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Logics for Data and Knowledge Representation

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									 Logics for Data and Knowledge
                 Representation
                                                      Modal Logic


   Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese
Outline
   Introduction
   Syntax
   Semantics
   Satisfiability and Validity
   Kinds of frames
   Correspondence with FOL




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Introduction
   We want to model situations like this one:
    1. “Fausto is always happy” circumstances”
    2. “Fausto is happy under certain


   In PL/ClassL we could have: HappyFausto


   In modal logic we have:
    1. □ HappyFausto
    2. ◊ HappyFausto


    As we will see, this is captured through the notion of “possible words” and
    of “accessibility relation”


3
Syntax
   We extend PL with two logical modal operators:
    □ (box) and ◊ (diamond)


    □P : “Box P” or “necessarily P” or “P is necessary true”
    ◊P : “Diamond P” or “possibly P” or “P is possible”


    Note that we define □P = ◊P, i.e. □ is a primitive symbol


   The grammar is extended as follows:
    <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ |
    <wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff>∨ <wff> |
               <wff>  <wff> | <wff>  <wff> | □ <wff> | ◊ <wff>

4
Different interpretations
Philosophy           □P : “P is necessary”
                     ◊P : “P is possible”
Epistemic            □aP : “Agent a believes P ” or “Agent a knows P”


Temporal logics      □P : “P is always true”
                     ◊P : “P is sometimes true”
Dynamic logics or    □aP : “P holds after the program a is executed”
logics of programs
Description logics   □HASCHILDMALE  ∀HASCHILD.MALE
                     ◊HASCHILDMALE  ∃HASCHILD.MALE




 5
Semantics: Kripke Model
   A Kripke Model is a triple M = <W, R, I> where:
      W is a non empty set of worlds
      R ⊆ W x W is a binary relation called the accessibility relation
      I is an interpretation function I: L  pow(W) such that to each
       proposition P we associate a set of possible worlds I(P) in which P holds


   Each w ∈ W is said to be a world, point, state, event, situation, class …
    according to the problem we model
   For "world" we mean a PL model. Focusing on this definition, we can see a
    Kripke Model as a set of different PL models related by an "evolutionary"
    relation R; in such a way we are able to represent formally - for example -
    the evolution of a model in time.
   In a Kripke model, <W, R> is called frame and is a relational structure.


6
Semantics: Kripke Model
   Consider the following situation:
                                                       BeingHappy

                                 BeingSad     1             2                3
                                                                     BeingNormal

                                                            4
                                                       BeingNormal
   M = <W, R, I>
    W = {1, 2, 3, 4}
    R = {<1, 2>, <1, 3>, <1, 4>, <3, 2>, <4, 2>}
    I(BeingHappy) = {2}      I(BeingSad) = {1}     I(BeingNormal) = {3, 4}




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Truth relation (true in a world)
   Given a Kripke Model M = <W, R, I>, a proposition P ∈ LML and a possible
    world w ∈ W, we say that “w satisfies P in M” or that “P is satisfied by w
    in M” or “P is true in M via w”, in symbols:
    M, w ⊨ P in the following cases:


    1. P atomic          w ∈ I(P)
    2. P = Q            M, w ⊭ Q
    3. P = Q  T M, w ⊨ Q and M, w ⊨ T
    4. P = Q  T M, w ⊨ Q or M, w ⊨ T
    5. P = Q  TM, w ⊭ Q or M, w ⊨ T
    6. P = □Q            for every w’∈W such that wRw’ then M, w’ ⊨ Q
    7. P = ◊Q            for some w’∈W such that wRw’ then M, w’ ⊨ Q
    NOTE: wRw’ can be read as “w’ is accessible from w via R”

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Semantics: Kripke Model
   Consider the following situation:
                                                       BeingHappy

                                 BeingSad     1             2                 3
                                                                      BeingNormal

                                                            4
                                                       BeingNormal
   M = <W, R, I>
    W = {1, 2, 3, 4}
    R = {<1, 2>, <1, 3>, <1, 4>, <3, 2>, <4, 2>}
    I(BeingHappy) = {2}      I(BeingSad) = {1}     I(BeingNeutral) = {3, 4}


    M, 2 ⊨ BeingHappy       M, 2 ⊨ BeingSad
    M, 4 ⊨ □BeingHappy      M, 1 ⊨ ◊BeingHappy     M, 1 ⊨ ◊BeingSad
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Satisfiability and Validity
   Satisfiability
    A proposition P ∈ LML is satisfiable in a Kripke model M = <W, R, I> if M, w
    ⊨ P for all worlds w ∈ W.


    We can then write M ⊨ P


   Validity
    A proposition P ∈ LML is valid if P is satisfiable for all models M (and by
    varying the frame <W, R>).


    We can write ⊨ P




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Satisfiability
   Consider the following situation:                BeingHappy


                                BeingSad    1             2                3
                                                                   BeingNormal

                                                          4
                                                     BeingNormal
   M = <W, R, I>
    W = {1, 2, 3, 4}
    R = {<1, 2>, <2, 2>, <3, 2>, <4, 2>}
    I(BeingHappy) = {2}      I(BeingSad) = {1}   I(BeingNormal) = {3, 4}


    M, w ⊨ □BeingHappy for all w ∈ W, therefore □BeingHappy is satisfiable in
    M.

