1 Introduction The topic of this paper can be viewed from dierent standpoints. A modal logician would probably say that we combine polymodal K with PD

Transcription

1 Dynamic description logics Frank Wolter and Michael Zakharyaschev Institut fur Informatik, Universitat Leipzig Augustus-Platz 10-11, Leipzig, Germany; Keldysh Institute for Applied Mathematics Russian Academy of Sciences Miusskaya Square 4, Moscow, Russia ( s: Section: algebraic and model-theoretic aspects of modal logic The work of the second author was supported by the Russian Fundamental Research Foundation. 1

2 1 Introduction The topic of this paper can be viewed from dierent standpoints. A modal logician would probably say that we combine polymodal K with PDL and prove the decidability of the resulting hybrid. In the eld of knowledge representation, the paper can be characterized as an attempt to introduce a dynamic dimension in concept description (alias terminological) logics. And nally, in a broader perspective, our concern is to construct and study formalisms for representing and processing knowledge in dynamic application domains that would be maximally expressive, on the one hand, and decidable, on the other. Concept description (or simply description) logics originate from practical knowledge representation systems (see e.g. [3, 8, 1]) which, in turn, can be traced back to the ideas of semantic networks and frames. An application domain is represented in the framework of a description logic by means of formulas which dene complex concepts out of primitive ones and assert that certain objects belong to certain concepts or are in certain relations to some other objects. Starting, for instance, from the primitive concepts child, grandma, wealthy, warm island and the binary relations (or roles) has, lives we can dene a compound concept fortunate child = child ^ 9has:(grandma ^ wealthy ^9lives:warm island) comprising all children whose grandmothers are wealthy and live on warm islands. The formulas John : fortunate child, Mary lives Bahamas assert that John is a fortunate child and that Mary lives on Bahamas. The relativized existential quantier 9R has the same semantic meaning as the possibility operator 3 interpreted by the accessibility relation R. This observation, rst made by Schild [13], establishes a close connection between description logics and polymodal K. Many other concept and role constructs used in description logics have their modal counterparts as well, for instance, number restrictions, nominals, and transitive reexive closures of roles. Description logics were originally designed for representing only static knowledge. To take into account changes in time or under certain actions and retain the relative simplicity of the language (say, decidability) it is natural to extend it by the corresponding modal operators and thereby keep its propositional modal status. It is known, however, that combinations of rather simple modal systems may result in very complex ones (see e.g. [18]). The rst temporal and epistemic description logics constructed in [16,14,9]were either too expressive and consequently undecidable or too weak (the temporal operators were applicable either only to formulas or only to concepts). A compromise was found by Baader and Laux [2] who combined the description logic ALC of [15] with polymodal K by allowing applications of modal operators to both formulas and concepts and showed the decidability of the satisability problem for the resulting language in models with expanding domains. In [19, 20] we have launched a systematic investigation of description logics with modal operators and proved the decidabilityofvarious epistemic and temporal description logics under the constant domain assumption. In this paper we combine description logics with propositional dynamic logic PDL. PDL was originally conceived (see [11, 7]) as a formalism for reasoning about the behaviour of non-deterministic iterative programs described by regular expressions over a set of atomic programs and tests. It turned out to be also useful as a basis for a logic of action and planning in articial intelligence [17, 12] and for deontic logic [10]. Many other types of modal logics can be regarded as fragments of PDL, for instance, polymodal K, S4, various epistemic logics with the common knowledge operator. We increase the expressive power of PDL by extending its propositional basis to the language of description logics. The modal operators of PDL can be applied 2

