Extensions of LTL
3.2 Fixpoint Operators
We still remark that each of the above induction rules could be used for an al- ternative axiomatization of LTL+b. The systematic pattern of the axiomatization de- scribed above was to take as axioms the fixpoint characterization of a binary operator and a formula expressing whether it is chosen in its strong or weak version, respec- tively. Another possibility would be to take the fixpoint characterization or, what is actually sufficient, even only “one direction” of it together with the respective rule.
For example, with the operator unless this would be the axiom (unless1’) AunlessB→ eB∨ e(A∧AunlessB)
and the rule (indunless). In the next section we will see that there is also an intuitive pattern which underlies this form of axiomatization.
We conclude this section by illustrating the new operators with the help of some more logical laws. We restrict ourselves to formulas involving the non-strict unless and the strict atnext operator. Analogous laws can easily be stated for the other oper- ators.
(Tb18) 2(¬B →A)→AunlB, (Tb19) e(AunlB)↔ eAunl eB,
(Tb20) (A∧B)unlC ↔AunlC∧BunlC, (Tb21) Aunl(B∨C)↔AunlB∨AunlC, (Tb22) Aunl(B∧C)→AunlB∧AunlC, (Tb23) Aunl(AunlB)↔AunlB,
(Tb24) (AunlB)unlB ↔AunlB, (Tb25) 2(B →A)→AatnextB, (Tb26) e(AatnextB)↔ eAatnext eB,
(Tb27) (A∧B)atnextC ↔AatnextC∧BatnextC, (Tb28) (A∨B)atnextC ↔AatnextC∨BatnextC, (Tb29) Aatnext(B∨C)→AatnextB∨AatnextC.
Note that “idempotency” laws like (Tb23) and (Tb24) hold only for non-strict oper- ators but not for the strict ones.
The laws can easily be verified semantically or by a derivation withinΣLTLb . As an example, we show how to derive (Tb25):
Derivation of (Tb25).
(1) 2(B→A)→ e(B →A) (T6)
(2) 2(B→A)→ e2(B→A) (prop),(ltl3)
(3) 2(B→A)→ e(¬B →2(B→A)) (prop),(T14),(2)
(4) 2(B→A)→ e(B →A)∧ e(¬B→2(B→A)) (prop),(1),(3)
(5) 2(B→A)→AatnextB (indatnext),(4)
always operator:
2A↔A∧ e2A.
This logical equivalence means, in a sense which will be made more precise shortly, that the formula2Acan be viewed as a “solution” of the “equality”
(i) u↔A∧ eu
with the “unknown”u. In the same way, 3A and AunlessB can be viewed as
“solutions” of (ii) u↔A∨ eu, (iii) u↔ eB∨ e(A∧u),
and similarly for the other connectives. However, given such an equivalence, the corresponding temporal operator may not be determined uniquely. For example, (iii) is also solved byu ≡AuntilB, while (ii) admits the solutionu≡true.
In order to analyse the situation more formally, let us provisionally extend the underlying alphabet by a set V of (propositional) variables and allow formulas to contain such variables. For example, the equivalences (i)–(iii) are then formu- las containing the variable u. The semantical notions are extended by valuations Ξ= (ξ0, ξ1, ξ2, . . .)which are infinite sequences of mappings
ξi:V → {ff,tt},
and the valueK(Ξ)i (F) ∈ {tt,ff} is inductively defined asKi(F)before, with the provision that
K(Ξ)i (u) =ξi(u) foru∈ V.
We also writeFΞK to denote the set of (indexes of) states ofKin which formulaF
“is true”:
FΞK = {i ∈N|K(Ξ)i (F) =tt}.
FΞK is a subset ofN, i.e., an element of the powerset2NofN.
