plan. According to the latter reasoning task, the system is able to recognize whic h t yp e of action/plan has tak en place at a certain time in terv al, giv en a set of observ ations of the w orld. Adv an tages of using Description Logics are their high expressivit y com bined with de- sirable computational prop erties { suc h as decidabilit y , soundness and completeness of de- duction pro cedures (Buc hheit, Donini, & Sc haerf, 1993; Sc haerf, 1994; Donini, Lenzerini, Nardi, & Sc haerf, 1994; Donini, Lenzerini, Nardi, & Nutt, 1995). The main purp ose of this w ork is to in v estigate a class of de cidable temp oral Description Logics, and to pro vide com- plete algorithms for computing subsumption. T o this aim, w e start with T L - F , a language b eing the comp osition of a temp oral logic T L { able to express in terv al temp oral net w orks { together with the non-temp oral Description Logic F { a F eature Description Logic (Smolk a, 1992). It turns out that subsumption for T L - F is an NP-complete problem. Then, w e sho w ho w to reason with more expressiv e languages: T LU - F U , whic h adds disjunction b oth at the temp oral and non-temp oral sides of the language, and T L - ALC F , whic h extends the non-temp oral side with set-v alued features (i.e., roles) and a prop ositionally complete De- scription Logic (Hollunder & Nutt, 1990). In b oth cases w e sho w that reasoning is decidable and w e supply sound and complete pro cedures for computing subsumption. The pap er is organized as follo ws. After in tro ducing the main features of Description Logics in Section 2, Section 3 organizes the in tuitions underlying our prop osal. The tec hnical bases are in tro duced b y briey o v erviewing the temp oral extensions of Description Logics relev an t for this approac h { together with the in ter-relationships with the in terv al temp oral mo dal logic { sp ecically in tended for time and action represen tation and reasoning. The 464 A Temporal Description Logic f or Reasoning about A ctions and Plans basic feature temp oral language ( T L - F ) is in tro duced in Section 4. The language syn tax is rst describ ed in Section 4.1, together with a w ork ed out example illustrating the informal meaning of temp oral expressions. Section 4.2 rev eals the mo del theoretic seman tics of T L - F , together with a formal denition of the subsumption and instance recognition problems. Section 5 sho ws that the temp oral language is suitable for action and plan represen tation and reasoning: the w ell kno wn c o oking domain and blo cks world domain are tak en in to consideration. The sound and complete calculus for the feature temp oral language T L - F is presen ted in details in Section 6. A pro of that subsumption for T L - F is an NP-complete problem is included. The calculus for T L - F forms the basic reasoning pro cedure that can b e adapted to deal with logics ha ving an extended prop ositional part. An algorithm for c hec king subsumption in presence of disjunction ( T LU - F U ) is devised in Section 7.1; while in Section 7.2 the non-temp oral part of the language is extended with roles and full prop ositional calculus ( T L - ALC F ). In b oth cases, the subsumption problem is still decidable. Op erators for homogeneit y and p ersistence are presen ted in Section 8 for an adequate represen tation of w orld states. In particular, a p ossible solution to the fr ame pr oblem , i.e., the problem to compute what remains unc hanged b y an action, is suggested. Section 9 surv eys the whole sp ectrum of extensions of Description Logics for represen ting and reasoning with time and action. This Section is concluded b y a comparison with State Change based approac hes b y briey illustrating the eort made in the situation calculus area to temp orally extend this class of formalisms. Section 10 concludes the pap er. 2. Description Logics Description Logics 1 are formalisms designed for a logical reconstruction of represen ta- tion to ols suc h as fr ames , semantic networks , obje ct-oriente d and semantic data mo dels { see (Calv anese, Lenzerini, & Nardi, 1994) for a surv ey . No w ada ys, Description Logics are also considered the most imp ortan t unifying formalism for the man y ob ject-cen tered represen tation languages used in areas other than Kno wledge Represen tation. Imp ortan t c haracteristics of Description Logics are high expressivit y together with decidabilit y , whic h guaran tee the existence of reasoning algorithms that alw a ys terminate with the correct answ ers. This Section giv es a brief in tro duction to a basic Description Logic, whic h will serv e as the basic represen tation language for our prop osal. As for the formal apparatus, the formal- ism in tro duced b y (Sc hmidt-Sc hau & Smolk a, 1991) and further elab orated b y (Donini, Hollunder, Lenzerini, Spaccamela, Nardi, & Nutt, 1992; Donini et al., 1994, 1995; Buc hheit et al., 1993; De Giacomo & Lenzerini, 1995, 1996) is follo w ed: in this w a y , Description Logics are considered as a structur e d fragmen t of predicate logic. ALC (Sc hmidt-Sc hau & Smolk a, 1991) is the minimal Description Logic including full negation and disjunction { i.e., prop ositional calculus, and it is a notational v arian t of the prop ositional mo dal logic K ( m ) (Halp ern & Moses, 1985; Sc hild, 1991). The basic t yp es of a Description Logic are c onc epts , r oles , fe atur es , and individuals . A concept is a description gathering the common prop erties among a collection of individuals; from a logical p oin t of view it is a unary predicate ranging o v er the domain of individu- 1. Description Logics ha v e b een also called F r ame-Base d Description L anguages , T erm Subsumption L an- guages , T erminolo gic al L o gics , T axonomic L o gics , Conc ept L anguages or KL-One -like languages . 465 Ar t ale & Franconi C ; D ! A j (atomic concept) > j (top) ? j (b ottom) : C j (complemen t) C u D j (conjunction) C t D j (disjunction) 8 P . C j (univ ersal quan tier) 9 P . C j (existen tial quan tier) p : C j (selection) p # q j (agreemen t) p " q j (disagreemen t) p " (undenedness) p; q ! f j (atomic feature) p q (path) Figure 1: Syn tax rules for the ALC F Description Logic. als. Prop erties are represen ted either b y means of roles { whic h are in terpreted as binary relations asso ciating to individuals of a giv en class v alues for that prop ert y { or b y means of features { whic h are in terpreted as functions asso ciating to individuals of a giv en class a single v alue for that prop ert y . The language ALC F , extending ALC with features (i.e., functional roles) is considered. By the syn tax rules of Figure 1, ALC F c onc epts (denoted b y the letters C and D ) are built out of atomic c onc epts (denoted b y the letter A ), atomic r oles (denoted b y the letter P ), and atomic fe atur es (denoted b y the letter f ). The syn tax rules are expressed follo wing the tradition of Description Logics (Baader, B urc k ert, Heinsohn, Hollunder, M uller, Neb el, Nutt, & Protlic h, 1990). The me aning of concept expressions is dened as sets of individuals, as for unary pred- icates, and the meaning of roles as sets of pairs of individuals, as for binary predicates. F ormally , an interpr etation is a pair I = ( I ; I ) consisting of a set I of individuals (the domain of I ) and a function I (the interpr etation function of I ) mapping ev ery concept to a subset of I , ev ery role to a subset of I I , ev ery feature to a partial function from I to I , and ev ery individual in to a dieren t elemen t of I { i.e., a I 6 = b I if a 6 = b (Unique Name Assumption) { suc h that the equations of the left column in Figure 2 are satised. The ALC F seman tics iden ties concept expressions as fragmen ts of rst-order predicate logic. Since the in terpretation I assigns to ev ery atomic concept, role or feature a unary or binary (functional) relation o v er I , resp ectiv ely , one can think of atomic concepts, roles and features as unary and binary (functional) predicates. This can b e seen as follo ws: an atomic concept A , an atomic role P , and an atomic feature f , are mapp ed resp ectiv ely to the op en form ulas F A ( ), P ( ; ), and F f ( ; ) with F f satisfying the functionalit y axiom 8 y ; z . F f ( x; y ) ^ F f ( x; z ) ! y = z { i.e., F f is a functional relation. The righ tmost column of Figure 2 giv es the transformational seman tics of ALC F ex- pressions in terms of F OL w ell-formed form ul, while the left column giv es the standard extensional seman tics. As far as the transformational seman tics is concerned, a concept C , a role P and a path p corresp ond to the F OL op en form ulae F C ( ), F P ( ; ), and F p ( ; ), 466 A Temporal Description Logic f or Reasoning about A ctions and Plans > I = I true ? I = ; false ( : C ) I = I n C I : F C ( ) ( C u D ) I = C I \ D I F C ( ) ^ F D ( ) ( C t D ) I = C I [ D I F C ( ) _ F D ( ) ( 9 P . C ) I = f a 2 I j 9 b . ( a; b ) 2 P I ^ b 2 C I g 9 x . F P ( ; x ) ^ F C ( x ) ( 8 P . C ) I = f a 2 I j 8 b . ( a; b ) 2 P I ) b 2 C I g 8 x . F P ( ; x ) ) F C ( x ) ( p : C ) I = f a 2 dom p I j p I ( a ) 2 C I g 9 x . F p ( ; x ) ^ F C ( x ) p # q I = f a 2 dom p I \ dom q I j p I ( a ) = q I ( a ) g ( 9 x . F p ( ; x ) ^ F q ( ; x ) ) p " q I = f a 2 dom p I \ dom q I j p I ( a ) 6 = q I ( a ) g ( 9 x; y . F p ( ; x ) ^ F q ( ; y ) ) ^ ( 8 x; y . F p ( ; x ) ^ F q ( ; y ) ! x 6 = y ) ( p " ) I = I n dom p I :9 x . F p ( ; x ) ( p q ) I = p I q I 9 x . F p ( ; x ) ^ F q ( x; ) Figure 2: The extensional and transformational seman tics in ALC F . resp ectiv ely . It is w orth noting that the extensional seman tics of the left column giv es also an in terpretation for the form ulas of the righ t column so that the follo wing prop osition holds. Prop osition 2.1 (Concepts vs. fol form ul) L et C b e an ALC F c onc ept expr ession. Then the tr ansformational semantics of Figur e 2 maps C into a lo gic al ly e quivalent rst or der formula. A terminolo gy or TBox is a nite set of terminolo gic al axioms . F or an atomic concept A , called dene d c onc ept , and a (p ossibly complex) concept C , a terminological axiom is of the form A : = C . An atomic concept not app earing on the left-hand side of an y terminological axiom is called a primitive c onc ept . A cyclic simple TBo xes only are considered: a dened concept ma y app ear at most once as the left-hand side of an axiom, and no terminological cycles are allo w ed, i.e., no dened concept ma y o ccur { neither directly nor indirectly { within its o wn denition (Neb el, 1991). An in terpretation I satises A : = C if and only if A I = C I . As an example, consider the unary relation (i.e., a concept) denoting the class of happ y fathers, dened using the atomic predicates Man, Doctor, Rich, Famous (concepts) and CHILD, FRIEND (roles): HappyFather : = Man u ( 9 CHILD . > ) u 8 CHILD . ( Doctor u 9 FRIEND . ( Rich t Famous )) i.e., the men whose c hildren are do ctors ha ving some ric h or famous friend. An ABox is a nite set of assertional axioms , i.e. predications on individual ob jects. Let O b e the alphab et of sym b ols denoting individuals ; an assertion is an axiom of the form C ( a ), R ( a; b ) or p ( a; b ), where a and b denote individuals in O . C ( a ) is satised b y an in terpretation I i a I 2 C I , P ( a; b ) is satised b y I i ( a I ; b I ) 2 P I , and p ( a; b ) is satised b y I i p I ( a I ) = b I . 467 Ar t ale & Franconi A know le dge b ase is a nite set of terminological and assertional axioms. An in terpre- tation I is a mo del of a kno wledge base i ev ery axiom of is satised b y I . lo gic al ly implies A v C (written j = A v C ) if A I C I for ev ery mo del of : w e sa y that A is subsume d b y C in . The reasoning problem of c hec king whether A is subsume d b y C in is called subsumption che cking . lo gic al ly implies C ( a ) (written j = C ( a )) if a I 2 C I for ev ery mo del of : w e sa y that a is an instanc e of C in . The reasoning problem of c hec king whether a is an instanc e of C in is called instanc e r e c o gnition . An acyclic simple TBo x can b e transformed in to an exp ande d TBo x ha ving the same mo dels, where no dened concept mak es use in its denition of an y other dened concept. In this w a y , the in terpretation of a dened concept in an expanded TBo x do es not dep end on an y other dened concept. It is easy to see that A is subsumed b y C in an acyclic simple TBo x if and only if the expansion of A with resp ect to is subsumed b y the expansion of C with resp ect to in the empt y TBo x. The expansion pro cedure recursiv ely substitutes ev ery dened concept o ccurring in a denition with its dening expression; suc h a pro cedure ma y generate a TBo x exp onen tial in size, but it has b een pro v ed (Neb el, 1990) that it w orks in p olynomial time under reasonable restrictions. The follo wing in terc hangeably refers either to reasoning with resp ect to a TBo x or to reasoning in v olving expanded concepts with an empt y TBo x. In particular, while devising the subsumption calculus for the logics considered here, it is alw a ys assumed that all dened concepts ha v e b een expanded. 