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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






Introduction
Modal logic in CS
Syntax of modal logic
Semantics of modal logic
Logics of knowledge and belief
Temporal logics
2
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.
3
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)
4
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





5
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?
6
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?
7
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?
8
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
9
Deduction in modal logic
 Axioms
The 3 axioms of PL
•
•
•
A1.   (  )
A2. (  (  ))  ((  )  (  ))
A3. ((¬)  (¬))  (  )
The axiom to specify distribution of necessity
• A4. ( )  (    ) Distribution of modality
10
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  »
11
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.
12
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 
13
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    iff M |=W  or M |=W 

M |=W  →  iff M |=W ¬ or M |=W 
M |=W  and M |=W 
(¬   is true in W)

M |=W   iff
w': R(w,w')  M |=W' 

M |=W   iff
w': R(w,w')  M |=W' 
14
Examples
p – I am rich
q – I am president of Romania
r – I am holding a PhD in CS
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
I(W0, p) = ?
I(W0, p) = ?
I(W0, r) = ?
I(W0, r) = ?
W2
I(W2,p) = f
I(W2,q) = f
I(W2,r) = f
15
Examples
w1
p, q, r
w0
p, q, r
w2
p, q, r
w3
p, q, r
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) = ?
16
Different modal logic systems
The modal logic K
• A1.   (  )
• A2. (  (  ))  ((  )  (  ))
• A3. ((¬)  (¬))  (  )
• A4. ( )  (    )

XX
“it is impossible for A to be true and B to be false”
 Here is an invalidating model:
R(w0,w1), I(w0,p)=f, I(w1,p)=t
M |=W   iff
w': R(w,w')  M |=W' 
17
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 serial iff
• (wW: (w'W: (w,w')R))
18
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 reflexive iff
• (wW: (w,w)R).
19
Different modal logic systems
The modal logic S4
Add axiom
 X
X
 An S4-model is a K-model whose accessibility
relation is reflexive and transitive.
 A relation R on W is transitive iff
• (w1,w2,w3 wW:
(w1,w2)R  (w2, w3)R  (w1,w3)R).
20
Different modal logic systems
The modal logic B
Add axiom
 X  X
 A B-model is a K-model whose accessibility
relation is reflexive and symmetric.
 A relation R on W is symmetric iff
• (w1,w2W: (w1,w2)R  (w2,w1)R)
21
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 
is false.
X
S5 is the system obtained if every possible world is possible relative to every
22
other world
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)
23
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
reflexive
symmetric
T
D
reflexive
serial
K
24
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
25
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
26
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 |=W K 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 |=W B iff
w': R(w,w')  Ma |=W' 
 The difference between B and K is given by their
properties
27
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, )  
28
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
29
Inference rules for knowledge
(R1) Epistemic necessitation
|-   K(a, )
modal rule of necessity |-   
(R2) Logical omniscience
   and K(a, )  K(a, )
problematic
30
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
31
Inference rules for belief
(R1) Epistemic necessitation
|-   B(a, )
problematic
modal rule of necessity |-   
(R2) Logical omniscience
   and B(a, )  B(a, )
usually NO
32
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, )
33
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)
2. KA(KB(WA  WB))
3. KA(KB(WB))
(1b)
(2a)
(3)
4. WA  KB(WA)
5. KB(WA  WB)
1, A2
2, A2
A2: K(a, )  
6. KB(WA)  KB(WB)
7. WA  KB(WB)
5, A1
4, 6
A1: K(a, )  (K(a,)  K(a,))
8. KB(WB)  WA
9. KA(WA)
contrapositive of 7
3, 8, R2
Proof
R2:    and K(a, ) infer K(a, )
34
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
35
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
36
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
F – one time point
G – each time point
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
A – all path
E – some path
37
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 |=t A 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,t X p iff M |=s,t+1 p)
38




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
r
s
r
s
p
s
q
AGs
EGr
r
s
q
F - eventually
G - always
A - inevitable
E - optional
EFp
AFq
s
r - Alice is in Italy
s – Paris is the capital of France
s
q
p -Alice visits Paris
q - It is spring time
39
 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
• Each arc emanating from
a chance node corresponds
to a possible world
Dice
Player 2
1/36
1/18
Chance nodes
Dice
Player 1
1/36
1/18
• Each arc emanating from
a decision node corresponds
to a choice available in a
possible world
40
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