3 to both concepts and formulas. For example, we can dene concepts like easy cured child = child ^9has:angina ^h(give honey [ give aspirin) i:9has:angina (i.e., the set of children suering from angina that can be cured by using honey and aspirin). Our aim is to develop a technique of proving the decidability of the satisability problem for the resulting logics (which do not in general enjoy the nite model property). For simplicity we will be dealing here only with ALC; however it can be replaced with more expressive description logics provided that they are decidable. 2 Syntax and semantics of PDLC We begin by dening the dynamic concept description language PDLC and its semantics. Denition 1 (alphabet). The primitive symbols of PDLC are: concept names C 0 ;C 1 ;:::; role names R 0 ;R 1 ;:::; object names a 0 ;a 1 ;:::; the booleans (say, ^, :, >) and the relativized existential quantier 9R i, for every role name R i ; action variables 0 ; 1 ;:::; action constructs: ; (composition), [ (alternation), (iteration),? (test). Now we dene by induction the notions of a concept, a formula and an action term. Concepts will usually be denoted by symbols C and D, formulas by ', and, while for action terms we reserve small Greek characters from the beginning of the alphabet,,, etc. Denition 2 (concept, formula, action term). Every concept name as well as > is an atomic concept. Every action variable is an atomic action. If C and D are concepts, a and b object names, R a role name, ' and formulas, and action terms, then C ^ D, :C, 9R:C, []Care concepts; a : C, arb, ' ^, :', []'are formulas (the rst two of them are atomic); ;, [,, '? are action terms. The pure description part of the language PDLC is the standard concept description language ALC (see [6]); it is interpreted in models of the form I = ;R I 0 ;:::;CI 0 ;:::;ai 0 ;::: ; where is a set of objects, Ri I is a binary relation on interpreting the role name R i, Ci I is a subset of interpreting the concept name C i, and a I i 2 interprets the object name a i. The dynamic component ofpdlc is the language of the well known propositional dynamic logic PDL (see e.g. [7]). It is interpreted in frames (or labelled transition systems) of the form T = hw;t 0 ;T 1 :::i; (1) 3

4 where W is a non-empty set of states and T i a binary relation on W interpreting transitions corresponding to the action variable i. By combining these two kinds of models we arrive at the following denition. Denition 3 (model). A model of PDLC based on a frame T of the form (1) is a pair M = ht;ii in which I is a function associating with each state w 2 W an ALC-model D E I(w) = ;R I(w) 0 ;:::;C I(w) 0 ;:::;a I(w) 0 ;::: such that a I(u) i = a I(v) i for any u; v 2 W. Note that the set of objects is the same for every state in W 1 ; it is called the domain of M. It is also worth emphasizing that the interpretation of the object names does not depend on the particular world, which means that we use rigid designators. With minor changes we could also take into account the unique object name assumption according to which a I(w) i 6= a I(w) j whenever i 6= j. Denition 4 (satisfaction). Given a PDLC-model M = ht;ii and a state w in it, the value C I(w) of a concept C in w, the truth-relation (M;w) j= ' (or simply w j= ' if M is understood), and the relation T, an action term, are dened inductively as follows: 1. > I(w) = and C I(w) = C I(w) i, for C = C i ; 2. (C ^ D) I(w) = C I(w) \ D I(w) ; 3. (:C) I(w) = C I(w) ; 4. x 2 (9R i :C) I(w) i 9y 2 C I(w) xr I(w) i y; 5. x 2 ([]C) I(w) i 8v 2 W (wt v ) x 2 C I(v) ); 6. w j= C = D i C I(w) = D I(w) ; 7. w j= a : C i a I(w) 2 C I(w) ; 8. w j= arb i a I(w) R I(w) b I(w) ; 9. w j= ' ^ i w j= ' and w j= ; 10. w j= :' i w 6j= '; 11. w j= []'i 8v 2 W (wt v ) v j= '); 12. T '? = fhw; wi : w j= 'g; 13. T ; = T T (the composition of T and T ); 14. T [ = T [ T ; 15. T = T (the transitive and reexive closure of T ). A formula ' is satisable if there are a PDLC-model M and a state w in M such that w j= '. The main goal of this paper is to develop a satisability checking algorithm for PDLC-formulas. It is to be noted that although both ALC and PDL have the nite model property, their hybrid dened above does not enjoy it: as follows from [19], there is a PDLC-formula satisable in an innite model but not in nite ones. The satisability checking algorithm we are going to construct in Section 4 is based on the mosaic technique and the representation of models in the form of quasimodels. 1 This means that we accept the constant domain assumption. 4