3. T o w ards a T emp oral Description Logics Sc hmiedel (1990) prop osed to extend Description Logics with an in terv al{based temp oral logic. The temp oral v arian t of the Description Logic is equipp ed with a mo del-theoretic seman tics. The underlying Description Logic is F LE N R (Donini et al., 1995): it diers from ALC F in that it do es not con tain the > and ? concepts, it do es not ha v e neither negation nor disjunction, and it has cardinalit y restrictions and conjunction o v er roles. The new temp oral term-forming op erators are the temp oral qualier at , the existen tial and univ ersal temp oral quan tiers sometime and alltime . The qualier op erator sp ecies the time at whic h a concept holds. The temp oral quan tiers in tro duce the temp oral v ariables constrained b y means of temp oral relationships based on Allen's in terv al algebra extended with metric constrain ts to deal with durations, absolute times, and gran ularities of in terv als. T o giv e an example of this temp oral Description Logic, the concept of Mortal can b e dened b y: Mortal : = LivingBeing u ( sometime ( x ) ( after x NO W ) ( at x ( : LivingBeing )) ) with the meaning of a LivingBeing at the reference in terv al NO W , who will not b e aliv e at an in terv al x sometime after the reference in terv al NO W . Sc hmiedel do es not prop ose an y algorithm for computing subsumption, but giv es some preliminary hin ts. Actually , Sc hmiedel's logic is argued to b e undecidable (Bettini, 1997), sacricing the main b enet of Description Logics: the p ossibilit y of ha ving decidable inference tec hniques. Sc hmiedel's temp oral Description Logic, when closed under complemen tation, con tains as a prop er fragmen t the temp oral logic H S prop osed b y Halp ern and Shoham (1991). The logic H S is a prop ositional mo dal logic whic h extends prop ositional logic with mo dal form ul of the kind h R i . and [ R ] . { where R is a basic Allen's temp oral relation and hi 468 A Temporal Description Logic f or Reasoning about A ctions and Plans and [] are the p ossibilit y and necessit y mo dal op erators. F or example, the mo dal form ula LivingBeing ^ h after i . : LivingBeing corresp onds to the ab o v emen tioned Mortal concept. Unfortunately , the H S logic is sho wn to b e undecidable, at least for most in teresting classes of temp oral structures: \ One gets de cidability only in very r estricte d c ases, such as when the set of temp or al mo dels c onsider e d is a nite c ol le ction of structur es, e ach c onsisting of a nite set of natur al numb ers. " (Halp ern & Shoham, 1991) W eida and Litman (1992, 1994) prop ose T-Rex , a lo ose h ybrid in tegration b et w een Description Logics and constrain t net w orks. Plans are dened as collections of steps together with temp oral constrain ts b et w een their duration. Eac h step is asso ciated with an action t yp e, represen ted b y a generic concept in K-Rep { a non-temp oral Description Logic. Th us a plan is seen as a plan network , a temp oral constrain t net w ork whose no des, corresp onding to time in terv als, are lab eled with action t yp es and are asso ciated with the steps of the plan itself. As an example of plan in T-Rex they sho w the plan of preparing spaghetti marinara: ( defplan Assemble- Spaghetti- Marinara (( step 1 Boil- Spaghetti ) ( step 2 Make- Marinara ) ( step 3 Put- Together- SM )) (( step 1 ( b efo re meets ) step 3) ( step 2 ( b efo re meets ) step 3)) ) This is a plan comp osed b y three actions, i.e., b oiling spaghetti, preparing marinara sauce, and assem bling all things at the end. T emp oral constrain ts b et w een the steps establish the temp oral order in doing the corresp onding actions. A structural plan subsumption algorithm is dened, c haracterized in terms of graph matc hing, and based on t w o separate notions of subsumption: pure terminological subsumption b et w een action t yp es lab eling the no des, and pure temp oral subsumption b et w een in terv al relationships lab eling the arcs. The plan library is used to guide plan recognition (W eida, 1996) in a w a y similar to that prop osed b y Kautz (1991). Ev en if this w ork has strong motiv ations, no formal seman tics is pro vided for the language and the reasoning problems. Starting from the assumption that an action has a duration in time, our prop osal con- siders an in terv al-based mo dal temp oral logic { in the spirit of Halp ern and Shoham (1991) { and reduces the expressivit y of (Sc hmiedel, 1990) in the direction of (W eida & Litman, 1992). While Sc hmiedel's w ork lac ks computational mac hinery , and Halp ern and Shoham's logic is undecidable, here an expressiv e decidable logic is obtained, pro viding sound and complete reasoning algorithms. Dieren tly from T-Rex whic h uses t w o dieren t languages to represen t actions and plans { a non temp oral Description Logic for describing actions and a second language to comp ose plans b y adding temp oral information { here an exten- sion of a Description Logic is c hosen in whic h time op erators are a v ailable directly as term constructors. This view implies an in tegration of a temp oral domain in the seman tic struc- ture where terms themselv es are in terpreted, giving the formal w a y b oth for a w ell-founded notion of subsumption and for pro ving soundness and completeness of the corresp onding pro cedure. As an example of the formalism, the plan for preparing spaghetti marinara in tro duced ab o v e is represen ted as follo ws: 469 Ar t ale & Franconi Assemble- Spaghetti- Marinara : = 3 ( y z w ) ( y ( b efo re ; meets ) w )( z ( b efo re ; meets ) w ) . ( Boil- Spaghetti @ y u Make- Marinara @ z u Put- Together- SM @ w ) Moreo v er, it is p ossible to build temp or al structur e d actions { as opp osed to the atomic actions prop osed in T-Rex { describing ho w the w orld state c hanges b ecause of the o ccur- rence of an action: in fact, our language allo ws for feature represen tation in order to relate actions to states of the w orld (see Section 5.2). This kind of expressivit y is not captured b y T-Rex , since it uses a non-temp oral Description Logic to represen t actions. The main application of T-Rex is plan recognition; according to the ideas of Kautz (1991) a Closed W orld Assumption (CW A) (W eida, 1996) is made, assuming that the plan library is com- plete and an observ ed plan will b e fully accoun ted for b y a single plan. CW A is not relied on here, follo wing the Op en W orld Seman tics c haracterizing Description Logics. W eak er, but monotonic, deductions are allo w ed in the plan recognition pro cess. Ho w ev er, their pro- cedures for recognizing a ne c essary , optional or imp ossible individual plan with resp ect to a plan description is still applicable, if the plan library is giv en a closed w orld seman tics. 4. The F eature T emp oral Language T L - F The feature temp oral language T L - F is the basic logic considered here. This language is comp osed of the temp oral Logic T L { able to express in terv al temp oral net w orks { and the non-temp oral F eature Description Logic F . Note that, eac h logic of the family of T emp oral Description Logics in tro duced in this pap er is iden tied b y a comp osed string in whic h the rst part refers to the temp oral part of the language while the other one refers to the non-temp oral part. 4.1 Syn tax Basic t yp es of the language are c onc epts , individuals , temp or al variables and intervals . Concepts can describ e en tities of the w orld, states and ev en ts. T emp oral v ariables denote in terv als b ound b y temp oral constrain ts, b y means of whic h abstract temp oral patterns in the form of constrain t net w orks are expressed. Concepts (resp. individuals) can b e sp ecied to hold at a certain temp oral v ariable (resp. in terv al). In this w a y , action typ es (resp. individual actions ) can b e represen ted in a uniform w a y b y temp orally related concepts (resp. individuals). F or the basic temp oral in terv al relations the Allen notation (Allen, 1991) (Figure 3) is used: b efore ( b ), meets ( m ), during ( d ), o v erlaps ( o ), starts ( s ), nishes ( f ), equal ( = ), after ( a ), met-b y ( mi ), con tains ( di ), o v erlapp ed-b y ( oi ), started-b y ( si ), nished-b y ( ). Conc ept expr essions (denoted b y C ; D ) are syn tactically built out of atomic c onc epts (denoted b y A ), atomic fe atur es (denoted b y f ), atomic p ar ametric fe atur es (denoted b y ? g ) and temp or al variables (denoted b y X ; Y ). T emp oral concepts ( C ; D ) are distinguished from non-temp oral concepts ( E ; F ), follo wing the syn tax rules of Figure 4. Names for atomic features and atomic parametric features are from the same alphab et of sym b ols; the ? sym b ol is not in tended as op erator, but only as dieren tiating the t w o syn tactic t yp es. 470 A Temporal Description Logic f or Reasoning about A ctions and Plans Relation A bbr. In v erse i j befo re ( i; j ) b a meets ( i; j ) m mi overlaps ( i; j ) o oi sta rts ( i; j ) s si during ( i; j ) d di f inishes ( i; j ) f Figure 3: The Allen's in terv al relationships. T emp oral v ariables are in tro duced b y the temp oral existen tial quan tier \ 3 " { excluding the sp ecial temp oral v ariable ] , usually called NO W , and in tended as the reference in terv al. V ariables app earing in temp oral constrain ts ( T c ) m ust b e declared within the same temp oral quan tier, with the exception of the sp ecial v ariable ] . T emp oral v ariables app earing in the righ t hand side of an \@" op erator are called bindable . Concepts m ust not include unb ound (a.k.a. fr e e ) bindable v ariables. Informally , a bindable v ariable is said to b e b ound in a concept if it is declared at the nearest temp oral quan tier in the b o dy of whic h it o ccurs; this a v oid the usual formal inductiv e denition of a b ound v ariable. Moreo v er, in c hained constructs of the form (( C [ Y 1 ]@ X 1 )[ Y 2 ]@ X 2 : : : ) non bindable v ariables { i.e., the ones on the left hand side of an \@" op erator { cannot app ear more than once. Note that, since Description Logics are a fragmen t of F OL with one free v ariable, the ab o v e men tioned restrictions force the temp oral side of the language to ha v e only one free temp oral v ariable, i.e., the reference time ] . As usual, terminological axioms for building simple acyclic T L - F TBo xes are allo w ed. While using in a concept expression a name referring to a dened concept, it is p ossible to use the substitutiv e qualier construct, to imp ose a coreference with a v ariable app earing in the denition asso ciated to the dened concept. The statemen t C [ Y ]@ X constrains the v ariable Y , whic h should app ear in the denition of the dened concept C , to corefer with X (see Section 5.2 for an example). A dra wbac k in the use of this op erator is the requiremen t to kno w the in ternal syn tactical form of the dened concept, namely , the names of its temp oral v ariables. Let O and O T b e t w o alphab ets of sym b ols denoting individuals and temp or al intervals , resp ectiv ely . An assertion { i.e., a predication on temp orally qualied individual en tities { is a statemen t of one of the forms C ( i; a ) ; p ( i; a; b ) ; ? g ( a; b ) ; R ( i; j ), where C is a concept, p is a feature, ? g is a parametric feature, R is a temp oral relation, a and b denote individuals in O , i and j denote temp oral in terv als in O T . 471 Ar t ale & Franconi T L C ; D ! E j (non-temp oral concept) C u D j (conjunction) C @ X j (qualier) C [ Y ]@ X j (substitutiv e qualier) 3 ( X ) T c . C (existen tial quan tier) T c ! ( X ( R ) Y ) j (temp oral constrain t) ( X ( R ) ] ) j ( ] ( R ) Y ) T c ! T c j T c T c R ; S ! R , S j (disjunction) s j mi j f j : : : (Allen's relations) X ; Y ! x j y j z j : : : (temp oral v ariables) X ! X j X X F E ; F ! A j (atomic concept) > j (top) E u F j (conjunction) p # q j (agreemen t) p : E (selection) p; q ! f j (atomic feature) ?g j (atomic parametric feature) p q (path) Figure 4: Syn tax rules for the in terv al Description Logic T L - F 4.1.1 A clarifying Example Let us no w informally see the in tended me aning of the terms of the language T L - F (for the formal details see Section 4.2). Concept expressions are in terpreted o v er pairs of temp or al intervals and individuals h i; a i , meaning that the individual a is in the extension of the con- cept at the in terv al i . If a concept is in tended to describ e an action, then its in terpretation can b e seen as the set of individual actions of that t yp e o ccurring at some in terv al. Within a concept expression, the sp ecial \ ] " v ariable denotes the curren t in terv al of ev aluation; in the case of actions, it is though t that it refers to the temp oral in terv al at whic h the action itself o c curs . The temp oral existen tial quan tier in tro duces in terv al v ariables, related to eac h other and p ossibly to the ] v ariable in a w a y dened b y the set of temp or al c onstr aints . T o ev aluate a concept at an in terv al X , dieren t from the curren t one, it is necessary to temp orally qualify it at X { written C @ X ; in this w a y , ev ery o ccurrence of 472 A Temporal Description Logic f or Reasoning about A ctions and Plans - - - OnTable(BLOCK) Basic- Stack(BLOCK) OnBlock(BLOCK) x ] y Figure 5: T emp oral dep endencies in the denition of the Basic- Stack action. ] em b edded within the concept expression C is in terpreted as the X v ariable 2 . The informal meaning of a concept with a temp oral existen tial quan tication can b e understo o d with the follo wing examples in the action domain. Basic- Stack : = 3 ( x y ) ( x m ] )( ] m y ) . ( ( ? BLOCK : OnTable )@ x u ( ? BLOCK : OnBlock )@ y ) Figure 5 sho ws the temp oral dep endencies of the in terv als in whic h the concept Basic- Stack holds. Basic- Stack denotes, according to the denition (a terminological axiom), an y action o ccurring at some in terv al in v olving a ? BLOCK that w as once OnTable and then OnBlock . The ] in terv al could b e understo o d as the o ccurring time of the action t yp e b eing dened: referring to it within the denition is an explicit w a y to temp orally relate states and actions o ccurring in the w orld with resp ect to the o ccurrence of the action itself. The temp oral constrain ts ( x m ] ) and ( ] m y ) state that the in terv al denoted b y x should meet the in terv al denoted b y ] { the o ccurrence in terv al of the action t yp e Basic- Stack { and that ] should meet y . The parametric feature ? BLOCK pla ys the role of formal parameter of the action, mapping an y individual action of t yp e Basic- Stack to the blo c k to b e stac k ed, indep enden tly from time. Please note that, whereas the existence and iden tit y of the ? BLOCK of the action is time in v arian t, it can b e qualied dieren tly in dieren t in terv als of time, e.g., the ? BLOCK is necessarily OnTable only during the in terv al denoted b y x . Let us commen t no w on the in tro duction of explicit temp oral v ariables. The absence of explicit temp oral v ariables w ould w eak en the temp oral structure of a concept since arbitrary relationships b et w een more than t w o in terv als could not b e represen ted an ymore. F or example, ha ving only implicit in terv als it is not p ossible to describ e the situation in whic h t w o concept expressions, sa y C and D , hold at t w o meeting in terv als (sa y x , y ) with the rst in terv al starting and the second nishing the reference in terv al (i.e., the temp oral pattern ( x meets y )( x sta rts ] )( y nishes ] ) cannot b e represen ted). More precisely , it is not p ossible to represen t temp oral relations b et w een more than t w o in terv als if they are not deriv able b y the temp oral propagation of the constrain ts imp osed on pairs of v ariables. While explicit v ariables go against the general thrust of Description Logics, the gained expressiv e p o w er together with the observ ation that the v ariables are limited only to the temp oral part of the language are the main motiv ations for using them. Ho w ev er, it is easy to drop them b y limiting the temp oral expressiv eness as prop osed b y Bettini (1997) (see also Section 9). An assertion of the t yp e Basic-Stack ( i; a ) states that the individual a is an action of the t yp e Basic-Stack o ccurred at the in terv al i . Moreo v er, the same assertion implies that a is related to a ? BLOCK , sa y b , whic h is of t yp e OnTable at some in terv al j , meeting i , and of t yp e OnBlock at another in terv al l , met b y i . 