Extensions of LTL
3.6 Combinations of Extensions
As in Sects. 2.6 and 3.4 we conclude with a remark on the Deduction Theorem.
According to the semantical considerations above this no longer holds in LTL+i in the general form of LTL. A possible modification could be formulated with the re- striction as in the semantical case. Another, more precise one is given as follows:
F ∪ {A} B ⇒ F 2A→B if the derivation ofBfromF ∪{A}
contains no application of the rule (init).
Its justification can be taken verbally from the proof of Theorem 2.3.3. The converse of the general form (i.e., without any restriction) still holds in LTL+i.
If past operators are combined with binary or fixpoint operators it would be rea- sonable to extend the “past aspect” to these operators. For example, enriching LTL+b by past operators (yielding a logic LTL+b+p) should not only introduce eand2as discussed in Sect. 3.4 but also the “past analogies” of the binary operators. Such binary past operators can be introduced, e.g., for the informal phrase
“Aheld in all preceding states since the last state in whichBheld”.
Again strict or non-strict, strong or weak interpretations are possible. We consider only strict versions and fix the following formal definitions.
• Ki(AsinceB) =tt ⇔ Kj(B) =tt for somej <iand Kk(A) =tt for everyk,j <k<i.
• Ki(AbacktoB) =tt ⇔ Kj(B) =tt for somej <iand Kk(A) =tt for everyk,j <k<i or
Kk(A) =tt for everyk <i.
The operators since and backto are obvious past analogies of until and unless. In the same way operators atlast (“Aheld in the last state in whichBheld”) and after (“Aheld afterBheld”) reflecting atnext and before can be defined as follows.
• Ki(AatlastB) =tt ⇔ Kj(B) =ff for everyj <i or
Kk(A) =tt for the greatestk <iwith Kk(B) =tt.
• Ki(AafterB) =tt ⇔ for everyj <iwith Kj(B) =tt
there is somek,j <k <i,with Kk(A) =tt.
The various relationships between the (future) binary operators and their connec- tions to e,2, and3can be systematically transferred to these new operators and e, e,2, and3−. We give only a few examples which should be compared with (Tb1) and (Tb5)–(Tb9).
• AsinceB↔ e3−B∧AbacktoB,
• AbacktoB↔Batlast(A→B),
• AatlastB ↔Bafter(¬A∧B),
• AafterB↔ ¬(A∨B)backto(A∧ ¬B),
• eA↔Aatlast true,
• 2A↔A∧Abackto false.
A sound and weakly complete axiomatization could be based on one of the op- erators. Depending on this choice, one of the axioms
(since) AsinceB↔ eB∨ e(A∧AsinceB), (backto) AbacktoB↔ eB∨ e(A∧AbacktoB), (atlast) AatlastB ↔ e(B→A)∧ e(¬B→AatlastB), (after) AafterB↔ e¬B∧ e(A∨AafterB)
being the analogies of (until1), (unless1), (atnext1), and (before1), respectively,
should be added to the other extending axioms. These are again fixpoint character- izations and it should be noted that the strong and weak versions since and backto of “since” now have different characterizations with respect to the involved previous operators.
Another remarkable fact is that none of the analogies of the respective axioms (until2), (unless2), (atnext2), and (before2) needs to be taken as a further axiom here. They can be derived in the formal systemΣLTLp augmented with one of the above axioms. Observe that for the binary future operators those additional axioms characterized strong and weak operator versions, i.e., as mentioned in Sect. 3.2, least or greatest fixpoints, of certain corresponding equivalences. So the derivability of their analogies also indicates that, in fact, the least and greatest fixpoints in the case of past operators coincide.
