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Transcript
```Ariel Jarovsky and Eyal Altshuler
8/11/07, 15/11/07
1
Today
• A short review
• Multi-Modal Logic
• First Order Modal Logic
• Applications of Modal Logic:
• Artificial Intelligence
• Program Verification
• Summary
2
3
Introduction
 Modal Logics are logics of qualified truth.
 (From the dictionary)
Modal – of form, of manner, pertaining to
mood, pertaining to mode
 Necessary, Obligatory, true after an action,
known, believed, provable, from now on,
since, until, and many more…
4
Syntax – Language
The formal language:

A non-empty set of propositions (as in
classical logic): P  { p , p , p , }
1
2
3
{¬, Ù, Ú, , , ,W,, à }

Operators:

Parentheses.

Some define the ◊ as:
The Modal
Operators
à A  def ¬W¬A
5
Syntax – Formulas
• Formulas are the only syntactic category of
Propositional Modal Logics, as in CPL.
• Every proposition p is a formula.
• If A, B are formulas, then the following are also
formulas:
•
¬A,
A Ù B,
A Ú B,
A  B,
AB
• If A is a formula then the following are also
formulas:
•
WA
àA
6
Modal Logics - Semantics
 Possible worlds semantics
(Kripke, 1959)
 The different possible worlds represent
the states of a given problem.
7
Semantics - Frame
 A frame is a pair (W,R) where W is a non-
empty set and R is a binary relation on W.
 W is the set of all possible worlds, or states.
 R determines which worlds are accessible
from any given world in W.
 We say that b is accessible from a iff (a,b)R.
 R is known as the accessibility relation.
8
Semantics – Model
 A Model is a triple M=(W,R,V) while (W,R) is a
frame and V is a valuation.
 A valuation is a function V : P  W  {T , F }.
Informally, V(p,w)=T is to be thought as p is
true at world w.
9
Semantics – Semantic Relation
 The relation
‘
between a pair (M,w) where
M is a model and w is a world, and a formula
A, is defined recursively as follows:
 M , w ‘ p, p  P  V ( p, w)  T

M , w ‘ pA Ù B  M , w ‘ A and M , w ‘ B
 Similar for the other classical logic connectors.


M , w ‘ WA  x W if wRx then M , x ‘ A
M , w ‘ à A  x W s.t. wRx Ù M , x ‘ A
10
Logics
 Given a language L(P) (P is a set of atoms) a
logic  is defined to be any subset of formulas
generated from P that satisfies:
  includes all tautologies;
 Closure under Modus Ponens.
 Closure under uniform substitution.
11
Normal Logic
 A logic  is said to be normal if it contains the
formula scheme:
K : W( A  B)  (WA WB)
 and if it is provided with the modal
inference necessitation rule:
Λ A
Λ WA
12
Axiomatic Systems
 An axiomatic system for a normal logic  is
made up of the following three components:
 An axiomatic system of CPL (as HPC)
 The axiom scheme denoted:
K : W( A  B)  (WA WB)
 The modal inference rule of necessitation:
Λ A
Λ WA
13
14
Multi-Modal Logics
 There exist logic languages with more than
one modal operator
 Why do you think?
 They may use:
 Collection of symbols {[i]}
 Each modal [i] has its dual, <i>
 <i>A= [i]A.
15
Multi-Modal Logics- Syntax
• Very similar to the syntax of uni-modal
•Every proposition p is a formula.
• If A, B are formulas, then the following are also
formulas:
•
¬A,
A Ù B,
A Ú B,
A  B,
AB
• If A is a formula then the following are also
formulas:
•
[i ] A
i A
16
Multi-Modal Logics- Semantics
 A frame F for multimodal language is define
as follows –
F=(W,{Ri | i})
 W is a non-empty set of worlds
 For each i, Ri is a binary relation on W.
 A model M is a tupple M=(W,{Ri | i},V)
 A valuation V is function
V : P  W  {T , F }
17
The Semantic Relation
 The relation
‘
between a pair (M,w) where
M is a model and w is a world, and a formula
A, is defined recursively as follows:
 M , w ‘ p, p  P  V ( p, w)  T

M , w ‘ pA Ù B  M , w ‘ A and M , w ‘ B
 Similar for the other classical logic connectors.


M , w ‘ [i] A  x W if (w, x)  Ri then M , x ‘ A
M , w ‘  i  A  x W s.t. (w, x)  Ri Ù M , x ‘ A
18
Multi-Modal Logics
 A Logic is defined as same as in uni-modal
logics (includes all tautologies and closed
under MP and substitution).