5 3 Quasimodels The aim of this section is to show that (modulo a given formula) every PDLCmodel can be represented as a structure, called a quasimodel, every state in which is nite. Let us x a PDLC-formula ' and denote by sub', con' and ob' the sets of all subformulas, concepts and object names occurring in ', respectively. Denition 5 (Fischer{Ladner closure). The Fischer{Ladner closure of ' is the pair h('); (')i in which (')sub' and (') con' are the smallest sets of formulas and concepts that are closed under subformulas and subconcepts, respectively, and satisfy the following conditions: [; ] 2 (') ) [][] 2 ('); [ [ ] 2 (') ) [] ; [] 2 ('); [ ] 2 (') ) [][ ] 2 ('); [?] 2 (') ) 2 ('); [; ]C 2 (') ) [][]C 2 ('); [ [ ]C 2 (') ) []C; []C 2 ('); [ ]C 2 (') ) [][ ]C 2 ('); [?]C 2 (') ) 2 ('). Denition 6 (quasistate). Consider a structure of the form q = hx q ;R q 0 ;:::;Cq 0 ;:::;([ 0]D 0 ) q ;:::;a q 0 ;:::;q i: (2) Here X q is a nite set, the domain of q, R q i X q X q for every role name R i in ', C q i X q for every C i 2 con', a q i 2 X q for every a i 2 ob', ([ i ]D j ) q X q for every [ i ]D j in ('), and q is a subset of ('). The value C q of a concept C 2 (') in q is computed almost in the same way as in Denition 4, the only dierence is that now the value ([]C) q is given in q directly as a value of an atomic concept. We call q a quasistate for ' if the following conditions hold: ([; ]C) q =([][]C) q ; ([ [ ]C) q =([]C) q \([]C) q ; ([ ]C) q = C q \ ([][ ]C) q ; ([?]C) q = fx 2 X q : 2 q ) x 2 C q g; C = D 2 q i C q = D q, for every C = D 2 ('); a : C 2 q i a q 2 C q, for every a : C 2 ('); arb 2 q i a q R q b q, for every arb 2 ('); ^ 2 q i 2 q and 2 q, for every ^ 2 ('); : 2 q i 62 q, for every : 2 ('); [; ] 2 q i [][] 2 q, for every [; ] 2 ('); [ [ ] 2 q i [] ; [] 2 q, for every [ [ ] 2 ('); [ ] 2 q i ; [][ ] 2 q, for every [ ] 2 ('); 5

6 [?] 2 q i 62 q or 2 q, for every [?] 2 ('). Instead of 2 q we will often write q j= and say that is true in q. Given a structure of the form (2), we can always eectively decide whether it is a quasistate for '. Let m = hq; T 1 ;:::;T k ibe a frame in which Q is a set of quasistates for ' and T i is a binary relation on Q for every action variable i in '. For an action term constructed from the action variables 1 ;:::; k, let T be the binary relation on Q dened by items 12{15 of Denition 4 (in which w should be replaced by q). Denition 7 (run). By a run in m = hq; T 1 ;:::;T k iwe will mean a set r which contains precisely one object from the domain X q of each quasistate q 2 Q let us denote this object by r(q) and such that for every concept []C 2 (') and every q 2 Q we have: r(q) 2 ([]C) q,8q 0 2Q(qT q 0 ) r(q 0 ) 2 C q0 ). If only the ())-part of this condition holds then r is called a weak run. Denition 8 (quasimodel). The frame m = hq; T 1 ;:::;T k iis a quasimodel for ' if for every q 2 Q and every x 2 X q there is a run r in m such that r(q) =x; for every a 2 ob', r a = fa q : q 2 Qg is a run in m; for every [] 2 (') and every q 2 Q, q j= [],8q 0 2Q(qT q 0 ) q 0 j= ): (3) A formula ' is satisable in m if q j= ' for some q 2 Q. If in this denition we replace runs with weak runs and require that only the ())-part of condition (3) holds then m will be called a weak quasimodel for '. Theorem 9 (quasimodel completeness). A formula ' is satisable i it is satisable in some quasimodel for '. Proof ()) Suppose ' is satisable in a model M = ht;ii based on a frame T of the form (1) and having a domain. With each w 2 W we associate a quasistate q w = hx qw ;R qw 0 ;:::;Cqw 0 ;:::;([ 0]D 0 ) qw ;:::;a qw 0 ;:::;qw i in the following way. For every x 2 let t w (x) =fc2(') :x2c I(w) g: Then X qw contains the objects a 2 ob' from (without loss of generality wemay assume a I(w) = a) and also one representative z =2 ob' from each class [x] w = fy 2 :t w (x)=t w (y)g,ifsuchzexists, xr qw i y i either one of x, y is not in ob' and x 0 R I(w) i y 0 for some x 0 2 [x] w, y 0 2 [y] w, or x; y 2 ob' and xr I(w) i y, x 2 C qw i i x 2 C I(w) i, x 2 ([ i ]D j ) qw i x 2 ([ i ]D j ) I(w), and qw = f 2 (') : w j= g. It is a matter of routine to check by induction that for every C 2 ('), we have C qw =C I(w) \X qw and so q w is a quasistate for '. The structure m = hq; T 0 1 ;:::;T 0 n i; where Q = fq w : w 2 W g and q u T 0 i q v i ut i v, for every action variable i in ' and all u; v 2 W, is then a quasimodel satisfying ' (to construct a run through a given x 2 X qu, one can take anyr(q w )2[x] w \X qw, for every w 2 W ). 6