2. Since an y concept is implicitly temp orally qualied at the sp ecial ] v ariable, it is not necessary to explicitly qualify concepts at ] . 473 Ar t ale & Franconi ( s ) E = fh [ u; v ] ; [ u 1 ; v 1 ] i 2 T ? < T ? < j u = u 1 ^ v < v 1 g ( f ) E = fh [ u; v ] ; [ u 1 ; v 1 ] i 2 T ? < T ? < j v = v 1 ^ u 1 < u g ( mi ) E = fh [ u; v ] ; [ u 1 ; v 1 ] i 2 T ? < T ? < j u = v 1 g : : : (me aning of the other A l len temp or al r elations) ( R , S ) E = R E [ S E h X ; T c i E = fV : X 7! T ? < j 8 ( X ( R ) Y ) 2 T c . hV ( X ) ; V ( Y ) i 2 R E g : Figure 6: The temp oral in terpretation function. Basic- Stack ( i; a ) = ) 9 b . ? BLOCK ( a; b ) ^ 9 j; l . ( OnTable ( j; b ) ^ OnBlock ( l ; b ) ^ m ( j; i ) ^ m ( i; l )) An individual action is an ob ject in the conceptual domain asso ciated with the relev an t prop erties { or states { of the w orld aected b y the individual action itself via a bunc h of fe atur es ; moreo v er, temp oral relations constrain time in terv als imp osing an ordering in the c hange of the states of the w orld. 4.2 Seman tics In this Section, a T arski-st yle extensional seman tics for the T L - F language is giv en, and a formal denition of the subsumption and recognition reasoning tasks is devised. Assume a linear, un b ounded, and dense temp oral structure T = ( P ; < ), where P is a set of time p oin ts and < is a strict partial order on P . In suc h a structure, giv en an in terv al X and a temp oral relation R , it is alw a ys p ossible to nd an in terv al Y suc h that ( X ( R ) Y ). The assumption of linear time { whic h means that for an y t w o p oin ts t 1 and t 2 suc h that t 1 t 2 the set of p oin ts f t j t 1 t t 2 g is totally ordered { ts the in tuition ab out the nature of time, so that the pair [ t 1 ; t 2 ] can b e though t as the closed in terv al of p oin ts b et w een t 1 and t 2 . The interval set of a structure T is dened as the set T ? < of all closed in terv als [ u; v ] : = f x 2 P j u x v ; u 6 = v g in T . A primitive interpr etation I : = hT ? < ; I ; I i consists of a set T ? < (the interval set of the selected temp oral structure T ), a set I (the domain of I ), and a function I (the primitive interpr etation function of I ) whic h giv es a meaning to atomic concepts, features and parametric features: A I T ? < I ; f I : ( T ? < I ) par tial 7 ! I ; ? g I : I par tial 7 ! I A tomic parametric features are in terpreted as partial functions; they dier from atomic features for b eing indep enden t from time. In order to giv e a meaning to temp oral expressions presen t in generic concept expres- sions, Figure 6 denes the temp or al interpr etation function . The temp or al interpr etation function E dep ends only on the temp oral structure T . The lab eled directed graph h X ; T c i { where X is the set of v ariables represen ting the no des, and T c is the set of temp oral con- strain ts represen ting the arcs { is called temp or al c onstr aint network . The in terpretation 474 A Temporal Description Logic f or Reasoning about A ctions and Plans A I V ;t; H = f a 2 I j h t; a i 2 A I g = A I t > I V ;t; H = I = > I ( C u D ) I V ;t; H = C I V ;t; H \ D I V ;t; H ( p # q ) I V ;t; H = f a 2 dom p I t \ dom q I t j p I t ( a ) = q I t ( a ) g = ( p # q ) I t ( p : C ) I V ;t; H = f a 2 dom p I t j p I t ( a ) 2 C I V ;t; H g ( C @ X ) I V ;t; H = C I V ; V ( X ) ; H ( C [ Y ]@ X ) I V ;t; H = C I V ;t; H[f Y 7!V ( X ) g ( 3 ( X ) T c . C ) I V ;t; H = f a 2 I j 9W . W 2 h X ; T c i E H[f ] 7! t g ^ a 2 C I W ;t; ; g f I t = ^ f t : I par tial 7 ! I j 8 a . ( a 2 dom ^ f t $ h t; a i 2 dom f I ) ^ ^ f t ( a ) = f I ( t; a ) ( p q ) I t = p I t q I t ?g I t = ? g I Figure 7: The in terpretation function. of a temp oral constrain t net w ork is a set of v ariable assignmen ts that satisfy the temp oral constrain ts. A variable assignment is a function V : X 7! T ? < asso ciating an in terv al v alue to a temp oral v ariable. A temp oral constrain t net w ork is c onsistent if it admits a non empt y in terpretation. The notation, h X ; T c i E f x 1 7! t 1 ;x 2 7! t 2 ;::: g , used to in terpret concept expressions, denotes the subset of h X ; T c i E where the v ariable x i is mapp ed to the in terv al v alue t i . It is no w p ossible to in terpret generic concept expressions. Consider the equations in tro duced in Figure 7. An interpr etation function I V ;t; H , based on a v ariable assignmen t V , an in terv al t and a set of constrain ts H = f x 1 7! t 1 ; : : : g o v er the assignmen ts of inner v ariables, extends the primitiv e in terpretation function in suc h a w a y that the equations of Figure 7 are satised. In tuitiv ely , the in terpretation of a concept C I V ;t; H is the set of en tities of the domain that are of t yp e C at the time in terv al t , with the assignmen t for the free temp oral v ariables in C giv en b y V { see ( C @ X ) I V ;t; H { and with the constrain ts for the assignmen t of v ariables in the scop e of the outermost temp oral quan tiers giv en b y H . Note that, H in terprets the v ariable renaming due to the temp oral substitutiv e qualier { see ( C [ Y ]@ X ) I V ;t; H { and it tak es eect during the c hoice of a v ariable assignmen t, as the equation for ( 3 ( X ) T c . C ) I V ;t; H sho ws. In absence of free v ariables in the concept expression { with the exception of ] { for notational simplication the natur al in terpretation function C I t ; b eing equiv alen t to the in terpretation function C I V ;t; H with an y V suc h that V ( ] ) = t and H = ; is in tro duced. The set of in terpretations f C I V ;t; H g obtained b y v arying I ; V ; t with a xed H is maximal wrt set inclusion if H = ; , i.e., the set of natural in terpretations includes an y set of in terpretations with a xed H . In fact, since H represen ts a constrain t in the assignmen t of v ariables, the unconstrained set is the larger one. Note that, for feature in terpretation only the natural one is used since it is not admitted to temp orally qualify them. 475 Ar t ale & Franconi - - - Make-Spaghetti Boil Boil-Spaghetti x ] ] Figure 8: T emp oral dep endencies in the denition of the Boil- Spaghetti plan. An in terpretation I satises the terminological axiom A : = C i A I t = C I t , for ev ery t . A concept C is subsume d b y a concept D ( C v D ) if C I t D I t for ev ery in terpretation I and ev ery in terv al t . An in terpretation I is a mo del for a concept C if C I t 6 = ; for some t . If a concept has a mo del, then it is satisable , otherwise it is unsatisable . Eac h T L - F concept expression is alw a ys satisable, with the pro viso that the temp oral constrain ts in tro duced b y the existen tial quan tiers are consisten t. This latter condition can b e easily c hec k ed during the reduction of the concept in to a normal form when the minimal temp oral net w ork (see Section 11, denition 6.5) is computed. It is in teresting to note that only the relations s, f, mi are really necessary , b ecause it is p ossible to express an y temp oral relationship b et w een t w o distinct in terv als using only these three relations and their transp ositions si, , m (Halp ern & Shoham, 1991). The follo wing equiv alences hold: 3 x ( x a ] ) . C @ x 3 xy ( y mi ] )( x mi y ) . C @ x 3 x ( x d ] ) . C @ x 3 xy ( y s ] )( x f y ) . C @ x 3 x ( x o ] ) . C @ x 3 xy ( y s ] )( x y ) . C @ x T o assign a meaning to ABo x axioms, the temp oral in terpretation function E is extended to temp oral in terv als so that i E is an elemen t of T ? < for eac h i 2 O T . The seman tics of assertions is the follo wing: C ( i; a ) is satised b y an in terpretation I i a I 2 C I i E ; p ( i; a; b ) is satised b y I i p I i E ( a I ) = b I ; ? g ( a; b ) is satised b y I i ? g I ( a I ) = b I ; and R ( i; j ) is satised b y I i h i E ; j E i 2 R E . Giv en a kno wledge base , an individual a in O is said to b e an instanc e of a concept C at the interval i if j = C ( i; a ). No w w e are able to giv e a seman tic denition for the reasoning task w e already called sp e cic plan r e c o gnition with r esp e ct to a plan description . This is an inference service that computes if an individual action/plan is an instance of an action/plan typ e at a certain in terv al, i.e., the task kno wn as instance recognition in the Description Logic comm unit y . Giv en a kno wledge base , an in terv al i , an individual a and a concept C , the instanc e r e c o gnition pr oblem is to test whether j = C ( i; a ). 5. Action and plan represen tation: t w o examples An action description represen ts ho w the w orld state ma y ev olv e in relation with the p ossible o ccurrence of the action itself. A plan is a complex action: it is describ ed b y means of temp orally related w orld states and simpler actions. The follo wing in tro duces examples of action and plan represen tations from t w o w ell kno wn domains, the co oking domain (Kautz, 476 A Temporal Description Logic f or Reasoning about A ctions and Plans - - - - - Make- Spaghetti Boil- Spaghetti Boil Make- Marinara Put- Together- SM x y y z w Figure 9: T emp oral dep endencies in the denition of Assemble- Spaghetti- Marinara . 1991; W eida & Litman, 1992) and the blo c k w orld (Allen, 1991), with the aim of sho wing the applicabilit y of our framew ork. 5.1 The Co oking Domain Let us in tro duce the plan Boil-Spaghetti : Boil- Spaghetti : = 3 x ( x b ] ) . ( Make- Spaghetti @ x u Boil ) Figure 8 sho ws the temp oral dep endencies of the in terv als in whic h the concept Boil-Spa- ghetti holds. The denition emplo ys the ] in terv al to denote the o ccurrence time of the plan itself; in this w a y , it is p ossible to describ e ho w dieren t actions or states of the w orld concurring to the denition of the plan are related to it. This is wh y the v ariable ] is explicitly presen t in the denition of Boil- Spaghetti , instead of a generic v ariable: the Boil action should tak e place at the same time of the plan itself, while Make- Spaghetti o ccurs b efore it. The denition of a plan can b e reused within the denition of other plans b y exploiting the full comp ositionalit y of the language. The plan dened ab o v e Boil- Spaghetti is used in the denition of Assemble- Spaghetti- Marinara : Assemble- Spaghetti- Marinara : = 3 ( y z w ) ( y b w )( z b w ) . ( Boil- Spaghetti @ y u Make- Marinara @ z u Put-Together- SM @ w ) In this case, precise temp oral relations b et w een the no des of t w o corresp onding temp oral constrain t net w orks are asserted: e.g., the action Put- Together- SM tak es place strictly after the Boil action (Figure 9). Observ e that the o ccurrence in terv al of the plan Assemble- Spa- ghetti- Marinara do es not app ear in the Figure b ecause it is not temp orally related with an y other in terv al. A plan subsuming Assemble- Spaghetti- Marinara is the more general plan dened b e- lo w, Prepare- Spaghetti , supp osing that the action Make- Sauce subsumes Make- Marinara . This means that among all the individual actions of the t yp e Prepare- Spaghetti there are all the individual actions of t yp e Assemble- Spaghetti- Marinara : Prepare- Spaghetti : = 3 ( y z ) () . ( Boil- Spaghetti @ y u Make- Sauce @ z ) 477 Ar t ale & Franconi - - - - Holding-Block(OBJ1) Clear-Block(OBJ2) ON(OBJ1, OBJ2) Clear-Block(OBJ1) Clear-Block(OBJ1) Stack(OBJ1, OBJ2) w x y v z ] Figure 10: T emp oral dep endencies in the denition of the Stack action. Ho w ev er, note that Boil- Spaghetti do es not subsume Prepare- Spaghetti , ev en if it is a conjunct in the denition of the latter. This could b e b etter explained observing ho w the denition of Prepare- Spaghetti plan is expanded: Prepare- Spaghetti : = 3 ( x y z ) ( x b y ) . ( Make- Spaghetti @ x u Boil @ y u Make- Sauce @ z ) Then, the Boil action o ccurs at the in terv al y { whic h can b e dieren t from the o ccurring time of Prepare- Spaghetti { as the eect of binding Boil- Spaghetti to the temp oral v ari- able y . On the con trary , in the denition of Boil- Spaghetti the Boil action tak es place ne c- essarily at the same time. Subsumption b et w een Prepare- Spaghetti and Boil- Spaghetti fails since dieren t temp oral relations b et w een the actions describing the t w o plans and the plans themselv es are sp ecied. In particular, observ e that the Boil- Spaghetti plan denotes a narro w er class than the plan expression 3 ( x y ) ( x b y ) . ( Make- Spaghetti @ x u Boil @ y ) whic h subsumes b oth Prepare- Spaghetti and Boil- Spaghetti itself. 5.2 The Blo c ks W orld Domain As a further example of the expressiv e p o w er of the temp oral language, it is no w sho wn ho w to represen t the Stack action in the blo c ks w orld, in a more detailed w a y than the previous simple Basic-Stack action used as a clarifying example. Th us a stac king action in v olv es t w o blo c ks, whic h should b e b oth clear at the b eginning; the cen tral part of the action consists of grasping one blo c k; at the end, the blo c ks are one on top of another, and the b ottom one is no longer clear (Figure 10). Our represen tation b orro ws from the Ra t Description Logic (Heinsohn, Kudenk o, Neb el, & Protlic h, 1992) the in tuition of represen ting action parameters b y means of partial functions mapping from the action itself to the in v olv ed action parameter (see Section 9). In the language, these functions are called p ar ametric fe atur es . F or example, the action Stack has the parameters ? OBJECT1 and ? OBJECT2 , represen ting in some sense the ob jects that are in v olv ed in the action indep enden tly from time. So, in the assertion \ ? OBJECT1 ( a; bl ock - a )", bl ock - a denotes the rst ob ject in v olv ed in the action a at an y in terv al. On the other hand, an assertion in v olving a (non-parametric) feature, e.g., \ ON ( i; bl ock - a; bl ock - b )", do es not imply an ything ab out the truth v alue at in terv als other than i . The concept expression, whic h denes the Stack action, mak es use of temp oral qualied concept expressions, including feature sele ctions and agr e ements : the expression ( ? OBJECT2 : Clear- Block )@ x means that the second parameter of the action should b e a Clear- Block 478 A Temporal Description Logic f or Reasoning about A ctions and Plans at the in terv al denoted b y x ; while ( ? OBJECT1 ON # ? OBJECT2 )@ y indicates that at the in terv al y the ob ject on whic h ? OBJECT1 is placed is ? OBJECT2 . The formal denition of the action Stack is: Stack : = 3 ( x y z v w ) ( x ] )( y mi ] )( z mi ] )( v o ] )( w f ] )( w mi v ) . ( ( ? OBJECT2 : Clear- Block )@ x u ( ? OBJECT1 ON # ? OBJECT2 )@ y u ( ? OBJECT1 : Clear- Block )@ v u ( ? OBJECT1 : Holding- Block )@ w u ( ? OBJECT1 : Clear- Block )@ z ) The ab o v e dened concept do es not state whic h prop erties are the prerequisites for the stac king action or whic h prop erties m ust b e true whenev er the action succeeds. What this action in tuitiv ely states is that ? OBJECT1 will