 A logic  is said to be normal if it contains the
schemata:
Ki : [i]( A  B)  ([i] A  [i]B)
 And satisfies the necessitation
rule for each i.
Λ A
Λ [i ] A
 The smallest normal logic is generally denoted
Ki.
19
Multi-Modal Logic - Example
(A)
(B)
Tomorrow, Dan will have 3 children.
 Let us look on the formula –
 A ÙB
 Intuitively, It has to be true only in the day in
which his third child was born.
20
Example
 Formally, we will define a frame to be- F  (W , R1 , R2 )
 W – the days during the year.
 R1 – all the pairs (dayi, dayi-1).
 R2 – all the pairs (dayi, dayi+1).
 A world w in model M in which A  B will
be true is-
R1
A–T
B-F
A–T
B-F
A–T
B-F
R2
A–T
B-T
A–F
B-T
Dan’s 3rd child
birthday
A–F
B-T
A–F
B-T
21
First Order Modal Logic
 Motivation:
 Every lecturer strikes.
 Yossi is a lecturer.
 Thus Yossi strikes.
 The formal language –
 There are two parts –
 A common part for all of the languages.
 A signature - unique for every language.
22
First Order Modal Logic
 The common part –
{¬, Ù, Ú, , ,W, à }
 Operators:
{, }
 Quantifies:
 Parentheses.
 Variables: v1,v2,…
• Syntactic Categories –
• formulas
Will be detailed
• terms
23
First Order Modal Logic
 Signature: the unique part of every language - 
 A non-empty set of function symbols.
 A (maybe empty) set of constants.
 A (maybe empty) set of predicate symbols.
 Terms:
 Every variable is a term.
 Every constant is a term.
 If f is a function symbol and t1,…,tn are terms,
then f(t1,…,tn) is also a term.
24
Definition of a formula
 If p is a predicate symbol and t1,…,tn are
terms, the p(t1,…,tn) is an atomic formula.
 If A, B are formulas then the following are
also formulas:
 A, AB, AB, AB, AB
 x.A, x.A
 A, A
25
First Order Logic- Semantics
 Let L(σ) be a first order language.
 When is a formula true?
 A Structure M is a pair M=<D,I>, such that –
 D – (domain) a non-empty set of objects.
 I – an interpretation function of σ:

I [c ]  D

I[ f ] Dn  D

I [ p]  D n
26
FOL – Valuations
 A valuation is a function from terms do the
domain
V :xD
 However, it is generalized to a function V : o  D
from terms to the domain and is defined as:
 V[c]= I[c]
 V[x] – given by V.
 V[f(t1,…,tn)]=I[f](V[t1],…,V[tn])
27
Domains in First Order Modal Logic
 This is a problematic issue. Why?
 “Tomorrow, everyone will be glad”.
 A new question is added- “Who is everyone?”
 On Sunday- Everyone includes Yossi,Dan and
Moshe.
 On Monday- Everyone includes Yossi,Dan,
 On Tuesday- Everyone includes Dan, Moshe
28
Domain- 3 natural definitions
1) The set of all individuals existing in the
actual world (D = a).
2) The set of all individuals existing in a given
possible world w (D = w).
3) The set of all the individuals existing in any
world (D = *=UwWw).
29
Domain- 3 natural definitions
 The quantifiers have different meanings,
according to the definition of the domain1)
 a x means- ‘for all x in the actual world’.
a x means- ‘for an x in the actual world’.
2)
 w x means- ‘for all x in the world w’.
w x means- ‘for an x in the world w’.
3)
* x means- ‘for all x’.
* x means- ‘for at least one x’.
30
31
Where is modal logic used?

Modal logic is a widely applicable method of
reasoning for many areas of computer
science.

Artificial Intelligence
Database theory
Distributed systems
Program verification
Cryptography theory




32
AI – Epistemic Logic
 Epistemic Logic is the modal logic that
 Philosophy, Artificial Intelligence, Distributed
Systems.
 Important: our examples in that part will be
quantifiers, more than one modal)
33
Epistemic Logic – Syntax
 Will be minimally defined, more details – next
lecture of the seminar.
 Suppose there are n agents.
 Let P  { p1 , p2 , p3 ,
}
be a non-empty
set of propositions.
 Operators: {¬, Ù, Ú, , ,[i ],  i }
 [i]φ- agent i knows φ.
 <i>φ- agent i knows that φ is true at some
state.
34
Epistemic Logic- Syntax
 Formulas are defined as usual.
agent knows, it may be helpful to reason
n
 Everyone knows:
Eφ  Ù[i]φ
 Common knowledge:
Cφ  Ù Eφ( k ) ,
k
i 1
Eφ(1)  Eφ , Eφ( k 1)  E( Eφ( k ) )
35
Applications of Epistemic Logic
(semantics)
 In a multi-agent system, there are n agents.
