Standard and Non-Standard Inferences in the Description Logic FL 0 Using Tree Automata

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1 EPiC Seies in Computing Volume 55, 2018, Pages 1 14 GCAI th Global Confeence on Atificial Intelligence Standad and Non-Standad Infeences in the Desciption Logic FL 0 Using Tee Automata Fanz Baade, Olive Fenández Gil, and Maximilian Pensel Theoetical Compute Science, TU Desden Desden, Gemany fistname.lastname@tu-desden.de Abstact Although being quite inexpessive, the desciption logic (DL) FL 0, which povides only conjunction, value estiction and the top concept as concept constuctos, has an intactable subsumption poblem in the pesence of teminologies (TBoxes): subsumption easoning w..t. acyclic FL 0 TBoxes is conp-complete, and becomes even ExpTimecomplete in case geneal TBoxes ae used. In the pesent pape, we use automata woking on infinite tees to solve both standad and non-standad infeences in FL 0 w..t. geneal TBoxes. Fist, we give an altenative poof of the ExpTime uppe bound fo subsumption in FL 0 w..t. geneal TBoxes based on the use of looping tee automata. Second, we employ paity tee automata to tackle non-standad infeence poblems such as computing the least common subsume and the diffeence of FL 0 concepts w..t. geneal TBoxes. 1 Intoduction In the ealy days of DL eseach, the inexpessive DL FL 0, which has only conjunction, value estiction and the top concept as concept constuctos, was consideed to be the smallest possible DL. In fact, when poviding a fomal semantics fo so-called popety edges of semantic netwoks in the fist DL system KL-ONE [12], value estictions wee used. Fo this eason, the language fo constucting concepts in KL-ONE and all of the othe ealy DL systems [11, 21, 19, 27] contained FL 0. It came as a supise when it was shown that subsumption easoning w..t. acyclic FL 0 teminologies (TBoxes) is conp-had [20]. The complexity inceases when moe expessive foms of TBoxes ae used: fo cyclic TBoxes to PSpace [1, 15] and fo geneal TBoxes consisting of geneal concept inclusions (GCIs) even to ExpTime [2]. Thus, w..t. geneal TBoxes, subsumption easoning in FL 0 is as had as in ALC, its closue unde negation. These negative complexity esults fo FL 0 wee one of the easons why the attention in DL eseach shifted fom FL 0 to EL, which is obtained fom FL 0 by eplacing value estiction with existential estiction as a constucto. In fact, subsumption easoning in EL stays polynomial even in the pesence of geneal TBoxes [13, 2]. In addition to standad infeences such as the subsumption and the instance poblem, also vaious non-standad infeences have been Suppoted by the DFG Reseach Taining Goup 1763 (QuantLA) and the DFG gant BA 1122/20-1. D. Lee, A. Steen and T. Walsh (eds.), GCAI-2018 (EPiC Seies in Computing, vol. 55), pp. 1 14

2 investigated in detail fo EL [28, 16, 18]. Fo FL 0, thee ae vey few esults that conside geneal TBoxes. Fo the ExpTime-completeness esult fo subsumption in [2], the ExpTime uppe bound is actually inheited fom the moe expessive DL ALC. Dedicated subsumption algoithms and nonstandad infeences have been consideed mostly fo the case without TBoxes o w..t. a esticted fom of TBoxes in FL 0 and its extension by numbe estictions [1, 7, 5, 6, 9]. In the pesent pape, we use automata woking on infinite tees to solve both standad and non-standad infeences in FL 0 w..t. geneal TBoxes. Fist, we intoduce the so-called least functional model of an FL 0 concept w..t. an FL 0 TBox, and pove that subsumption coesponds to inclusion of least functional models. Then we show that such least functional models can be epesented using so-called looping tee automata (LTAs), and use this esult to educe the subsumption poblem to the emptiness poblem fo LTAs. Since the constucted automata ae of exponential size and the emptiness poblem fo LTAs can be decided in linea time, this yields an altenative poof of the ExpTime uppe bound fo subsumption in FL 0 w..t. geneal TBoxes. Note that, in contast to the case of acyclic o cyclic TBoxes, whee automata on finite o infinite wods can be employed to decide subsumption [1], GCIs equie the use of automata woking on tees. In ode to deal also with non-standad infeences such as computing the least common subsume (lcs) [8, 17, 28] o the diffeence [24, 14] of two FL 0 concepts w..t. a geneal FL 0 TBox, the automata constuctions need to be extended consideably. Instead of constucting an automaton that epesents the least functional model of a fixed FL 0 concept C, we ae now inteested in building an automaton accepting all tees epesenting least functional models of FL 0 concepts. Fo this task, simple LTAs (which do not have an acceptance condition) ae not sufficient; instead we use so-called paity tee automata (PTAs). We show that the constucted automaton can be used to decide whethe the lcs o the diffeence of two given FL 0 concepts w..t. a geneal FL 0 TBox exists, and to actually compute the lcs o diffeence concept in case the answe is affimative. Fo the DL EL, the poblem of deciding the existence of the lcs w..t. a geneal TBox was investigated in [28]. Due to the space constaints, we cannot povide all technical details and poofs in this pape. They can be found in the technical epot [4]. 2 Least functional models fo FL 0 TBoxes In Desciption Logics, concept constuctos ae used to build complex concepts out of concept names (unay pedicates) and ole names (binay pedicates). A paticula DL is detemined by the available constuctos. Stating with fixed finite sets N C and N R of concept and ole names, espectively, the set of FL 0 concepts is inductively defined as follows: (top concept) and evey concept name A N C is an FL 0 concept, if C, D ae FL 0 concepts and N R is a ole name, then C D (conjunction) and.c (value estiction) ae FL 0 concepts. The semantics of FL 0 is defined using fist-ode intepetations I = ( I, I) consisting of a non-empty domain I and an intepetation function I that assigns a set A I I to each concept name A and a binay elation I I I to each ole name. This function is extended to FL 0 concepts as follows: 2 I = I and (C D) I = C I D I, (.C) I = {x I y I : (x, y) I y C I }.