b e on ? OBJECT2 in a situation where b oth ob jects are clear at the start of the action. Note that the w orld states describ ed at the in terv als denoted b y v ; w ; z are the result of an action of gr asping a previously clear blo c k: Grasp : = 3 ( x w z ) ( x o ] )( w f ] )( w mi x )( z mi ] ) . ( ( ? OBJECT1 : Clear- Block )@ x u ( ? OBJECT1 : Holding- Block )@ w u ( ? OBJECT1 : Clear- Block )@ z ) The Stack action can b e redened b y making use of the Grasp action: Stack : = 3 ( x y u v ) ( x ] )( y mi ] )( u f ] )( v o ] ) . ( ( ? OBJECT2 : Clear- Block )@ x u ( ? OBJECT1 ON # ? OBJECT2 )@ y u ( Grasp [ x ]@ v )@ u ) The temp oral substitutiv e qualier ( Grasp [ x ]@ v ) r enames within the dened Grasp action the v ariable x to v and it is a w a y of making coreference b et w een t w o temp oral v ariables, while the temp oral constrain ts p eculiar to the renamed v ariable x are inherited b y the substituting in terv al v . F urthermore, the eect of temp orally qualifying the grasping action at u is that the ] v ariable asso ciated to the grasping action { referring to the o ccurrence time of the action itself { is b ound to the in terv al denoted b y u . Because of this binding on the o ccurrence time of the grasping action, the ] v ariable in the grasping action and the ] v ariable in the stac king action denote dieren t time in terv als, so that the grasping action o ccurs at an in terv al nishing the o ccurrence time of the stac king action. No w it is sho wn ho w from a series of outside observ ations action recognition can b e p erformed { i.e., the task called sp e cic plan r e c o gnition with r esp e ct to a plan description . The follo wing ABo x describ es a situation in whic h blo c ks can b e cle ar , gr asp e d and/or on eac h other, and in whic h a generic individual action a is taking place at time in terv al i a ha ving the blo c ks blo ck-a and blo ck-b as its parameters: ? OBJECT1 ( a; bl ock - a ) ; ? OBJECT2 ( a; bl ock - b ) ; o ( i 1 ; i a ) ; Clear- Block ( i 1 ; bl ock - a ) ; ( i 2 ; i a ) ; Clear- Block ( i 2 ; bl ock - b ) ; mi ( i 3 ; i 1 ) ; f ( i 3 ; i a ) ; Holding- Block ( i 3 ; bl ock - a ) ; mi ( i 4 ; i a ) ; Clear- Block ( i 4 ; bl ock - a ) ; mi ( i 5 ; i a ) ; ON ( i 5 ; bl ock - a; bl ock - b ) The system deduces that, in the con text of a kno wledge base comp osed b y the ab o v e ABo x and the denition of the Stack concept in the TBo x, the individual action a is of t yp e Stack at the time in terv al i a , i.e., j = Stack ( i a ; a ). 479 Ar t ale & Franconi C @ X u D @ X ! ( C u D )@ X ( C @ X 1 )@ X 2 ! C @ X 1 ( C @ X 1 u D )@ X 2 ! C @ X 1 u D @ X 2 C u 3 ( X ) T c . D ! 3 ( X ) T c . ( C u D ) if C do esn't con tain free v ariables 3 ( X ) T c 1 . ( C u 3 ( Y ) T c 2 . D [ Y 1 ]@ X 1 : : : [ Y p ]@ X q @ X ) ! 3 ( X ] [ Y 1 =X 1 ] ::: [ Y p =X q ] Y ) T c 1 [ T c 2+ [ ] =X ] . ( C u D + @ X ) if D do esn't con tain existen tial temp oral quan tiers p : ( q : C ) ! ( p q ) : C p : ( C u D ) ! p : C u p : D p : ( q 1 # q 2 ) ! p q 1 # p q 2 Pr escriptions: X ] [ Y 1 =X 1 ] ::: [ Y p =X q ] Y returns the union of the t w o sets of v ariables X and Y , where eac h o ccurrence of Y 1 ; : : : ; Y p is substituted b y X 1 ; : : : ; X q , resp ectiv ely , while all the other elemen ts of Y o ccurring in X are renamed with fresh new iden tiers. Z + is in tended to b e the expression Z where the same substitution or renaming has tak en place. The condition on the last rule forces application to start from the last nested existen tial temp oral qualied concept. Figure 11: Rewrite rules to transform an arbitrary concept in to an existen tial concept. 6. The Calculus for T L - F This Section presen ts a calculus for deciding subsumption b et w een temp oral concepts in the Description Logic T L - F . The calculus is based on the idea of separating the inference on the temp oral part from the inference on the Description Logic part. This is ac hiev ed b y rst lo oking for a normal form of concepts. Concept subsumption in the temp oral language is then reduced to concept subsumption b et w een non-temp oral concepts and to subsumption b et w een temp oral constrain t net w orks. 6.1 Normal F orm Ev ery T L - F concept expression can b e reduced to an equiv alen t existential c onc ept of the form: 3 ( X ) T c . ( Q 0 u Q 1 @ X 1 u : : : u Q n @ X n ), where eac h Q is a non-temp oral concept, i.e., it is an elemen t of the language F . A concept in existen tial form can b e seen as a c onc eptual temp or al c onstr aint network , i.e., a lab eled directed graph h X ; T c ; Q @ X i where arcs are lab eled with a set of arbitrary temp oral relationships { represen ting their disjunction { no des are lab eled with non-temp oral concepts and, for eac h no de X , the temp oral relation ( X = X ) is implicitly true. Moreo v er, since the normalized concepts do not con tain free v ariables or substitutiv e qualiers, in the follo wing the natural in terpretation function (see Section 4.2) is used. Prop osition 6.1 (Equiv alence of EF) Every c onc ept C c an b e r e duc e d in line ar time into an e quivalent existential c onc ept ( ef C ) , by exhaustively applying the set of r ewrite rules of Figur e 11. 480 A Temporal Description Logic f or Reasoning about A ctions and Plans Pro cedure Covering( h X ; T c i ; y ): mid ; ; result ; ; Z = f z 2 X j ( z (= ; : : : ) y ) 2 T c g ; 8 s 2 } ( Z ) do if j s j 2 and the graph h X ; T c i obtained b y deleting the \=" temp oral relation b et w een the no de y and eac h of the no des in s is inconsisten t then mid mid [ f s g ; 8 s 2 mid do if :9 t 2 mid . t s then result result [ f s g ; return result . Figure 12: Pro cedure whic h computes a co v ering. Note that ( ef C ) mak es explicit all the p ossible c hains of features b y reducing eac h non- temp oral concept Q to a conjunction of atomic concepts, feature selections restricted to atomic concepts and feature agreemen ts { i.e., eac h Q is a feature term expression (Smolk a, 1992). The normalization pro ceeds b y disco v ering all the p ossibles in teractions b et w een no des with the in ten tion of making explicit all the implicit information. A crucial temp oral in- teraction o ccurs when a no de is alw a ys coinciden t with a set of no des in ev ery p ossible in terpretation of the temp oral net w ork. Denition 6.2 (Co v ering) Given a temp or al c onstr aint network h X ; T c i , let y 2 X and Z = f z 1 ; z 2 ; : : : ; z p g X , with p 1 , and y 62 Z . Z is a Covering for y if 8V 2 h X ; T c i E , V ( y ) 2 fV ( z 1 ) ; V ( z 2 ) ; : : : ; V ( z p ) g and for e ach W Z , W is not a c overing for y . If Z = ; , then y is c al le d unc over e d, otherwise y is said c over e d by Z . Prop osition 6.3 (Co v ering pro cedure) Given a temp or al c onstr aint network h X ; T c i in minimal form (se e, e.g., (van Be ek & Manchak, 1996)) and a no de y 2 X then the pr o c e dur e describ e d in Figur e 12 r eturns al l the p ossible c overings for y with size 2 . The idea b ehind the co v ering is that whenev er a set of no des f z 1 ; z 2 ; : : : ; z p g is a co v ering for y the disjunctiv e concept expression ( Q z 1 t : : : t Q z p ) should b e conjunctiv ely added to the concept expression Q y . Actually , since in T L - F concept disjunction is not allo w ed it will b e sucien t to add to the no de y the L e ast Commom Subsumer ( lcs ) of ( Q z 1 t : : : t Q z p ) as dened b elo w. Denition 6.4 ( lcs) L et Q 1 ; : : : ; Q n ; Q; C b e F c onc ept expr essions. Then, the c onc ept Q = lcs f Q 1 ; : : : ; Q n g is such that: Q 1 v Q ^ : : : ^ Q n v Q and ther e is no C such that Q 1 v C ^ : : : ^ Q n v C ^ C < Q . Giv en a concept in existen tial form, the temp oral completion of the constrain t net w ork is computed as describ ed b elo w. Denition 6.5 (Completed existen tial form) The temp or al c ompletion of a c onc ept in existential form { the Complete d Existential F orm, CEF { is obtaine d by se quential ly ap- plying the fol lowing steps: 481 Ar t ale & Franconi ( closure ) The tr ansitive closur e of the A l len temp or al r elations in the c onc eptual temp or al c onstr aint network is c ompute d, obtaining a minimal temp or al network (se e, e.g., (van Be ek & Manchak, 1996)). ( = collapsing ) F or e ach e quality temp or al c onstr aint, c ol lapse the e qual no des by applying the fol lowing r ewrite rule: 3 ( X ) T c ( x i = x j ) . Q ! ( 3 ( X n f x j g ) T c [ x j =x i ] . Q [ x j =x i ] if x i 6 = x j and x j 6 = ] . 3 ( X n f x i g ) T c [ x i = ] ] . Q [ x i = ] ] if x i 6 = x j and x j = ] . Then apply exhaustively the rst rule of Figur e 11. ( co v ering ) F or e ach y 2 X let c ompute the c overing = f Z 1 ; : : : ; Z n g fol lowing the pr o c e dur e showe d by pr op osition 6.3. Whenever the c overing is not empty, tr anslate Q y applying the fol lowing r ewrite rule: Q y ! Q y u i =1 ::: n lcs f Q i 1 ; : : : ; Q i m g wher e Z i = f z i 1 ; : : : ; z i m g , and Q i j @ z i j 2 h X ; T c ; Q @ X i . ( parameter in tro duction ) New information is adde d to e ach no de b e c ause of the pr es- enc e of p ar ameters, as the fol lowing rules show. The ; symb ol is intende d so that, e ach time the c onc ept expr ession in the left hand side app e ars in some no de of the temp or al c onstr aint network, p ossibly c onjoine d with other c onc epts, then the right hand side r epr esents the c onc ept expr ession that must b e c onjunctively adde d to al l the other no des; squar e br ackets p oint out optional p arts; the letters f ( ?f ) and g ( ?g ), p ossibly with subscripts, denote atomic (p ar ametric) fe atur es while p and q stand for generic fe atur es. ?g 1 : : : ?g n [ f [ p ]] : C ; ?g 1 : : : ?g n : > . ?g 1 : : : ?g n [ f [ p ] ] # g [ q ] ; ?g 1 : : : ?g n : > . ?g 1 : : : ?g n # ?f 1 : : : ?f m ; ?g 1 : : : ?g n # ?f 1 : : : ?f m . ?g 1 : : : ?g n g [ p ] # ?f 1 : : : ?f m [ f [ q ]] ; ?g 1 : : : ?g n : > u ?f 1 : : : ?f m : > . Prop osition 6.6 (Equiv alence of CEF) Every c onc ept in existential form c an b e r e- duc e d into an e quivalent c omplete d existential c onc ept. Both the c overing and the p ar ameter intr o duction steps can b e computed indep enden tly after the =-c ol lapsing step and then conjoining the resulting concept expressions. Observ e that, to obtain a completed existen tial concept, the steps of the normalization pro cedure require linear time with the exception of the computation of the transitiv e closure of the temp oral relations, and the co v ering step. Both these steps in v olv e NP-complete temp oral constrain t problems (v an Beek & Cohen, 1990). Ho w ev er, it is p ossible to devise reasonable subsets of Allen's algebra for whic h the problem is p olynomial (Renz & Neb el, 1997). The most relev an t prop erties of a concept in CEF is that all the admissible in terv al temp oral relations are explicit and the concept expression in eac h no de is no more renable without c hanging the o v erall concept meaning; this is stated b y the follo wing prop osition. Prop osition 6.7 (No de indep endence of CEF) L et h X ; T c ; Q @ X i b e a c onc eptual tem- p or al c onstr aint network in its c omplete d form (CEF); then, for al l Q 2 Q and for al l 482 A Temporal Description Logic f or Reasoning about A ctions and Plans F c onc ept expr essions C such that C 6w Q , ther e exists an interpr etation I such that h X ; T c ; ( Q u C )@ X i I t 6 = h X ; T c ; Q @ X i I t , for some interval t . Pr o of. The prop osition states that the information in eac h no de of the CEF is indep enden t from the information in the other no des. In fact, h X ; T c ; ( Q u C )@ X i I t = h X ; T c ; Q @ X i I t if the concept expression in one no de implies new information in some other no de. Tw o cases can b e distinguished. i) Cover e d No des . Both the (= c ol lapsing) rule and the (c overing) rule pro vide to restrict a co v ered no de with the most sp ecic F concept expression. Indeed, the (= c ol lapsing) rule pro vides collapsing t w o con temp orary no des conjoining the concept expressions of eac h of them. On the other hand, the (c overing) rule adds to the co v ered no de the most sp ecic F concept expression that subsumes the disjunctiv e concept expression that is implicitly true at the co v ered no de. Note that, thanks to the (Closur e) rule, all the p ossible equal temp oral relations are made explicit. So these t w o normalization rules co v er all the p ossible cases of temp oral in teractions b et w een no des. ii) No c oincident no des . Ev ery time-in v arian t information should spread o v er all the no des. Both parametric features and the > concept ha v e a time-in v arian t seman tics: the only time- in v arian t concept expressions are > , ?g 1 : : : ?g n : > , ?g 1 : : : ?g n # ?f 1 : : : ?f m , with n; m 1, or an arbitrary conjunction of these terms. The (p ar ameter intr o duction) rule captures all the p ossible syn tactical cases of completion concerning time-in v arian t concept expressions. By induction on the syn tax, it can b e pro v en that adding to a no de an y other concept expression c hanges the o v erall in terpretation. 2 The last normalization pro cedure eliminates no des with redundan t information. This nal normalization step ends up with the concept in the essential gr aph form , that will b e the normal form used for c hec king concept subsumption. Denition 6.8 (Essen tial graph) The sub gr aph of the CEF of a c onc eptual temp or al c on- str aint network T = h X ; T c ; Q @ X i obtaine d by deleting the no des lab ele d only with time- invariant c onc ept expr essions { with the exc eption of the ] no de { is c al le d essen tial graph of T : ( ess T ). Prop osition 6.9 (Equiv alence of essen tial graph) Every c onc ept in c omplete d existen- tial form c an b e r e duc e d in line ar time into an e quivalent essential gr aph form. Theorem 6.10 (Equiv alence of normal form) Every c onc ept expr ession c an b e r e duc e d into an e quivalent essential gr aph form. If a p olynomial fr agment of A l len 's algebr a is adopte d, the r e duction takes p olynomial time. As an example, the normal form is sho wn { i.e., the essen tial graph { of the previously in tro duced Stack action (see Section 5.2): Stack : = 3 ( x y v w z )( x ] )( y mi ] )( z mi ] )( w f ] )( v o ] )( y mi x )( z mi x )( w f x ) ( v ( o ; d ; s ) x )( z (= ; s ; si ) y )( w m y )( v b y )( w m z )( v b z )( w mi v ) . ( ( ? OBJECT2 : Clear- Block u ? OBJECT1 : > )@ x u ( ? OBJECT1 ON # ? OBJECT2 )@ y u ( ? OBJECT1 : Clear- Block u ? OBJECT2 : > )@ v u ( ? OBJECT1 : Hold- Block u ? OBJECT2 : > )@ w u ( ? OBJECT1 : Clear- Block u ? OBJECT2 : > )@ z ) 483 Ar t ale & Franconi In this example, the essen tial graph is also the CEF of Stack since there are no redundan t no des. 6.2 Computing Subsumption A concept subsumes another one just in case ev ery p ossible instance of the second is also an instance of the rst, for ev ery time in terv al. Thanks to the normal form, concept subsump- tion in the temp oral language is reduced to concept subsumption b et w een non-temp oral concepts and to subsumption b et w een temp oral constrain t net w orks. A similar general pro- cedure w as rst presen ted in (W eida & Litman, 1992), where the language for non-temp oral concepts is less expressiv e { it do es not include features or parametric features. T o compute subsumption b et w een non-temp oral concepts { whic h ma y p ossibly include lcs concepts { w e refer to (Cohen, Borgida, & Hirsh, 1992). In the follo wing, w e will write \ w F " for subsumption b et w een non-temp oral F concepts taking in to accoun t lcs concepts. Denition 6.11 (V ariable mapping) A v ariable mapping M is a total function M : X 1 7! X 2 such that M ( ] ) = ] . We write M ( X ) to intend fM ( X ) j X 2 X g , and M ( T c ) to intend f ( M ( X ) ( R ) M ( Y )) j ( X ( R ) Y ) 2 T c g . Denition 6.12 (T emp oral constrain t subsumption) A temp or al c onstr aint ( X 1 ( R 1 ) Y 1 ) is said to subsume a temp or al c onstr aint ( X 2 ( R 2 ) Y 2 ) under a generic variable mapping M , written ( X 1 ( R 1 ) Y 1 ) w M ( X 2 ( R 2 ) Y 2 ) , if M ( X 1 ) = X 2 , M ( Y 1 ) = Y 2 and ( R 1 ) E ( R 2 ) E for every temp or al interpr etation E . Prop osition 6.13 (TC subsumption algorithm) ( X 1 ( R 1 ) Y 1 ) w M ( X 2 ( R 2 ) Y 2 ) if and only if M ( X 1 ) = X 2 , M ( Y 1 ) = Y 2 and the disjuncts in R 1 ar e a sup erset of the disjuncts in R 2 . Pr o of. F ollo ws from the observ ation that the 13 temp oral relations are m utually disjoin t and their union co v ers the whole in terv al pairs space. 