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Master of Science in Artificial Intelligence, 2009-2011. Knowledge Representation and Reasoning. University "Politehnica" of Bucharest Department of Computer Science Fall 2009 Adina Magda Florea http://turing.cs.pub.ro/krr_09 curs.cs.pub.ro. Lecture 4. Modal Logic Lecture outline

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Knowledge Representation and Reasoning

University "Politehnica" of Bucharest

Department of Computer Science

Fall 2009

http://turing.cs.pub.ro/krr_09

curs.cs.pub.ro

Modal Logic

Lecture outline

• Introduction

• Modal logic in CS

• Syntax of modal logic

• Semantics of modal logic

• Logics of knowledge and belief

• Temporal logics

• In first order logic a formula is either true or false in any model

• In natural language, we distinguish between various “modes of truth”, e.g, “known to be true”, “believed to be true”, “necessarily true”, “true in the future”

• “Barack Obama is the president of the US” is currently true but it will not be true at some point in the future.

• “After program P is executed, A hold” is possibly true if the program performs what is intended to perform.

• Classical logic is truth-functional = truth value of a formula is determined by the truth value(s) of its subformula(e) via truth tables for ,, ¬, and →.

• Lewis tried to capture a non-truth-functional notion of “A Necessarily Implies B” (A → B)

• We can take A → B to mean “it is impossible for A to be true and B to be false”

• He chose a symbol, P, and wrote PA for “A is possible”; then:

• ¬PA is “A is impossible”

• ¬P¬A is “not-A is impossible”

• Then he used the symbol N to stand for ¬P and expressed

• NA := ¬P¬A “A is necessary”

• Because → is logical implication, we can transform it like this:

• A → B := N(A → B) = ¬P¬(A → B) = ¬P¬(¬A  B) = ¬P(A  ¬B)

• P - “possibly true”

• N - “necessarily true”

• Modal logics - modes of truth: 

• Basic modal logic:  - box, and  - diamond

• The necessity / possibility - necessary, and  - possible

• Logics about knowledge - what an agent knows / believes

• Deontic logic -  - it is obligatory that, and  - it is permissible that

• Temporal logic

• Dynamic logic

• Logic of knowledge and belief

• Model problems and complex reasoning

The Lady and the Tiger Puzzle

• There are two rooms, A and B, with the following signs on them:

• A: In this room there is a lady, and in the other room there is a tiger”

• B: “In one of these rooms there is a lady and in one of them there is a tiger”

• One of the two signs is true and the other one is false.

Q: Behind which door is the lady?

The King's Wise Men Puzzle

• The King called the three wisest men in the country.

• He painted a spot on each of their foreheads and told them that at least one of them has a white spot on his forehead.

• The first wise man said: “I do not know whether I have a white spot”

• The second man then says “I also do not know whether I have a white spot”.

• The third man says then “I know I have a white spot on my forehead”.

Q: How did the third wise man reason?

Mr. S. and Mr. P Puzzle

• Two numbers m and n are chosen such that 2  m  n  99.

• Mr. S is told their sum and Mr. P is told their product.

• Mr. P: "I don't know the numbers. "

• Mr. S: "I knew you didn't know. I don't know either."

• Mr. P: "Now I know the numbers."

• Mr  S: "Now I know them too."

Q: In view of the above dialogue, what are the numbers?

• Atomic formulae: p ::= p0 | p1 | p2 | q …. where pi , q are atoms in PL

• Formulae: ::= p | ¬ |   |   |    |    |  →  where  and  are a wffs in PL

Examples:

• p → q

• p → q

•  (p1 → p2) → ((p1) → (p2))

• Schema:

•  → 

•  →  

• ( →  ) → ( →  )

• Schema Instances: Uniformly replace the formula variables with formulae (inference)

Examples:

• p → p is an instance of  →  but

• p → q is not

• Axioms

The 3 axioms of PL

• A1.  ()

• A2. ( ( ))  (( )  ( ))

• A3. ((¬)  (¬))  ( )

The axiom to specify distribution of necessity

• A4. ( )  (    ) Distribution of modality

• Inference rules

• Substitution (uniform)   ’

• Modus Ponens  , (  )  

• The modal rule of necessity |-   

• « for any formula , if  was proved then we can infer  »

• Nonlinear model

• The semantics of modal logic is known as the Kripke Semantics, also called the Possible World approach

Directed graph (V, E)

• Vertices V = {v, v1, v2, …}

• Directed edges {(s1,t1), (s2,t2),…} from source vertex si V to the target vertex tiV for i = 1,2,…

Cross product of a set V, V x V

• {(v,w) | vV and wV} the set of all ordered pairs (v,w), where v and w are from V.

Directed graph

- a pair (V,E), where V = {v, v1, v2, …} and E  V x V is a binary relation over V.

• A Kipke frame is a directed graph <W, R>, where:

• W is a non-empty set of worlds (points, vertices) and

• R  W x W is a binary relation over W, called the accessibility relation.

• An interpretation of a wff in modal logic on a Kripke frame <W, R> is a function I : W x L → {t,f} which tells the truth value of every atomic formula from the language L at every point (in every word) in W.

