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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

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
Lecture 4

Modal Logic

Lecture outline

  • Introduction
  • Modal logic in CS
  • Syntax of modal logic
  • Semantics of modal logic
  • Logics of knowledge and belief
  • Temporal logics
1 introduction
1. Introduction
  • 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.
history
History
  • 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)
modal operators
Modal operators
  • 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
2 modal logic in cs
2. Modal logic in CS
  • 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?

modeling modal reasoning
Modeling modal reasoning

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?

modeling modal reasoning1
Modeling modal reasoning

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?

3 modal logic syntax
3. Modal logic - Syntax
  • 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
deduction in modal logic
Deduction in modal logic
  • Axioms

The 3 axioms of PL

    • A1.  ()
    • A2. ( ( ))  (( )  ( ))
    • A3. ((¬)  (¬))  ( )

The axiom to specify distribution of necessity

    • A4. ( )  (    ) Distribution of modality
deduction in modal logic1
Deduction in modal logic
  • Inference rules
  • Substitution (uniform)   ’
  • Modus Ponens  , (  )  
  • The modal rule of necessity |-   
    • « for any formula , if  was proved then we can infer  »
4 semantics of modal logic
4. Semantics of modal logic
  • 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.

semantics of modal logic
Semantics of modal logic
  • 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 
semantics of modal logic1
Semantics of modal logic
  • 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\' 
slide15

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) = ?

slide16

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) = ?

different modal logic systems
Different modal logic systems

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\' 

different modal logic systems1
Different modal logic systems

The modal logic D

Add axiom

  • 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))
different modal logic systems2
Different modal logic systems

The modal logic T

Add axiom

  • 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).
different modal logic systems3
Different modal logic systems

The modal logic S4

Add axiom

  • 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).

different modal logic systems4
Different modal logic systems

The modal logic B

Add axiom

  • 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)
different modal logic systems5
Different modal logic systems

The modal logic S5

Add the axiom

  • 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

different modal logic systems6
Different modal logic systems

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 systems7

reflexive

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

5 logics of knowledge and belief
5. Logics of knowledge and belief
  • 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
logics of knowledge and belief
Logics of knowledge and belief
  • 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

logics of knowledge and belief1
Logics of knowledge and belief
  • 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
properties of knowledge
Properties of knowledge

(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, )  

properties of knowledge1
Properties of knowledge

(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

inference rules for knowledge
Inference rules for knowledge

(R1) Epistemic necessitation

|-  K(a, )

modal rule of necessity |-   

(R2) Logical omniscience

   and K(a, ) K(a, )

problematic

properties of belief
Properties of belief

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

inference rules for belief
Inference rules for belief

(R1) Epistemic necessitation

|-  B(a, ) problematic

modal rule of necessity |-   

(R2) Logical omniscience

   and B(a, ) B(a, )

usually NO

some more axioms for beliefs
Some more axioms for beliefs

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, )

slide34

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

34

6 temporal logic
6. Temporal logic
  • 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

branching time logic ctl
Branching time logic - CTL
  • 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
branching time logic ctl1
Branching time logic - CTL

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

slide38
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)

38

slide39
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

39

slide40
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

40

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