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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: 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

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 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 Chains

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ACompatibility Relation Cis a set ofchainssuch that:ca Bfiffc C,ca=caimpliesca f

Chains and Satisfiability

Given a Compatibility Relation C and a formula fLa, Ca :f (read f is true in C) is defined as follows:

Ca:fiffc=<ce,cBi,cBiBj ,…,ca,…,cab>C,ca f

MATL: Logical Consequence

Definition: A set of MATL formulae Glogically entailsa:f

G a : f

if for every Compatibility Relation C and every chain cC:

- 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

- We useCTL structureson thelanguagesof the correspondingviewsaslocal modelsof the views

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 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 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 statesofS , there isac Cwithca= < f , s >

MATL vs Modal Logic

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

MATL vs Modal Logic

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

Restrictions:

- Assume La=Lb for all views a,bB*
- Assume each ais associated with the set of all the propositional models of La

MATL vs Modal Logic

Theorem: For any formulae f,y Laand view aB*

a: BX(f y) (BXf BXy)

Theorem:For any view aB* and set of formulae G,fLa

a : G a : f impliesa : BXG a : BXf

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

MATL vs Modal Logic

Theorem: For any formulae f,y Laand view aB*

a: BX(f y) (BXf BXy)

Theorem: For any view aB* and set of formulae G,fLa

a : G a : f impliesa : BXG a : BXf

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

Theorem:For any view aB* and set of formulae G,fLe

e : G e : f iffa : G a : f

MATL vs Modal Logic

Theorem: For any view aB* and formula f Le

Kf iff a : f

(where Kdenotes satisfiability in Modal K)

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