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Validity
   Prove that P: □A  ◊A is valid                            A

                                                      1       2


                                                              3
                                                              A
   In all models M = <W, R, I>,
    (1) □A means that for every w∈W such that wRw’ then M, w’ ⊨ A
    (2) ◊A means that for some w∈W such that wRw’ then M, w’ ⊨ A


    It is clear that if (1) then (2) in the example
    (as we will see this is valid in serial frames)


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Kinds of frames
   Given the frame F = <W, R>, the relation R is said to be:

        Serial              iff for every w ∈ W, there exists w’ ∈ W s.t. wRw’
        Reflexive           iff for every w ∈ W, wRw
        Symmetric           iff for every w, w’ ∈ W, if wRw’ then w’Rw
        Transitiveiff for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’
          then wRw’’
        Euclidian iff for every w, w’, w’’ ∈ W, if wRw’ and wRw’’
          then w’Rw’’


   We call a frame <W, R> serial, reflexive, symmetric or transitive
    according to the properties of the relation R



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Kinds of frames
   Serial: for every w ∈ W, there exists w’ ∈ W s.t. wRw’

         1             2              3


   Reflexive: for every w ∈ W, wRw

         1             2


   Symmetric: for every w, w’ ∈ W, if wRw’ then w’Rw


         1             2              3


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Kinds of frames
   Transitive: for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’ then wRw’’

         1             2             3


   Euclidian: for every w, w’, w’’ ∈ W, if wRw’ and wRw’’ then w’Rw’’

         1             2



                       3




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Valid schemas
   A schema is a formula where I can change the variables
   THEOREM. The following schemas are valid in the class of indicated frames:
    D:   □A  ◊A         valid for serial frames
    T:   □A  A          valid for reflexive frames
    B:   A  □◊A         valid for symmetric frames
    4:   □A  □□A        valid for transitive frames
    5:   ◊A  □◊A        valid for Euclidian frames


    NOTE: if we apply T, B and 4 we have an equivalence relation


   THEOREM. The following schema is valid:
    K:    □(A  B)  (□A  □B)            Distributivity of □ w.r.t. 


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Proof for T: □ A  A valid for reflexive frames
 Assuming M, w ⊨ □A, we want to prove that M, w ⊨ A.


 From the assumption M, w ⊨ □A, we have that for every w’∈W such that
 wRw’ we have that M, w’ ⊨ A (1).

 Since R is reflexive we also have w’Rw, we then imply that M, w ⊨ A (by
 substituting w to w’ in (1))



                  □A, A

      1              2




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Proof for B: A  □◊A valid for symmetric frames
 Assume M, w ⊨ A. To prove that M, w ⊨ □◊A we need to show that for
 every w’ ∈ W such that wRw’ then M, w ⊨ ◊A.
 M, w ⊨ ◊A is that for some w’’∈W such that w’Rw’’ then M, w’’ ⊨ A.
 Therefore we need to prove that for every w’∈W such that wRw’ and for
 some w’’∈W such that w’Rw’’ then M, w’’ ⊨ A

 Since R is symmetric, from wRw’ it follows that w’Rw. For w’’∈W such that
 w’’ = w, we have that w’Rw’’ and M, w’’ ⊨ A.
 Hence M, w ⊨ A.


     A, □◊A        ◊A

       1           2             3


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Reasoning services: EVAL
   Model Checking (EVAL)
    Given a (finite) model M = <W, R, I> and a proposition P ∈ LML we want to
    check whether M, w ⊨ P for all w ∈ W


                              M, w ⊨ P for all w ?



                                                     Yes
                 M, P               EVAL
                                                     No




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Reasoning services: SAT
   Satisfiability (SAT)
    Given a proposition P ∈ LML we want to check whether there exists a
    (finite) model M = <W, R, I> such that M, w ⊨ P for all w ∈ W


                           Find M such that M, w ⊨ P for all w



                                                       M
                   P                     SAT
                                                       No




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Reasoning services: UNSAT
   Unsatisfiability (unSAT)
    Given a (finite) model M = <W, R, I> and a proposition P ∈ LML we want to
    check that does not exist any world w such that M, w ⊨ P


                        Verify that ∃ w such that M, w ⊨ P



                                                    w
                  M, P               VAL
                                                    No




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Reasoning services: VAL
   Validity (VAL)
    Given a a proposition P ∈ LML we want to check that M, w ⊨ P for all
    (finite) models M = <W, R, I> and w ∈ W

                         Verify that M, w ⊨ P for all M and w



                                                      Yes
                     P                 VAL
                                                      No




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Correspondence between □ and ∀ (◊ and ∃)
   We can define a translation function T: LML  LFO as follows:

    1. T(P) = P(x) for all propositions P in LML
    2. T(P) = T(P) for all propositions P
    3. T(P * Q) = T(P) * T(Q) for all propositions P, Q and *∈{,,}
    4. T(□P) = ∀x T(P) for all propositions P
    5. T(◊P) = ∃x T(P) for all propositions P


    THEOREM:
    For all propositions P in LML, P is modally valid iff T(P) is valid in FOL.




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