Finite State Machines

1
Finite State Machines

• A Finite State Machine on the set of atoms P is a
  tuple
• f ltS,J,R,Lgt
• where
• S is a finite set of states
• J ? S is the set of initial states
• R ? S is the Transition Relation
• L S ?2P is the Labelling Function

2
Finite State Machines

• Let f ltS,J,R,Lgt be a FSM and f a CTL formula.
  Then f,s ? f is defined as follows
• f,s ? p iff p ? L(s)
• f,s ? ?y iff f,s ? y
• f,s ? f?y iff f,s ? f and f,s ? y

3
Finite State Machines

• Let f ltS,J,R,Lgt be a FSM and f a CTL formula.
  Then f,s ? f is defined as follows
• f,s ? p iff p ? L(s)
• f,s ? ?y iff f,s ? y
• f,s ? f?y iff f,s ? f and f,s ? y
• f,s ? EX f iff f,s? ? f for some s? with s R
  s?
• f,s ? E(f U y) iff for some path
  x(ss1,s2,,sn) and some k?1
• f,sk ? y and for all 1? jlt k, f,sj ? f
• f,s ? A(f U y) iff for all path
  x(ss1,s2,,sn) and some k?1
• f,sk ? y and for all 1? jlt k, f,sj ? f

4
FSM Satisfiability
y
f
y
f
f
s
s
s
y
y
f
f
f
y
f,s ? E(f U y)
f,s ? A(f U y)
f,s ? EX f