though, note that there is such a result for the temporal DLTDL-Lite#kromF [Art+07] (see Section 3.4 for a description of that formalism). Specifically, we show it for DL-LTL whereDLis a DL for which the satisfiability problem of conjunctions ofDLliterals can be decided in NP. As described in Section 3.3, the direct application of the approach applied in the related work [BGL12; BBL15b], captured in Lemma 3.13, yields only a nondeterministic exponential time procedure, because of the exponential size of the set W.
The idea of our Algorithm 3.1 is to integrate the LTL and the DL satisfiability testing:
we test the satisfiability of the LTL formula Φpa as it is done in Algorithm 2.1 and ensure in parallel that the guessed worlds are such that the respective conjunctions of DLliterals from Definition 3.12 are satisfiable. Since we disregard both rigid names and the TKB, the DL part can be solved in this way, by checking the satisfiability of the k conjunctions in χi separately, in exponentially many (possibly nondeterministic) tests, each requiring only polynomial time. We then can consider W to consist of all worlds guessed, although we never had to construct it in total.
Definition 4.3 Given a DL-LTL formula Φ and disregarding rigid symbols, satisfia-bility can be decided using Algorithm 3.1 by taking h∅,∅i as input TKB, assuming GUESSDATA to do nothing, and defining TESTRSAT(Φ,O,Ai, d, i, s, p, W) to return true iff the below conjunction of DLaxioms is satisfiable:
^
pj∈W
ϕj∧ ^
pj∈W
¬ϕj.
♦ Observe that, in this special case without rigid symbols and TKB, we do not need additional data and have n = 0. The latter means that our additional adaptations regarding nin Algorithm 3.1 do not have any effect w.r.t. Algorithm 2.1.
Lemma 4.4 The nondeterministic algorithm described in Definition 4.3 decides satis-fiability in DL-LTL by using only polynomial space if NRC =∅ and NRR=∅.
Proof. We defineW as the union of all setsW encountered while running the algorithm.
Then, the correctness follows from Lemma 3.13, Fact 3.15, the correctness of the LTL algorithm (see Lemma 2.21), Definition 3.12 regarding the empty TKB and no rigid sym-bols (i.e., the kconjunctionsχi do not share concept or role names), Lemma 3.16, and the fact that our adaptation leads to a negative result iff one of the latter conjunctions is not satisfiable.
The space complexity is a consequence of Lemma 2.21, the fact that the size of the conjunction considered in the r-satisfiability test is linear in that of Φ, and (P2).
The complexity of satisfiability in DL-LTL is then obtained by combining Lemma 4.4 and the well-known result of Savitch [Sav70, Thm. 1].
Corollary 4.5 Satisfiability in DL-LTL is in PSpace if NRC =∅ and NRR =∅.
4.3 With Rigid Names
In this section, we show that rigid concept symbols change the picture. In a nutshell, this is the case because, now, interactions are possible: we cannot only use the LTL
features to discern exponentially many time points and to nondeterministically choose a specific axiom at each of them, but we can also apply the DL part to correspondingly discern exponentially many (rigid) concepts, instantiated by different individuals, and thus “save” the LTL choices invariant to time, at the respective of those individuals—
via rigid names. If the axioms chosen are associated with specific individuals in this way, then the DL part may additionally constrain the choice. We prove that it needs a NExpTime Turing machine to decide satisfiability in this setting.
The corresponding containment result directly follows from Lemma 3.17 forDL-LTL because of (P2). Lemmas 4.1 and 4.2 thus let us state the following more specific result.
Corollary 4.6 Satisfiability in EL-LTL and DL-Lite [[horn|bool]|H] -LTL is in NExpTime, even if NRR 6=∅.
NExpTime-hardness can be shown already for the case with rigid concept names, by reducing the 2n+1-bounded domino problem.
Definition 4.7 A domino system is a triple D= (T, H, V), where T is a finite set of domino types and H, V ⊆ T ×T are the horizontal and vertical matching conditions.
