Understanding MATL Semantics and Local Models in Modal Logic Frameworks

MATL: Semantics e Local Models BA BB BABA BABB BBBB BBBA . . . . . . . . . . . .

MATL: Semantics e BA BB BABA BABB BBBB BBBA Each viewa is associated with a set of local models (e.g. CTL structures) of the corresponding language La and a (local)satisfiability relation. . . . . . . . . . . . .

MATL: Semantics e ce BABBf BA BB cBA BBf BABA BABB BBBB BBBA f . . . . . . . . . . . .

MATL: Semantics e ce BABBf BA BB cBA BBf BABA BABB BBBB BBBA cBABB f . . . . . . . . . . . .

MATL: Semantics e ce BABBf BA BB cBA BBf BABA BABB BBBB BBBA cBABB f Achain clinks local models which assign the same truthvalue to formulae with the same intended meaning . . . . . . . . . . . .

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 e ce= ce BABBf BA cBA BBf BBf cBA

Compatibility Chains e ce= ce BABBf BA cBA BBf BBf cBA ACompatibility Relation Cis a set ofchainssuch that:ca Bfiffc C,ca=caimpliesca f

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: Semantics e Chains BA BB BABA BABB BBBB BBBA . . . . . . . . . . . .

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 • 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 statesofS , 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,bB* • 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 aB* a: BX(f  y)  (BXf  BXy)

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 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 Logic Theorem: For any view aB* and formula f  Le Kf iff a : f (where Kdenotes satisfiability in Modal K)

Understanding MATL Semantics and Local Models in Modal Logic Frameworks