In this page, we try to understand the semantics of complex CTL* formulas.
It is easy to understand the semantics if you break the formulas down into subformulas.

graph TB START --> s0 s0 --> s1 s0 --> s2 s1 --> s3 --> dots0 s3 --> dots1 s1 --> s4 --> dots2 s2 --> s5 --> dots3 style START fill:#FFFFFF, stroke:#FFFFFF style dots0 fill:#FFFFFF, stroke:#FFFFFF style dots1 fill:#FFFFFF, stroke:#FFFFFF style dots2 fill:#FFFFFF, stroke:#FFFFFF style dots3 fill:#FFFFFF, stroke:#FFFFFF START(( )) dots0((π0)) dots1((π1)) dots2((π2)) dots3((π3)) s0(("s0
{a,b}")) s1(("s1
{b,c}")) s2(("s2
{c}")) s3(("s3
{a,b}")) s4(("s4
{c}")) s5(("s5
{c}"))

\( M, s_0 \models \bm{EF} (b \land c) \)

Step.1: \( \bm{F} (b \land c) \)

\( \bm{F} (b \land c) \supseteq \{ \pi_0, \pi_1, \pi_2, \pi_0^1, \pi_1^1, \pi_2^1 ... \} \)
( This means \( \pi_0, \pi_1 \models \bm{F} (b \land c) \))

Step.2: \( \bm{E} \{ \pi_0, \pi_1 \} \)

\( \bm{E} \{ \pi_0, \pi_1, \pi_2 \} \supseteq \{ s_0 \} \)
( This means \( s_0 \models \bm{E} \{ \pi_0, \pi_1, \pi_2 \} \subseteq \bm{EF} (b \land c) \))

\( M, s_0 \models \bm{A} (b \bm{U} c) \)

Step.1: \( b \bm{U} c \)

\( b \bm{U} c \supseteq \{ \pi_0, \pi_1, \pi_2, \pi_3, \pi_0^1, \pi_1^1, \pi_2^1, \pi_3^1, ... \} \)

Step.2: \( \bm{A} \{ \pi_0, \pi_1, \pi_2, \pi_3 \} \)

\( \bm{A} \{ \pi_0, \pi_1, \pi_2, \pi_3 \} \supseteq \{ s_0 \} \)

\( M, s_0 \models \bm{EXEF} (b \land c) \)

Step.1: \( \bm{EF} (b \land c) \)

\( \bm{EF} (b \land c) \supseteq \{ s_0, s_1 \} \)

Step.2: \( \bm{EX} \{ s_1 \} \)

\( \bm {EX} \{ s_1 \} \supseteq \{ s_0 \} \)

\( M, s_0 \models \bm{AXA} (b \bm{U} c) \)

Step.1: \( \bm{A} (b \bm{U} c) \)

\( \bm{A} (b \bm{U} c) \supseteq \{ s_0, s_1, s_2 \} \)

Step.2: \( \bm{AX} \{ s_1, s_2 \} \)

\( \bm{AX} \{ s_1, s_2 \} \supseteq \{ s_0 \} \)

Question: \( s_0 \, ? \, \bm{EXAX} (a \land b) \)

Step.1: \( \bm{X} ( a \land b ) \) \( \bm{X} (a \land b) \supseteq \{ \pi_0^1, \pi_1^1 \} \)
Step.2: \( \bm{AX} (a \land b) \) \( \bm{A} \{ \pi_0^1, \pi_1^1 \} \supseteq \{ \} \)
Step.3: \( \bm{XAX} (a \land b) \) \( \bm{X} \{ \} \supseteq \{ \} \)
Step.4: \( \bm{EXAX} (a \land b) \) \( \bm{E} \{ \} \supseteq \{ \} \)
So, \( s_0 \not\models \bm{EXAX} (a \land b) \)