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 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:

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

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

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

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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

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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

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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

Validity Prove that P: □A ◊A is valid

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)12

1 2

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A

A

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 Transitive iff 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’

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’’

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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 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 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.

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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 ∃ 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

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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