• A Kripke model M of a formula  (an interpretation which makes the formula true) is

• the triple <W, R, I>, where I is an interpretation of the formula on a Kripke frame <W,R> which makes the formula true.

• This is denoted by M |=W 

• Using the model, we can define the semantics of formulae in modal logic and can compute the truth value of formulae.

• M |=W  iff M |=/W  (or M |=W ¬)

• M |=W   iff M |=W  and M |=W 

• M |=W    iff M |=W  or M |=W 

• M |=W  →  iff M |=W ¬ or M |=W 

(¬   is true in W)

• M |=W   iff w': R(w,w')  M |=W' 

• M |=W   iff w': R(w,w')  M |=W' 

W1

I(W1,p) = f

I(W1,q) = f

I(W1,r) = a

W0

I(W0,p) = f

I(W0,q) = f

I(W0,r) = f

W2

I(W2,p) = f

I(W2,q) = f

I(W2,r) = f

Examples

p – I am rich

q – I am president of Romania

r – I am holding a PhD in CS

I(W0, p) = ? I(W0, p) = ?

I(W0, r) = ? I(W0, r) = ?

w1

p, q, r

w2

p, q, r

w0

p, q, r

w3

p, q, r

Examples

p -Alice visits Paris

q - It is spring time

r - Alice is in Italy

I(W0, p) = ? I(W0, p) = ?

I(W0, q) = ? I(W0, q) = ?

I(W0, r) = ? I(W0, r) = ?

I(W1, p) = ? I(W1, p) = ?

The modal logic K

• A1.  ()

• A2. ( ( ))  (( )  ( ))

• A3. ((¬)  (¬))  ( )

• A4. ( )  (    )

• X  X

• Here is an invalidating model:

R(w0,w1), I(w0,p)=f, I(w1,p)=t

• “it is impossible for A to be true and B to be false”

M |=W   iff w': R(w,w')  M |=W' 

The modal logic D

• X X

• In fact, D-models are K-models that meet an additional restriction: the accessibility relation must be serial.

• A relation R on W is serialiff

• (wW: (w'W: (w,w')R))

The modal logic T

• X  X

• A T-model is a K-model whose accessibility relation is reflexive.

• A relation R on W is reflexiveiff

• (wW: (w,w)R).

The modal logic S4

• X  X

• An S4-model is a K-model whose accessibility relation is reflexiveand transitive.

• A relation R on W is transitiveiff

• (w1,w2,w3 wW:

(w1,w2)R  (w2, w3)R  (w1,w3)R).

The modal logic B

• X  X

• A B-model is a K-model whose accessibility relation isreflexive and symmetric.

• A relation R on W is symmetriciff

• (w1,w2W: (w1,w2)R  (w2,w1)R)

The modal logic S5

• X   X

• An S5-model is a K-model whose accessibility relation is reflexive, symmetric, and transitive.

• That is, it is an equivalence relation

• Exercise: Find an S5-model in which X  X is false.

S5 is the system obtained if every possible world is possible relative to every other world

The modal logic S5

• X   X

• A relation is euclidian iff (w1,w2,w3W: (w1,w2)R 

(w1, w3)R  (w2,w3)R)

Different modal logic systems

D = K + D

T = K + T

S4 = T + 4

B = T + B

S5 = S4 + B

S5

symmetric

transitive

S4

B

transitive

symmetric

T

D

reflexive

serial

K

• Used to model "modes of truth" of cognitive agents

• Distributed modalities

• Cognitive agents  characterise an intelligent agent using symbolic representations and mentalistic notions:

• knowledge - John knows humans are mortal

• beliefs - John took his umbrella because he believed it was going to rain

• desires, goals - John wants to possess a PhD

• intentions - John intends to work hard in order to have a PhD

• commitments - John will not stop working until getting his PhD

• How to represent knowledge and beliefs of agents?

• FOPL augmented with two modal operators K and B

K(a,) - a knows 

B(a,) - a believes 

with LFOPL, aA, set of agents

• Associate with each agent a set of possible worlds

• Kripke model Ma of agent a for a formula 

• Ma =<W, R, I>

with R  A x W X W

and I - interpretation of the formula on a Kripke frame <W,R> which makes the formula true for agent a

• An agent knows a propositions in a given world if the proposition holds in all worlds accessible to the agent from the given world

Ma |=WK iff w': R(w,w')  Ma |=W' 

• An agent believes a propositions in a given world if the proposition holds in all worlds accessible to the agent from the given world

Ma |=WB iff w': R(w,w')  Ma |=W' 

• The difference between B and K is given by their properties

(A1) Distribution axiom:

K(a, ) K(a,   )  K(a, )

"The agent ought to be able to reason with its knowledge"

( )  (  )(Axiom of distribution of modality)

K(a, )  ( K(a,)  K(a,) )

(A2) Knowledge axiom: K(a, )  

"The agent can not know something that is false"

 (T) - satisfied if R is reflexive

K(a, )  

(A3) Positive introspection axiom

K(a, )  K(a, K(a, ))

