Lemma 2.15 For all u%RC ∈∆IuK and D∈S(O), we haveu%RC ∈DIK iff
• O |=C vD, if K is in EL, and
• O |=∃R−vD, if K is in DL-LiteHhorn.
Lastly, we consider so-called simulations, which in [Baa03] are described as binary relations between nodes of two so-called EL description graphs that respect the labels and edges of those graphs. Such an EL description graph is obtained for an interpreta-tion I by regarding I as a graph such that the domain elements are the nodes, labeled by the concept names the elements satisfy; and the (labeled) edges are given by the roles connecting the elements in I. We define the notion of simulation directly w.r.t.
two interpretations.
Definition 2.16 A relationσ⊆∆I×∆J is asimulation(ofIbyJ), writtenσ:I → J, iff the following hold for all (d, e)∈σ:
• d∈AI impliese∈AJ for all A∈NC, and
• (d, d0) ∈ RI implies that there is an element e0 ∈ ∆J such that (d0, e0) ∈ σ and
(e, e0)∈RJ for all R∈NR. ♦
It is easy to inductively construct a simulation of the finite canonical interpretation of a KB K by any other model ofK.
Lemma 2.17 For every model J of an EL knowledge base K, there is a simulation σ of IKf by J such that (a, aJ)∈σ for all a∈NI(K).
2.2 Propositional Linear Temporal Logic
Propositional linear temporal logic (LTL5), also known as propositional temporal logic, extends propositional logic with modal operators to represent past and future moments in formulas. Accordingly, the signatures are as in propositional logic, sets of proposi-tional variables.
Definition 2.18 (Syntax of Propositional LTL) Let P = {p1, . . . , p`} be a finite propositional logic signature. The set of propositional LTL formulas over P is defined by the following grammar:
ϕ::=p| ¬ϕ1 |ϕ1∧ϕ2|#Fϕ1 |#Pϕ1 |ϕ1Uϕ2|ϕ1Sϕ2
where p∈ P, and ϕ1 andϕ2 are formulas, in their turn. ♦ Observe that LTL allows for Boolean negation and conjunction. The operators #F and #P are called “next” and “previous”, respectively. The formulaϕ1Uϕ2 stands for
“ϕ1 until ϕ2”, and ϕ1Sϕ2 represents its dual, read “ϕ1 since ϕ2”. We may use#iF to denote a sequence of i#F-operators, and similar for the previous operator. We assume
5For simplicity, we often drop the prefix “propositional” and simply refer to linear temporal logic (LTL) throughout the work.
Operator Definition Name ϕ1∨ϕ2 ¬(¬ϕ1∧ ¬ϕ2) disjunction ϕ1 →ϕ2 ¬(ϕ1∧ ¬ϕ2) implication
3Fϕ trueUϕ eventually (some time in the future) 2Fϕ ¬3F¬ϕ always in the future
3Pϕ trueSϕ historically (some time in the past) 2Pϕ ¬3P¬ϕ always in the past
Figure 2.2: Definitions of derived operators for propositional LTL.
true to denote an arbitrary but fixed propositional tautology (e.g., p∨ ¬p, where p is a propositional variable) and false to denote its negation. As usual, further derived operators can be defined as in Figure 2.2. We further use ϕ1 ↔ϕ2 as an abbreviation for (ϕ1 →ϕ2)∧(ϕ1 ←ϕ2).
The operators#F and U are called thefuture operators, and#P and S are the past operators. Together, they represent the temporal operators. In accordance with that, a propositional LTL formula is called afuture formulaif it contains no past operators and a past formula if it contains no future operators. An LTL formula is calledseparated if no future operators occur in the scope of past operators and vice versa. A subformula of a separated LTL formula ϕis a top-level future (past) formula of ϕ if it is of one of the forms in 2.1 (2.2) and occurs in ϕ at least once in the scope of no other temporal operator:
#F ϕ1,¬(#Fϕ1), ϕ1Uϕ2,¬(ϕ1Uϕ2) (2.1)
#P ϕ1,¬(#Pϕ1), ϕ1Sϕ2,¬(ϕ1Sϕ2) (2.2) The set of all subformulas of an LTL formula ϕis denoted by S(ϕ).6 The set Clo(F) denotes the closure under negation of Sϕ∈FS(ϕ).
In propositional LTL, the flow of time is considered to be bounded with respect to the past, discrete, and, as the name suggests, linear. It is represented by the sequence of natural numbers, such that every point in time (alsotime point or moment) is rep-resented by one number. An LTL interpretation then is a corresponding structure: a sequence of propositional interpretations which, respectively, determine the propositions that are true at the corresponding points in time.
Definition 2.19 (Semantics of Propositional LTL)LetP ={p1, . . . , p`}be a finite propositional logic signature. Apropositional LTL structure forP is an infinite sequence W= (wi)i≥0 of worlds wi ⊆ P.
