MATL: Semantics

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MATL Semantics
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MATL Semantics
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Each view a is associated with a set of local
models (e.g. CTL structures) of the corresponding
language La and a (local) satisfiability relation.
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MATL Semantics
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MATL Semantics
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MATL Semantics
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A chain c links local models which assign the
same truth value to formulae with the same
intended meaning
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Compatibility Chains

• Chains are finite sequences of local models of
  the form
• c ltce ,cBi ,cBiBj ,,ca gt
• where
• each element ca is a local model of La
• a bg (i.e. b is a prefix of a)

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Compatibility Chains

• Chains are finite sequences of local models of
  the form
• c ltce ,cBi ,cBiBj ,,ca gt
• where
• each element ca is a local model of La
• a bg (i.e. b is a prefix of a)
• Chains can go through different modalities
  express how different nested modalities affect
  each other.

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Compatibility Chains
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Compatibility Chains
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A Compatibility Relation C is a set of chains
such that ca ? Bf iff ?c?? C,
c?aca implies c?a ? f
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Chains and Satisfiability

• Given a Compatibility Relation C and a formula
  f?La , C ? a f (read f is true in C) is defined
  as follows
• C ? a f iff ?cltce ,cBi ,cBiBj ,,ca ,,cab
  gt?C, ca ? f

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MATL Semantics
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MATL Logical Consequence

• Definition A set of MATL formulae G logically
  entails a f
• G ? a f
• if for every Compatibility Relation C and every
  chain c?C
• if for every prefix b of a (i.e. a bg for some
  g)
• cb ? Gb
• then
• ca ? f

where Gb f b f belongs to G
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MATL Structure

• We use CTL structures on the languages of the
  corresponding views as local models of the views

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MATL Structure

• We use CTL structures on the languages of the
  corresponding views as local models of the views
• Satisfiability in CTL is defined with respect to
  a CTL structure and a state.
• Therefor we take as local models pairs of the
  form
• lt f , s gt
• where
• f lt S,J,R,Lgt is a CTL structure
• s is a state of f (i.e. s belongs to S)

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MATL Structure

• We use pairs ltCTL structure,stategt as local
  models of each views
• A MATL structure is a Compatibility Relation C
  such that
• 1 for any chain c ? C, ca lt f , s gt
• - where f lt S,J,R,Lgt is a CTL structure and
  
• - s is a state in S

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MATL Structure

• We use pairs ltCTL structure,stategt as local
  models of each views
• A MATL structure is a Compatibility Relation C
  such that
• 1 for any chain c ? C, ca lt f , s gt
• - where f lt S,J,R,Lgt is a CTL structure and
  
• - s is a state in S
• 2 for any state s? of S , there is a c?? C with
  ca lt f , s? gt

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MATL vs Modal Logic

• Under appropriate restrictions, MATL is
  equivalent to Modal Logic K (n).

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MATL vs Modal Logic

• Under appropriate restrictions, MATL is
  equivalent to Modal Logic K(n).
• Restrictions
• Assume La Lb for all views a,b?B
• Assume each a is associated with the set of all
  the propositional models of La

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MATL vs Modal Logic

• Theorem For any formulae f,y ? La and view a?B
• ? a BX(f ? y) ? (BXf ? BXy)

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MATL vs Modal Logic

• Theorem For any formulae f,y ? La and view a?B
• ? a BX(f ? y) ? (BXf ? BXy)
• Theorem For any view a?B and set of formulae
  G,f?La
• a G ? a f implies a BXG ? a BXf
• (BXG BXy y is a formula in G)

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MATL vs Modal Logic

• Theorem For any formulae f,y ? La and view a?B
• ? a BX(f ? y) ? (BXf ? BXy)
• Theorem For any view a?B and set of formulae
  G,f?La
• a G ? a f implies a BXG ? a BXf
• (BXG BXy y is a formula in G)
• Theorem For any view a?B and set of formulae
  G,f?Le
• e G ? e f iff a G ? a f

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MATL vs Modal Logic

• Theorem For any view a?B and formula f ? Le
• ?K f iff ? a f
• (where ?K denotes satisfiability in Modal K)