Consider now, e.g., the equivalence (ii), assume thatAdoes not contain the vari- ableu, and fix some arbitrary K andΞ. With the “right-hand side” A∨ eu we associate the mappingΥA∨cu: 2N→2Nwith
ΥA∨cu:M→A∨ euΞ[u:M]K
where Ξ[u:M] denotes the valuation (ξ0, ξ1, ξ2, . . .) that agrees with Ξ for all variables except foru, for which it is given byξi(u) = tt⇔i ∈M. The formulas true and3Aare solutions of the equivalence. For true we have
trueΞK =N.
With Ξ[u:M] = (ξ0, ξ1, ξ2, . . .) such that ξi(u) = tt ⇔ i ∈ trueΞK, i.e., ξi(u) = tt for everyi ∈ N, we obtainK(Ξ[u:i M])(A∨ eu) = tt for everyi ∈ N and therefore
A∨ euΞ[u:K M]=N.
This means that
ΥA∨cu(trueΞK) =trueΞK and similarly one can find that
ΥA∨cu(3AΞK) =3AΞK.
Generally, for a solutionC of the equivalence,CΞK is a fixpoint ofΥA∨cu, i.e., a setM ⊆Nsuch thatΥA∨ cu(M) = M. Moreover, the representationsCΞK of solutions can be compared by set inclusion. For example,3AΞK ⊆trueΞK holds for anyKandΞ, and we summarize all this by simply saying that true and3Aare fixpoints of (ii) and3Ais a smaller fixpoint than true.
An equivalence may have many fixpoints, and extremal (least or greatest) fix- points among them are usually of particular interest. In case of (ii),trueΞK =N, so true is obviously the greatest fixpoint (for anyKandΞ) and, in fact,3Ais the least one. To see this, assume thatM⊆Nis some set such that
(∗) A∨ euΞ[u:M]K =M
holds. It then suffices to prove that3AΞK ⊆M. To this end, assume thati ∈/ Mfor somei ∈N. Inductively, we show thatj ∈/Mholds for allj ≥i: the base case holds by assumption, and ifj ∈/ Mthen equation(∗)implies thatK(Ξ[u:M])j (A∨ eu) =ff, which meansξj+1(u) =K(Ξ[u:j M])(eu) = ff; hencej ∈/ euΞ[u:K M]and therefore j + 1 ∈/ M. Moreover, equation(∗)analogously implies that, for every j ∈/ M, K(Ξ)j (A) =K(Ξ[u:M])j (A) =ff; hencej ∈/ AΞK. Together we obtain thatj ∈/ AΞK for allj ≥i, which meansK(Ξ)i (3A) = ff, i.e.,i ∈/ 3AΞK and so concludes the proof of3AΞK ⊆M.
Similarly, it can be shown that false and2Aare the least and greatest fixpoints of the equivalence (i) above, and thatAuntil B andAunless B are the least and greatest fixpoints of (iii).
We now generalize these considerations to an extension of LTL. We want to in- troduce new (unary) logical operators which, applied to a formulaF(generally con- taining a variableu), provide formulas which are the least and the greatest fixpoints of the equivalence
u↔F;
more precisely: the semantical evaluationK(Ξ)i of the formulas is determined by the least and greatest fixpoints of the mappingΥF : 2N→2N(i.e., the least and greatest subsetsM⊆NwithΥF(M) =M) where
ΥF :M→FΞ[u:K M],
as exemplified above forF ≡A∨ eu.
However, we must take some care: not all equivalences need have solutions. A simple example is the equivalenceu ↔ ¬u which obviously does not admit any solutions. But, as shown by the following example, even if fixpoints exist there need not be least and greatest ones.
Example. For a propositional constantv ∈V, consider the formula F ≡v↔ eu
and letK = (η0, η1, η2, . . .) be a temporal structure such thatηi(v) = tt if and only ifiis even. We will show that the functionΥF has precisely two incomparable fixpoints with respect toK. In fact,Mis a fixpoint if and only if, for arbitraryΞ,
M=ΥF(M)
=v↔ euΞ[u:K M]
={i∈N|ηi(v) =tt⇔i+ 1∈M}
={2j|2j+ 1∈M} ∪ {2j+ 1|2j+ 2∈/M}.