3 An FL 0 TBox T is a finite set of geneal concept inclusions (GCIs), which ae expessions of the fom C D fo FL 0 concepts C, D. The intepetation I is a model of T if it satisfies all the GCIs in T, i.e., C I D I holds fo all GCIs C D in T. Given an FL 0 TBox T and two FL 0 concepts C, D, we say that C is subsumed by D (denoted as C T D) if C I D I fo all models I of T. These two concepts ae equivalent (denoted as C T D) if C T D and D T C. If the TBox is empty, we wite C D and C D instead of C D and C D. Fo FL 0, the subsumption poblem is ExpTime-complete: the complexity uppe bound is inheited fom the moe expessive DL ALC [23], and the lowe bound was shown in [2]. In the next section, we will actually descibe an altenative way fo showing the uppe bound. It is based on the chaacteization of subsumption intoduced in the emainde of this section. In FL 0, subsumption and equivalence can be nicely chaacteized using language inclusion. This chaacteization elies on tansfoming FL 0 concepts into an appopiate nomal fom as follows. Fist, the semantics given to concept constuctos in FL 0 implies that value estictions distibute ove conjunction, i.e., fo all FL 0 concepts C, D and oles it holds that.(c D).C.D. Using this equivalence as a ewite ule fom left to ight, each FL 0 concept can be tanslated into an equivalent one that is eithe o a conjunction of concepts of the fom 1... n.a, whee { 1,..., n } N R and A N C. Such concepts can be abbeviated as w.a, whee w epesents the wod 1... n. Note that n = 0 means that w is the empty wod ε, and thus ε.a coesponds to A. Futhemoe, a conjunction of the fom w 1.A... w m.a can be witten as L.A whee L N R is the finite language {w 1,..., w m }. We use the convention that.a coesponds to the top concept. Thus, if N C = {A 1,..., A l }, then any two FL 0 concepts C, D can be epesented as C K 1.A 1... K l.a l, D L 1.A 1... L l.a l, whee K 1, L 1,..., K l, L l ae finite languages ove the alphabet of ole names N R, i.e., finite subsets of N R. We call this epesentation the language nomal fom (LNF) of C, D. Using the LNF, it was shown in [9] that C D iff L i K i fo all i, 1 i l. In the pesence of a non-empty TBox T, a simila chaacteization of subsumption and equivalence can be obtained [22], but now the definition of the languages needs to take the GCIs in T into account. Given an FL 0 concept C and a TBox T, we define L T (C) := {(w, A) N R N C C T w.a}, and call this set the value estiction set of C with espect to T. Poposition 1 ([22]). Let T be an FL 0 TBox and C, D FL 0 concepts. Then C T D iff L T (D) L T (C). Chaacteizing an FL 0 concept C whose LNF is C K 1.A 1... K l.a l by its value estiction set L T (C) genealizes detemining the finite languages K i (1 i l) in the nomalization step. Indeed, it is easy to see that, fo the empty TBox, we have L (C) = {(w, A i ) w K i, 1 i l}. Fo a geneal TBox T, L T (C) may be infinite, as illustated by the TBox T = {A.A}, whee e.g. L T (A) = {(ε, A), (, A), (, A),...}. Theefoe, to detemine subsumption of two concepts by compaing the espective value estiction sets, inclusion between infinite sets must be consideed. In the next section, we will show how such infinite sets can be epesented using infinite tees, and how tee automata can be used to test inclusion. But fist, we intoduce a semantic equivalent of the value estiction set, called least functional model. (1) 3

4 Definition 2. An intepetation I = ( I, I) is called a functional intepetation if I = N R and I = {(u, u) u N R } fo all N R. We call such a functional intepetation a functional model of the FL 0 concept C w..t. the FL 0 TBox T if it is a model of T such that ε C I. Calling such intepetations and models I functional is justified by the fact that oles ae indeed intepeted as (total) functions: fo evey u N R and evey N R, the wod u is the unique -successo of u. As an immediate consequence of this functional intepetation of oles we have w ( u.a) I iff wu A I. (2) We define inclusion and intesection of functional intepetations as follows: I J if A I A J fo all A N C ; I J is the unique functional intepetation that satisfies A I J = A I A J fo all A N C. It is easy to see that functional models ae closed unde intesection, i.e., if I and J ae both functional models of C w..t. T, then so is thei intesection I J. This actually not only holds fo binay intesection, but also fo abitay intesection of functional models [22], which implies that thee must exist a least functional model, i.e., a functional model J of C w..t. T such that J I holds fo all functional models I of C w..t. T. Hee we descibe a diffeent and moe constuctive way of showing the existence of a least functional model, which is based on the use of the value estiction set. Definition 3. Given