LDK R Logics for Data and Knowledge Representation Modal Logic Originally by Alessandro Agostini and...

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

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

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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