A propositional LTL structureW = (wi)i≥0 satisfies a propositional LTL formula ϕ at (time point) i≥0, writtenW, i|=ϕ, if the corresponding condition of Figure 2.3 is satisfied. The fact that W, i|=ϕdoes not hold is denoted byW, i6|=ϕ.
IfW,0|=ϕ, thenW is amodel ofϕ.
A propositional LTL-formulaϕissatisfiableif it has a model. Two propositional LTL formulas ϕ1 and ϕ2 areequivalent, writtenϕ1≡ϕ2, if they have the same models. ♦
6Recall that the same notation is used to denote the set of subconcepts of a concept.
2.2 Propositional Linear Temporal Logic
Formulaϕ Condition for W, i|=ϕ
p p∈wi
¬ϕ1 W, i6|=ϕ1
ϕ1∧ϕ2 W, i|=ϕ1 and W, i|=ϕ2
#Fϕ1 W, i+ 1|=ϕ1
#Pϕ1 i >0 andW, i−1|=ϕ1
ϕ1Uϕ2 there is a k≥i, such thatW, k|=ϕ2 and, for all j,i≤j < k, we have W, j |=ϕ1 ϕ1Sϕ2 there is a k, 0≤k≤i, such thatW, k|=ϕ2
and, for all j,k < j≤i, we have W, j |=ϕ1
Figure 2.3: Semantics of propositional LTL formulas for an interpretation W= (wi)i≥0 for a signatureP, assumingp∈ P.
The empty conjunction and disjunction are interpreted as true and false, respectively.
Again, the signature is in the following generally not mentioned explicitly if it is irrele-vant or clear from the context.
Above, the operatorsU and S are defined in their non-strict version. The semantics of the strict operatorsU< andS<differs in that the parameters jand kin Figure 2.19 must not equal i. This means thatW, i|=ϕ2, which implies W, i|=ϕ1Uϕ2, does not imply W, i |= ϕ1U<ϕ2; and similar for the since operator. However, in the presence of the #F-operator, U and U< can be expressed in terms of each other. Specifically, the formula ϕ1Uϕ2 is equivalent to ϕ2 ∨(ϕ1U<ϕ2), and ϕ1U<ϕ2 is equivalent to ϕ1∧#F(ϕ1Uϕ2); and similar for S and S<.
Further, note that the above definition of propositional LTL extends the usual defini-tion of that logic, which only considers the temporal operators #F and U [Pnu77]. For that reason, this extended logic is often referred to as Past-LTL. An important result for this logic, the so-called separation theorem [Gab87, Thm. 2.4], is given below.
Lemma 2.20 ([Gab87, Thm. 2.4]) Every propositional LTL formula ϕis equivalent to a propositional LTL formula that is separated.
Note that [Gab87] actually consider a slightly different temporal logic allowing only S< and U< as temporal operators. However, it is well-known that #F and #P can be simulated in this setting: #Fϕ≡ falseU<ϕ and #Pϕ≡falseS<ϕ. Moreover, the non-strict versions of the operators can be expressed in terms of the strict ones and the other way around while retaining separation, as shown above. Thus, Lemma 2.20 holds also for the logic we focus on. The size of the resulting separated LTL formula then however may be non-elementary in the size of the original formula (i.e., specifically, the number of stacked exponents is determined by the number of alternations between past operators and future operators) [Gab87]. But we use this result in a context where the size of the formula is irrelevant.
We close this section on LTL by describing the procedure for deciding LTL satis-fiability originally proposed in [SC85, Sec. 4] in Algorithm 2.1, which we extend in Chapters 4, 5, and 6 to obtain our results. In particular, [SC85] propose a nonde-terministic algorithm which runs in an amount of space polynomially bounded by the
Algorithm 2.1: Procedure for Deciding LTL Satisfiability Input: LTL formulaϕ
Output: true ifϕis satisfiable, otherwisefalse
1 i:= 0
2 s:= Guess a number ≤2|ϕ| and >0
3 p:= Guess a number≤4|ϕ|
4 Fnext :=∅,Fs:=∅,FU :=∅
5 Fpres:= Guess a subset of Clo({ϕ})
6 if not CONSISTENT(Fpres) ornot INITIAL(Fpres) orϕ6∈ Fpres then
7 returnfalse
8 while i≤s+p do
9 if i >0 thenFpres:=Fnext
10 Fnext := Guess a set of subformulas of ϕ
11 if not CONSISTENT(Fnext) or not TCONSISTENT(Fpres,Fnext) then
12 return false
13 if i=sthen
14 Fs:=Fpres
15 FU := All formulas of the formϕ1Uϕ2 ∈ Fs
16 if i≥sthen
17 FU := All formulas of the formϕ1Uϕ2 ∈ FU such that ϕ2 6∈ Fpres
18 i:=i+ 1
19 if FU =∅ and TCONSISTENT(Fpres,Fs) then
20 returntrue
21 returnfalse
2.2 Propositional Linear Temporal Logic
formulaϕconsidered; we consider |ϕ|to denote the number of symbols occurring inϕ.