This means that, for everyj ∈N,
2j ∈M⇔2j+ 1∈M and 2j+ 1∈M ⇔2j+ 2∈/ M which is obviously the case if and only if either
M={0,1,4,5,8,9, . . .}={n ∈N|n mod 4∈ {0,1}}
or
M={2,3,6,7,10,11, . . .}={n ∈N|n mod 4∈ {2,3}}.
So these two sets are the only fixpoints ofΥF. One is the complement of the other;
in particular, they are incomparable.
To pursue our approach, let us now first note the trivial fact that, if a least fixpoint exists then it is unique, and the same holds for the greatest fixpoint. Furthermore, a well-known sufficient condition that least and greatest fixpoints exist at all in sit- uations like the one given here is that of monotonicity: for any set D, a function Υ : 2D→2Dis called monotone ifΥ(E1)⊆Υ(E2)holds wheneverE1 ⊆E2, for arbitraryE1,E2⊆D. It is called anti-monotone ifE1⊆E2impliesΥ(E1)⊇Υ(E2).
Theorem 3.2.1 (Fixpoint Theorem of Tarski). Assume thatDis some set and that Υ : 2D→2Dis a monotone function. Then
a)μΥ=
{E⊆D|Υ(E)⊆E} is the least fixpoint ofΥ. b)νΥ =
{E⊆D|E⊆Υ(E)} is the greatest fixpoint ofΥ.
Proof. a) We writeΥ for the set {E ⊆ D | Υ(E) ⊆ E}. Let E ∈ Υ. Because
μΥ =
Υ, we certainly haveμΥ ⊆ E, and by monotonicity ofΥ it follows that Υ(μΥ) ⊆Υ(E). By definition ofΥ, we know thatΥ(E) ⊆E. Thus,Υ(μΥ) ⊆ E holds for allE∈Υ, which implies thatΥ(μΥ)⊆
Υ =μΥ.
Again by monotonicity ofΥ, we obtain thatΥ(Υ(μΥ))⊆Υ(μΥ), and therefore Υ(μΥ)∈Υ. This impliesμΥ =
Υ ⊆Υ(μΥ), so altogether we have shown that Υ(μΥ) =μΥ, and thusμΥ is a fixpoint ofΥ.
To see thatμΥ is the least fixpoint ofΥ, assume thatE ⊆ Dis some arbitrary fixpoint, i.e.,Υ(E) =E. In particular,Υ(E)⊆E, and thusE∈Υ. By definition of μΥ, it follows thatμΥ ⊆E, which completes the proof.
b) The proof of this part is dual, exchanging⊆and
by⊇and
.
In the present context, we can apply Theorem 3.2.1 to functions ΥF, and it is easy to see that the polarity of (the occurrences of) the variableuin the formulaF helps us determine the monotonicity ofΥF. Roughly speaking,uoccurs with positive or negative polarity depending on which side of an implicationuoccurs. Formally, polarity is inductively defined as follows:
• uoccurs with positive polarity in the formulau.
• An occurrence ofuin a formulaA→Bis of positive polarity if it is of positive polarity inBor of negative polarity inA; otherwise it is an occurrence of negative polarity.
• The operators eand2preserve the polarity of variable occurrences.
For the derived operators, it follows that∧,∨, and epreserve the polarity, whereas
¬reverses the polarity of occurrences. As for formulasA↔B, every occurrence of u is both of positive and negative polarity because it appears on both sides of an implication. For example, the variableu has a positive polarity in the formula v → eu, a negative polarity inv → ¬eu and occurrences of both positive and negative polarity in the formulav ↔ eu of the above example (v ∈ Vin each case).
