Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Modal Logic

Originally by Alessandro Agostini and Fausto
Giunchiglia Modified by Fausto Giunchiglia, Rui
Zhang and Vincenzo Maltese
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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

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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
• ltAtomic Formulagt A B ... P Q ...
  ? ?
• ltwffgt ltAtomic Formulagt ltwffgt ltwffgt?
  ltwffgt ltwffgt? ltwffgt
• ltwffgt ? ltwffgt ltwffgt ? ltwffgt ?
  ltwffgt ? ltwffgt

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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 logics of programs ?aP P holds after the program a is executed
Description logics ?HASCHILDMALE ? ?HASCHILD.MALE ?HASCHILDMALE ? ?HASCHILD.MALE
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Semantics Kripke Model

• A Kripke Model is a triple M ltW, R, Igt 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, ltW, Rgt is called frame and is
  a relational structure.

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Semantics Kripke Model

• Consider the following situation
• M ltW, R, Igt
• W 1, 2, 3, 4
• R lt1, 2gt, lt1, 3gt, lt1, 4gt, lt3, 2gt, lt4, 2gt
• I(BeingHappy) 2 I(BeingSad) 1
  I(BeingNormal) 3, 4

BeingHappy
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2
3
BeingSad
BeingNormal
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BeingNormal
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Truth relation (true in a world)

• Given a Kripke Model M ltW, R, Igt, 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 ? T M, 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
• M ltW, R, Igt
• W 1, 2, 3, 4
• R lt1, 2gt, lt1, 3gt, lt1, 4gt, lt3, 2gt, lt4, 2gt
• 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

BeingHappy
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2
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BeingSad
BeingNormal
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BeingNormal
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Satisfiability and Validity

• Satisfiability
• A proposition P ? LML is satisfiable in a Kripke
  model M ltW, R, Igt 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 ltW, Rgt).
• We can write ? P

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Satisfiability

• Consider the following situation
• M ltW, R, Igt
• W 1, 2, 3, 4
• R lt1, 2gt, lt2, 2gt, lt3, 2gt, lt4, 2gt
• I(BeingHappy) 2 I(BeingSad) 1
  I(BeingNormal) 3, 4
• M, w ? ?BeingHappy for all w ? W, therefore
  ?BeingHappy is satisfiable in M.

BeingHappy
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2
3
BeingSad
BeingNormal
4
BeingNormal
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Validity

• Prove that P ?A ? ?A is valid
• In all models M ltW, R, Igt,
• (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)

A
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A
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Kinds of frames

• Given the frame F ltW, Rgt, 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
  wRw
• Transitive iff for every w, w, w ? W, if wRw
  and wRw then wRw
• Euclidian iff for every w, w, w ? W, if wRw
  and wRw then wRw
• We call a frame ltW, Rgt 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
• Reflexive for every w ? W, wRw
• Symmetric for every w, w ? W, if wRw then wRw

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Kinds of frames

• Transitive for every w, w, w ? W, if wRw and
  wRw then wRw
• Euclidian for every w, w, w ? W, if wRw and
  wRw then wRw

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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 wRw, we then
  imply that M, w ? A (by substituting w to w in
  (1))

?A, A
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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
  wRw then M, w ? A. Therefore we need to
  prove that for every w?W such that wRw and for
  some w?W such that wRw then M, w ? A
• Since R is symmetric, from wRw it follows that
  wRw. For w?W such that w w, we have that
  wRw and M, w ? A.
• Hence M, w ? A.

A, ??A
?A
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Reasoning services EVAL

• Model Checking (EVAL)
• Given a (finite) model M ltW, R, Igt and a
  proposition P ? LML we want to check whether M, w
  ? P for all w ? W
• M, w ? P for all w ?

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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 ltW, R,
  Igt such that M, w ? P for all w ? W
• Find M such that M, w ? P for all w

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Reasoning services UNSAT

• Unsatisfiability (unSAT)
• Given a (finite) model M ltW, R, Igt 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

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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 ltW, R,
  Igt and w ? W
• Verify that M, w ? P for all M and w

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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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