X  X (S4) - satisfied if R is transitive

K(a, )  K(a, K(a, ))

(A4) Negative introspection axiom

K(a, )  K(a, K(a, ))

X   X (S5) - satisfied if R is euclidian

(R1) Epistemic necessitation

|-  K(a, )

modal rule of necessity |-   

(R2) Logical omniscience

   and K(a, ) K(a, )

problematic

Distribution axiom: B(a, ) B(a,   )  B(a, )

YES

Knowledge axiom: B(a, )   NO

Positive introspection axiom

B(a, )  B(a, B(a, ))

YES

Negative introspection axiom

B(a, )  B(a, B(a, )) problematic

(R1) Epistemic necessitation

|-  B(a, ) problematic

modal rule of necessity |-   

(R2) Logical omniscience

   and B(a, ) B(a, )

usually NO

Knowing what you believe

B(a, ) K(a, B(a, ))

Believing what you know

K(a, ) B(a, )

Have confidence in the belief of another agent

B(a1, B(a2,)) B(a1, )

8. KB(WB)  WA contrapositive of 7

9. KA(WA) 3, 8, R2

R2:    and K(a, ) inferK(a, )

Two-wise men problem - Genesereth, Nilsson

(1) A and B know that each can see the other's forehead. Thus, for example:

(1a) If A does not have a white spot, B will know that A does not have a white spot

(1b) A knows (1a)

(2) A and B each know that at least one of them have a white spot, and they each know that the other knows that. In particular

(2a) A knows that B knows that either A or B has a white spot

(3) B says that he does not know whether he has a white spot, and A thereby knows that B does not know

1. KA(WA  KB( WA) (1b)

2. KA(KB(WA WB)) (2a)

3. KA(KB(WB)) (3)

Proof

4. WA  KB(WA) 1, A2A2: K(a, )  

5. KB(WA  WB) 2, A2

6. KB(WA)  KB(WB) 5, A1A1: K(a, )  (K(a,)  K(a,))

7. WA  KB(WB) 4, 6

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• The time may be linear or branching; the branching can be in the past, in the future of both

• Time is viewed as a set of moments with a strict partial order, <, which denotes temporal precedence.

• Every moment is associated with a possible state of the world, identified by the propositions that hold at that moment

Modal operators of temporal logic (linear)

p U q - p is true until q becomes true - until

Xp - p is true in the next moment - next

Pp - p was true in a past moment - past

Fp - p will eventually be true in the future - eventually

Gp - p will always be true in the future – always

Fp  true U p

Gp  F p

F – one time point

G – each time point

• Temporal structure with a branching time future and a single past - time tree

• CTL – Computational Tree Logic

• In a branching logic of time, a path at a given moment is any maximal set of moments containing the given moment and all the moments in the future along some particular branch of <

• Situation - a world w at a particular time point t, wt

• State formulas - evaluated at a specific time point in a time tree

• Path formulas - evaluated over a specific path in a time tree

CTL Modal operators over both state and path formulas

From Temporal logic (linear)

Fp - p will sometime be true in the future - eventually

Gp - p will always be true in the future - always

Xp - p is true in the next moment - next

p U q - p is true until q becomes true - until

(p holds on a path s starting in the current moment t until q comes true)

Modal operators over path formulas(branching)

Ap - at a particular time moment, p is true in all paths emanating from that point - inevitable p

Ep - at a particular time moment, p is true in some path emanating from that point - optional p

F – one time point

G – each time point

A – all path

E – some path

LB- set of moment formula

LS - set of path-formula

Semantics

M = <W, T, <, | |, R> - every tT has associated a world wtW

M |=t  iff t||

 is true in the set of moments for which  holds

M |=t pq iff M |=t p and M |=t q

M |=t p iff M |=/t p

M |=s,t pUq iff (t': tt' and M |=s,t' q and

(t": t  t" t'  M |=s,t" p))

p holds on a path s starting in the current moment t until q comes true

Fp  true Up

Gp  F p

M |=tA p iff (s: sSt  M |=s,t p)Ep  A p

s is a path, St - all paths starting at the present moment

M |=s,tX p iff M |=s,t+1 p)

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• s is true in each time point (G) and in all path (A)

• r is true in each time point (G) in some path (E)

• p will eventually (F) be true in some path (E)

• q will eventually (F) be true in all path (A)

s

p

s

q

F - eventually

G - always

A - inevitable

E - optional

AGs

EGr

EFp

AFq

r

s

r

s

r

s

q

s

q

s

r - Alice is in Italy p -Alice visits Paris

s – Paris is the capital of France q - It is spring time

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• Each situation has associated a set of accessible words - the worlds the agent believes to be possible. Each such world is a time tree.

• Within these worlds, the branching future represents the choices (options) available to the agent in selecting which action to perform

• Similar to a decision tree in a game of chance

Decision nodes

Player 1

Dice

• Each arc emanating from

• a chance node corresponds

• to a possible world

Player 2

1/18

1/36

Chance nodes

Dice

• Each arc emanating from

• a decision node corresponds

• to a choice available in a

• possible world

Player 1

1/36

1/18

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