2 Denition 6.14 (T emp oral constrain t net w ork subsumption) A temp or al c onstr aint network h X 1 ; T c 1 i subsumes a temp or al c onstr aint network h X 2 ; T c 2 i under a variable map- ping M : X 1 7! X 2 , written h X 1 ; T c 1 i w M h X 2 ; T c 2 i , if hM ( X 1 ) ; M ( T c 1 ) i E h X 2 ; T c 2 i E for every temp or al interpr etation E . Prop osition 6.15 (TCN subsumption algorithm) h X 1 ; T c 1 i w M h X 2 ; T c 2 i i, after c omputing the temp or al tr ansitive closur e, ther e exists a variable mapping M : X 1 7! X 2 such that for al l X 1 i ; Y 1 j 2 X 1 exist X 2 m ; Y 2 n 2 X 2 which satisfy ( X 1 i ( R 1 i;j ) Y 1 j ) w M ( X 2 m ( R 2 m;n ) Y 2 n ) . Pr o of. \ ( " Since from denition 6.12 ( X 1 i ( R 1 i;j ) Y 1 j ) w M ( X 2 m ( R 2 m;n ) Y 2 n ) implies that ( R 1 i;j ) E ( R 2 m;n ) E for ev ery E , then, from the denition of in terpretation of a temp oral constrain t net w ork, it is easy to see that eac h assignmen t of v ariables V in the in terpretation of h X 2 ; T c 2 i is also an assignmen t in the in terpretation of hM ( X 1 ) ; M ( T c 1 ) i . \ ) " Supp ose that one is not able to nd suc h a mapping; then, b y h yp othesis, for eac h p ossible v ariable mapping there exists some i; j suc h that R 1 i;j is not a sup erset of R 2 m;n . 484 A Temporal Description Logic f or Reasoning about A ctions and Plans Since, b y assumption, the temp oral constrain t net w orks are minimal, the temp oral relation R 2 m;n cannot b e further restricted. So, for eac h v ariable mapping and eac h temp oral in ter- pretation E , w e can build an assignmen t V suc h that hV ( X 2 m ) ; V ( X 2 n ) i 2 ( R 2 m;n ) E while hV ( X 1 i ) ; V ( X 1 j ) i 62 ( R 1 i;j ) E . No w, w e can extend the assignmen t V in suc h a w a y that V 2 ( h X 2 ; T c 2 i ) E while V 62 ( hM ( X 1 ) ; M ( T c 1 ) i ) E . This con tradicts the assumption that h X 1 ; T c 1 i w M h X 2 ; T c 2 i . 2 Denition 6.16 (S-mapping) A n s-mapping fr om a c onc eptual temp or al c onstr aint net- work h X 1 ; T c 1 ; Q @ X 1 i to a c onc eptual temp or al c onstr aint network h X 2 ; T c 2 ; Q @ X 2 i is a variable mapping S : X 1 7! X 2 such that the non-temp or al c onc ept lab eling e ach no de in X 1 subsumes the non-temp or al c onc ept lab eling the c orr esp onding no de in S ( X 1 ) , and h X 1 ; T c 1 i w S h X 2 ; T c 2 i . The algorithm for c hec king subsumption b et w een temp oral concept expressions reduces the subsumer and the subsumee in essen tial graph form, then it lo oks for an s-mapping b e- t w een the essen tial graphs b y exhaustiv e searc h. T o pro v e the completeness of the o v erall subsumption pro cedure it will b e sho w ed that the in tro duction of lcs 's preserv es the sub- sumption. A mo del-theoretic c haracterization of the lcs will b e giv en for sho wing this prop ert y . Let's start to build an Herbrand mo del for an F concept. Let C 0 ( x ) denote the rst order form ula corresp onding to a concept C (see prop osition 2.1), while the function- alit y of features can b e expressed with a set of form ul F . By syn tax induction it easy to sho w that C 0 ( x ) is an existen tially quan tied form ula with one free v ariable. Moreo v er, the matrices of suc h form ula is a conjunction of p ositiv e predicates. F [ f C 0 ( x ) g is logically equiv alen t to F [ f C 00 ( x ) g where the functionalit y axioms allo w to map ev ery subform ula V y 9 y . F f ( x; y ) in to 9 ! y . F f ( x; y ). Then C 00 ( x ) is suc h that all the existen tial quan tiers in C 0 ( x ) (whic h come from the rst order con v ersion of features) are replaced b y 9 ! quan tiers. No w, F [ f C 000 ( a ) g { where a is a constan t substituting the free v ariable x and C 000 ( a ) is obtained b y sk olemizing the 9 ! quan tied v ariables { is a set of denite Horn clauses. Denition 6.17 (Herbrand mo del) L et C b e an F c onc ept expr ession. Then we dene its Minimal Herbrand Mo del H C as the Minimal Herbrand Mo del of the ab ove mentione d set of denite Horn clauses F [ f C 000 ( a ) g . Lemma 6.18 ( F concept subsumption) L et C ; D b e F c onc ept expr essions, and H C ; H D their minimal Herbr and mo dels obtaine d by skolemizing the rst or der set F [f C 000 ( a ) ; D 000 ( a ) g . Then, C v D i H D H C . Pr o of. C v D i F [ f C 0 ( x ) g j = D 0 ( x ), i F [ f C 00 ( x ) g j = D 00 ( x ), where C 00 and D 00 are obtained b y applying the functionalit y axioms to the set f C 0 ( x ) ; D 0 ( x ) g (i.e., uni- fying the v ariables in the functional predicates) and then replacing all the existen tial quan tiers b y 9 ! quan tiers. No w, C 000 ( x ) and D 000 ( x ) are obtained b y sk olemizing the 9 ! quan tied v ariables in the follo wing w a y: let C 00 ( x ) = 9 ! y 1 ; : : : ; y n ( x; y 1 ; : : : ; y n ) and let D 00 ( x ) = 9 ! y 1 ; : : : ; y k ; z 1 ; : : : ; z m ( x; y 1 ; : : : ; y k ; z 1 ; : : : ; z m ), with 0 k n , then sk olemize the form ula: = 9 ! y 1 ; : : : ; y n ; z 1 ; : : : ; z m ( x; y 1 ; : : : ; y n ) ^ ( x; y 1 ; : : : ; y k ; z 1 ; : : : ; z m ), and let 0 ( x ) indicate its sk olemized form. Then, C 000 ( x ) = 0 ( x ) and D 000 ( x ) = 0 ( x ). No w, since ev ery existen tial quan tication in C 00 ( x ) ; D 00 ( x ) w as of t yp e 89 ! then the thesis is true 485 Ar t ale & Franconi i F [ f C 000 ( a ) g j = D 000 ( a ), where a is a constan t substituting the free v ariable x (see (v an Dalen, 1994)). No w, as sho w ed b y lemma 6.17, b oth C 000 ( a ) and D 000 ( a ) ha v e minimal Her- brand mo dels H C ; H D that v erify the lemma h yp othesis. Then, F [ f C 000 ( a ) g j = D 000 ( a ) i H D H C . 2 W e are no w able to giv e a mo del-theoretic c haracterization of the lcs that will b e crucial to pro v e the subsumption-preserving prop ert y . Lemma 6.19 ( lcs mo del prop ert y) L et Q 1 ; : : : ; Q n b e F c onc ept expr essions, and H Q 1 ; : : : ; H Q n their minimal Herbr and mo dels obtaine d by skolemizing the rst or der set F [ f Q 000 1 ( a ) ; : : : ; Q 000 n ( a ) g . Then, Q = lcs f Q 1 ; : : : ; Q n g i H Q = H Q 1 \ : : : \ H Q n . Pr o of. First of all, let sho w that H Q is the minimal Herbrand mo del of a concept Q in the language F . Ev ery H Q i can b e seen as a ro oted directed acyclic graph where no des are lab elled with (p ossible empt y) set of atomic concepts and arcs with atomic features while equalit y constrain ts b et w een no des corresp ond to features agreemen t. Whithout loss of generalit y let us consider the case where H Q = H Q 1 \ H Q 2 . It is sucien t to sho w that H Q is a ro oted directed acyclic graph. Let a b e the ro ot of H Q 1 ; H Q 2 , then will b e pro v ed b y induction that if F i ( a i 1 ; a i ) 2 H Q (where F i is the rst order translation of a feature, a i 1 ; a i are obtained as a result of the sk olemization pro cess, and a 0 = a ) then f F 1 ( a; a 1 ) ; : : : ; F i ( a i 1 ; a i ) g H Q . The case i = 1 is trivial. Let i > 1. No w, F i ( a i 1 ; a i ) 2 H Q i F i ( a i 1 ; a i ) 2 H Q 1 \ H Q 2 . But a i 1 is uniquely dened b y the sk olem function f F i 1 (where, the function sym b ols f F i are newly generated for eac h feature F i b y the sk olemization pro cess). Then, F i ( a i 1 ; a i ) 2 H Q 1 \ H Q 2 i F i ( a i 1 ; f F i 1 ( a i )) 2 H Q 1 \ H Q 2 i F i 1 ( a i 2 ; f F i 1 ( a i )) 2 H Q 1 \ H Q 2 . Then the thesis is true b y induction. Let us no w pro v e the \ ( " direction. Supp ose b y absurd that there is an F concept C suc h that: Q 1 v C ^ Q 2 v C ^ C < Q . Then, Q 1 v C i H C H Q 1 , and Q 2 v C , i H C H Q 2 . But then H C H Q 1 \ H Q 2 , i.e., H C H Q . Then Q v C whic h con tradicts the h yp othesis. The \ ) " direction can b e pro v ed with analogous considerations. 2 Prop osition 6.20 ( lcs subsumption-preserving prop ert y) L et A; B ; C ; D b e F c on- c epts, then A u ( B t C ) v D i A u lcs f B ; C g v D . Pr o of. A u ( B t C ) v D i A u B v D and A u C v D . No w, A u B v D i F [ f A 000 ( a ) ; B 000 ( a ) g j = D 000 ( a ) i H A [ H B j = D 000 ( a ) i H D H A [ H B . F or the same reasons, A u C v D i H D H A [ H C . But then, H D H A [ H B and H D H A [ H C , i.e., H D H A [ ( H B \ H C ), i.e., H D H A [ H lcs f B ;C g . But, H D H A [ H lcs f B ;C g i A u lcs f B ; C g v D . 2 The follo wing theorem pro vides a sound and complete pro cedure to compute subsump- tion. The completeness pro of tak es in to accoun t that the temp oral structure is dense and un b ounded. This allo ws us to in tro duce an y new no de to a conceptual temp oral constrain t net w ork without c hanging its meaning. Remem b er that, for eac h of these redundan t no des, time-in v arian t information holds. Theorem 6.21 ( T L - F concept subsumption) A c onc ept C 1 subsumes a c onc ept C 2 i ther e exists an s-mapping fr om the essential gr aph of C 1 to the essential gr aph of C 2 . 486 A Temporal Description Logic f or Reasoning about A ctions and Plans Pr o of. Let T 1 = h X 1 ; T c 1 ; Q @ X 1 i b e the essen tial graph of C 1 , and T 2 = h X 2 ; T c 2 ; Q @ X 2 i b e the essen tial graph of C 2 . \ ( " (Soundness). F ollo ws from the fact that the essen tial graph form is logically equiv- alen t to the starting concept, and from the soundness of the pro cedures for computing b oth the TCN subsumption (prop osition 6.15) and the subsumption b et w een non-temp oral concepts (Cohen et al., 1992). \ ) " (Completeness). Supp ose that suc h an s-mapping do es not exist. Tw o main cases can b e distinguished. i) There is not a mapping M suc h that h X 1 ; T c 1 i w M h X 2 ; T c 2 i . By adding redundan t no des to T 2 , an equiv alen t conceptual temp oral constrain t net w ork T 2 = h X 2 ; T c 2 ; Q @ X 2 i ma y b e obtained. Let us consider suc h an extended net w ork in a w a y that there exists a v ariable mapping M suc h that h X 1 ; T c 1 i w M h X 2 ; T c 2 i . No w, for all p ossible M , there is a no de X 1 i 2 X 1 suc h that M ( X 1 i ) = X 2 j with X 2 j 62 X 2 . No w, Q 1 i 6w F Q 2 j , since X 2 j cannot coincide with other no des in X 2 neither can ha v e a co v ering otherwise the h yp othesis that the mapping M do es not exist w ould b e con tradicted. Then from prop osition 6.7 Q 2 j is in a time-in v arian t no de, whereas Q 1 i is not since T 1 is an essen tial graph. Then, although the construction of M allo ws for the existence of a unique V 3 for b oth net w orks (follo ws from prop osition 6.15), it is p ossible to build an instance of T 2 that is not an instance of T 1 . ii) F or eac h p ossible mapping M suc h that h X 1 ; T c 1 i w M h X 2 ; T c 2 i there will b e alw a ys t w o no des X 1 i and X 2 j suc h that M ( X 1 i ) = X 2 j and Q 1 i 6w F Q 2 j . No w, the concept ex- pression Q 2 j cannot b e rened (lo oking for a subsumption relationship with Q 1 i ) b y adding to it an F concept since from prop osition 6.7 this w ould c hange the o v erall in terpretation. On the other hand, the lcs in tro duction { whic h w ould substitute the more sp ecic con- cept disjunction implicitly presen ts b ecause of a no de co v ering { is a subsumption-in v arian t concept substitution, as sho w ed b y lemma 6.20. Both cases con tradict the assumption that T 1 subsumes T 2 . 2 6.2.1 Complexity of Subsumption No w it is sho wn that c hec king subsumption b et w een T L - F concept expressions in the es- sen tial graph form is an NP-complete problem. Therefore, a p olynomial reduction from the NP-complete problem of deciding whether a graph con tains an isomorphic subgraph is pre- sen ted. It is then sho wn that the subsumption computation, as prop osed in theorem 6.21, can b e done b y a non-deterministic algorithm that tak es p olynomial time in the size of the concepts in v olv ed. First of all let us consider the complexit y of computing subsumption b et w een non-temp oral concepts. Lemma 6.22 ( F subsumpion complexit y) L et C ; D b e F c onc ept expr essions that c an c ontain lcs 's. Then, che cking whether C v F D takes p olynomial time. Pr o of. See (Cohen et al., 1992). 2 Here the problem of sub gr aph isomorphism is briey recalled. Giv en t w o graphs, G 1 = ( V 1 ; E 1 ) and G 2 = ( V 2 ; E 2 ), G 1 con tains a subgraph isomorphic to G 2 if there exists a 3. Since subsumption is computed with resp ect to a xed ev aluation time, V maps the dieren t o ccurrences of ] to the same in terv al; this justies the c hoice that M ( ] ) = ] . 487 Ar t ale & Franconi subset of the v ertices V 0 V 1 and a subset of the edges E 0 E 1 suc h that j V 0 j = j V 2 j , j E 0 j = j E 2 j , and there exists a one-to-one function f : V 2 7! V 0 satisfying f u; v g 2 E 2 i f f ( u ) ; f ( v ) g 2 E 0 . Giv en a graph G = ( V ; E ), with V = f v 1 ; : : : ; v n g asso ciate a temp oral concept expression: C : = 3 ( v 1 ; : : : ; v n ) : : : ( v i ( b ; a ) v j ) : : : . ( A @ v 1 u : : : u A @ v n ), where A is an atomic concept and f v i ; v j g 2 E . This transformation allo ws us to pro v e that the problem of subgraph isomorphism can b e reduced to the subsumption of temp oral concepts. Prop osition 6.23 Given two gr aphs G 1 and G 2 , G 1 c ontains a sub gr aph isomorphic to G 2 i C 2 w C 1 , wher e C 1 and C 2 ar e the c orr esp onding temp or al c onc epts expr essions. Pr o of. A temp oral net w ork with edges lab eled only with the ( b efo re _ after ) relation is alw a ys consisten t, minimal and non-directed 4 (Gerevini & Sc h ub ert, 1994). Then, eac h temp oral concept is in the essen tial graph form. No w the pro of easily follo ws since, ev ery time G 2 is an isomorphic subgraph of G 1 the one-to-one function f is also an s-mapping from C 2 to C 1 , and it is true that C 2 w C 1 . On the other hand, the s-mapping that giv es rise to the subsumption is also the one-to-one isomorphism from G 2 to G 1 . 