Lemma 3.2.2. LetF be a formula,u∈ V be a propositional variable, and the func- tionΥF : 2N→2Nbe given by
ΥF(M) =FΞ[u:M]K .
a)ΥF is monotone if every occurrence ofuinF has positive polarity.
b)ΥF is anti-monotone if every occurrence ofuinFhas negative polarity.
Proof. Both parts a) and b) are proved simultaneously by structural induction on the formulaF.
1. F ≡v∈V,F ≡false orF ≡u¯∈ V,u¯≡u: Thenudoes not occur inF. This impliesΥF(M) =ΥF(M)for arbitraryMandM; soΥFis both monotone and anti-monotone.
2. F ≡u: Then the only occurrence ofuinFis of positive polarity. So, part b) is trivial, and part a) follows since we have
ΥF(M) ={i∈N|K(Ξ[u:i M])(u) =tt}=M for everyM; soΥF is monotone.
3. F ≡A→B: Then
ΥF(M) = A→BΞ[u:K M]
= {i∈N|K(Ξ[u:M])i (A→B) =tt}
= {i∈N|K(Ξ[u:i M])(A) =ff} ∪ {i ∈N|K(Ξ[u:i M])(B) =tt}
= (N\ΥA(M))∪ΥB(M)
for everyM. Let now M1 ⊆ M2. If every occurrence ofu inF is of posi- tive polarity then every occurrence ofuinAis of negative polarity and every occurrence of u in B is of positive polarity. By induction hypothesis,ΥA is anti-monotone andΥB is monotone; thus we have ΥA(M1) ⊇ ΥA(M2)and ΥB(M1) ⊆ ΥB(M2)and therefore obtainΥF(M1) ⊆ ΥF(M2)which proves part a). If every occurrence of u in F is of negative polarity then we con- clude analogously thatΥAis monotone andΥBis anti-monotone which provides part b).
4. F ≡ eA: Then
ΥF(M) =eAΞ[u:M]K ={i∈N|i+ 1∈ΥA(M)}
for everyM. LetM1⊆M2. If every occurrence ofuinF is of positive polarity then so it is inA. By induction hypothesis,ΥB(M1)⊆ΥB(M2); so we obtain part a) because of
{i∈N|i+ 1∈ΥA(M1)} ⊆ {i ∈N|i+ 1∈ΥA(M2)}, and the argument for part b) is analogous.
5. F ≡2A: Then
ΥF(M) ={i∈N|j ∈ΥA(M)for everyj ≥i}
for everyM, and the assertions a) and b) are found analogously as in the previous
case.
These observations now suggest how to define the extension of LTL announced above: we introduce a new operatorμwith the informal meaning thatμuAdenotes the least fixpoint of the equivalenceu↔A. (A second operator ν for the greatest fixpoint can be derived fromμ.) In order to ensure the existence of the fixpoints, we restrict the application ofμtoAby requiring that all occurrences ofuinAmust be of positive polarity.
The propositional variableubecomes bound by the new (fixpoint) operator, just as quantifiers bind variables of first-order logic: an occurrence of a propositional variableu in a formula A is called bound if it appears in some subformula μuB ofA; otherwise it is called free. A formula is closed if it does not contain any free
propositional variables. The formulaAu(B)results fromAby substituting the for- mulaB for all free occurrences of the propositional variableu. When carrying out this substitution, we tacitly assume that no free occurrences of propositional vari- ables inB become bound by this substitution. (As in first-order logic, this can be achieved by renaming the bound propositional variables ofAif necessary.)
We denote this extension of LTL by LTL+μ. Its languageLμLTL is formally ob- tained fromLLTL by adding a denumerable setV of propositional variables to the alphabet, extending the syntax rules ofLLTLby the two clauses
• Every propositional variable ofVis a formula,
• IfAis a formula andu ∈ V is a propositional variable all of whose free occur- rences inAare of positive polarity thenμuAis a formula,
and extending the polarity definition by fixing that the polarity of every free occur- rence of a propositional variable inμuAis the same as the polarity of the occurrence inA.