LetT be a domino system andI =t0, . . . , tn−1 ∈Tn be an initial condition, which is a sequence of domino types of lengthn >0. A mappingτ: [0,2n+1−1]×[0,2n+1−1]→T is a 2n+1-bounded solution of D respecting the initial condition I iff the following hold for all x, y <2n+1:
• ifτ(x, y) =tand τ((x+ 1) mod 2n+1, y) =t0, then (t, t0)∈H;
• ifτ(x, y) =tand τ(x,(y+ 1) mod 2n+1) =t0, then (t, t0)∈V;
• τ(i,0) =ti fori < n. ♦
The complexity we target is established in [BGG97, Thm. 6.1.2], where it is shown that there is a domino system D = (T, H, V) such that, given an initial condition I = t0, . . . , tn−1 ∈ Tn, the problem of deciding if D has a 2n+1-bounded solution re-spectingI is NExpTime-hard.
For the reduction, we apply the features outlined above. The exponentially many different time points, each associated with a specific rigid concept and individual in-stantiating it, represent the positions in the plane of the domino. To tile the plane, we represent the domino types as flexible concepts and require a specific named individual to always satisfy one of them, by nondeterministically choosing the corresponding asser-tion. The ontology is used to transfer that domino choice to the entire domain (i.e., by using local GCIs of the form> v. . .) and to save it via a rigid concept at the individual associated to the current time point. The rigid concepts ensure that all positions and the chosen types are instantiated inevery world, which allows us to enforce the matching conditions.
Theorem 4.8 Satisfiability inDL-LTL isNExpTime-hard ifNRC 6=∅, even ifNRR =∅.
Proof. Our proof consists of two steps: we first reduce the 2n+1-bounded domino prob-lem for a given domino Dwith initial conditionI to checking the satisfiability of a DL-LTL formula ΦD,I applying ⊥, and then dispose of the⊥constructor by using qualified
4.3 With Rigid Names
existential restriction. We thus cover both kinds of DLs DL we consider (see (P1), in the beginning of the chapter). For the reduction, we adapt the NExpTime-hardness proof forALC-LTL w.r.t. rigid concepts [BGL12, Lem. 6.2]; we point out the differences to that reduction at the end of the proof. We assume a domino system D= (T, H, V) with initial condition I to be given and construct a DL-LTL formula ΦD,I such that ΦD,I is satisfiable iffDhas a 2n+1-bounded solution respectingI. Subsequently, we first list and describe the symbols we use and then specify ΦD,I.
We discernglobal1 concept names that are flexible and either satisfied by all domain elements or by none of them; local concept names are rigid and used to identify specific individuals. Global names are used to transfer values such as domino types from a named individualato all other elements, and local names are used to save these values at some of these elements. Altogether, we use the following symbols:
• an individual namea;
• flexible (global) concept names Z0, . . . , Z2n+1, Z0h. . . , Z2n+1h , Z0v, . . . , Z2n+1v , to realize three binary counters modulo 22n+2 (Z,Zh, and Zv) over the positions in the plane, by iterating over time;
• rigid (local) concept names X0, . . . , Xn and Y0, . . . , Yn, to similarly realize two binary counters modulo 2n+1, where the X-counter describes the horizontal and theY-counter the vertical position in the plane;
• flexible (global) concept names Gt, Ght,Gvt, and a rigid (local) concept name Lt for all t∈T, to describe the tiling;
• rigid (local) concept names X0, . . . , Xn,Y0, . . . , Yn, and flexible (global) concept namesZ0, . . . , Z2n+1,Zh0, . . . , Zh2n+1, andZv0, . . . , Zv2n+1 representing the comple-ments of the above counters;
• auxiliary flexible concept namesN,E0h, . . . , E2n+1h ,E0v, . . . , E2n+1v .