2 Theorem 6.24 (NP-hardness) Conc ept subsumption b etwe en T L - F c onc ept expr essions in normal form is an NP-har d pr oblem. Pr o of. F ollo ws from prop osition 6.23 and the reduction b eing clearly p olynomial. 2 No w the NP-completeness is pro v en. Theorem 6.25 (NP-completeness) Conc ept subsumption b etwe en T L - F c onc ept expr es- sions in normal form is an NP-c omplete pr oblem. Pr o of. T o pro v e NP-completeness it is necessary to sho w that the prop osed calculus can b e solv ed b y a nondeterministic algorithm that tak es p olynomial time. No w, giv en t w o temp oral concepts, T 1 and T 2 , in their essen tial graph form, let j X 1 j = N 1 and j X 2 j = N 2 . Then, to c hec k whether T 1 w T 2 , the algorithm guesses one of the N N 1 2 v ariable mapping from T 1 to T 2 and v eries whether it is an s-mapping, to o. This last step can b e done in deterministic p olynomial time since, giv en a mapping M , it is p ossible to determine whether h X 1 ; T c 1 i w M h X 2 ; T c 2 i b y c hec king at most N 1 ( N 1 1) = 2 edges lo oking for subsumption b et w een the corresp onding temp oral relations (solv ed b y a set inclusion pro cedure); while the N 1 non-temp oral concept subsumptions can b e computed in p olynomial time. 2 7. Extending the Prop ositional P art of the Language The prop ositional part of the temp oral language can b e extended to ha v e a more p o w erful, but still decidable, Description Logic. It is p ossible either to add full disjunction, b oth at the temp oral and non-temp oral lev els ( T LU - F U ), or to ha v e a prop ositionally complete language at the non-temp oral lev el only ( T L - ALC F ). Please note that in these languages it is not p ossible to express full negation, and in particular the negation of the existen tial temp oral quan tier. This is crucial, and it 4. If ( v i ( b ; a ) v j ) then ( v j ( b ; a ) v i ), to o. 488 A Temporal Description Logic f or Reasoning about A ctions and Plans ( C t D )@ X ! C @ X t D @ X p : ( C t D ) ! p : C t p : D ( C 1 t C 2 ) u D ! ( C 1 u D ) t ( C 2 u D ) 3 ( X ) T c . ( C t D ) ! 3 ( X ) T c . C t 3 ( X ) T c . D Figure 13: Rewrite rules for computing the disjunctiv e form. mak es the dierence with other logic-based approac hes (Sc hmiedel, 1990; Bettini, 1997; Halp ern & Shoham, 1991). The dual of 3 (i.e., the univ ersal temp oral quan tier 2 ) mak es the satisabilit y problem { and the subsumption { for prop ositionally complete languages undecidable in the most in teresting temp oral structures (Halp ern & Shoham, 1991; V enema, 1990; Bettini, 1993). F or the represen tation of actions and plans in the con text of plan recognition, the univ ersal temp oral quan tier is not strictly necessary . This limitation mak es these languages decidable, with nice computational prop erties, and capable of supp orting other kinds of useful extensions. The examples sho wn throughout the pap er ma y serv e as a partial v alidation of the claim. Section 8.1 prop oses the in tro duction of a limited univ ersal temp oral quan tication that main tains decidabilit y of subsumption. 7.1 Disjunctiv e Concepts: T LU - F U The language T LU - F U adds to the basic language T L - F the disjunction op erator { with the usual seman tics { b oth at the temp oral and non-temp oral lev els: C ; D ! T L j C t D ( T LU ) E ; F ! F j E t F ( F U ) Before sho wing ho w to mo dify the calculus to c hec k subsumption, let us b egin with a clarifying example. The gain in expressivit y allo ws us to describ e the alternativ e realizations that a giv en plan ma y ha v e. Let us consider a scenario with a rob ot mo ving in an empt y ro om that can mo v e only either horizon tally or v ertically . Let's call Rect-Move that whic h in v olv es a simple sequence of the t w o basic mo ving actions. Then, to describ e a Rect-Move plan w e can mak e use of the disjunction op erator: Rect- Move : = 3 ( x y ) ( ] m x )( x m y ) . ( Hor- Move @ x u Ver- Move @ y ) t 3 ( x y ) ( ] m x )( x m y ) . ( Ver- Move @ x u Hor- Move @ y ) 7.1.1 The Calculus f or T LU - F U Normal F orm In computing subsumption, a normal form for concepts is needed. The normalization pro- cedure is similar to that rep orted in Section 6.1. Let us start b y reducing eac h concept expression in to an equiv alen t disjunctive c onc ept of the form: ( 3 ( X 1 ) T c 1 . G 1 ) t t ( 3 ( X n ) T c n . G n ) t Q 1 t t Q m 489 Ar t ale & Franconi where G i are conjunctions of concepts of the form Q i k @ X i k , and eac h Q do es not con tain neither temp oral information, nor disjunctions, i.e., it is an elemen t of the language F . Prop osition 7.1 (Equiv alence of disjunctiv e form) Every c onc ept C c an b e r e duc e d into an e quivalent disjunctive form ( df C ) , by exhaustively applying the set of r ewrite rules of Figur e 13 in addition to the rules intr o duc e d in Figur e 11. It is no w p ossible to compute the c omplete d disjunctive normal form ( cdnf C ). Eac h dis- junct of suc h normal form has some in teresting prop erties, whic h are crucial for the pro of of the theorem 7.4 on concept subsumption: temp oral constrain ts are alw a ys explicit, i.e., an y t w o in terv als are related b y a basic temp oral relation; there is no disjunction, either implicit or explicit, neither in the conceptual part nor in the temp oral part, i.e., it is a T L - F concept; the information in eac h no de is indep enden t of the information in the other no des and it do es not con tain time-in v arian t (i.e., redundan t) no des. Denition 7.2 (Completed disjunctiv e normal form) Given a c onc ept in disjunctiv e form , the completed disjunctiv e normal form is obtaine d by applying the fol lowing r ewrite rules to e ach disjunct: ( T emp oral completion ) The rules of denition 6.5 ar e applie d to e ach disjunct with the exclusion of the co v ering step, which is r eplac e d by the t -in tro duction step. If a disjunct is unsatisable { i.e., the temp or al c onstr aint network asso ciate d with it is inc onsistent { then eliminate it. ( Essen tial form ) The rules of denition 6.8 ar e applie d to e ach disjunct. ( t in tro duction ) R e duc e to c onc epts c ontaining only b asic temp or al r elationships: 3 ( X ) ( X 1 ( R , S ) X 2 ) T c . C ! 3 ( X )( X 1 R X 2 ) T c . C t 3 ( X )( X 1 S X 2 ) T c . C Prop osition 7.3 (Equiv alence of CDNF) Every c onc ept expr ession c an b e r e duc e d into an e quivalent c omplete d disjunctive normal c onc ept. Subsumption The theorem 7.4 reduces subsumption b et w een CDNF concepts in to subsumption of disjun- ction-free concepts, suc h that the results of theorem 6.21 can b e applied. The follo wing theorem giv es a terminating, sound, and complete subsumption calculus for T LU - F U . Theorem 7.4 ( T LU - F U concept subsumption) L et C = C 1 t t C m and D = D 1 t t D n b e T LU - F U c onc epts in CDNF. Then, C v D if and only if 8 i 9 j . C i v D j . Pr o of. Since it is easy to sho w that C 1 t : : : t C n v D i 8 i . C i v D w e need only to pro v e the restricted thesis: C i v D 1 t t D n i C i v D 1 _ : : : _ C i v D n . Ev ery concept expression in CDNF corresp onds to an existen tial quan tied form ula with t w o free v ariables. Moreo v er, the matrices of suc h form ul are conjunctions of p ositiv e predicates. Let us denote the form ula corresp onding to a concept C as C 0 ( t; x ). No w, the restricted thesis holds i it is true that F [ f C 000 i ( a; b ) g j = D 000 1 ( a; b ) _ D 000 2 ( a; b ). No w, let H B the minimal Herbrand mo del of F [ f C 000 i ( a; b ) g . Then, F [ f C 000 i ( a; b ) g j = D 000 1 ( a; b ) _ D 000 2 ( a; b ) i H B j = D 000 1 ( a; b ) _ D 000 2 ( a; b ). Since w e are talking of a single mo del, D 000 1 ( a; b ) _ D 000 2 ( a; b ) is v alid in H B if and only if either D 000 1 ( a; b ) or D 000 2 ( a; b ) is v alid in H B . This pro v es the theorem. 5 2 5. The pro of of this theorem comes from an idea of W erner Nutt. 490 A Temporal Description Logic f or Reasoning about A ctions and Plans As a consequence of the theorems 6.25, 7.4 the follo wing complexit y result holds. Theorem 7.5 ( T LU - F U subsumption complexit y) Conc ept subsumption b etwe en T LU - F U c onc ept expr essions in normal form is an NP-c omplete pr oblem. 7.2 A Prop ositionally Complete Language: T L - ALC F T L - ALC F uses the prop ositionally complete Description Logic ALC F (Hollunder & Nutt, 1990) for non-temp oral concepts b y c hanging the syn tax rules for T L - F in the follo wing w a y: E ; F ! F U j ? j : E j p " q j p "j 8 P . E j 9 P . E ( ALC F ) The in terpretation functions are extended to tak e in to accoun t roles: P I T ? < I I P I t = ^ P t I I j 8 a; b . h a; b i 2 ^ P t $ h t; a; b i 2 P I As seen in Section 2, ALC F adds to F full negation { th us in tro ducing disagreemen t ( p " q ) and undenedness ( p " ) for features, and role quan tication ( 8 P . E ; 9 P . E ). As an example of the expressiv e p o w er gained, let us rene the description of the w orld states in v olv ed in the Stack action (see Section 5.2). Supp ose that a blo c k is describ ed b y sa ying that it has LATERAL-SIDE s (role) and BOTTOM- and TOP-SIDE s (features). Then, the prop ert y of b eing clear could b e represen ted as follo ws: Clear- Block : = Block u 8 LATERAL- SIDE . Clear u TOP- SIDE : HAS- ABOVE " whic h sa ys that, in order to b e clear, eac h LATERAL-SIDE has to b e clear and nothing has to b e o v er the TOP-SIDE . No w, the situation in whic h a blo c k in v olv ed in a Stack action is on top of another one is reform ulated with the follo wing concept expression: ( ? OBJECT1 TOP- SIDE HAS-ABOVE # ? OBJECT2 ) F urthermore, giv en the ab o v e denition of Clear- Block s, it can b e deriv ed that: ( ? OBJECT1 TOP- SIDE HAS-ABOVE # ? OBJECT2 ) v ( ? OBJECT1 : : Clear- Block ) i.e., an ob ject, ha ving another ob ject on top of it, is no more a clear ob ject. In T L - ALC F it is p ossible to describ e states with some form of incomplete kno wledge b y exploiting the disjunction among non-temp oral concepts. F or example, let us sa y that the agen t of an action can b e either a h uman b eing or a mac hine: ? AGENT : ( Person t Robot ). 7.2.1 The Calculus f or T L - ALC F This Section presen ts a calculus for deciding subsumption b et w een temp oral concepts in the Description Logic T L - ALC F . Again, the calculus is based on the idea of separating the inference on the temp oral part from the inference on the Description Logic part (\ v ALC F "), and adopting standard pro cedures dev elop ed in the t w o areas. Normal F orm Once more, the subsumption calculus is based on a normalization pro cedure. The rst step reduces a concept expression in to an equiv alen t existential form { 3 ( X ) T c . ( Q 0 u Q 1 @ X 1 u : : : u Q n @ X n ) { b y applying the rewrite rules of Figure 11 augmen ted with the 491 Ar t ale & Franconi :> ! ? :? ! > : ( C u D ) ! : C t : D : ( C t D ) ! : C u : D : : C ! C :8 P . C ! 9 P . : C :9 P . C ! 8 P . : C : f : C ! f " t f : : C : p : C ! f " t f : ( : q : C ) if p = f q : p # q ! p " t q " t p " q : p " q ! p " t q " t p # q ( f p ) " ! f " t f : ( p " ) Note: By f w e denote b oth an atomic feature and an atomic parametric feature. Figure 14: Rewrite rules to transform an arbitrary concept in to a simple concept. rule: p : ( q 1 " q 2 ) ! p q 1 " p q 2 . Eac h Q is a non-temp oral concept, i.e., it is an elemen t of the language ALC F . In the follo wing normalization step there will b e a need to v erify concept satisabilit y for non-temp oral concept expressions. An ALC F concept E is unsatisable i E v ALC F ? . Algorithms for c hec king satisabilit y and subsumption of concepts terms in ALC F are w ell kno wn (Hollunder & Nutt, 1990). Denition 7.6 (Completed existen tial form) The temp oral completion of a c onc ept in existential form { the Completed Existen tial F orm , CEF { is obtaine d by se quential ly applying the fol lowing steps: ( closure, collapsing, co v ering ) As r ep orte d in denition 6.5. As for the co v ering , tr anslate the c onc ept expr ession Q y applying the r ewrite rule: Q y ! Q y u i =1 ::: n ( Q i 1 t : : : t Q i m ) . ( parameter in tro duction ) This r e quir es two phases. 