L ogics for D ata and K nowledge R epresentation

LLogics for DData and KKnowledgeRRepresentation

Modal Logic

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

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Outline Introduction Syntax Semantics Satisfiability and Validity Kinds of frames Reasoning services Theorem of equivalence with FOL Theorem of equivalence with DL Tableau calculus

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Introduction We want to model situations like this one:

1. “Fausto is always happy”

2. “Fausto is happy under certain circumstances”

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:

<Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ |

<wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff>∨ <wff> |

<wff> <wff> | <wff> <wff> | □ <wff> | ◊ <wff>

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SYNTAX :: SEMANTICS :: SATISFIABILITY AND VALIDITY :: FRAMES :: REASONING SERVICES :: THEOREMS :: TABLEAU

Different interpretations

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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


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.

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Semantics: Kripke Model Consider the following situation:

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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1 2 3

4

BeingHappy

BeingSad

BeingNormal

BeingNormal


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 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 = <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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1 2 3

4

BeingHappy

BeingSad

BeingNormal

BeingNormal


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:

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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1 2 3

4

BeingHappy

BeingSad

BeingNormal

BeingNormal


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 w’Rw

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1 2 3

1 2

1 2 3


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

wRw’’

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

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

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1 2 3

1 2

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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 D: □A ◊A valid for serial frames In all serial frames M = <W, R>, we have that if (1) then (2)

(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

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1 2

3


□A, ◊A

□A, ◊A, A

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))

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□A, A

1 2


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 accessible world w’ ∈ W, i.e. 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.

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1 2 3

A, □◊A ◊A


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 ?

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EVALM, PYes

No


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

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SATPM

No


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 does not exist w such that M, w ⊨ P

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VALM, Pw

No


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

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VALPYes

No


Equivalence Modal logics – First Order Logic We can define a translation function : LML LFO as

follows:

(P) = P(x) for all propositions P in LML

(P) = (P) for all propositions P

(P * Q) = (P) * (Q) for all propositions P, Q and *∈{,,}

(□P) = ∀x (P) for all propositions P

5(◊P) = ∃x (P) for all propositions P

THEOREM:

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

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Equivalence Modal logics – Description logics Take ALCEU:

<Atomic> ::= A | B | ... | P | Q | ... | ⊥ | ⊤

<wff> ::= <Atomic> | ¬ <wff> | <wff> ⊓ <wff> | <wff> ⊔ <wff> | ∀R.C | ∃R.C

We can define an equivalent multi-modal logic with a mapping function as follows:

(A) = A for A atomic

(¬C) = ¬ (C)

(C ⊓ D) = (C) (D)

(C ⊔ D) = (C) (D)

(∃R.C) = ◊R (C)

(∀R.C) = □R (C)

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

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Modal logics Tableau: introduction

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Recall modal logics semantics:P = □Q for every w’∈W such that wRw’ then M, w’ ⊨ Q

P = ◊Q for some w’∈W such that wRw’ then M, w’ ⊨ Q

Each time we use □ or ◊ we state something about accessible worlds!

Recall satisfiability:

A proposition P ∈ LML is satisfiable if there exist a Kripke model in which it is true.

Therefore the key idea in the modal logics tableau is:If M, w ⊨ □Q then Q must be present in all w’ accessible from w

If M, w ⊨ ◊Q then Q must be present in some w’ trees accessible from w

For all other formulas follow the rules of PL tableau


Modal logics Tableau: rules

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We indicate with , i the fact that must be true in world i ∈W

Given the formula in input we apply the rules below by verifying that not all branches are closed:

() (P Q), i | P, i and Q, i

() (P Q), i | P, i or Q, i (two branches)

(◊) ◊P, i | iRj P, j given any (i,j)∈R to denote that P is true | in j given that it is accessible from i

(□) iRj □P, i | P, j

(duality) □P, i | ◊P, i

(duality) ◊P, i | □P, i

We start by convention with , 0


Modal logics Tableau Example (I)

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( A B) B satisfiable?

() ( A B) B , 0

/ \

() ( A B) , 0 B , 0

| (open)

A , 0

|

B , 0

(open)


Modal logics Tableau Example (II)

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◊A satisfiable?

(duality) ◊A , 0

|

(□) □P , 0

0R1

|

P , 1

(open)


Modal logics Tableau Example (III)

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□A □A valid? We negate and check whether ALL branches are closed.

The negation is: (□A □A) □A □A

() □A □A , 0

|

(duality) □A , 0

|

(□) □A , 0

0R1

|

(◊) ◊A , 0

|

A , 1

|

0R1 A , 1

(closed)


Modal logics Tableau: additional rules

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We have extra rules to convey frame properties:

(reflexivity) * | iRi

(symmetry) iRj | jRi

(transitivity) iRj jRk | iRk

(seriality) * | iRj

Euclidian properties can be given as a combination of the first three.


Modal logics Tableau Example (IV)

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□A A valid in reflexive frames?

The negation is: (□A A) (□A A) □A A

() □A A , 0

|

A , 0

|

(□) (reflexive) □A , 0

0R1

|

A , 1

|

0R0 A , 0

(closed)


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