The first n+ 1 bits of the Z, Zh and Zv-counters are used to represent the 2n+1 x-coordinates, and the second n+ 1 bits are used to represent the y-coordinates of the plane. We count with the Z-counter up to 22n+2 by iterating over time and use the time points to realize all positions (x, y) ∈ [0,2n+1 −1]×[0,2n+1 −1]. Specifically, we enforce a to satisfy the concepts from the corresponding subset of {Z0, . . . , Z2n+1} for every value of the Z-counter.2 By modeling the concept names Zi as global, we ensure that the value of the counter is always transferred to all domain elements. For every position represented by the Z-counter, the Zh and Zv-counters represent the top and right neighbor position, respectively. Note however that, regarding a specific world, these three counters only allow us to enforce that the position they currently represent—regarding the Z-counter, we call it the current position—is instantiated, by all individuals.
To be able to address arbitrary positions independently of the current time point, we use the rigid concept names X0, . . . , Xn and Y0, . . . , Yn. In particular, we ensure
1Not to be confused withrigidoralways(in time).
2For simplicity, we refer toainstead of to “the domain element representinga”.
that there is always (at least) one individual whose X and Y-values match the value of the global Z-counter, such that every position (x, y)∈[0,2n+1−1]×[0,2n+1−1] is represented by at least one individual in every world.
To tile the plane, we ensure that asatisfies exactly one current global domino type Gtin every world, which represents the typetchosen for the current position. The local concept Lt is used to represent this choice independently of time, similar to the use of the X/Y-counter w.r.t. the positions represented by the Z-counter. Specifically, we do not only ensure that there is an individual whose X and Y-values match the value of the global Z-counter, but it also must satisfy Lt if Gt is the current global type. In this way, we ensure that all positions and the chosen types are instantiated in every world, which allows us to enforce the matching conditions. For the latter, we explicitly represent the types chosen for the two important neighbor positions as global types Ght and Gvt and ensure that they match the local types of the respective positions.
We now formalize these descriptions and the initial condition and construct the DL-LTL formula ΦD,I as a conjunction of theDL-LTL formulas listed in the following.
• Every value of the Z-counter is realized in some world by a, which means thata satisfies the concepts from the corresponding subset of {Z0, . . . , Z2n+1}:
2F
This formula requires the i-th bit of the Z-counter to be flipped from one world to the next iff all preceding bits are true. Thus, the value of the Z-counter in the next world is equal to the value in the current world incremented by one.
• In every world, the Zh and Zv-counter are synchronized with the Z-counter and also realized by a. This is described similar to the Z-counter in two steps. (i) The x-coordinate represented by the Zh-counter equals the one represented by theZ-counter plus 1: (ii) The y-coordinates represented by the two counters are the same:
2F
• Regarding a, the interpretation of the concept nameZi as the complement of Zi
is enforced as follows; corresponding conjuncts are used to model Zhi andZvi, the complements of Zih andZiv, respectively:
2F
^
0≤i≤2n+1
Zi(a)↔ ¬Zi(a).
4.3 With Rigid Names
• The values of the three global counters Z, Zh, and Zv (and their complements) are shared by all individuals in every world. Again, the global formulas for Zh and Zv, and also those for the three complements, are analogous to the ones for theZ counter:
2F ^
0≤i≤2n+1
((> vZi)∨(Ziv ⊥)).
• In every world, also one combination of the X and the Y-counter is realized by (at least) one individual. To ensure that all combinations are covered, we consider the one matching the globalZ-counter (in this world) and, to make sure that they are realized, we require the concept N to be instantiated in every world. N is then used to model the former:
In the same way, we enforce the correct interpretation of the complements of the local counters. That is, we use a formula as the above one where Zi, Xi, and Yi−(n+1) are replaced byZi, Xi,and Yi−(n+1), respectively.
Altogether, the above descriptions ensure that every position of the plane is realized in every world by some individual. Note that we need this to enforce the matching conditions given in Dwith the help of global CIs, since they address different positions at the same time. Next, we describe the admissible tilings.
