STRL STRL
© STRL 1996-2008







3 Propositional Interval Temporal Logic (PITL)

PITL (5)

Propositional Interval Temporal Logic (PITL) is a

  • discrete,
  • linear temporal logic
  • for both finite and infinite time which includes
  • a basic construct for sequential composition and
  • an analog of Kleene star and Omega star

Syntax of PITL (6)

f ::=  p | ¬f | f ∨ f | skip | f ;f | f *
               1    2        1  2

where

  • skip  is an interval (sequence) of 2 states.
  • f1 ;f2  is called ‘f1  chop f2  ’ and denotes sequential composition of two intervals.
  • f* is called ‘f chopstar’ and denotes (in)finite iteration of an interval.

Derived formulae (7)

Cf      ^=   skip;f        next
cf      ^=   ¬ C ¬f        weaknext
more    ^=   C true         intervalw ith≥ 2 states
empty   ^=   ¬ more        onestateinterval
inf      ^=   true ;false     infinite interval
finite    ^=   ¬ inf          finiteinterval
fm ore    ^=   m ore ∧ finite  finitew ith≥ 2 states
 f      ^=   finite ;f       sometimes
!f      ^=   ¬   ¬f        alw ays
if      ^=   f ;true        someinitialsubinterval
If      ^=   ¬ (i¬f )      allinitialsubintervals
af      ^=   finite ;f ;true  somesubinterval
Af      ^=   ¬ (a¬f )      allsubintervals
etc.

Semantics of PITL (8)

The main semantic notion is interval which is a sequence of states.

Let

  • Σ denotes the set of states.
  •   +
Σ  denote the set of non-empty finite sequences of states.
  • Σ ω denote the set of infinite sequences of states.
  • σ denote an interval, σ ∈ Σ+  ∪ Σω .
  • Let M be the “meaning” (semantic) function from   +    ω
Σ   ∪ Σ to {tt,ff} .

Semantics of PITL (9)

  • M  σ[[p]] = ttiffσ0 (p ) = tt
  • M  σ[[¬f]] = ttiff not(M σ [[f ]] = tt)
  • M  σ[[f1 ∨ f2]] = ttiff M σ[[f1]] = ttorM σ[[f2]] = tt
  • M  σ[[skip]] = ttiff |σ | = 1 where |σ| denotes length of σ and is defined as number of states minus 1
  • M  σ[[f1 ;f2]] = ttiff  (exists k, s.t.M σ0...σk[[f1]] = ttandM  σk...σ|σ|[[f2]] = tt) or (σ isinfi niteandM  σ[[f1]] = tt)

Semantics of PITL (10)

  • M   [[f*]] = ttiff
   σ  ifσ isfinitethen  (existl ,...,l s.t.l = 0 andl =  |σ |and
      0     n    0         n  forall0 ≤ i < n,l ≤  l   andM          [[f ]] = tt)
                i   i+1       σli...σli+1 else  (existl0,...,ln s.t.l0 = 0and  M  σln...|σ|[[f]] = ttand  forall0 ≤ i < n,li ≤ li+1 andM σli...σli+1[[f]] = tt) or  (existaninfinitenum ber of li s.t.l0 = 0and  forall0 ≤ i,li ≤ li+1 andM  σli...σli+1[[f]] = tt)

Axiom/proof system for PITL (11)

PropAx        Allaxiom sforpropositionallogic
ChopAssoc     ⊢  (f0 ;f1) ;f2 ≡ f0 ;(f1 ;f2)
OrChopImp     ⊢  (f0 ∨ f1) ;f2 ⊃ (f0 ;f2) ∨ (f1 ;f2)
ChopOrImp     ⊢  f0 ;(f1 ∨ f2) ⊃ (f0 ;f1) ∨ (f0 ;f2)
Em ptyChop     ⊢  empty;f1  ≡  f1
ChopEm pty     ⊢  f1 ;em pty ≡ f1
BiBoxChop     ⊢  I(f0 ⊃  f1) ∧ !(f2 ⊃  f3) ⊃  (f0 ;f2) ⊃  (f1 ; f3)
StateIm pBi     ⊢  p ⊃  I p
NextIm pW Next ⊢  Cf0  ⊃  ¬C ¬f0
SkipA nd       ⊢  (skip ∧ f0);true  ⊃  ¬((skip ∧ ¬f0) ;true)
BoxInduct      ⊢  f0 ∧ ! (f0 ⊃ ¬ C¬f0 ) ⊃  ! f0
ChopStarEqv    ⊢  f*0 ≡  (em pty ∨ ((f0 ∧ m ore );f*0))
ChopStarInduct ⊢  (inf ∧ f0 ∧ !(f0 ⊃ (f1 ∧ fmore);f0)) ⊃  f*1
MP           ⊢  f0 ⊃  f1 and ⊢ f0 implies ⊢ f1
BoxG en        ⊢  f0 implies ⊢ ! f0
BiGen         ⊢  f0 implies ⊢ I f0







December 5, 2008
Home | Training | Research | People | About | News | ITL home