Basic Propositional Linear Temporal Logic
2.2 Temporal Logical Laws
In any logic, the valid formulas and consequence relationships express “logical laws”. An example from classical logic is the tautology
(A→B)∧(B →C)→(A→C)
mentioned in Sect. 1.1. According to the semantical definitions in the previous sec- tion we should expect that such tautologies remain valid in temporal logic where we may substitute formulas ofLLTLforAandB, e.g.,
(eC →2D)∧(2D →3E)→(eC →3E).
Let us confirm this expectation formally:
Definition. A formula ofLLTLis called tautologically valid if it results from a tautol- ogyAofLPLby consistently replacing the propositional constants ofAby formulas ofLLTL.
Theorem 2.2.1. Every tautologically valid formula is valid.
Proof. LetV={v1, . . . ,vn}be a set of propositional constants, and letA1, . . . ,An
be formulas ofLLTL. For any formulaAofLPL(V), letA∗ denote the formula of LLTLwhich results fromAby replacing every occurrence of a propositional constant vj ∈V inAbyAj. LetKbe a temporal structure (for the propositional constants ofLLTL) andi ∈N. We define a (classical) valuationBforV byB(vj) =Ki(Aj) forj = 1, . . . ,nand claim that
B(A) =Ki(A∗),
which proves the theorem. Indeed, ifBis tautologically valid, and thereforeB≡A∗ for some classical tautologyA, thenKi(B) =B(A) =tt. The proof of the claim runs by structural induction onA.
1. A≡vj ∈V: ThenA∗≡Aj; henceB(A) =B(vj) =Ki(Aj) =Ki(A∗).
2. A≡false: ThenA∗≡false andB(A) =ff=Ki(A∗).
3. A≡B→C: ThenA∗≡B∗→C∗, and with the induction hypothesis we get B(A) =tt ⇔ B(B) =ff or B(C) =tt
⇔ Ki(B∗) =ff or Ki(C∗) =tt
⇔ Ki(B∗→C∗) =Ki(A∗) =tt.
Clearly, the transfer of classical logical laws to LTL can be extended to the re- lation. Suppose a formulaB is a consequence of some setFof formulas in PL.
Again, if we (consistently) substitute formulas of LTL in the formulas ofFandB, we should not destroy the logical relationship. For example,
eC →2D,2D →3E eC →3E
should hold because of the classical A→B,B→C A→C.
For a simple formulation of this fact (restricted to finite setsF) we remember that A1, . . . ,An B ⇔ A1→(A2→. . .→(An →B). . .)
in PL, and so we may define:
Definition. LetA1, . . . ,An,B (wheren ≥ 1) be formulas ofLLTL.B is called a tautological consequence ofA1, . . . ,An ifA1 →(A2 →. . .→(An →B). . .)is tautologically valid.
Theorem 2.2.2. A1, . . . ,An B whenever B is a tautological consequence of A1, . . . ,An.
Proof. LetBbe a tautological consequence ofA1, . . . ,An. Then, by Theorem 2.2.1, A1→(A2 →. . .→(An →B). . .). Applying Theorem 2.1.7ntimes (starting
withF=∅) we getA1, . . . ,AnB.
Example. As a simple application of theorems 2.2.1 and 2.2.2 we may show that logical equivalence∼=of formulas ofLLTLis an equivalence relation (i.e., a reflexive, symmetrical, and transitive relation): since A ↔A is tautologically valid we have the reflexivity assertion A∼=A, i.e.,
A↔A
with Theorem 2.2.1. Second,A ↔ B is a tautological consequence of B ↔ A, so we have A↔B B↔A by Theorem 2.2.2, and this implies the symmetry A∼=B⇒B∼=A, i.e.,
A↔B ⇒ B↔A
with Theorem 2.1.8. An analogous argument establishes
A↔B and B↔C ⇒ A↔C
expressing the transitivity of ∼=.
With the Theorems 2.2.1 and 2.2.2 we know of logical laws in LTL coming from the “classical basis” of the new logic. Let us now turn to proper temporal logical laws concerning the temporal operators. We give quite an extensive list of valid formulas, proving the validity only for a few examples. Many of these laws describe logical equivalences.
Duality laws
(T1) ¬eA↔ e¬A, (T2) ¬2A↔3¬A,
(T3) ¬3A↔2¬A.
(T2) and (T3) express the duality of the operators2 and3. (T1) asserts that eis self-dual and was proved in the previous section.
Reflexivity laws (T4) 2A→A, (T5) A→3A.
These formulas express the fact that “henceforth” and “sometime” include the
“present”.
Laws about the “strength” of the operators (T6) 2A→ eA,
(T7) eA→3A, (T8) 2A→3A,
(T9) 32A→23A.