an FL 0 concept C and an FL 0 TBox T, we define I C,T = (N R, IC,T ) to be the functional intepetation satisfying A I C,T = {w N R (w, A) L T (C)} fo all A N C. In [4] we show that I C,T is indeed the least functional model of C w..t. T. Theoem 4. Let C be an FL 0 concept and T an FL 0 TBox. Then the functional intepetation I C,T is the least functional model of C w..t. T. As an immediate consequence of Definition 3 we obtain that L T (D) L T (C) iff I D,T I C,T. Thus, the chaacteization of subsumption fomulated in Poposition 1 can be efomulated in tems of least functional models as follows. Coollay 5. Let T be an FL 0 TBox and C, D FL 0 concepts. Then C T D iff I D,T I C,T. In the next section we show how least functional models can be epesented using tee automata. In paticula, this will allow us to educe the subsumption poblem in FL 0 to a well-know decision poblem fo tee automata. 3 Least functional models as tees Given a non-empty, finite set of symbols Σ = {σ 1,..., σ k } and a finite set of labels L, an L-labeled Σ-tee is a mapping t : Σ L that assigns a label t(w) L to evey node w Σ. Intuitively, the nodes of a Σ-tee t coespond to finite wods in Σ, whee the empty wod ε epesents the oot of t and evey node w has k childen coesponding to the wods 4

5 wσ 1,..., wσ k. Since fo a non-empty alphabet Σ the set Σ of all wods ove Σ is infinite, Σ-tees ae by definition infinite. The set of all L-labeled Σ-tees is denoted by T ω Σ,L. Functional intepetations can be epesented as 2 N C -labeled N R -tees and vice vesa. In fact, let I = (N R, I) be a functional intepetation. Then we define t I : N R 2N C as t I (w) := {A N C w A I } fo all w N R. Convesely, let t be a 2N C -labeled N R -tee. Then we define the functional intepetation I t = (N R, It ) as A It := {w N R A t(w)} fo all A N C. Obviously, these two tansfomations ae bijections that ae invese to each othe. If I = I C,T is the least functional model of the FL 0 concept C w..t. the FL 0 TBox T, then we use T C,T to denote the coesponding tee t I. In case the set of ole names is non-empty (which we will always assume in the following), the tee T C,T is infinite. Ou goal is now to give a finite epesentation of such tees using tee automata [26]. Definition 6 (Looping tee automaton (LTA)). A looping tee automaton is a tuple A = (Σ, Q, L, Θ, I) whee Σ = {σ 1,..., σ k } is a finite set of symbols, Q is a finite set of states, L is a finite set of labels, Θ Q L Q k is the tansition elation and I Q is a set of initial states. A un of this automaton on an L-labeled Σ-tee t is a Q-labeled Σ-tee ρ : Σ Q such that ρ(ε) I and (ρ(w), t(w), ρ(wσ 1 ),..., ρ(wσ k )) Θ fo all w Σ. The tee language L(A) ecognized by A is the set of all L-labeled Σ-tees t such that A accepts t, i.e., such that A has a un on t. In geneal, LTAs ecognize sets of tees. In ode to uniquely epesent the tee T C,T, we conside LTAs ecognizing singleton sets. Definition 7. Let A = (Σ, Q, L, Θ, I) be a looping tee automaton. We say that A epesents the L-labeled Σ-tee t if L(A) = {t}. As shown in [3], the tees that can be epesented in this way ae exactly the egula tees, whee an infinite tee is egula if (up to isomophism) it contains only finitely many distinct subtees [25]. To constuct an automaton that epesents the tee T C,T, we fist constuct an LTA that accepts exactly the tee epesentations of functional models of C w..t. T. We assume without loss of geneality that C and all concepts occuing in T ae in wod nomal fom (WNF), i.e., they ae conjunctions of value estictions w.a whee w N R and A N C. Given an FL 0 concept E = w 1.A 1... w n.a n in WNF, we define Ê := { w 1.A 1,..., w n.a n } and val(e) := { u i.a i 1 i n and u i is a suffix of w i }. The latte definition is extended to TBoxes by setting val(t ) := val(e) val(f ). E F T Definition 8. Let C be an FL 0 concept and T an FL 0 TBox such that C and all concepts occuing in T ae in WNF. We set V C,T := val(c) val(t ) and define the LTA A C,T = (Σ, Q, L, Θ, I) as follows: Σ := N R = { 1,..., k }, Q := 2 V C,T, and L := 2 N C ; I := {X Q Ĉ X}; Θ := {(q, σ, q 1,..., q k ) the popeties (1),...,(4) hold}, whee 5