Theνoperator is introduced as the abbreviation νuA ≡ ¬μu¬Au(¬u);
we will see below thatνuAdenotes the greatest fixpoint of the equivalenceu ↔A.
The substitution of¬ufor the free occurrences ofu ensures that all occurrences of u are of positive polarity in the formula to which the fixpoint operator is applied.
Clearly, the polarities of all free occurrences of propositional variables inνuAare as inA.
The semantics of LTL+μ has to take into account the valuation of proposi- tional variables. As indicated already, the earlierKi(F)therefore takes now the form K(Ξ)i (F)whereΞ = (ξ0, ξ1, ξ2, . . .)is a sequence of valuationsξi :V → {ff,tt}of the propositional variables. The clauses of the inductive definition forK(Ξ)i (F)are as forKi(F)before, extended by
• K(Ξ)i (u) =ξi(u) foru∈ V,
• K(Ξ)i (μuA) =tt ⇔ i ∈μΥA
and the definition of validity inKis adapted accordingly:KF ifK(Ξ)i (F) = ttfor everyiandΞ. Expanding the representation ofμΥAgiven in Theorem 3.2.1 and the definition ofΥA, the semantic clause forμuAcan be restated more explicitly as
• K(Ξ)i (μuA) =tt ⇔ i ∈Mfor allM⊆Nsuch thatAΞ[u:K M]⊆M.
In order to be sure that this definition really corresponds to our intention of defin- ing the least fixpoint of the mappingΥA, even for nested fixpoints, we have to extend the proof of Lemma 3.2.2 for the case whereF ≡ μ¯uA. Ifu ≡ u¯ thenu has no free occurrence inF, and thereforeΥF is both monotone and anti-monotone. So let u≡u, assume for part a) that every free occurrence of¯ uinF, hence inA, is of posi- tive polarity, and letM1,M2,M⊆NwhereM1⊆M2. The assertions of the lemma
are to be understood for arbitraryKandΞ. So the induction hypothesis, applied for the valuationΞ[¯u:M], implies that
AΞ[¯Ku:M][u:M1]⊆AΞ[¯Ku:M][u:M2].
We have to show thatΥF(M1)⊆ΥF(M2)where ΥF(Mi) =
{M⊆N|AΞ[u:MK i][¯u:M] ⊆M}.
Assume thati ∈/ ΥF(M2)for some i ∈ N; then there exists someM ⊆ Nsuch thatAΞ[u:MK 2][¯u:M] ⊆Mandi ∈/ M. Becauseu and¯uare different propositional variables,Ξ[u:M2][¯u:M] = Ξ[¯u:M][u:M2], and the induction hypothesis yields AΞ[¯Ku:M][u:M1] ⊆AΞ[¯Ku:M][u:M2] ⊆M, and therefore we findi ∈/ ΥF(M1). Be- causeiwas chosen arbitrarily, this provesΥF(M1)⊆ΥF(M2), completing the proof of part a) of the lemma. The arguments for part b) are similar.
From this completed proof of Lemma 3.2.2 and Theorem 3.2.1 we have shown thatμuAΞK defines the least fixpoint of the mappingΥAas intended.
For the derivedνoperator, the semantics is given by
• K(Ξ)i (νuA) =tt ⇔ i∈νΥA or, again somewhat more explicitly, by
• K(Ξ)i (νuA) =tt ⇔ M⊆AΞ[u:K M]for someM⊆Nsuch thati ∈M.