Proof of (T9). We have to show that, for arbitraryKandi ∈ N,Ki(32A) = tt impliesKi(23A) =tt:
Ki(32A) =tt ⇒ Kj(2A) =tt for somej ≥i
⇒ Kk(A) =tt for somej ≥iand everyk ≥j
⇒ Kk(A) =tt for somek ≥j with arbitraryj ≥i
⇒ Kj(3A) =tt for everyj ≥i
⇒ Ki(23A) =tt.
Idempotency laws (T10) 22A↔2A, (T11) 33A↔3A.
Proof of (T10). Here we have to show thatKi(22A) =Ki(2A)for arbitraryKand i∈N:
Ki(22A) =tt ⇔ Kj(2A) =tt for everyj ≥i
⇔ Kk(A) =tt for everyj ≥iand everyk≥j
⇔ Kk(A) =tt for everyk ≥i
⇔ Ki(2A) =tt.
Commutativity laws (T12) 2eA↔ e2A, (T13) 3eA↔ e3A.
These logical equivalences state the commutativity of ewith2and3.
Proof of (T12). For arbitraryKandi∈N:
Ki(2eA) =tt ⇔ Kj(eA) =tt for everyj ≥i
⇔ Kj+1(A) =tt for everyj ≥i
⇔ Kj(A) =tt for everyj ≥i+ 1
⇔ Ki+1(2A) =tt
⇔ Ki( e2A) =tt.
Distributivity laws
(T14) e(A→B) ↔ eA→ eB, (T15) e(A∧B) ↔ eA∧ eB, (T16) e(A∨B) ↔ eA∨ eB, (T17) e(A↔B) ↔ (eA↔ eB), (T18) 2(A∧B) ↔ 2A∧2B, (T19) 3(A∨B) ↔ 3A∨3B, (T20) 23(A∨B) ↔ 23A∨23B, (T21) 32(A∧B) ↔ 32A∧32B.
(T14)–(T17) express the distributivity of eover all (binary) classical operators. Ac- cording to (T18) and (T19),2is distributive over ∧and3is distributive over ∨.
Finally, (T20) and (T21) assert that “infinitely often” distributes over∨and “almost always” distributes over∧.
Proof of (T14). For arbitraryKandi∈N: Ki(e(A→B)) =tt ⇔ Ki+1(A→B) =tt
⇔ Ki+1(A) =ff or Ki+1(B) =tt
⇔ Ki(eA) =ff or Ki(eB) =tt
⇔ Ki(eA→ eB) =tt.
Weak distributivity laws
(T22) 2(A→B) → (2A→2B), (T23) 2A∨2B → 2(A∨B), (T24) (3A→3B) → 3(A→B), (T25) 3(A∧B) → 3A∧3B, (T26) 23(A∧B) → 23A∧23B, (T27) 32A∨32B → 32(A∨B).
These formulas state that at least “some direction” of further distributivities of2,3, 23, and32hold.
Proof of (T23). For arbitraryKandi∈N:
Ki(2A∨2B) =tt ⇒ Ki(2A) =tt or Ki(2B) =tt
⇒ Kj(A) =tt for everyj≥i or Kj(B) =tt for everyj≥i
⇒ Kj(A) =tt or Kj(B) =tt for everyj ≥i
⇒ Kj(A∨B) =tt for everyj ≥i
⇒ Ki(2(A∨B)) =tt.
Fixpoint characterizations of 2and3 (T28) 2A↔A∧ e2A,
(T29) 3A↔A∨ e3A.
(T28) is a recursive formulation of the informal characterization of2Aas an “infinite conjunction”:
2A ↔ A∧ eA∧ eeA∧ eeeA∧. . .,
and (T29) is analogous for3A. These formulas are therefore also called recursive characterizations. The relationship with “fixpoints” will be reconsidered in a more general context in Sect. 3.2.
Proof of (T28). For arbitraryKandi∈N:
Ki(A∧ e2A) =tt ⇔ Ki(A) =tt and Ki(e2A) =tt
⇔ Ki(A) =tt and Kj(A) =tt for everyj ≥i+ 1
⇔ Kj(A) =tt for everyj ≥i
⇔ Ki(2A) =tt.
Monotonicity laws
(T30) 2(A→B) → (eA→ eB), (T31) 2(A→B) → (3A→3B).
It may be observed that this list of laws is (deliberately) redundant. For example, (T30) can be established as a consequence of the laws (T6) and (T14). We now give a direct proof.
Proof of (T30). For arbitraryKandi∈N:
Ki(2(A→B)) =tt ⇒ Kj(A→B) =tt for everyj ≥i
⇒ Kj(A) =ff or Kj(B) =tt for everyj ≥i
⇒ Ki+1(A) =ff or Ki+1(B) =tt
⇒ Ki(eA) =ff or Ki( eB) =tt
⇒ Ki(eA→ eB) =tt.
Frame laws
(T32) 2A → ( eB→ e(A∧B)),
(T33) 2A → (2B →2(A∧B)), (T34) 2A → (3B→3(A∧B)).
These formulas mean that ifAholds forever then it may be “added” (by conjunction) under each temporal operator.