6 1. Ê q implies F q fo all E F T, 2. i u.a q implies u.a q i fo all i, 1 i k, 3. u.a q i and i u.a V C,T implies i u.a q fo all i, 1 i k, 4. A σ iff ε.a q fo all A N C. Intuitively, the set Q consists of all possible sets of elevant value estictions that may be satisfied by a node of a functional model of C w..t. T. In paticula, the definition of I ensues that only tees of functional intepetations fo which ε belongs to C ae accepted. Popety (1) in the definition of Θ ensues that the GCIs of T ae satisfied by the functional intepetation coesponding to the tee. Popeties (2) and (3) basically ealize the fact that, in a functional intepetation, an element w belongs to the value estiction i u.a iff its unique i successo w i belongs to u.a. Finally, popety (4) expesses that A and ε.a ae equivalent. With this intuition, it is not had to show that A C,T is the automaton we ae looking fo (see [22] fo details). Lemma 9. L(A C,T ) = {t I I is a functional model of C w..t. T }. In ode to obtain an automaton that epesents T C,T, we estict the automaton A C,T to the tansitions with minimal states, whee states ae odeed using set inclusion. Howeve, befoe we can do this, we need to emove states that cannot occu in a un. Intuitively, this is necessay to avoid using a minimal state that leads into a dead-end. We say that a state q of an LTA A is inactive if thee is no un of A that contains q. As shown in [10], the set of inactive states of an LTA can be computed in linea time. Definition 10. Let A C,T = (Σ, Q, L, Θ, I) be the LTA defined in Definition 8, Q + Q be the states of A C,T that ae not inactive, and Θ + := Θ (Q + L (Q + ) k ) the tansitions of A C,T that do not use inactive states. We define the automaton ÂC,T = (Σ, Q +, L, Θ, Î) as follows: Î := {q I Q + q q fo all q I Q + }; Θ := {(q, σ, q 1,..., q k ) Θ + q 1 q 1,..., q k q k fo all (q, σ, q 1,..., q k ) Θ+ }. The eason why the least states w..t. set inclusion equied by the definitions of Î and Θ exist is basically that uns ae closed unde intesection, i.e., if ρ 1, ρ 2 ae uns of A C,T on some tees, then ρ 1 ρ 2 with (ρ 1 ρ 2 )(w) = ρ 1 (w) ρ 2 (w) is also a un of A C,T. The automaton ÂC,T is in fact deteministic: it has exactly one initial state and fo evey pai (q, σ) Q + L exactly one tansition with these two fist components. Togethe with popety (4) in Definition 8 this implies that ÂC,T accepts exactly one tee, and it is not had to show that this tee is T C,T (see [22] fo details). Theoem 11. The automaton ÂC,T epesents the tee T C,T that coesponds to the least functional model of C w..t. T. In paticula, this implies that T C,T is a egula tee. By Coollay 5, subsumption coesponds to inclusion between least functional models. The latte can be checked using a poduct constuction on the coesponding automata. Assume that ÂC,T = (Σ, Q C, L, Θ C, I C ) and ÂD,T = (Σ, Q D, L, Θ D, I D ) ae the automata espectively epesenting the tees T C,T and T D,T, as constucted in Definition 10. We define a new automaton P C,D,T = (Σ, Q, { }, Θ, I) that accepts the infinite { }-labeled Σ-tee t iff I D,T I C,T : 6 Q := Q C Q D and I := I C I D ;

7 ((q (1), q (2) ),, (q (1) σ (1) σ (2), 1, q(2) 1 ),..., (q(1) k, q(2) k )) Θ iff thee ae σ(1), σ (2) Σ such that (q (1), σ (1), q (1) 1,..., q(1) k ) Θ C and (q (2), σ (2), q (2) 1,..., q(2) k ) Θ D. We have L(P C,D,T ) iff L(P C,D,T ) = {t } iff I D,T I C,T. Since the emptiness poblem fo looping tee automata is decidable in linea time [10] and the automata ÂC,T and ÂD,T, and thus also P C,D,T, have a size that is exponential in the size of C, D, T, this yields an exponential-time algoithm fo checking subsumption. Coollay 12. Subsumption in FL 0 w..t. TBoxes is in ExpTime. A poduct constuction simila to the one employed above can also be used to obtain an automaton epesenting the intesection of two least functional models. Given least functional models I C,T and I D,T, thei intesection is the functional intepetation I = (N R, I) that satisfies A I = A I C,T A I D,T fo all A N C. Fo the coesponding tees this means that t I (w) = T C,T (w) T D,T (w) fo all w N R. It is easy to see that I is again a model of T. Howeve, it need not be the least functional model of some FL 0 -concept (see Section 5 fo an example). As a fist step towads deciding whethe it is o not, we show in [4] that t I can be epesented by a looping automaton. Lemma 13. Thee is an LTA PC,D,T of size exponential in the size of T, C, D such that L(PC,D,T ) = {t I} whee I is the intesection of I C,T and I D,T. The functional intepetation I obtained as the intesection of I C,T and I D,T may o may not be the least functional model of some FL 0 concept E w..t. T. In the next section, we show how this can be decided by constucting an automaton that accepts exactly the least functional models w..t. T, i.e., the tee language LF(T ) := {T C,T C is an FL 0 concept}. In Section 5 we use this esult to show that the existence of the least common subsume of two FL 0 concepts w..t. an FL 0 TBox is decidable. 