This can be seen by observing that withM =N\Mwe obviously have
¬Au(¬u)Ξ[u:M]K =N\AΞ[u:MK ] and therefore
¬Au(¬u)Ξ[u:K M]⊆M andi∈/M ⇔ M⊆AΞ[u:K M]andi∈M. So we obtain in fact
K(Ξ)i (νuA) =tt ⇔ K(Ξ)i (¬μu¬Au(¬u)) =tt
⇔ ¬Au(¬u)Ξ[u:M]K ⊆M andi∈/Mfor someM ⊆N
⇔ M ⊆AΞ[u:MK ]for someM ⊆Nsuch thati ∈M. Example. Assuming that u does not occur in A, let us verify that the formula 2A↔νu(A∧ eu)is valid. This claim can obviously be proved by showing that
2AΞK =νu(A∧ eu)ΞK
holds for any K and Ξ. For the direction “⊆” of this set equation, assume that i ∈ 2AΞK. Writing M for 2AΞK, we will proveM ⊆ A∧ euΞ[u:M]K in or- der to obtaini ∈ νu(A∧ eu)ΞK by the above semantic clause. Indeed, for any
j ∈M, we find thatj ∈AΞK and thatj + 1∈M. Becauseudoes not occur inA, we may conclude thatj ∈AΞ[u:K M]∩euΞ[u:K M], and thusj∈A∧ euΞ[u:K M].
For “⊇” we show thatM⊆2AΞK holds for anyMwithM⊆A∧ euΞ[u:M]K . Sinceνu(A∧ eu)ΞK is defined as the union of all such setsM, the assertion then follows. So assume thatM ⊆A∧ euΞ[u:M]K and thati ∈M. Clearly, we obtain thati∈AΞ[u:M]K ; hence alsoi∈AΞK, becauseudoes not occur inA. Moreover, we havei+ 1∈M. Continuing inductively, we find thatj ∈AΞK for allj ≥i, that
is,i∈2AΞK.
Similarly, we find that the other temporal operators of LTL+b can be expressed in LTL+μby noting the following equivalences, where the propositional variableu is again assumed not to occur inAorB. (Writing down these formulas we presuppose a suitable language which results from extending LTL by both “b” and “μ”.) (Tμ1) 2A ↔ νu(A∧ eu),
(Tμ2) 3A ↔ μu(A∨ eu),
(Tμ3) AuntilB ↔ μu(eB∨ e(A∧u)), (Tμ4) AunlessB ↔ νu(eB∨ e(A∧u)), (Tμ5) AuntB ↔ μu(B∨(A∧ eu)), (Tμ6) AunlB ↔ νu(B∨(A∧ eu)),
(Tμ7) AatnextB ↔ νu( e(B→A)∧ e(¬B→u)), (Tμ8) AbeforeB ↔ νu(e¬B∧ e(A∨u)).
The shape of these laws follows the fixpoint characterizations (T28), (T29), and (Tb11)–(Tb16). The difference between strong and weak binary operators is pre- cisely reflected by the choice of the least or greatest fixpoint.
We thus find that the logic LTL+μ provides uniform syntactic means for the definition of all the temporal operators that we have encountered so far, although formulas written in that language may quickly become difficult to read: compare the formulas2(A→3B)and
νu1((A→μu2(B∨ eu2))∧ eu1).
We will study in more depth the expressiveness of LTL+μin Chap. 4 where we show that LTL+μcan express many more temporal relations than the logics LTL or LTL+b.
The uniform definition of the languageLμLTLis mirrored by a simple and uniform axiomatization of LTL+μ. A sound and weakly complete formal system ΣLTLμ is obtained as an extension ofΣLTLby the following axiom and rule:
(μ-rec) Au(μuA)→μuA,
(μ-ind) Au(B)→B μuA→B if there is no free occurrence ofuinB.
The axiom (μ-rec) is “one direction” of the equivalence μuA ↔Au(μuA)which asserts thatμuAis a fixpoint. The rule (μ-ind) expresses thatμuAis smaller than any other fixpointB. For formulas involving greatest fixpoints, the following formula and rule can be derived:
(ν-rec) νuA→Au(νuA),
(ν-ind) B→Au(B) B→νuA if there is no free occurrence ofuinB.
As in first-order logic, the formulation of the deduction theorem requires some care. Still, we have
F ∪ {A} B ⇒ F 2A→B ifAis a closed formula.