• Every world gets exactly one (global) domino type that represents the choice for the current position and is shared by all individuals:
2F _
t∈T
(> vGt)∧ ^
t0∈T\{t}
(Gt0 v ⊥). (4.2)
Again, we similarly consider the global domino types Ght andGvt representing the choices for the current right and, respectively, top neighbor position (represented by Zh andZv).
• Given the domino types chosen for the neighbor positions, the horizontal and vertical matching condition can be enforced easily: type t∈T represented by Ght in the current world equals that represented by Gt in the world where the Z-counter is equal to the currentZh-counter), we use the local (rigid) domino types Lt. First, we ensure that the local type satisfied by
the individual representing the current position equals the current global type Gt with CIs as follows for all t∈T:
2F ^
t∈T
(N uGtvLt)∧(NuLtvGt).
Note that, because of the second of the above CIs and the fact that every world has exactly one global domino type Gt, satisfied by all individuals (see (4.2)), every individual satisfyingN in some world also has exactly one local domino type: the global type of that world, in which it represents the current position (see (4.1), which also ensures that such an individual exists for every world).
To synchronize the choices of domino types, especially those represented byGt, Ght and Gvt, we employ the auxiliary concept names Eih and Eiv: interpretation of E0hu · · · uE2n+1h must include all those domain elements that satisfy the one combination of X and Y-values which matches the current Zh -counter. This particularly includes the individual instantiating N that is forced to exist in the corresponding world by (4.1). Moreover, it allows us to ensure that the global domino type Ght matches the local domino type Lt at all domain elements satisfyingE0hu· · ·uE2n+1h ; analogous arguments and CIs apply regarding the vertical direction:
This conjunction identifies a particular x-position in theZ-counter. If the y-coordinate represented by theZ-counter is 0 additionally, as enforced above, then the corresponding domino type of the initial condition is enforced. This finishes the definition of the DL-LTL formula ΦD,I, which consists of the conjuncts specified and mentioned above.
Observe that the size of ΦD,I is polynomial in n. Moreover, it can be checked that ΦD,I
is satisfiable w.r.t. hOD,I,∅iiffD has a 2n+1-bounded solution respecting I.
We lastly describe how the bottom constructor can be eliminated. We apply an idea of [BBL05] (see the proof of Theorem 7), use a freshrigid concept nameLand a flexible role name R, and require the following formula to be satisfied:
ΦL:=¬L(a)∧2F(∃R.LvL).
By replacing the negated CI ¬(N v ⊥) in ΦD,I by > v ∃R.N, we ensure that
4.3 With Rigid Names
• in all the exponentially many worlds, a has an R-successor that satisfies N and, because of (4.1), represents the position associated to the corresponding world;
and
• aas well as the above mentioned individuals do not satisfy L in any world, since L is rigid.
We hence can useLinstead of⊥without changing the semantics everywhere else in the formula ΦD,I. The reason for this is that it suffices to enforce the CIs with ⊥ on the right-hand side to hold regarding individuals representing the 22n+2 relevant positions, instead of for all domain elements. We denote by Φ0D,I the formula resulting from these replacements and get that Φ0D,I ∧ΦL is satisfiable iff D has a 2n+1-bounded solution respecting I.
Given the above observations, it is easy to see that the polynomial size and cor-rectness are retained. This finishes the reduction, and NExpTime-hardness follows from (P1) and theNExpTime-hardness of the 2n+1-bounded domino problem [BGG97, Thm. 6.1.2].
We point out main differences to the proof forALC-LTL, which similarly reduces the domino problem. The disjunction allowed in ALC CIs allows to enforce the matching conditions by using only a single global type Gt, while we additionally needGht andGvt together with a complex synchronization with the help of the auxiliary concepts. More specifically, ALC-LTL allows to require that the current global type and the local types of the elements representing the important neighbor positions fulfill some condition of H and V (for details, we refer to the original proof). In contrast, we have to ensure the matching using the nondeterminism of LTL and hence the DL-LTL formula and the individual a. Another difference is the presence of the concept names explicitly representing the complements of the various counters (e.g., Xi). In ALC, this can be directly expressed using negation.