4 An automaton accepting all least functional models Given an FL 0 TBox T, we want to constuct a tee automaton ÂT ecognizing the tee language LF(T ) of all least functional models w..t. T. In the following, we assume that T is an abitay, but fixed FL 0 TBox. The looping automata employed in the pevious section ae not sufficient to constuct ÂT since we need to expess things like finiteness, which equies an appopiate acceptance condition. It tuns out that paity tee automata [26] ae well-suited fo ou pupose since they have the equied closue popeties (intesection, complement, pojection) and thee exists a fine-gained analysis of the complexity of these opeations. Definition 14 (Paity tee automaton (PTA)). A paity tee automaton is a tuple A = (Σ, Q, L, Θ, I, c), whee (Σ, Q, L, Θ, I) is an LTA and c : Q N is a mapping that assigns to each state in Q a numbe, called its pioity. A un ρ of A on a tee t is called accepting if fo all paths in the tee ρ the maximum of the pioities appeaing infinitely often in the path is an even numbe. The tee language L(A) ecognized by A is the set of all L-labeled Σ-tees t such that A has an accepting un on t. 7

8 t (1) : (, {A}) t (2) : (, {A}) t (3) : (, {A}) ({A}, {A}) ({A}, {A}) (, {A}) (, {A}) ({A}, {A}) Figue 1: Tees belonging to L T In addition to the usual set opeations complementation and intesection, we will need pojection. A pojection is a mapping h fom an alphabet Σ to an alphabet Σ. It is applied to a tee t by applying it to the label of each node in t. Given a set S T ω Σ,L, the application of h to S yields the set h(s) = {h(t) t S}, whee h(t) stands fo the composition h t of the functions h and t. We say that h(s) is obtained fom S by pojection. Poposition 15 ([26]). The class of languages ecognized by paity tee automata is closed unde complement, intesection and pojection. Ou fist step towads constucting ÂT is to constuct a PTA A T that ecognizes FL 0 concepts C togethe with (not necessaily least) functional models of C w..t. T. Hee togethe with means that we epesent C and its functional model within a single tee, which has tuples as labels. To be moe pecise, we conside infinite L -labeled Σ-tees t whee the label set is L = 2 N C 2 N C, i.e., the nodes of t ae labeled with tuples of the fom (σ c, σ i ) whee each component is a set of concept names. By applying the pojections h c (σ c, σ i ) = σ c and h i (σ c, σ i ) = σ i to such a tee t, we obtain the L-labeled tees t c and t i, whee L = 2 N C. In addition, if t c (w) fo only finitely many w N R, then t c induces an FL 0 concept C(t c ) as follows: C(t c ) := w.a. (3) w N R A t c(w) Let T fin N R,L denote the subset of T ω N R,L consisting of the tees satisfying this finiteness estiction. Ou goal is to constuct a PTA A T that ecognizes L T := {t T fin N R,L t i epesents a functional model of C(t c ) w..t. T }. Example 16. Let N C = {A} and N R = {}, and conside the FL 0 TBox T = {.A A}. In addition, conside the thee tees depicted in Fig. 1, whee the symbol stands fo the infinite N R -tee whose nodes ae labeled with (, ). Obviously, these tees belong to T fin N R,L, and thus the concepts induced by the fist components of the labels ae well-defined. Indeed, we have C(t (1) c ) =.A = C(t (2) c ) and C(t (3) c ) =.A. The functional intepetations epesented by t (1) i, t (2) i, and t (3) i satisfy A I t (1) i = {ε, }, A I t (2) i = {ε,, } = A I t (3) i. It is easy to see that I t (1) i is the least functional model of.a w..t. T. The functional intepetations I t (2) i and I t (3) i ae identical. This intepetation is a functional model of.a w..t. T, but not the least one, wheeas it is the least functional model of.a w..t. T. Ou constuction of A T is simila to the one of A C,T, but now the concept C is not given, but needs to be guessed. This is done in the fist component of the states of A T. The second 8

9 component togethe with the paity acceptance condition ensues that only finite concepts ae guessed. The thid component of the states basically coesponds to a state of A C,T, and thus ensues the functional model condition. Definition 17. Let T be an FL 0 TBox such that all concepts occuing in T ae in WNF. We set V NC,T := N C val(t ) and define the PTA A T := (Σ, Q T, L, Θ T, I T, c) as follows: Q T := {(q c, l, q) 2 N C {0, 1} 2 V N C,T q c q (l = 0 = q c = )}; Σ := N R = { 1,..., k } and I T := Q T ; Θ T := {((q c, l, q), (σ c, σ i ), (q c1, l 1, q 1 ),..., (q ck, l k, q k )) the popeties (1), (2), (3) hold}, whee 1. (q, σ i, q 1,..., q k ) Θ (see Definition 8 with V C,T eplaced by V NC,T ), 2. l = 0 implies l 1 = 0,..., l k = 0, 3. q c = σ c ; c(q c, l, q) = l, fo all (q c, l, q) Q T. Assume that ρ is an accepting un of A T on a tee t. Then, by König s lemma, (2) togethe with the paity acceptance condition ensues that only finitely many nodes in the un ae labeled with a state whose middle component is 1. In combination with the condition l = 0 q c = in the definition of Q T this ensues that only finitely many nodes in ρ ae labeled with a state whose fist component is non-empty. 