1. Each Q is tr anslate d in disjunctiv e normal form . First the simple form 6 is ob- taine d by tr ansforming e ach Q fol lowing the r ewrite rules r ep orte d in Figur e 14. The disjunctiv e normal form is then obtaine d by r ewriting e ach Q { which is now in simple form { using the fol lowing rules, which c orr esp ond to the rst or der rules for c omputing the disjunctive normal form of lo gic al formul: ( C 1 t C 2 ) u D ! ( C 1 u D ) t ( C 2 u D ) p : ( C t D ) ! p : C t p : D 6. A simple c onc ept con tains only complemen ts of the form : A , where A is a primitiv e concept, and no sub-concepts of the form p " , where p is not an atomic (parametric) feature { this corresp onds to a rst order logical form ula in negation normal form. 492 A Temporal Description Logic f or Reasoning about A ctions and Plans ?g 1 : : : ?g n [ f [ p ]] : C ! ?g 1 : : : ?g n : > . ?g 1 : : : ?g n [ f [ p ]] # g [ q ] ! ?g 1 : : : ?g n : > . ?g 1 : : : ?g n # ?f 1 : : : ?f m ! ?g 1 : : : ?g n # ?f 1 : : : ?f m . ?g 1 : : : ?g n g [ p ] # ?f 1 : : : ?f m [ f [ q ]] ! ?g 1 : : : ?g n : > u ?f 1 : : : ?f m : > . ?g 1 : : : ?g n " ?f 1 : : : ?f m ! ?g 1 : : : ?g n " ?f 1 : : : ?f m . ?g " ! ?g " . ?g 1 : : : ?g n : ( ?g n +1 " ) ! ?g 1 : : : ?g n : ( ?g n +1 " ) . ?g 1 : : : ?g n [ f [ p ]] " g [ q ] ! ?g 1 : : : ?g n : > . ?g 1 : : : ?g n g [ p ] " ?f 1 : : : ?f m [ f [ q ]] ! ?g 1 : : : ?g n : > u ?f 1 : : : ?f m : > . Figure 15: Rewrite rules that compute the parameter in tro duction step. 2. F or e ach Q j = E j 1 t : : : t E j n , on c ompute its time-invariant p art (let us indic ate this p articular c onc ept expr ession as ~ Q j ). This gives ~ Q j by c omputing for e ach disjunct E j i in Q j its time-invariant information ~ E j i . If E j i v ALC F ? , then ~ E j i = ? . Otherwise, r ewrite every c onjunct in E j i as showe d in Figur e 15, while the c onjuncts not c onsider e d ther e ar e r ewr ote to > . Now, unless ther e is an ~ E j i = > , ~ Q j = ~ E j 1 t : : : t ~ E j n must b e c onjunctively adde d to al l the other no des. Prop osition 7.7 (Equiv alence of CEF) Every c onc ept in existential form c an b e r e- duc e d into an e quivalent c omplete d existential c onc ept. As for the T L - F case, b oth c overing and p ar ameter intr o duction can b e computed inde- p enden tly . As a consequence of the ab o v e normalization phase, the prop osition 6.7 (no de indep endence) is no w true for T L - ALC F concepts in CEF. Observ e that, to obtain a CEF concept, the steps of the normalization pro cedure require the computation of the transitiv e closure of the temp oral relations { whic h is an NP-complete problem (v an Beek & Co- hen, 1990) { and the computation of ALC F subsumption { whic h is a PSP A CE-complete problem (Hollunder & Nutt, 1990). Before the presen tation of the last normalization phase, whic h will eliminate redundan t no des, it is no w p ossible to c hec k whether a concept expression is satisable. Prop osition 7.8 (Concept satisabilit y) A T L - ALC F c onc ept in CEF, h X ; T c ; Q @ X i , is satisable (with the pr oviso that the temp or al c onstr aints ar e satisable) if and only if the non-temp or al c onc epts lab eling e ach no de in X ar e satisable. Che cking satisability of a T L - ALC F c onc ept in CEF is a PSP A CE-c omplete pr oblem. Pr o of. Is a direct consequence of the no de indep endence established b y prop osition 6.7, whic h is true also for T L - ALC F concepts in CEF. 2 The normalization pro cedure no w go es on b y rewriting unsatisable concepts to ? and then computing the essential gr aph form for satisable concepts. This last phase is more 493 Ar t ale & Franconi complex than for the other temp oral languages considered in this pap er essen tially b ecause ALC F can express the > concept b y means of a concept expression (e.g., > = A t : A ). F rom this consideration it follo ws that in T L - ALC F a redundan t no de can b e deriv ed from a complex concept expression (e.g., b oth A t : A , and ?g : A t ?g : : A are redundan t no des). The k ey idea is that all the time-in v arian t information is presen t in the ] no de thanks to the CEF. Th us it is needed only to extract this information from the ] no de b y computing the disjunctiv e normal form of Q ] , applying the ~ translation, and then testing whether ~ Q ] v ALC F Q i , for a giv en no de x i . Denition 7.9 (Essen tial graph) The sub gr aph of the CEF of a T L - ALC F c onc eptual temp or al c onstr aint network T = h X ; T c ; Q @ X i obtaine d by deleting the no des x i such that ~ Q ] v ALC F Q i { with the exc eption of the ] no de { is c al le d essen tial graph of T : ( ess T ). Prop osition 7.10 (Equiv alence of essen tial graph) Every CEF c onc ept c an b e r e duc e d into an e quivalent essential gr aph form (and, obviously, every c onc ept c an b e r e duc e d into an e quivalent essential gr aph form). Subsumption The o v erall normalization pro cedure reduces the subsumption problem in T L - ALC F to the subsumption b et w een ALC F concepts. Theorem 7.11 ( T L - ALC F concept subsumption) A c onc ept C 1 subsumes a c onc ept C 2 if and only if ther e exists an s-mapping fr om the essential gr aph of C 1 to the essential gr aph of C 2 . The ab o v e theorem giv es a sound and complete algorithm for computing subsumption b e- t w een T L - ALC F concepts (the pro of is the same as the one for theorem 6.21). The sub- sumption problem is no w PSP A CE-hard, since satisabilit y and subsumption for ALC F concepts w ere pro v en to b e PSP A CE-complete (Hollunder & Nutt, 1990). 8. Extending the Expressivit y for States The follo wing suggests ho w to extend the basic language to cop e with imp ortan t issues in the represen tation of states. (i) Homogeneit y allo ws us to consider prop erties of the w orld { p eculiar to states { whic h remain true in eac h subin terv al of the in terv al in whic h they hold. (ii) P ersistence guaran tees that a state holding as an eect of an action con tin ues to hold unless there is no evidence of its falsit y at some time. An approac h to the fr ame problem is then presen ted, sho wing a p ossible solution to one of the most (in)famous problems in AI literature. The follo wing subsections shall b e in terested more in seman tically c haracterizing actions and states than on computational prop erties. The extensions prop osed no w to the temp oral languages are for ha ving a full edged Description L o gic for time and action . 8.1 Homogeneit y In the temp oral literature homo geneity c haracterizes the temp oral b eha vior of w orld states: when a state holds o v er an in terv al of time t , it also holds o v er subin terv als of t . Th us, if 494 A Temporal Description Logic f or Reasoning about A ctions and Plans - - - r OnTable(BLOCK) Simple-Stack(BLOCK) r OnBlock(BLOCK) x ] y Figure 16: T emp oral dep endencies in the denition of the Simple-Stack action. a blo c k is on the table for a whole da y , one can conclude that it is also on the table in the morning. On the other hand, actions are not necessarily homogeneous. In the linguistic literature a dierence is made b et w een activity and p erformanc e v erbs. The distinction comes out in the fact that activit y v erbs do ha v e sub-ev en ts that are denoted b y the same v erb, whereas p erformance v erbs do not. Generally , activit y v erbs represen t ongoing ev en ts, for example to e at and to run , and can b e describ ed as homogeneous predicates; whereas p erformance v erbs represen t ev en ts with a w ell dened gran ularit y in time, suc h as to pr ep ar e sp aghetti . P erformance v erbs are an example of an ti-homogeneous ev en ts: if they o ccur o v er an in terv al of time t , then they do not o ccur o v er a subin terv al of t , as they w ould not y et b e completed. The language is extended b y in tro ducing the Homo geneity op erator: C ; D ! r C (homogeneous concept) The seman tics of homogeneous concepts is easily giv en in terms of the seman tics of the temp oral univ ersal quan tier: r C 2 x ( x (= ; s ; d ; f ) ] ) . C @ x . This means that r C is an homogeneous concept if and only if when it holds at an in terv al it remains true at eac h subin terv al. In particular, 2 x univ ersally qualies the temp oral v ariable x , while the temp oral constrain t ( x (= ; s ; d ; f ) ] ) imp oses that x is a generic in terv al con tained in ] . Moreo v er, it is alw a ys true that r C v C , i.e., r C is a more sp ecic concept than C . Let us consider as an example a more accurate denition of the Basic-Stack action (see Section 4.1.1): Simple-Stack : = 3 ( x y )( x m ] )( ] m y ) . ( ( ? BLOCK : r OnTable )@ x u ( ? BLOCK : r OnBlock )@ y ) Figure 16 sho ws the temp oral dep endencies of the in terv als in whic h the Simple-Stack holds. The dierence with the Basic-Stack action is the use of the homogeneit y op erator. In fact, since the predicates OnTable and OnBlock denote states, their homogeneit y should b e explicitly declared. The assertion Simple-Stack ( i; a ) sa ys that a is an individual action of t yp e Simple-Stack o ccurred at in terv al i . Moreo v er, the same assertion implies that a is related to a ? BLOCK , sa y b , whic h is of t yp e OnTable at some in terv al j { meeting i { and at all in terv als included in j , while it is of t yp e OnBlock at another in terv al l { met b y i { and at all in terv als included in l : Simple-Stack ( i; a ) = ) 9 b . ? BLOCK ( a; b ) ^9 j; l . m ( j; i ) ^ m ( i; l ) ^ 8 ^ ; ^ l . (= ; s ; d ; f )( ^ ; j ) ^ (= ; s ; d ; f )( ^ l ; l ) ! OnTable ( ^ ; b ) ^ OnBlock ( ^ l ; b ) : 495 Ar t ale & Franconi - - - r OnTable(BLOCK) Instant- Stack(BLOCK) r OnBlock(BLOCK) z ] y Figure 17: T emp oral dep endencies in the denition of the Instant- Stack action. Note that the Simple-Stack action subsumes the Instant- Stack action, whose temp oral dep endencies are depicted in Figure 17: Instant- Stack : = 3 ( z y )( ] f z )( ] m y ) . ( ( ? BLOCK : r OnTable )@ z u ( ? BLOCK : r OnBlock )@ y ) Subsumption holds b ecause the class of in terv als { obtained b y homogeneit y of the state OnTable as dened in the Simple-Stack action { including x and all its subin terv als is a subset of the class of in terv als o v er whic h the blo c k is kno wn to b e on the table, according to the denition of Instant- Stack { this latter class includes all the subin terv als of z . If the Instant- Stack action had b een dened without the r op erator, then it w ould not sp ecialize an y more the Simple-Stack action. In fact, according to suc h a w eak er denition of Instant- Stack , sp ecifying that the ob ject is on the table at z do es not imply that the ob ject is on the table at subin terv als of z ; in particular, it is not p ossible to deduce an y more that the ob ject is on the table at x and its subin terv als, as sp ecied in the denition of Simple-Stack action. Moreo v er, the we ak Instant- Stack action t yp e w ould not sp ecialize the we ak Simple-Stack action t yp e { i.e., Basic- Stack { to o. Th us, homogeneit y helps us to dene states and actions in a more accurate w a y , suc h that imp ortan t inferences are captured. As seen ab o v e, the denition of homogeneit y mak es use of univ ersal temp oral quan- tication. Remem b er that subsumption in a prop ositionally complete Description Logic with b oth existen tial and univ ersal temp oral quan tication is undecidable and it is still an op en problem if it b ecomes decidable in absence of negation (Bettini, 1993). The homo- geneit y op erator is a restricted form of univ ersal quan tication. An ev en more restricted form in terests us here, where the concept C in r C do es not con tain an y other temp oral op erator (called simple homo gene ous c onc ept ). The expressiv eness of the resulting logic is enough, for example, to correctly represen t the homogeneous nature of states. In (Artale, Bettini, & F ranconi, 1994) an algorithm to compute subsumption in T L - F augmen ted with the homogeneit y op erator is prop osed. Ev en if a formal pro of is still not a v ailable, go o d argumen ts are discussed to conjecture its completeness. This w ould also pro v e decidabilit y of this logic and of the corresp onding mo dal logics. 8.2 P ersistence This Section sho ws ho w our framew ork can b e successfully extended in a general w a y to cop e with inertial prop erties. In the basic temp oral language, a prop ert y holding, sa y , as a p ost-condition of an action at a c ertain in terv al, is not guaran teed to hold an ymore at other included or subsequen t in terv als. This is the reason wh y w e prop ose an extended 496 A Temporal Description Logic f or Reasoning about A ctions and Plans - - - Load(GUN) ] Fire(GUN,TARGET) ] Loaded(GUN) x Dead(TARGET) y := Loaded(GUN) x : Loaded(GUN) z or Figure 18: Denitions of the actions Load and Fire . formalism, in whic h states can b e represen ted as homogeneous and p ersisten t concepts. As a motiv ation for in tro ducing the p ossibilit y of represen ting p ersisten t prop erties in the language, this Section considers ho w to solv e the fr ame pr oblem , and in particular the famous example of the Y ale T urk ey Sho oting Scenario (Sandew all, 1994; Allen & F erguson, 1994), formerly kno wn as the Y ale Sho oting Pr oblem . An inertia op erator \ = " is in tro duced here. In tuitiv ely , = C is curren tly true if it w as true at a preceding in terv al { sa y i { and there is no evidence of the falsit y of C at an y in terv al b et w een the curren t one and i . Th us, the prop ert y of an individual of b eing of t yp e C p ersists o v er time, unless a con tradiction arises. The formalization of the inertia op erator mak es use of the epistemic op erator K (Donini, Lenzerini, Nardi, Sc haerf, & Nutt, 1992), in whic h K C denotes the set of individuals known to b e instances of the concept C 7 . Denition 8.1 (Inertia) = C ( j; a ) i 9 i . star t ( i ) star t ( j ) ^ C ( i; a ) ^ 8 h . star t ( h ) end ( i ) ^ end ( h ) end ( j ) ! : K : C ( h; a ) . where start and end are t w o functions giving resp ectiv ely the starting and the ending p oin t of an in terv al { conditions on endp oin ts are simpler and more readable than their equiv alen ts on in terv al relations; : K : C ( h; a ) means that it is not kno wn that a is not of t yp e C at in terv al h . F urthermore, the follo wing relation holds: 8 a; j . C ( j; a ) ! = C ( j; a ); i.e., = C subsumes C . The ab o v e denition can b e captured b y a temp oral language equipp ed with the epistemic op erator { K { and the homogeneit y op erator { r : = C C t 3 ( x y ) ( x ( b ; m ; o ; ; di ) ] )( x ( s ; si ) y )( y ] ) . ( C @ x u r ( : K : C )@ y ) Tw o action t yp es are dened, Load { with the parameter ? GUN { and Fire { with the parameters ? GUN and ? TARGET (Figure 18): Load : = 3 x ( ] m x ) . ? GUN : Loaded @ x Fire : = 3 ( x y z ) ( ] f x )( ] m y )( ] m z ) . ( ? GUN : := Loaded @ x t ? TARGET : Dead @ y ) u ? GUN : : Loaded @ z The action Load describ es loading a gun. The action Fire describ es ring the gun against a target: eects of ring are that the gun b ecomes unloaded and either the target is dead 7. An epistemic interpr etation is a pair ( I , W ) in whic h I is an in terpretation and W is a set of in terpreta- tions suc h that ( K C ) I ; W = T J 2W ( C J ; W ). 497 Ar t ale & Franconi - - - Load( gun ) Fire( gun , fr e d ) Loaded( gun ) Dead( fr e d ) = Loaded( gun ) : Loaded( gun ) i i 1 j 0 j j 1 j 2 Figure 19: Actions instances in the Y ale Sho oting Problem. or the gun w as not loaded { p ossibly b y inertia { b efore ring. The Y ale Sho oting Problem considers the situation describ ed b y the follo wing set of assertions (ABo x): Load ( i; lo ad - action ) ; ? GUN ( lo ad - action ; gun ) ; a ( j; i ) ; Fire ( j; r e - action ) ; ? GUN ( r e -action ; gun ) ; ? TARGET ( r e - action ; fr e d ) : i.e., at the b eginning the gun is loaded; then, the action of ring the gun against the target f r ed is p erformed. According to the seman tics of the language, logical consequences of the kno wledge base are: j = 9 i 1 . m ( i; i 1 ) ^ Loaded ( i 1 ; gun ) j = 9 j 1 . m ( j; j 1 ) ^ : Loaded ( j 1 ; gun ) j = 9 j 0 . f ( j; j 0 ) ^ = Loaded ( j 0 ; gun ) j = 9 j 2 . m ( j; j 2 ) ^ Dead ( j 1 ; fr e d ) : i.e., (see also Figure 19) (i) the Load action mak es the gun loaded; (ii) the Fire action mak es the gun unloaded at the end ; (iii) since there is no evidence to the con trary , the gun is still loaded at j 0 b y inertia; (iv) since the gun is not unloaded at j 0 , the target f r ed m ust b e dead. Since the inertia op erator is useful to describ e the b eha vior of pr op erties , whic h are c haracterized as homogeneous concepts, a simple w a y of represen ting p ersistence in the con text of homogeneous concepts is prop osed. Prop osition 8.2 L et P b e a pr op erty { i.e., P : = r P 0 is an homo gene ous c onc ept { and a know le dge b ase such that 6j = P ( j; a ) . = P ( j; a ) is true in { i.e., j = = P ( j; a ) { if and only if two intervals i; k exist such that: j = ( star t ( i ) star t ( j ) ^ P ( i; a ) ) and [ f s ( i; k ) ; f ( j; k ) ; P ( k ; a ) g is satisable. Pr o of. The en tailmen t test v eries the rst part of the denition of inertia, while the satisabilit y test v eries that, b et w een the in terv al at whic h the system kno ws that the individual a b elongs to P { i { and the in terv al at whic h P ( a ) is deduced b y inertia { j { do es not exist an in terv al h at whic h the system kno ws that P ( a ) is false. Indeed, suc h in terv al h w ould b e related to the in terv al k b y the relation in and since it is supp osed that P is homogeneous, the kno wledge base with : P ( h; a ) ^ P ( k ; a ) ^ in ( h; k ) w ould b e inconsisten t. 2 The deduction P ( j; a ) ! = P ( j; a ) can b e obtained as a particular case of the ab o v e stated prop osition. 