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4 Algebraic semantics for PITL

Algebraic semantics for PITL (12)

Can we give the semantic domain an algebraic structure?  [5mm] Let [[f ]] denote the set of intervals for which M  σ[[f]] = tt  , i.e.,

[[f]] ^= {σ |M σ[[f]] = tt}

The ∨ of two PITL formula is then

[[f1 ∨ f2]] =
— definition of[[]]
{σ|M σ [[f1 ∨ f2]] = tt}
— definition ofM  σ[[f1 ∨ f2]]
{σ|M σ [[f1]] = ttorM σ [[f2 ]] = tt}
— settheory,let ∪ denoteunion
{σ|M σ [[f1]] = tt} ∪ {σ|M σ[[f2]] = tt}
— definition of[[]]
[[f1]]∪ [[f2]]

Algebraic semantics for PITL (13)

So we need algebraic operators that correspond to ¬,∨,skip,; and *

  • ∨ corresponds to union (∪ ) of sets of intervals
  • ¬ corresponds to complement, i.e.,
    [[¬f ]] =
—  definitionof[[]]
{σ |M  σ[[¬f ]] = tt}
—  definitionofM  σ[[¬f ]]
{σ |not(M  σ[[f]] = tt)}
—--settheory,let--denotesetcomplement
{σ |M  σ[[f]] = tt}
—--definitionof[[]]
[[f]]

Algebraic semantics for PITL (14)

What about chop (‘; ’)?  [2mm] Let ⋅ denote the fusion of two intervals σ1,σ2 ∈ Σ * ∪ Σ ω , i.e.,

Let a,b ∈ Σ (a and b are not the same), v,w ∈ Σ * and s,t ∈ Σ ω

         (
         ||  vaw   if   σ1 = va,  σ2 = aw
         ||||  ∅     if   σ1 = va,  σ2 = bw
         {  vas   if   σ1 = va,  σ2 = as
σ1 ⋅σ2 ^= ||  ∅     if   σ1 = va,  σ2 = bs
         ||||  s     if   σ1 = s,   σ2 = aw
         (  s     if   σ1 = s,   σ2 = t

Let S, T ⊆ Σ * ∪ Σ ω then

S ⋅T =^ {σ1 ⋅ σ2|σ1 ∈ S andσ2 ∈ T }

Algebraic semantics for PITL (15)

  • ; ’ corresponds to fusion ‘⋅ ’, i.e.,
    [[f1 ;f2]] =
— definition of[[]]
{σ|M σ [[f1 ;f2]] = tt}
— definition ofM  σ[[f1 ;f2]]
{σ|(existsk, s.t. M σ0...σk[[f1]] = ttandM  σk...σ|σ|[[f2]] = tt)
   or(σ isinfiniteandM σ[[f1]] = tt)}
— definition of⋅
{σ|M σ [[f1 ]] = tt} ⋅{σ |M σ[[f2]] = tt}
— definition of[[]]
[[f1]]⋅[[f2]]

Algebraic semantics for PITL (16)

What about em pty  ?

[[em pty]] =
—  definitionof[[]]
{σ |M  σ[[em pty]] = tt}
—  definitionofM  σ[[em pty]]
{σ |σ isa1 state interval}
—  definitionofΣ
Σ

Algebraic semantics for PITL (17)

What about skip  ?

skip  can be defined as ----------
Σ  ∪Σ  ⋅Σ

---------
Σ ∪ Σ ⋅Σ =
— De-M-organforsettheory
Σ-∩ Σ-⋅Σ-

--
Σ is the set of intervals containing ≥  2 states

-- --
Σ ⋅Σ is the set of intervals containing ≥ 3 states

-----
Σ-⋅Σ- is the set of intervals containing ≤ 2 states

Σ-∩ Σ-⋅Σ- is the set of intervals containing exactly 2 states

Algebraic semantics for PITL (18)

What about a state formula, i.e., a formula without temporal operators?  [5mm] A state formula only constrains the first state of an interval. Let p be a state formula. Then the following holds

[[p]] = ([[p ]]∩ Σ) ⋅T

where

T=^ [[true]] = Σ* ∪ Σω
∅=^ [[false]] = ∅

Algebraic semantics for PITL (19)

What about chopstar ‘* ’?  [2mm] In the semantics of ‘* ’ both finite and infinite iteration are considered simultaneously. Let’s define separate algebraic operators for them.

Let S * and S ω denote respectively finite and infinite iteration of a set S ⊆  Σ+ ∪ Σ ω and can be defined as follows

S0   ^=   Σ
Si   ^=   {⋃σ1 ⋅ σ2|σ1 ∈ Si- 1 andσ2 ∈ S }
S *  ^=   ⋃i∈N Si
S ω  ^=    iSi

Then we have

[[f*]] = ...= [[f ]]* ∪ [[f]]ω







December 5, 2008
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