1 Consequently, (3) implies that t T fin N R,L, and thus C(t c ) is a well-defined FL 0 concept. It emains to see why t i is a functional model of C(t c ) w..t. T. Basically, this is a consequence of Lemma 9 since, in the thid component, A T behaves like A C,T fo C = C(t c ). Note that the condition q c q in the definition of Q T togethe with (3) ealizes the condition Ĉ X in the definition of the set of initial states of A C,T. Given this intuition, it is not had to see that A T ecognizes the tee language L T (see [4] fo a fomal poof). Poposition 18. Let T be an FL 0 TBox. Then, L(A T ) = L T. At fist sight one might think that going fom A T to ÂT can be achieved in a way simila to ou constuction of  C,T out of A C,T in the pevious section. Howeve, the minimization appoach employed in Section 3 does not wok if the concept C is not fixed fom the outset, but is guessed duing the un of the automaton. In fact, assume that ρ is a un of A T. Whethe a concept name appeaing in the thid component of a state ρ(w) violates minimality o not also depends on what is guessed in the fist component of states ρ(wv) labeling successos wv of w. Instead of tying to minimize A T, we follow a diffeent appoach that uses the closue popeties of PTAs. Basically, given A T, it is not had to constuct an automaton A nl that accepts all FL 0 concepts C togethe with thei non-least functional models. By applying complementation and the othe closue popeties, we can then use A nl and A T to constuct the desied automaton ÂT. To be moe pecise, ou constuction of  T poceeds as follows: 1 Note that dispensing with the middle component of the states and defining c(q) = 0 iff the fist component of q is empty would not wok. In fact, a state with empty fist component may have a successo node whee this component is non-empty. Thus, though in evey path thee would be only finitely many states with non-empty fist component, one could still have infinitely many such states in the whole tee, e.g., at all nodes n s fo n 0. 9

10 1. The automaton A nl needs to accept exactly those L -labeled N R -tees t such that t i epesents a non-least functional model of C(t c ) w..t. T. To achieve this, we fist constuct an automaton that accepts all FL 0 concepts C togethe with two functional models of C w..t. T, whee one is a witness fo the fact that the othe is not the least one. Fom this, we obtain A nl by pojecting away the component coesponding to the witness. Details of this constuction can be found in [4]. 2. Once we have A nl, we can obtain an automaton A nl ecognizing the complement language L(A nl ). Notice that, although L(A nl ) contains all tees t such that t i epesents the least functional model of C(t i ), it also contains many othe tees that do not encode a concept togethe with a functional model (e.g., tees that violate the finiteness estiction fo the fist component o whee the second component does not encode a functional model of the concept epesented by the fist component). Howeve, to filte out such ough tees, we can simply intesect the language accepted by A nl with the one accepted by A T. 3. Closue unde intesection of the class of languages accepted by PTAs thus yields a PTA fo the language L(A nl ) L(A T ), which consists exactly of those tees t T fin N R,L such that t i epesents the least functional model of C(t c ). Thus, the desied automaton ÂT is obtained by applying closue unde the pojection h i (σ c, σ i ) = σ i to the intesection automaton. Summing up, we thus have poved the following main esult of this pape. Theoem 19. Let T be an FL 0 TBox. Then we can effectively constuct a PTA ÂT such that L(ÂT ) = {T C,T C is an FL 0 concept}. The complexity of closue opeations and decision pocedues fo paity tee automata is detemined not only by the numbe of states, but also by the numbe of diffeent pioities used. Using known esults on the complexity of closue opeations on PTAs, we can show that  T has the following size. Coollay 20. The numbe of states of  T is double exponential and the numbe of pioities single exponential in the size of T. Basically, this is due to the fact that the automaton A T has an exponential numbe of states in the size of T and a constant numbe of pioities. The second exponential blowup fo the numbe of states comes fom the complement opeation as does the exponential blowup fo the numbe of pioities (see [4] fo details). 5 Non-standad infeences As an application of the automaton ÂT constucted above, we conside two non-standad infeences fo FL 0 w..t. geneal TBoxes, the least common subsume and the diffeence. Least common subsumes Let T be an FL 0 TBox and C, D FL 0 concepts. The FL 0 concept E is a least common subsume (lcs) of C and D w..t. T if 10 C T E and D T E, and fo all FL 0 concepts F such that C T F and D T F we have E T F.

