### III.3 Semantic Preliminaries

#### Interval and Length [Slide 93]

An interval σ is a ﬁnite sequence of states

σ : σ0σ1σ2

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 = σ0 |σ| = 0 σ = σ0σ1 |σ| = 1 σ = σ0σ1…σn |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 : σ0σ1 be an interval where

 σ0(Q) = tt σ1(Q) = ﬀ σ0(q) = ﬀ σ1(q) = ﬀ

Q is a state variable and q is a static variable.

#### Preﬁx, Suﬃx and Sub Interval [Slide 95]

Let σ = σ0σ1σ2 be an interval then

• σ0σk(where 0 k |σ|) denotes a preﬁx interval of σ
• σkσ|σ|(where 0 k |σ|) denotes a suﬃx interval of σ
• σkσl(where 0 k l |σ|) denotes a sub interval of σ

Example 28.

Let σ = σ0σ1σ2σ3 be an interval then

 σ0σ1 is a preﬁx interval of σ σ1σ2σ3 is a suﬃx interval of σ σ1σ2 is a sub interval of σ

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