![](bg2.png)
bounded versions of the standard operators of LTL.
Lower-case letters p, q, ... are used to denote proposi-
tional variables, and Greek lower-case letters α, β, ...
are used to denote formulas. We write A ≡ B to in-
dicate the syntactical identity between A and B. The
symbol ω is used to represent the set of natural num-
bers. Lower-case letters i, j and k are used to denote
any natural numbers. The symbol ≥ or ≤ is used to
represent a linear order on ω. Let l be a fixed positive
integer. Then, the symbol ω
l
is used to denote the set
{i ∈ ω | i ≤ l}. In the following discussion, l is fixed
as a certain positive integer.
Definition 2.1 (BLTL). Let S be a non-empty set of
states. A structure M := (σ, I) is a model if
1. σ is an infinite sequence s
0
, s
1
, s
2
, ... of states in S,
2. I is a mapping from the set Φ of propositional
variables to the power set of S.
A satisfaction relation (M, i) |= α for any formula
α, where M is a model (σ, I) and i (∈ ω) represents
some position within σ, is defined inductively by
1. for any p ∈ Φ, (M, i) |= p iff s
i
∈ I(p),
2. (M, i) |= α∧ β iff (M, i) |= α and (M, i) |= β,
3. (M, i) |= α∨ β iff (M, i) |= α or (M, i) |= β,
4. (M, i) |= α→β iff (M, i) |= α implies (M, i) |= β,
5. (M, i) |= ¬α iff not-[(M, i) |= α],
6. for any i ≤ l − 1, (M, i) |= Xα iff (M, i+ 1) |= α,
7. for any i ≥ l, (M, i) |= Xα iff (M, l) |= α,
8. (M, i) |= Gα iff ∀ j ≥ i with j ∈ ω
l
[(M, j) |= α],
9. (M, i) |= Fα iff ∃ j ≥ i with j ∈ ω
l
[(M, j) |= α],
10. for any m ∈ ω, (M, l + m) |= α iff (M, l) |= α.
A formula α is valid in BLTL if (M, 0) |= α for any
model M := (σ, I). A formula α is satisfiable in LTL
if (M, 0) |= α for some model M := (σ, I).
Since BLTL depents on l, it should precisely be
named e.g., BLTL[l], but, for the sake of simplicity,
we use the name BLTL in the following.
An expression α ↔ β means (α→β) ∧ (β→α).
Expressions
V
C and
W
C are used to represent the fi-
nite conjunction and disjunction of the formulas in C,
respectively.
Proposition 2.2. The following formulas are valid in
BLTL:
1. X(α◦ β) ↔ Xα◦ Xβ where ◦ ∈ {→, ∧, ∨},
2. X(¬α) ↔ ¬(Xα),
3. Gα→α,
4. Gα → Xα,
5. Gα → XGα,
6. Gα → GGα,
7. α∧ G(α→Xα)→Gα (temporal induction),
8. for any m ∈ ω, X
l+m
α ↔ X
l
α,
9. Gα ↔
^
{X
i
α | i ∈ ω
l
},
10. Fα ↔
_
{X
i
α | i ∈ ω
l
}.
Proof. We show some critical cases. Let M be an
arbitrary model and |= be an arbitrary satisfaction re-
lation on M.
(7): We show (M, 0) |= α ∧ G(α→Xα)→Gα.
Suppose (M, 0) |= α ∧ G(α→Xα), i.e., (a): (M, 0) |=
α and (b): (M, 0) |= G(α→Xα). We will show
(M, 0) |= Gα, i.e., ∀ j ∈ ω
l
[(M, i) |= α]. From (b),
we obtain:
(M, 0) |= G(α→Xα)
iff ∀ j ∈ ω
l
[(M, j) |= α→Xα]
iff ∀ j ∈ ω
l
[(M, j) |= α =⇒ (M, j) |= Xα]
iff (c): ∀j ∈ ω
l
[(M, j) |= α =⇒ (M, j + 1) |=
α (if j ≤ l − 1) or (M, l) |= α (if j ≥ l)].
We now show the required fact ∀i ∈ ω
l
[(M, i) |= α]
by mathematical induction on i. Base step: We have
(M, 0) |= α by (a). Induction step: Suppose (M, k) |=
α with k ≤ l − 1. Then, we obtain (M, k+ 1) |= α by
(c). Suppose (M, k) |= α with k ≥ l. Then, we obtain
(M, l) |= α by (c), and hence obtain (M, k + 1) |= α
where k+ 1 = l + m with m ∈ ω.
(8): We obtain: (M, 0) |= X
l+n
α iff (M, l) |= α iff
(M, 0) |= X
l
α.
(9): We obtain: (M, 0) |= Gα iff ∀ j ∈
ω
l
[(M, j) |= α] iff ∀ j ∈ ω
l
[(M, 0) |= X
j
α] iff
(M, 0) |=
^
{X
j
α | j ∈ ω
l
}.
Remark that 8, 9 and 10 in Proposition 2.2 are re-
garded as characteristic axioms concerning the time
bound l. Note that 9 and 10 in Proposition 2.2 be-
come the axioms of LTL if ω
l
is replaced by ω. Thus,
BLTL is quite natural as a bounded time formalism.
Formulas of classical logic (CL) are constructed
from (countably many) propositional variables, →, ¬,
V
(finite conjunction) and
W
(finite disjunction).
Definition 2.3 (CL). Let Θ be a finite (non-empty) set
of formulas. V is a mapping from the set Φ of propo-
sitional variables to the set {t, f} of truth values. V is
called a valuation. A satisfaction relation V |= α for
any formula α is defined inductively by
1. V |= p iff V(p) = t for any p ∈ Φ,
2. V |= ¬α iff not-(V |= α),
3. V |= α→β iff V |= α implies V |= β,
4. V |=
V
Θ iff V |= α for any α ∈ Θ,
5. V |=
W
Θ iff V |= α for some α ∈ Θ.
A formula α is valid (satisfiable) in CL if V |= α
for any (some) valuation V.
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