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4 Algebraic semantics for PITL

Can we give the semantic domain an algebraic structure?

Let [[f ]] denote the set of intervals for which [[ f ]] σ = tt  , i.e.,

[[f]] ^= {σ | [[ f]] σ = tt}

The ∨ of two PITL formula is then

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

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 |[ [ ¬f ]] σ = tt}
—  definitionof[[¬f]]σ
{σ |not([[ f ]] σ = tt)}
—--settheory,let--denotesetcomplement
{σ |[ [ f ]] σ = tt}
—--definitionof[[]]
[[f ]]

What about chop (‘; ’)?

Let ⋅ denote the fusion of two intervals           +    ω
σ1, σ2 ∈ Σ  ∪ Σ , i.e.,

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

         (|  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 }

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

What about empty  ?

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

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

What about a state formula, i.e., a formula without temporal operators?

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+ ∪ Σ ω
∅ ^= [[false]] = ∅

What about chopstar ‘* ’?

In the semantics of ‘* ’ both finite and infinite iteration are considered simultaneously. Let’s define separate algebraic operators for them.

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

f(X )    =^  Σ ∪ S ⋅X  |g(X )     ^=  S ⋅X
f0(X )   =^  X         |g0(X )    ^=  X
fi+1(X ) =^  f(f i(X )) |gi+1(X )  ^=  g(gi(X ))
S*       =^  ⋃ f i(∅)   |Sω        ^=  ⋂ gi(T)
               i       |               i

Then we have

[[f *]] = ...= [[f ]]* ∪ [[f ]]ω







August 27, 2009
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