498 A Temporal Description Logic f or Reasoning about A ctions and Plans 9. Related W orks The original formalism devised b y Allen (1991) forms, in its v ery basis, the foundation for our w ork. It is a predicate logic in whic h in terv al temp oral net w orks can b e in tro duced, prop erties can b e asserted to hold o v er in terv als, and ev en ts can b e said to o c cur at in ter- v als. His approac h is v ery general, but it suers from problems related to the seman tic formalization of the predicates hold and occur (Blac kburn, 1992). Moreo v er, computa- tional prop erties of the formalism are not analyzed. The study of this latter asp ect w as, on the con trary , our main concern. In the Description Logic literature, other approac hes for represen ting and reasoning with time and action w ere prop osed. In the b eginning the approac hes based on an explicit notion of time are surv ey ed, and then the Strips -lik e approac hes are considered. This Section ends b y illustrating some of the approac hes dev oted to temp orally extend the situation calculus. Bettini (1997) suggests a v ariable-free extension with b oth existen tial and univ ersal temp oral quan tication. He giv es undecidabilit y results for a class of temp oral languages { resorting to the undecidabilit y results of Halp ern and Shoham's temp oral logic { and in- v estigates appro ximated reasoning algorithms. Basically , he extends the ALC N description logics with the existen tial and univ ersal temp oral quan tiers, but, unlik e our formalism, explicit in terv al v ariables are not allo w ed. The temp oral quan tication mak es use of a set of temp oral constrain ts on t w o implicit in terv als: the reference in terv al and the curren t one. In this framew ork, the concept of Mortal can b e dened as: Mortal : = LivingBeing u 3 ( after ) . ( not LivingBeing ) Sc hild (1993) prop oses the em b edding of p oint -based tense op erators in a prop ositionally closed Description Logic. He pro v ed that satisabilit y in ALC T , the p oin t-based temp oral extension of ALC , in terpreted on a linear, un b ounded and discrete temp oral structure, is PSP A CE-complete. His ideas w ere applied b y (Fisc her, 1992; Neu wirth, 1993) in the Ba ck system. Note that a p oin t-based temp oral on tology is unable to express all the v ariet y of relations b et w een in terv als. Baader and Laux (1995) in tegrate mo dal op erators for time and b elief in a terminological system lo oking for an adequate seman tics for the resulting com bined language. The ma jor p oin t in this pap er is the p ossibilit y of using mo dal op erators not only inside concept expressions but also in fron t of concept denitions and assertions. The follo wing example sho ws the notion of Happy-father , where dieren t mo dalities in teract: [ BEL- JOHN ]( Happy- father : = 9 MARRIED- TO . ( Woman u [ BEL-JOHN ] Pretty ) u h future i8 CHILD . Graduate ) In this case, it is John 's b elief that a Happy-father is someone married to a w oman b eliev ed to b e prett y b y John , and whose c hildren will b e graduates sometime in the future. The seman tics has a Kripk e-st yle: eac h mo dal op erator is in terpreted as an accessibilit y relation on a set of p ossible w orlds, while the domain of ob jects is split in to (p ossible) dieren t domain ob jects, eac h one dep ending on a giv en w orld. This latter c hoice captures the case of dieren t denitions for the same concept { suc h as [ BEL- JOHN ]( A : = B ) and [ BEL-PETER ]( A : = C ) { since the t w o form ul are ev aluated in dieren t w orlds. The main restriction is that all the mo dal op erators do not satisfy an y sp ecic axioms for b elief or time. On the other hand, the language is pro vided with a complete and terminating algorithm that should 499 Ar t ale & Franconi serv e, as the authors prop ose, \...as a basis for satisabilit y algorithms for more complex languages". There are Description Logics in tended to represen t and reasoning ab out actions follo wing the Strips tradition. Heinsohn, Kudenk o, Neb el and Protlic h (1992) describ e the Ra t system, used in the Wip pro ject at the German Researc h Cen ter for AI (DFKI). They use a Description Logic to represen t b oth the w orld states and atomic actions. A second formalism is added to comp ose actions in plans and to reason ab out simple temp oral relationships. No explicit temp oral constrain ts can b e expressed in the language. Ra t actions are dened b y the c hange of the w orld state they cause, and they are instan taneous as in the Strips -lik e systems, while plans are linear sequences of actions. The most imp ortan t service oered b y Ra t is the simulate d exe cution of part of a plan, c hec king if a giv en plan is fe asible and, if so, computing the global pre- and p ost-conditions. The feasibilit y test is similar to the usual consistency c hec k for a concept description: they temp or al ly pr oje ct the pre- and p ost-conditions of individual actions comp osing the plan, resp ectiv ely bac kw ard and forw ard. If this do es not lead to an inconsisten t initial, nal or in termediate state, the plan is feasible and the global pre- and p ost-conditions are determined as a side eect. Dev an bu and Litman (1991, 1996) describ e the Clasp system, a plan-b ase d kno wledge represen tation system extending the notion of subsumption and classication to plans, to build an ecien t information retriev al system. In particular, Clasp w as used to repre- sen t plan-lik e kno wledge in the domain of telephone switc hing soft w are b y extending the use of the soft w are information system lassie (Dev an bu, Brac hman, Selfridge, & Ballard, 1991). Clasp is designed for represen ting and reasoning ab out large collections of plan descriptions, using a language able to express temp oral, conditional and lo oping op erators. F ollo wing the Strips tradition, plan descriptions are built starting from states and actions, b oth represen ted b y using the Classic (Brac hman, McGuiness, P atel-Sc hneider, Resnic k, & Borgida, 1991) terminological language. Since plans constructing op erators corresp ond to regular expressions, algorithms for subsumption in tegrate w ork in automata theory with w ork in concept subsumption. The temp oral expressiv e p o w er of this system can capture to sequences, disjunction and iterations of actions and eac h action is instan taneous. F ur- thermore, state descriptions are restricted to a simple conjunction of primitiv e Classic concepts. Lik e Ra t , Clasp c hec ks if an instan tiated plan is w ell formed, i.e., the sp ecied sequence of individual actions are able to transform the giv en initial state in to the goal state b y using the Strips rules. W e end up b y rep orting on the eorts made b y researc hers in the situation calculus eld to o v ercome the strict sequen tial p ersp ectiv e inheren t to this framew ork. Recen t w orks enric h the original framew ork to represen t prop erties and actions ha ving dieren t truth v alues dep ending not only on the situation but also on time. The w ork of Reiter (1996), mo ving from the results sho w ed b y Pin to (1994) and b y T erno vsk aia (1994), pro vides a new axiomatization of the situation calculus able to capture concurren t actions, prop erties with con tin uous c hanges, and natural exogenous actions { those under nature's con trol. The notion of uent { whic h mo dels prop erties of the w orld { and situation are main tained. Eac h action is instan taneous and resp onsible for c hanging the actual situation to the subsequen t one. Concurren t actions are simply sets of instan taneous actions that m ust b e coheren t, i.e., the action's collection m ust b e non empt y and all the actions o ccur at the same time. Pin to (1994) and Reiter (1996) in tro duce the time dimension essen tially to capture b oth 500 A Temporal Description Logic f or Reasoning about A ctions and Plans the o ccurrence of the natural actions, due to kno wn la ws of ph ysics { i.e., the ball b ouncing at times prescrib ed b y motion's equations { and the dynamic b eha vior of ph ysical ob jects { i.e., the p osition of a falling ball. This is realized b y in tro ducing a time argumen t for eac h action function, while prop erties of the w orld are divided in to t w o dieren t classes: classical uen ts that hold or do not hold throughout situations, and c ontinuous p ar ameters that ma y c hange their v alue during the time spanned b y the giv en situation. More dev oted to ha v e a situation calculus with a time in terv al on tology is the w ork of T erno vsk aia (1994). In order to describ e pr o c esses { i.e., actions extended in time { she in tro duces durationless actions that initiate and terminate those pro cesses. As a matter of fact, pro cesses b ecome uen ts, with instan taneous ev en ts { Start(Fluent) and Finish(Fluent) { whic h resp ectiv ely mak e true or false the corresp onding uen t, and with p ersistence assumptions that mak e the uen t true during the in terv al. F or example, in a blo c ks w orld the picking-up pro cess is treated as a uen t with Start(picking-up(x)) and Finish(picking- up(x)) instan taneous actions that enable or falsify the picking-up uen t. 10. Conclusions The main ob jectiv e of this pap er w as the design of a class of logical formalisms for uni- formly represen ting time, actions and plans. According to this framew ork, an action has a duration in time, it can ha v e parameters, whic h are the ties with the temp oral ev olution of the w orld, and it is p ossibly asso ciated o v er time with other actions. A mo del-theoretic seman tics including b oth a temp oral and an ob ject domain w as dev elop ed, for giving b oth a meaning to the language form ul and a w ell founded denition of the v arious reasoning services, allo wing us to pro v e soundness and completeness of the corresp onding algorithms. The p eculiar computational prop erties of this logic mak e it an eectiv e represen tation and reasoning to ol for plan recognition purp oses. An action taxonom y based on subsumption can b e set up, and it can pla y the role of a plan library for plan retriev al tasks. This pap er con tributes to exploration of the decidable realm of in terv al-based temp oral extensions of Description Logics. It presen ted complete pro cedures for subsumption rea- soning with T L - F , T LU - F U and T L - ALC F . In addition, the subsumption problem for T L - F w as pro v en an NP-complete problem. The subsumption pro cedures are based on an in terpretation preserving transformation that op erates a separation b et w een the tem- p oral and the non-temp oral parts of the formalism. Th us, the calculus can adopt distinct standard pro cedures dev elop ed in the Description Logics comm unit y and in the temp oral constrain ts comm unit y . T o obtain decidable languages the k ey idea w as to restrict the tem- p oral expressivit y b y eliminating the univ ersal quan tication on temp oral v ariables. While a prop ositionally complete Description Logic with b oth existen tial and univ ersal temp oral quan tication is undecidable, it is still an op en problem if it b ecomes decidable in absence of negation. With the in tro duction of the homogeneit y op erator in v estigation of the impact of a restricted form of temp oral univ ersal quan tication in the language T L - F w as b egun. Sev eral extensions w ere prop osed to the basic temp oral language. With the p ossibilit y to sp ecify homogeneous predicates the temp oral b eha vior of w orld states can b e describ ed in a more natural w a y , while the in tro duction of the non-monotonic inertial op erator giv es rise to some forms of temp oral prediction. Another extension { not considered in this pap er { deals with the p ossibilit y of relating an action to more elemen tary actions, de c omp osing 501 Ar t ale & Franconi it in partially ordered steps (Artale & F ranconi, 1995). This kind of reasoning is found in hierarc hical planners lik e Nonlin (T ate, 1977), Sipe (Wilkins, 1988) and F orbin (Dean, Firb y , & Miller, 1990). Ac kno wledgemen ts This pap er is a substan tial extension and revision of (Artale & F ranconi, 1994). The w ork w as partially supp orted b y the Italian National Researc h Council (CNR) pro ject \On tologic and Linguistic T o ols for Conceptual Mo deling", and b y the \F oundations of Data W arehouse Qualit y" ( D WQ ) Europ ean ESPRIT IV Long T erm Researc h (L TR) Pro ject 22469. The rst author wishes to ac kno wledge also LADSEB-CNR of P ado v a and the Univ ersit y of Firenze for ha ving supp orted part of his w ork. Some of the w ork carried on for this pap er w as done while the second author w as w orking at ITC-irst, T ren to. This w ork o w es a lot to our colleagues Claudio Bettini and Alfonso Gerevini, for ha ving in tro duced us man y y ears ago to the temp or al maze . Sp ecial thanks to Ac hille C. V arzi, for taking time to review the tec hnical details of the pap er and for his insigh tful commen ts on the philosoph y of ev en ts, and to F austo Giunc higlia, for useful discussions and feedbac k. Thanks to P aolo Bresciani, Nicola Guarino, Eugenia T erno vsk aia and Andrea Sc haerf for enligh tening commen ts on earlier drafts of the pap er. W erner Nutt and Luciano Serani help ed us to ha v e a deep er insigh t in to logic. W e w ould also lik e to thank Carsten Lutz for the helpful discussions w e had with him ab out temp oral represen tations. Man y anon ymous referees c hec k ed out man y errors of previous v ersions of the pap er. All the errors of the pap er are, of course, our o wn. References Allen, J. F. (1991). T emp oral reasoning and planning. In Allen, J. F., Kautz, H. A., P ela vin, R. N., & T enen b erg, J. D. (Eds.), R e asoning ab out Plans , c hap. 1, pp. 2{68. Morgan Kaufmann. Allen, J. F., & F erguson, G. (1994). Actions and ev en ts in in terv al temp oral logic. Journal of L o gic and Computation , 4 (5). Sp ecial Issue on Actions and Pro cesses. Artale, A., Bettini, C., & F ranconi, E. (1994). Homogeneous concepts in a temp oral de- scription logic. 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