7 (() Now let m = hq; T 1 ;:::;T n i be a quasimodel for ' such that q j= ' for some q 2 Q. Construct a standard model M = hm;ii based on the frame m by taking, for every q 2 Q, I(q) =h;r I(q) 0 ;:::;C I(q) 0 ;:::;r a0 ;:::i; where is the set of all runs in m, rr I(q) i r 0 i r(q)r q i r0 (q), and r 2 C I(q) i i r(q) 2 C q i. By a straightforward induction one can show that for all C 2 ('), 2 ('), q 2 Q and r 2, we haver2c I(q) i r(q) 2 C q, and (M;q) j= i 2 q. Therefore, ' is satised in M. 2 It is worth noting that as a consequence of the proof of ()) we obtain Corollary 10. A formula ' is satisable i it is satisable in a quasimodel containing at most ](') =2 2j(')j 2 j(')j job'j2 j(')j pairwise non-isomorphic quasistates the cardinality of the domains in which does not exceed [(') =2 j(')j + job'j : As is well known, PDL is complete with respect to models based on intransitive trees. We remind the reader that a frame hw;ri is an intransitive tree if it is rooted, cycle-free, and contains no distinct paths of the form xry 1 R:::Ry n Ry and xrz 1 R:::Rz m Ry. Denition 11 (tree quasimodel). A(weak) quasimodel m = hq; T 1 ;:::;T k i for ' is called a tree (weak) quasimodel if Tm = S ft i :1ikgis an intransitive tree order on Q and T i \ T j = ; whenever i 6= j. If no quasistate of a tree (weak) quasimodel m dierent from its root has more than one Tm-successor then m is called a bouquet (weak) quasimodel. Theorem 12 (tree quasimodel completeness). A formula ' is satisable i it is satisable inatree quasimodel for ' the domains of quasistates in which are of cardinality [('). 4 Eective satisability criterion Assume again that we have xed a PDLC-formula '. Denition 13 (block). Let b = hq; T 1 ;:::;T k i be a nite bouquet weak quasimodel with root q 0. Say that a weak run r in b is root-saturated if 8[]C 2 (') (r(q 0 )=2([]C) q 0 )9q0 2Q(q 0 T q 0 &r(q 0 ) =2C q0 )): We call b a block for ' if (a) for every q 2 Q and every x 2 X q there is a root-saturated weak run r in b such that r(q) = x; (b) every weak run r a, a 2 ob', is root-saturated; (c) 8[] 2 (') (q 0 6j= [] )9q 0 2Q(q 0 T q 0 &q 0 6j= )). 7

