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