Proof of (T32). For arbitraryKandi∈N:
Ki(2A) =tt ⇒ Ki+1(A) =tt
⇒ Ki+1(B) =ff or
Ki+1(B) =tt and Ki+1(A) =tt
⇒ Ki(eB) =ff or Ki( e(A∧B)) =tt
⇒ Ki(eB→ e(A∧B)) =tt.
Temporal generalization and particularization laws (T35) 2(2A→B)→(2A→2B),
(T36) 2(A→3B)→(3A→3B).
Proof of (T35). For arbitraryKandi ∈N, assume thatKi(2(2A→B)) =tt, i.e., Kj(2A → B) = ttfor everyj ≥ i. To proveKi(2A →2B) = tt, assume also thatKi(2A) = tt. This meansKk(A) =ttfor everyk ≥i; henceKk(A) = ttfor everyk ≥ j and everyj ≥ i and thereforeKj(2A) = ttfor every j ≥ i. With Lemma 2.1.1 we obtainKj(B) =ttfor everyj ≥i, which meansKi(2B) =ttand
proves the claim.
In Theorem 2.1.6 we stated the fundamental relationship between implication and consequence in LTL. In the presence of this theorem, laws of the form
2A→B
can also be written as a consequence relationship in the form AB,
for example:
(T22) A→B 2A→2B, (T30) A→B eA→ eB, (T32) A eB→ e(A∧B),
(T35) 2A→B 2A→2B,
(T36) A→3B 3A→3B.
This notation also explains why (T30) and (T31) are called monotonicity laws: they express a kind of monotonicity of eand3with respect to→(viewed as an order relation). The same property of2is noted as a weak distributivity law in (T22) but could also occur here.
The reformulations of (T35) and (T36) show their correspondence to the general- ization and particularization rules of classical first-order logic (cf. Sect. 1.2) accord- ing to the informal meaning of2and3as a kind of “for all” and “for some” relating these temporal operators to the classical quantifiers∀and∃, respectively. According to this relationship we call – following the notion of universal closure – a formula 2A the temporal closure ofA.
We end this section with some examples of how to use this collection of temporal logical laws. Firstly, assume that two formulas AandB are logically equivalent.
Then so are eAand eB, in symbols:
A∼=B ⇒ eA∼= eB.
The detailed arguments for this fact could be as follows: assume A ↔ B. Both A → B andB → Aare tautological consequences ofA ↔ B, and so we obtain both A→B and B →A by the Theorems 2.2.2 and 2.1.8. Applying (T30) and again Theorem 2.1.8, we conclude eA→ eBand eB→ eA. Finally, the formula eA ↔ eB is a tautological consequence of eA → eB and eB → eA, and another application of Theorems 2.2.2 and 2.1.8 yields eA↔ eB.
In analogous ways we could use (T22) and (T31) to show A∼=B ⇒ 2A∼=2B
and
A∼=B ⇒ 3A∼=3B
which altogether mean a kind of substitutivity of logically equivalent formulas under the temporal operators e,2, and3.
As another application of the logical laws we finally show a remarkable conse- quence and generalization of the idempotency laws (T10) and (T11). These seem to imply that, e.g., the formula
2223233A is logically equivalent to
2323A,
informally: the2-3-prefix2223233can be reduced to the shorter 2323. In fact, we will show that the formula is actually logically equivalent to
23A.
and, more generally, any2-3-prefix is reducible to one of the four cases2,3,32, or23. In preparation, we state two more laws of temporal logic.
Absorption laws
(T37) 323A↔23A,
(T38) 232A↔32A.
These laws assert that in a series of three alternating operators2and3, the first is
“absorbed” by the remaining two operators.
Proof. First,23A→323Ais just an instance of (T5). On the other hand, (T9) yields 323A→233A, and 323A→23Athen follows from (T11) and the substitutivity principle mentioned above. Taken together, we obtain (T37).
For the proof of (T38) we observe the following chain of logical equivalences:
232A ≡ 2¬2¬2A (by definition of3)
∼= ¬323¬A (by substitutivity from (T2) and (T3))
∼= ¬23¬A (by substitutivity from (T37))
∼= 32A (by substitutivity from (T2) and (T3)).
Theorem 2.2.3. LetA≡12. . .nB,n ≥1, be a formula ofLLTLwhere every i,1≤i≤n, is either2or3. Then
A∼=prefB
where pref is one of the four2-3-prefixes2,3,32, or23.
Proof. The theorem is proved by induction onn. The casen= 1is trivial since then A≡2BorA≡3B. Ifn >1then we have by induction hypothesis that
1. . .n−1nB ∼= prefnB
with pref being as described. If pref is2 or3then prefnB ∼= prefB, for some2-3-prefix pref of admissible form, can be established with the help of (T10) and (T11). Otherwise, we distinguish four different combinations of pref, which can be32or23, andn, which can be2or3. Any of these combinations can be reduced to an admissible prefix with the help of (T10), (T11), (T37), and (T38), and
the substitutivity principle.