11 As an immediate consequence of this definition we obtain that least common subsumes of C, D w..t. T ae unique up to equivalence T, if they exist. This justifies talking about the lcs of C, D w..t. T and to denote it (i.e., some element of the equivalence class) as E = lcs T (C, D). We will see below that the lcs need not always exist. The following lemma, whose poof can be found in [4], chaacteizes the cases whee it does. Lemma 21. Let C, D, E be FL 0 concepts and T a geneal FL 0 TBox. Then, E is the lcs of C and D w..t. T iff I E,T = I C,T I D,T. In paticula, the lcs of C and D w..t. T exists iff thee is an FL 0 concept E such that I E,T = I C,T I D,T. To see that in geneal the lcs w..t. geneal FL 0 TBoxes need not exist, conside the TBox T := {A.(A C), B.(B C)} whee A, B, C N C, and assume that we ae inteested in the lcs of A and B w..t. T. It is easy to see that I := I A,T I B,T satisfies A I = = B I and C I = { n n 1}. To show that I is not the least functional model of some FL 0 concept, assume that E is an FL 0 concept such that I is a functional model of E. Then thee is a finite subset W { n n 1} such that E = w W w.c. But then the least functional model I E,T of E w..t. T actually satisfies C I E,T = W, and thus I I E,T. Thus, thee is no FL 0 concept E such that I E,T = I A,T I B,T, which shows that A, B do not have an lcs w..t. T. Theoem 22. Let C, D be FL 0 concepts and T a geneal FL 0 TBox. Then we can effectively decide whethe C, D have an lcs w..t. T o not. In case the answe is yes, we can effectively compute this lcs. Poof. By Lemma 13 we can effectively constuct an LTA P C,D,T that epesents I C,T I D,T. By applying the intesection constuction to PC,D,T and ÂT and then testing emptiness of the esulting automaton, we can check whethe thee is an FL 0 concept E that has I C,T I D,T as its least functional model w..t. T. In ode to actually compute the lcs, we need to use a modified vesion Âc T of  T whee the concept component is not pojected away. Instead of applying the intesection constuction, one then needs to use a simila poduct constuction that checks whethe to Âc T and P C,D,T the second component of the tee accepted by Âc T coincides with the tee accepted by P C,D,T. When applying the emptiness test to the esulting automaton one then extacts a egula tee that witnesses non-emptiness. Fom the fist components in this tee, the lcs E can easily be ead off. Regading the complexity of the decision poblem, note that the numbe of states of the LTA P C,D,T is exponential in the size of C, D, and T, and that the numbe of states of  T is double exponential in the size of T. The poduct constuction thus again yields an automaton of double exponential size. Since the emptiness poblem fo paity tee automata is in NP co- NP, this shows that we can decide the existence of the lcs in 2NExpTime co-2nexptime (see [4] fo details). Concept diffeence In DLs, the notion of diffeence has ecently mostly been consideed fo TBoxes athe than concepts [16]. Given two TBoxes T 1, T 2, the diffeence between T 1 and T 2 is a TBox T 1 T 2 that has, as its consequences, exactly the consequences of T 1 that ae not consequences of T 2. In geneal, such a TBox need not exist, and thus it is inteesting to decide whethe it does. Hee, we modify this appoach by consideing a fixed TBox T and two concepts C, D. The consequences of C and D w..t. T ae then the value estictions that espectively follow fom these two concepts. 11

12 Definition 23. Let T be an FL 0 TBox and C, D FL 0 concepts. The concept diffeence of C and D w..t. T is an FL 0 concept E such that L T (E) = L T (C) \ L T (D). By Poposition 1, the concept diffeence is unique up to equivalence w..t. T, if it exists. In this case, we denote it as C T D. Deciding whethe the concept diffeence exists o not can be done in the same way as fo the lcs. The only change is that, instead of an LTA epesenting the intesection of two functional models, we now need to constuct an LTA epesenting thei diffeence. Given least functional models I C,T and I D,T, thei diffeence is the functional intepetation I = (N R, I) that satisfies A I = A I C,T \ A I D,T fo all A N C. Given automata epesenting the tees T C,T and T D,T, an automaton epesenting the tee t I whee I is the diffeence of I C,T and I D,T can be constucted similaly to the automaton fo the intesection. Theoem 24. Let C, D be FL 0 concepts and T a geneal FL 0 TBox. Then we can effectively decide whethe the diffeence of C and D w..t. T exists o not. In case the answe is yes, we can effectively compute C T D. The complexity of the decision pocedue is obviously the same as fo the lcs, i.e., in 2NExpTime co-2nexptime. The diffeence of concepts has been consideed befoe in the liteatue, but fo DLs othe than FL 0 and without a TBox [24, 14]. In [24] this is esticted to the case whee C D, wheeas in [14] no subsumption elationship between C and D is equied. In the moe geneal setting in [14], the set of diffeence candidates C D := {E C L D E C D} is consideed, whee C L is the set of concepts definable in the DL L unde consideation. In [24] it is agued that, semantically, the