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6 Interval Temporal Algebra

Interval Temporal Algebra (23)

    ω
(K,  ) is an interval temporal algebra iff Let       ----------
skip ^= Σ ∪ Σ ⋅ Σ ,           ----
finite=^ T ⋅∅ and       --------
!a  ^= finite ⋅a

P1
(K, ω) is a Boolean left (lazy) omega algebra
P2
a ⋅(b∪ c) ⊆ a ⋅b∪ a ⋅c
P3
(skip ∩ a) ⋅c∩ (skip ∩ b)⋅d = (skip ∩ a∩ b) ⋅(c∩ d)
P4
c ⋅(skip ∩ a)∩ d ⋅(skip ∩b) = (c∩ d) ⋅(skip ∩ a∩ b)
P5
(Σ ∩ a) ⋅c∩ (Σ ∩ b)⋅ d= (Σ ∩ a ∩ b)⋅(c ∩ d)
P6
c ⋅(Σ ∩ a)∩ d ⋅(Σ ∩ b)= (c ∩ d)⋅(Σ ∩ a ∩ b)
P7
finite ⊆ skip*
P8
-------
skip⋅a = Σ ∪ skip⋅a
P9
-------
a ⋅skip = Σ ∪ a ⋅skip

Axiom/proof system for PITL (24)

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

Examples of proof in ITA (25)

Example 1.

    finite ⋅Σ
    — a ⋅Σ = a
=   finite

Example 2.

    (T⋅ ∅)⋅∅
    — a ⋅(b⋅c) = (a⋅ b)⋅c
=   T ⋅(∅⋅∅)
    — ∅ ⋅a = ∅
=   T ⋅∅

ITA vs PITL (26)

Theorem 3. PITL is an interval temporal algebra

Theorem 4. PITL’s axiom/proof system can be derived from ITA’s axiom system







December 5, 2008
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