8 Denition 14 (satisfying set). A set S of blocks for ' is called a satisfying set for ' if (i) it contains a block with root q 0 such that q 0 j= ' and (ii) for every quasistate q in every block insthere exists a block inshaving q as its root. Our aim is to show that ' is satisable i there is a satisfying set for ' whose blocks contain at most N quasistates, for some N <!eectively determined by '. Denote by jj the length of an action term which is dened inductively as follows j i j =1,j?j=1; j[j= maxfjj; jjg; j; j = jj + jj; j j = jj +1. Now for every n 0we put where i (n) = i,?(n) =?; ( [ )(n) =(n)[(n); (; )(n) =(n); (n); (n) = n (n), n = >? [ [ (; ) [[(;:::;): {z } n In other words, (n) results from by replacing every occurrence of an action term of the form (which is not in the scope of a test?) with n. In particular, (n) contains no occurrence of. Finally, let l(') = maxfj([(') ]('))j :[]C2(') or[] 2(')g: We are in a position now toprove the main result of the paper. Theorem 15 (satisability criterion). A PDLC-formula ' is satisable i there is a satisfying set for ' each block in which contains at most N = l(') (j(')j +2[(')j(')j) quasistates whose domains are of cardinality [('). Proof ()) Suppose ' is satisable. Then, by Theorem 12, there is a tree quasimodel m = hq; T 1 ;:::;T k i satisfying ' at its root and having quasistates of size [('). We begin our construction of a satisfying set for ' by associating with each quasistate q in m a block b q = hq q ;T q 1 ;:::;T q k i. First, for every formula [] 2 (') such that q 6j= [] we select a quasistate q 0 2 Q for which qt q 0, q 0 6j=, and put it in Sel(q) (at the very beginning Sel(q) =;). Then, for every x 2 X q we x a run r in m coming through x (if x = a, for a 2 ob', then r = r a ) and for every concept []C 2 (') such that r(q) = x =2 ([]C) q, select a quasistate q 0 2 Q for which qt q 0, r(q 0 ) =2 C q0 and put it in Sel(q) together with its copy q 00. (Formally, taking the copy q 00 means that we duplicate the subtree of m generated by q 0 and connect it with the immediate predecessor q y of q 0 by the same relation T i that connects q y with q 0. The resulting structure is clearly again a tree quasimodel satisfying '.) 8

9 The number of selected quasistates does not exceed j(')j +2[(')j(')j; without loss of generality wemay assume these quasistates to be pairwise distinct (otherwise we can duplicate them as above). For each selected q 0 there is a unique Tm-path from q to q 0, namely (q; q 0 ) = fq 1 : qtm q 1Tm q0 g. Again, without loss of generality we assume that distinct paths (q; q 0 ) and (q; q 00 ) (for q 0 6= q 00 ) have no common quasistates save q (otherwise the duplication technique will do the job). Finally, we dene Q q to be the set of all quasistates in the paths (q; q 0 ), for q 0 2 Sel(q), and T q i to be the restriction of T i to Q q (taking into account all the duplications, of course). The constructed structure b q is a block. Indeed, it is clearly a nite bouquet weak quasimodel for ' with root q (for it may be considered as a subquasimodel of m) satisfying (b) and (c) by the construction. Suppose now that q 1 2 Q q, x 2 X q1, and let r be a weak run in b q coming through x. Consider the object r(q) and the set C of concepts []C 2 (') for which r(q) =2 ([]C) q. For each of them there is a weak run r []C such that r []C (q) =r(q), r []C (q []C ) =2 C q []C, for some q []C 2 Sel(q), and q 1 =2 (q; q []C ). Using these weak runs and r we construct a set r 0 by taking, for every q 0 2 Q q, r 0 (q 0 r(q0 ) )= r []C (q 0 ) if q 0 =2 (q; q []C ), for any []C2C otherwise Clearly, r 0 is a root-saturated weak run in b q coming through x, which establishes (a). Although the number of branches in b q does not exceed j(')j +2[(')j(')j, they may be too long. Our next step is to extract from b q a substructure a q which is still a block for ' and whose branches are of length l('). We will do this by cutting out certain fragments of branches in b q. Consider a branch (q; q 0 ) and suppose that q 0 was selected to \saturate" [] in q or []C in some x 2 X q. If the action term contains no occurrence of then, since qt q 0, the length of (q; q 0 ) is at most jj; in this case we leave this branch as it is. Suppose now that contains iteration. The \truncation" is conducted by induction on the construction of. If = ; then qt q qy T q q0, for some q y 2 (q; q 0 ), and we proceed by truncating (q; q y ) and (q y ;q 0 ). If = [ then either qt q q0 or qt q q0 which reduces the complexity of.finally, let =. Then there are quasistates q 1 ;:::;q n 2 (q; q 0 ), n<!, such that q = q 1 T q q 2T q :::Tq q n = q 0. If n ](') [(') then we proceed by considering the fragments (q i ;q i+1 ) for 1 i<n. Otherwise let r be the weak run in b q such that r(q) = x and r(q 0 ) =2 C q0 ; if q 0 \saturates" [] in q then r may beanyweak run in b q. Since n>](')[('), there must be two isomorphic quasistates q i and q j,1i<jn, such that t(r(q i )) = t(r(q j )). Then we cut out from (q; q 0 ) all the quasistates in the interval (q i ;q j )saveq i and put q i T q qy i q j T q qy, for every action variable. It should be clear that the resulting structure is still a block for ', and so by deleting repeating quasistates in the branches of b q we can construct a block a q for ' whose branches are of length l('). The satisfying set for ' we are looking for can be constructed now by taking the blocks a q for all non-isomorphic quasistates in m. (() Let S be a satisfying set for '. We are going to construct a quasimodel m satisfying ' as the limit of a sequence of weak quasimodels m n = hq n ;R n 1 ;:::;R n k i; n=1;2;::: the rst of which, m 1,isablockinSsatisfying ' at its root. Suppose now that we have already constructed a weak quasimodel m n. For every quasistate q 2 Q n Q n 1 9

