III.3 Semantic Preliminaries
Interval and Length [Slide 93]
An interval σ is a ﬁnite sequence of states
σ : σ_{0}σ_{1}σ_{2}…
Let Σ^{+} denote the set of all ﬁnite intervals with at least 1 state. The
length of an interval σ is denoted by σ and is the number of states minus
1.
Example 26.
σ = σ_{0} 
σ = 0 
σ = σ_{0}σ_{1}  σ = 1 
σ = σ_{0}σ_{1}…σ_{n} 
σ = n 
Static and State Variables [Slide 94]
 Static variables don’t change their values within an interval.
 State variables can change their values within an interval.
Var = StateVar ⋃
StaticVar and 
StateVar ⋂
StaticVar = ∅. 
State variables are denoted by capital ﬁrst symbols and static variables
are denoted by small symbols.
Example 27.
Let σ : σ_{0}σ_{1} be an interval where
σ_{0}(Q) 
= 
tt 
σ_{1}(Q) 
= 
ﬀ 
σ_{0}(q)  =  ﬀ  σ_{1}(q)  =  ﬀ 

Q is a state variable and q is a static variable.
Preﬁx, Suﬃx and Sub Interval [Slide 95]
Let σ = σ_{0}σ_{1}σ_{2}… be an interval then
 σ_{0}…σ_{k}(where 0 ≤ k ≤σ) denotes a preﬁx interval of σ
 σ_{k}…σ_{σ}(where 0 ≤ k ≤σ) denotes a suﬃx interval of σ
 σ_{k}…σ_{l}(where 0 ≤ k ≤ l ≤σ) denotes a sub interval of σ
Example 28.
Let σ = σ_{0}σ_{1}σ_{2}σ_{3} be an interval then
σ_{0}σ_{1} 
is a preﬁx interval of σ 
σ_{1}σ_{2}σ_{3}  is a suﬃx interval of σ 
σ_{1}σ_{2} 
is a sub interval of σ 
