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MATL: Semantics. e. Local Models. B A. B B. B A B A. B A B B. B B B B. B B B A. MATL: Semantics. e. B A. B B. B A B A. B A B B. B B B B. B B B A.

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Matl semantics
MATL: Semantics

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Local Models

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Matl semantics1
MATL: Semantics

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Each viewa 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 semantics2
MATL: Semantics

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Matl semantics3
MATL: Semantics

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Matl semantics4
MATL: Semantics

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Achain clinks local models which assign the same truthvalue to formulae with the same intended meaning

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Compatibility chains
Compatibility Chains

Chainsarefinite sequences of local modelsof the form:

c = <ce ,cBi,cBiBj ,…,ca >

where

  • eachelementca is a local model of La

  • a = bg (i.e. b is a prefixof a)


Compatibility chains1
Compatibility Chains

Chainsarefinite sequences of local modelsof the form:

c = <ce ,cBi,cBiBj ,…,ca >

where

  • eachelementca is a local model of La

  • a = bg (i.e. b is a prefixof a)

    Chains can go through different modalities: express how different nested modalities affect each other.


Compatibility chains2
Compatibility Chains

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ce= ce

BABBf

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Compatibility chains3
Compatibility Chains

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ce= ce

BABBf

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ACompatibility Relation Cis a set ofchainssuch that:ca Bfiffc C,ca=caimpliesca f


Chains and satisfiability
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:fiffc=<ce,cBi,cBiBj ,…,ca,…,cab>C,ca f


Matl semantics5
MATL: Semantics

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Chains

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Matl logical consequence
MATL: Logical Consequence

Definition: A set of MATL formulae Glogically entailsa: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

whereGb = {f | b:f belongs to G}


Matl structure
MATL Structure

  • We useCTL structureson thelanguagesof the correspondingviewsaslocal modelsof the views


Matl structure1
MATL Structure

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

  • Satisfiability in CTLis defined with respect to a CTL structure and a state.

    Therefor we take as local models pairs of the form

    < f , s >

    where

    • f = < S,J,R,L> is a CTL structure

    • s is a state of f (i.e. s belongs to S)


Matl structure2
MATL Structure

  • We use pairs <CTL structure,state> as local models of each views

  • AMATL structure is a Compatibility Relation C such that:

    1 for any chainc  C, ca= < f , s >

    - where f = < S,J,R,L> is a CTL structure and

    - s is a state inS


Matl structure3
MATL Structure

  • We use pairs <CTL structure,state> as local models of each views

  • AMATL structure is a Compatibility Relation C such that:

    1 for any chainc  C, ca= < f , s >

    - where f = < S,J,R,L>is a CTL structure and

    - s is a state inS

    2for any statesofS , there isac Cwithca= < f , s >


Matl vs modal logic
MATL vs Modal Logic

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


Matl vs modal logic1
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 ais associated with the set of all the propositional models of La


Matl vs modal logic2
MATL vs Modal Logic

Theorem: For any formulae f,y  Laand view aB*

a: BX(f  y)  (BXf  BXy)


Matl vs modal logic3
MATL vs Modal Logic

Theorem: For any formulae f,y  Laand 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 impliesa : BXG  a : BXf

(BXG = {BXy| yis a formula in G})


Matl vs modal logic4
MATL vs Modal Logic

Theorem: For any formulae f,y  Laand 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 impliesa : BXG  a : BXf

(BXG = {BXy| yis a formula in G})

Theorem:For any view aB* and set of formulae G,fLe

e : G  e : f iffa : G  a : f


Matl vs modal logic5
MATL vs Modal Logic

Theorem: For any view aB* and formula f  Le

Kf iff a : f

(where Kdenotes satisfiability in Modal K)


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