As an example we derive the formula e2¬B→AatlastB
which corresponds to (atnext2). The derivation also makes use of the laws (Tp1), (Tp7), and (Tp8) listed in Sect. 3.4:
(1) ¬(AatlastB)→
¬e(B→A)∨ ¬e(¬B→AatlastB) (prop),(atlast) (2) ¬(AatlastB)→
e(B∧ ¬A)∨ e(¬B∧ ¬(AatlastB)) (prop),(prev),(pltl2),(1) (3) ¬(AatlastB)→ ¬efalse (prop),(Tp1),(2) (4) ¬(AatlastB)∧ e2¬B→ ¬ efalse (prop),(3) (5) ¬(AatlastB)→
(eB∧ e¬A)∨(e¬B∧ e¬(AatlastB)) (prop),(Tp8),(2)
(6) 2¬B→ ¬B∧ e2¬B (pltl3)
(7) e(2¬B → ¬B∧ e2¬B) (prev),(6)
(8) e2¬B→ e(¬B∧ e2¬B) (prop),(pltl2),(7)
(9) e2¬B→ ¬eB∧ ee2¬B (prop),(Tp7),(8)
(10) ¬(AatlastB)∧ e2¬B→
e¬(AatlastB)∧ ee2¬B (prop),(5),(9) (11) e¬(AatlastB)→ e¬(AatlastB) (prop),(pltl1) (12) ¬(AatlastB)∧ e2¬B→
e(¬(AatlastB)∧ e2¬B) (prop),(Tp7),(10),(11) (13) ¬(AatlastB)∧ e2¬B→2¬efalse (indpast),(4),(12) (14) 3−efalse→(e2¬B→AatlastB) (prop),(13)
(15) e2¬B→AatlastB (mp),(pltl4),(14)
Finally we note that the extensions may also be combined with initial validity semantics. In this case the axiomatizations have to be adjusted in a similar way to the one in whichΣLTL0 results fromΣLTL.
Second Reading
As sketched out in the Second Reading paragraph in Sect. 2.3, temporal logic is a special branch of modal logic. Its intention is to formalize reasoning about statements “in the flow of time” and it is particularly designed for applications in computer science.
Capturing aspects of time is quite generally of interest in logics, and another field of pos- sible applications is encountered by the relationship between logic and (natural) languages.
In fact, there is also a “modal approach” to this topic, called tense logic, which is very close to temporal logic as described here.
“Basic” tense logic is an extension of classical propositional logic by unary tense opera- tors for building formulas of the form
2A (“It will always be the case thatA”), 2A (“It has always been the case thatA”)
(we use, because of the close relationship, the operator symbols of temporal logic) and 3A (“It will be the case thatA”),
3−A (“It has been the case thatA”)
with the duality relationship that3Acan be identified as¬2¬Aand3−Aas¬2¬A.
An extended version adds the binary tense operators until and since, i.e., formulas of the kind
AuntilB (“It will be the case thatB, andAup to then”), AsinceB (“It has been the case thatB, andAsince then”) to the basic equipment.
As to the language, this tense logic is “LTL+b+p without nexttime and previous opera- tors”. The semantics, however, is a “most general one” in the lines of modal logic (and many investigations then again address the questions of whether and how particular restrictions can be characterized by formulas of the logic). Adopting the notion of a Kripke structure K= ({ηι}ι∈K,)as defined for modal logic in the Second Reading paragraph of Sect. 2.3 (and informally understanding nowKas a set of “time points” andas the relation “earlier than”) the semantics of the tense operators is given by the clauses
Kι(2A) =tt ⇔ Kκ(A) =tt for everyκwithικ, Kι(2A) =tt ⇔ Kκ(A) =tt for everyκwithκι,
Kι(AuntilB) =tt ⇔ Kκ(B) =tt for someκwithικand Kk(A) =tt for everykwithιkandkκ and analogously for the other operators.
The definition for2Ais just as in modal logic and taken together, the clauses show that the semantical difference to temporal logic (as introduced so far) is given by the fact that in the latterKis fixed to be the setNof natural numbers andis<(which also means that 2and2are here defined in a “non-reflexive” version).
Bibliographical Notes
The until operator was originally investigated in tense logic (cf. the above Second Reading paragraph of Sect. 3.6) by Kamp [70] and introduced into the context of program analysis in [53]. The before operator can be found in [96], the atnext oper- ator was introduced in [82].
The logic LTL+μwas introduced in [11], based on the modalμ-calculus, which was studied in depth by Pratt [124] and Kozen [75]. Walukiewicz [157] proved the completeness of Kozen’s axiomatization of theμ-calculus, on which that of LTL+μ is based.
Quantification over propositional constants was investigated, e.g., in [138]. The first axiomatization of quantified propositional temporal logic (with future and past operators) is due to Kesten and Pnueli [73]; our presentation follows French and Reynolds [51]. The since operator was already used in [70]. Past operators in the form presented here were reintroduced in [92]. Our account of syntactic anchoring in Sect. 3.5 is related to [145].