The idea for constructing an LTL structureWsatisfyingϕis to iteratively guess subsets of Clo({ϕ}) that represent the subformulas satisfied in W at each point in time. More precisely, every such set induces a unique world containing exactly the propositional variables that are true in the guessed set. In what follows, we describe that procedure, given as Algorithm 2.1, in more detail. We thereby rely on the subprocedures below.
• CONSISTENT: Given a setF of LTL formulas, it checks the Boolean consistency of the latter by returning true iff the following hold for all ϕ∈Clo(F):
– ϕ=ϕ1∧ϕ2 ∈ F iff ϕ1, ϕ2∈ F, – ϕ=¬ϕ1 ∈ F iff ϕ1 6∈ F.
• INITIAL: Given a setF of LTL formulas, it checks ifF can describe the formulas satisfied at time point 0 in an LTL structure by returning true iff the following hold for allϕ∈Clo(F):
– ϕ=ϕ1Sϕ2∈ F iffϕ2 ∈ F – ϕ=#Pϕ16∈ F.
• TCONSISTENT: Given two setsFpres and Fnext of LTL formulas, it checks if they can be satisfied in an LTL structure at consecutive time points by returning true iff the following hold for all ϕ∈Clo(F):
– ϕ=#Fϕ1∈ Fpres iffϕ1 ∈ Fnext, – ϕ=#Pϕ1∈ Fnext iff ϕ1 ∈ Fpres,
– ϕ=ϕ1Uϕ2 ∈ Fpres iffϕ2 ∈ Fpres or (ϕ1 ∈ Fpres and ϕ1Uϕ2 ∈ Fnext), – ϕ=ϕ1Sϕ2∈ Fnext iff ϕ2 ∈ Fnext or (ϕ1 ∈ Fnext and ϕ1Sϕ2 ∈ Fpres).
The procedure is based on the fact that, if ϕ is satisfiable, then there must be a periodic model ofϕwith a period that starts at a time point at most exponential in the size ofϕand is of length at most exponential in the size ofϕ[SC85, Theorem 4.7]. This means that the algorithm can iterate over all time points until the end of the period using a counter ithat can be represented in polynomial space. The startsand lengthp of the period are guessed in the beginning.7This approach works specifically because the algorithm does not store all the sets of subformulas guessed during the iteration.
Instead, it focuses on only four such sets Fpres, Fnext, Fs, and FU, which are updated during processing and occupy only a polynomial amount of memory. Fpres specifies W w.r.t. the present time point i,Fnextdescribes the world for the next time point, Fs the one for time points, and FU is used as auxiliary set. Fpres is guessed in the beginning and it is checked if this set can describe the formulas satisfied at time point 0 in an LTL structure that is a model of ϕ. Subsequently, the counter is continuously incremented and, until it reaches the beginning of the period s, in each step: Fpres and Fnext are updated, which means that a set of subformulas of ϕ is guessed for the latter; further, the Boolean consistency of the new set, and its consistency with the set for the present time point (according to the temporal operators) are checked. Regarding the latter,
7For simplicity, we additionally requires >0 in Algorithm 2.1; this is clearly without loss of generality.
note that the satisfiability test for subformulas of the form ϕ1Uϕ2 may be deferred to the next iteration step if ϕ1 ∈ Fpres. At the beginning of the period, the current set Fpres is stored in Fs and, until the end of the period, the algorithm continues the iteration as before. In addition, it however has to consider the U-subformulas deferred at time point sto make sure that they are satisfied within the period. If it has reached the end of the period, it checks if the latter is the case and if Fs, guessed for describing W at the beginning of the period, can indeed fulfill that function.
Note that [SC85] do not regard the#P-operator, which is considered by us. However, we can obviously assume that the sets of subformulas that are guessed also include subformulas that include this operator and adapt the tests correspondingly (see the above specifications of INITIAL and TCONSISTENT). In particular, this does not affect the space requirements of the algorithm because the period that has to be guessed is still exponential in the size of the considered formula. Furthermore, we present the algorithm adapted to our setting, where satisfiability is to be decided w.r.t. time point 0.
The original algorithm checks satisfiability at a time point given as argument.
We recapitulate the correctness and space complexity of the procedure.
Lemma 2.21 ([SC85, Thm. 4.1 and 4.7]) Algorithm 2.1 nondeterministically de-cides the satisfiability of a given LTL formula ϕ by using an amount of space that is polynomially bounded in |ϕ|.