10 (Q 0 = ;) select a block b q 2 S with root q. Without loss of generality we may assume all the selected blocks and the weak quasimodel m n to be disjoint. The weak quasimodel m n+1 is then the result of hooking the selected blocks b q to m n by identifying their roots q with q 2 Q n Q n 1. Dene the limit m = hq; T 1 ;:::;T k i, of the constructed sequence by taking Q = [ fq n : n 1g; T i = [ ft n i : n 1g: It follows from the construction and the denition of a block that m is a quasimodel for '. 2 As a consequence of this criterion we obtain the following: Theorem 16. The satisability problem for PDLC-formulas is decidable. Remark 17. ALC, the underlying concept description logic we have considered in this paper is only one representative of the extensive family of description logics (see e.g. [6, 4,5]) that can be combined with PDL. And for many of them the developed technique is able to provide satisability checking algorithms. For instance, we can base PDL on the rather expressive logic CIQ of [4, 5] which has means for constructing inverses, unions, compositions and transitive reexive closures of roles as well as for restricted quantication of concepts. CIQ does not enjoy the nite model property but is decidable, and this is enough to establish decidability of its hybrid with PDL. References [1] F. Baader and B. Hollunder. A terminological knowledge representation system with complete inference algorithms. In Proceedings of the workshop on Processing Declarative Knowledge, PDK-91, pages 67{86. Lecture Notes in Articial Intelligence, No Springer Verlag, [2] F. Baader and A. Laux. Terminological logics with modal operators. In Proceedings of the 14th International Joint Conference on Articial Intelligence, pages 808{814, Montreal, Canada, Morgan Kaufman. [3] R.J. Brachman and J.G. Schmolze. An overview of the KL-ONE knowledge representation system. Cognitive Science, 9:171{216, [4] G. de Giacomo. Decidability of Class-Based Knowledge Representation Formalisms. PhD thesis, Univ. di Roma, [5] G. de Giacomo and M. Lenzerini. TBox and ABox reasoning in expressive description logics. In Proceedings of the fth Conference on Principles of Knowledge Representation and Reasoning, Montreal, Canada, Morgan Kaufman. [6] F. Donini, M. Lenzerini, D. Nardi, and A. Schaerf. Reasoning in description logics. In G. Brewka, editor, Principles of Knowledge Representation, pages 191{236. CSLI Publications, [7] D. Harel. Dynamic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 497{604. Reidel, Dordrecht, [8] T.S. Kaczmarek, R. Bates, and G. Robins. Recent developments in NIKL. In Proceedings of the 5th Nat. Conf. on Articial Intelligence (AAAI-86), pages 978{985,

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