Appendix B: Limitedness of Reset Vector Addition Systems

D
N
n 5 A = 〈S, Σ, ∆, s0, F 〉
S
Σ
∆ ⊆ S × Σ × {0, +1, −1, r} n × S
s0 ∈ S
F ⊆ S

0 +1
−1 r
t, t1, . . . {0, +1, −1, r} n
(s1, a1, t1, s2)(s2, a2, t2, s3), . . . ,
(sm, am, tm, sm+1) ∀1 ≤ i ≤ m.(si, ai, ti, si+1) ∈ ∆
s0
a, (1, 0)
a, (0, 1)
b, (0, 1)
s1
s2
b, (r, r)
a, (0, r)
⊕
k ∈ N k ⊕ 0 = k k ⊕ +1 = k + 1 k ⊕ −1 = k − 1
k ⊕ r = 0 n
A = 〈S, Σ, ∆, s0, F 〉
A = 〈 ˆ S, Σ, T, ˆs0〉
ˆ S 〈s, (c1, . . . , cn)〉 s ∈ S, ci ∈ N 1 ≤ i ≤ n
ˆs0 = 〈s0, (0, . . . , 0)〉
(〈s, (c1, . . . , cn)〉, a, 〈s, (c ′ 1, . . . , c ′ n)〉) ∈ T 〈s, a, t, s ′ 〉 ∈ ∆
(c ′ 1, . . . , c ′ n) = (c1, . . . , cn) ⊕ t
−→ 〈s, (c ′ 1, . . . , c ′ n)〉 (〈s, (c1, . . . , cn)〉, a, 〈s, (c ′ 1, . . . , c ′ n)〉) ∈
〈s, (c1, . . . , cn)〉 a
T 〈s, (c1, . . . , cn)〉 w
−→ 〈s, (c ′ 1, . . . , c ′ n)〉
w ∈ Σ +
D
D ∈ N ˆ SD
D ˆ SD = {〈s, (c1, . . . , cn)〉|
〈s, (c1, . . . , cn)〉 ∈ ˆ S (c1, . . . , cn) ≤ (D, . . . , D)} ≤
A D A AD A

a
s0 , (0, 0) s1 , (1, 0)
b
s 1 , (1, 1)
a b b b b
s 0 , (0, 1) s 0 , (0, 0)
a
a
b
s 1 , (1, 2)
b
s 1 , (1, 3)
ˆ a
SD 〈s, (c1, . . . , cn)〉 −→D 〈s, (c ′ 1, . . . , c ′ n)〉
AD 〈s, (c1, . . . , cn)〉 w
−→D
〈s, (c ′ 1, . . . , c ′ n)〉 w ∈ Σ +
2
a
s0 , (0, 0) s1 , (1, 0)
b
s 1 , (1, 1)
a b b b
s 0 , (0, 1) s 0 , (0, 0)
a
a
b
s 1 , (1, 2)
2
−→ −→D
s, s ′ ∈ S w ∈ Σ + s w
−→ s ′
(s, a1, t1, s1) (s1, a2, t2, s2) . . . (s |w|−1, a |w|, t |w|, s ′ )
w = a1 · a2 · · · a |w| D ∈ N s w
−→D s ′ 1 ≤ i ≤ |w|
t1 ⊕ t2 ⊕ · · · ⊕ ti ≤ (D, . . . , D)
D
ρ A l(ρ)
ρ = 〈s0, (0, . . . , 0)〉 −→ ∗ 〈s, (c1, . . . , cn)〉
s ∈ F
A L(A) = {l(ρ)|ρ A} D ∈ N D
A LD(A) = {l(ρ)|ρ AD}
ab∗a∗
2 a(ɛ + b + bb + bbb)a∗
A D
D LD(A) = ∅ LD(A) = Σ∗ LD(A) = L(A)
A L2(A) = Σ∗

, 0 b, 0
s0
a, r
a, 1
s1
s2
b, 0
a, 1
ω
ab ω + ab ∗ a ω 2ω
a(ɛ + b + bb + bbb)a ω
{0, +1, r} n
{0, +1, −1} n ci = D
1 ≤ i ≤ n ci D
ci = D ci
D
Y N
? ?Y
D ∃D
LD(A) = ∅ LD(A) = Σ ∗ LD(A) = L(A)
?Y ?Y

ω
L ω D(A) = ∅ L ω D(A) = Σ ∗ L ω D(A) = L ω (A)
? ?
ω

ω
L ω D(A) = ∅ L ω D(A) = Σ ∗ L ω D(A) = L ω (A)
? ?
ω