diffeence should be as lage as possible, which motivates consideing the elements of C D that ae maximal w..t. subsumption as diffeence concepts. The following poposition (whose poof can be found in [4]) shows that, in case of an empty TBox, ou definition of the diffeence of FL 0 concepts (Definition 23) coincides with these ealie definitions. Poposition 25. Let C, D be FL 0 concepts. Then C D always exists, and it is the unique subsumption maximal FL 0 concept in C D. 6 Conclusion We have shown that least functional models of FL 0 concepts w..t. geneal FL 0 TBoxes can be used to chaacteize subsumption, and that automata ÂC,T and ÂT can be constucted that espectively accept (i) exactly the least functional model of a fixed concept C w..t. a TBox T ; (ii) all least functional models w..t. a TBox T. We have used these automata to show that subsumption in FL 0 w..t. geneal TBoxes is in ExpTime and that the existence of the lcs and the diffeence of two FL 0 concepts is decidable. The complexity of the latte decision pocedues is quite high, in pat because of the use of complementation to constuct ÂT. One topic fo futue eseach will be to lowe this complexity o to show matching lowe bounds. One might think that the use of altenating automata could avoid the complementation in the constuction of  T, and thus allow us to guess a concept C and veify minimality of the intepetation using the same automaton. Howeve, it is not clea how one could enfoce that the pats of the automaton that check minimality at diffeent places in the tee ae all based on the same guessed concept C. Anothe topic fo futue eseach is to investigate what othe non-standad infeences fo FL 0 can be tackled with the appoach intoduced in this pape. In paticula, the unification 12

13 poblem fo FL 0 w..t. the empty TBox has been solved by using tee automata that basically guess a unifie [9]. Howeve, ensuing that diffeent occuences of the same vaiable ae eplaced with the same concept is only possible with tee automata if the wods occuing in the value estictions ae evesed. Unfotunately, in this evesed epesentation of value estictions, checking the satisfaction of a GCI is no longe a local popety, and thus cannot be done with a tee automaton. Consequently, it is not clea how these two appoaches could be combined. Refeences [1] Fanz Baade. Using automata theoy fo chaacteizing the semantics of teminological cycles. Ann. of Mathematics and Atificial Intelligence, 18: , [2] Fanz Baade, Sebastian Bandt, and Casten Lutz. Pushing the EL envelope. In Poc. of the 19th Int. Joint Conf. on Atificial Intelligence (IJCAI 2005), pages , Edinbugh (UK), Mogan Kaufmann, Los Altos. [3] Fanz Baade, Olive Fenández Gil, and Pavlos Maantidis. Appoximation in desciption logics: How weighted tee automata can help to define the equied concept compaison measues in FL 0. In Poc. of the 11th Int. Conf. on Language and Automata Theoy and Applications (LATA 2017), volume of Lectue Notes in Compute Science, pages Spinge-Velag, [4] Fanz Baade, Olive Fenández Gil, and Maximilian Pensel. Standad and non-standad infeences in the desciption logic FL 0 using tee automata. LTCS-Repot LTCS-18-04, Chai fo Automata Theoy, Institute fo Theoetical Compute Science, TU Desden, Gemany, See [5] Fanz Baade and Ralf Küstes. Computing the least common subsume and the most specific concept in the pesence of cyclic ALN -concept desciptions. In Poc. of the 22nd Geman Annual Conf. on Atificial Intelligence (KI 98), volume 1504 of Lectue Notes in Compute Science, pages Spinge-Velag, [6] Fanz Baade, Ralf Küstes, Alex Bogida, and Deboah L. McGuinness. Matching in desciption logics. J. of Logic and Computation, 9(3): , [7] Fanz Baade, Ralf Küstes, and Ralf Molito. Stuctual subsumption consideed fom an automata theoetic point of view. In Poc. of the 1998 Desciption Logic Wokshop (DL 98). CEUR Electonic Wokshop Poceedings, [8] Fanz Baade, Ralf Küstes, and Ralf Molito. Computing least common subsumes in desciption logics with existential estictions. In Poc. of the 16th Int. Joint Conf. on Atificial Intelligence (IJCAI 99), pages , [9] Fanz Baade and Paliath Naendan. Unification of concept tems in desciption logics. J. of Symbolic Computation, 31(3): , [10] Fanz Baade and Stephan Tobies. The invese method implements the automata appoach fo modal satisfiability. In Poc. of the Int. Joint Conf. on Automated Reasoning (IJCAR 2001), volume 2083 of Lectue Notes in Atificial Intelligence, pages Spinge-Velag, [11] Ronald J. Bachman, Deboah L. McGuinness, Pete F. Patel-Schneide, Loi Alpein Resnick, and Alexande Bogida. Living with CLASSIC: When and how to use a KL-ONE-like language. In John F. Sowa, edito, Pinciples of Semantic Netwoks, pages Mogan Kaufmann, Los Altos, [12] Ronald J. Bachman and James G. Schmolze. An oveview of the KL-ONE knowledge epesentation system. Cognitive Science, 9(2): , [13] Sebastian Bandt. Polynomial time easoning in a desciption logic with existential estictions, GCI axioms, and what else? In Poc. of the 16th Eu. Conf. on Atificial Intelligence (ECAI 2004), pages ,

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