We illustrate the use ofΣLTLμ (together with laws from LTL+b) by deriving (Tμ8):
Derivation of (Tμ8). LetF ≡νu(e¬B∧ e(A∨u))andunot be free inA,B.
(1) AbeforeB→ e¬B∧ e(A∨AbeforeB) (prop), (Tb16)
(2) AbeforeB→F (ν-ind),(1)
(3) F → e¬B∧ e(A∨F) (ν-rec)
(4) F → e¬B∧ e(F∨A) (prop),(3)
(5) F →AbeforeB (indbefore),(4)
(6) AbeforeB↔νu(e¬B∧ e(A∨u)) (prop),(2),(5) It is instructive to observe the special cases of the axioms and rules for the 2 operator: by law (Tμ1),2Ais justνu(A∧ eu), and therefore (ν-rec) and (ν-ind) can be rewritten as
2A→A∧ e2A and B →A∧ eB B→2A,
the first of which is (ltl3) whereas the second one is just a reformulation of the in- duction rule (ind) ofΣLTL. We could therefore drop (ltl3) and (ind) from the system ΣLTLμ if2were understood as a derived operator inLμLTL.
Similarly, the special cases for the binary operators show that the systematic pattern of the alternative axiomatization of LTL+b indicated at the end of the previous section is just the pattern described here. For example, (Tμ4) shows thatAunlessB is ν(eB∨ e(A∧u)); so (ν-rec) becomes the axiom
(unless1’) AunlessB→ eB∨ e(A∧AunlessB) of Sect. 3.1, and (ν-ind) becomes the rule
C → eB∨ e(A∧C) C →AunlessB
which is a reformulation of (indunless). So this latter rule determines AunlessBto be a greatest fixpoint.
Second Reading
In a Second Reading paragraph in Sect. 2.3 we mentioned some relationships between tem- poral and modal logic. The idea of introducing fixpoint operators may also be applied to
“normal” modal logic; the result is known as modalμ-calculus MμC.
This logic contains a (unary) modal operator2together with the fixpoint operatorμ.
Formulas are built analogously as in LTL+μ, including the constraint concerning polarity.
The operators3andνare introduced as before. As indicated in Sect. 2.3, a Kripke structure K = ({ηι}ι∈K,)for an underlying setVof propositional constants consists of a non- empty setK, valuationsηι:V→ {ff,tt}for allι∈K, and a binary accessibility relation . Using an analogous notation as in the above main text with a valuationΞ = (ξι)ι∈K
(whereξι:V→ {ff,tt}forι∈K), the semantics of the operator2is given by Kι(2A) =tt ⇔ Kκ(A) =tt for everyκwithικ
which provides
Kι(3A) =tt ⇔ Kκ(A) =tt for someκwithικ
for the dual operator3. For the semantics ofμone defines, for any formulaF, the mapping ΥF: 2K →2K,
ΥF:E→FΞ[u:K E]
whereFΞK ={ι∈K |K(Ξ)ι (F) =tt}andΞ[u:E]denotes the valuation(ξι)ι∈K with ξι(u) =tt⇔ι∈Eandξι(u) =ξι(u)for all variablesuother thanu. Then
K(Ξ)ι (μuA) =tt ⇔ ι∈μΥA
and
K(Ξ)ι (νuA) =tt ⇔ ι∈νΥA
whereμΥAandνΥAare the least and greatest fixpoints ofΥA, respectively (which can be shown to exist as in the case of LTL+μ).
From these definitions, the fixpoint characterization (Tμ1) for the temporal always oper- ator, and recalling the discussion in the above-mentioned Second Reading paragraph, it is evident that LTL+μcan be viewed as a special instant of MμC based on the operator d(with distinguished Kripke structures). However, there is also another more general relationship between MμC and temporal logics (including even others outside the “LTL family”) that can all be “embedded” into MμC. This makes MμC a simple common “framework” for all such logics. We will briefly come back to this aspect in Sect. 10.4.