 Each agent i has it’s local environment, that
consists of information of what i’s local state is in
the system.
 In addition there is a global environment, that
includes information that agents might not
necessarily know but is still important for the
system to run (this information is categorized as
seen from a “bird’s eye” view of the system).
36
Examples (1)
 A scrabble game:
 Agents i’s local environment:
 The letters i contained in its hand.
 The letters that have been currently played.
 Which words were played by each player.
 The current score.
 The global environment may contains The letters that haven’t been chosen by any
player.
37
Examples (2)
 A distributed system.
 Each process is an agent.
 The local environment of a process might
contain messages i has sent or received, the
values of local variables, the clock time.
 The global environment might include the
number of process, a log file of all the
process’ operations, etc.
38
Applying epistemic logic using
possible worlds semantics
 The environments defines a global state.
 A global state is a set (se,s1,…,sn) of
environments
 Se is the global environment.
 Each si is the local environment of agent i.
 A run is defined as a function from time to
global states.
 A point is a pair (r,m) where r is a run at
some time m (assume time to be the natural
numbers).
39
Applying epistemic logic using
possible worlds semantics
 A system is defined as a set of runs. Thus, our
description of a system entails a collection of
interacting agents.
 Intuitively, a system is the set of all possible
runs.
 At point (r,m), system is in some global state
r(m). Let ri(m) be the local environment for
agent i.
40
Applying epistemic logic using
possible worlds semantics
 Note that a system can be viewed in terms of
a frame.
 W = a set of points.
 Ri = the relation for agent i.
Ri  {((r , m),(r ', m ')) | ri (m)  r 'i (m ')}
 This means that agent i considers (r’,m’)
possible at point (r,m) if I has the same local
environment at both point.
 This means, intuitively, that if agent i runs in r
at time m, then he could continue running in r’
at time m’.
41
Applying epistemic logic using
possible worlds semantics
 Let  be a set of propositions.
 These propositions describe facts about the
system as “the system is deadlocked” or “the
value of variable x is 5”.
 An interpreted system is a tuple (S,V), where
S is a system and V is a function that maps
propositions in , V(p,s){true, false}, where
p is a proposition and s is a state.
42
Applying epistemic logic using
possible worlds semantics
 We associate I=(S,V) with the modal structure
M=(W,R1,…,Rn,V). Thus, agents’ knowledge is
determined by their local environment.
 What it means for a formula  to be true at
point (r,m) in I?
 By applying earlier definitions we get:
(I, r , m) ‘ φ  ( M , (r , m)) ‘ φ
43
44
Applying epistemic logic
using axiomatic systems
• Martha puts a spot of mud on the forehead of
each child.
•Each child can see the forehead of the otherA knows that B’s forehead is muddy, and
conversely.
•Neither child knows whether their own
45
Applying epistemic logic
using axiomatic systems
• Martha announces, “At least one of you
[Martha said] Ka Kb ( A Ú B)
• Then she asks, “does either of you know
[b sees a] Ka ( Kb A Ú Kb ¬A)
• Neither child answers. [b doesn't know] Ka ¬Kb B
• She asks the same question again, and
this time both children answer- “I know
mine is”.
[We want] Ka A
• How did it happen?
46
Definitions
 In order to proof the conclusion we have to
take an axiomatic system of classical logic (as
HPC) and add some axioms and rules of
inference:
 Distributivity
Ka ( X  Y )  ( K a X  K aY )
 Truth
Ka X  X
(Semantically, R is reflexive)
 Rule N
X
Ka X
X Y
 Rule R
K a X  K aY
47
Proof
K a Kb (¬A  B)
2. Kb (¬A  B)  ( Kb ¬A  Kb B)
3. Ka Kb (¬A  B)  Ka ( Kb ¬A  Kb B)
4. Ka ( Kb ¬A  Kb B)
1.
[Martha said]
Distributivity
Rule R 2
MP 1,3
It means that A knows that if B knows that A’s
forehead is not muddy then B knows his
Dist.: K a ( X  Y )  ( K a X  K aY )
Truth: Ka X  X
X Y
Rule R:
K a X 48K aY
Proof
K a Kb (¬A  B)
[Martha said]
2. Kb (¬A  B)  ( Kb ¬A  Kb B)
Distributivity
3. Ka Kb (¬A  B)  Ka ( Kb ¬A  Kb B)
Rule R 1
4. Ka ( Kb ¬A  Kb B)
MP 1,3
5. ( Kb ¬A  Kb B)  (¬Kb B  ¬Kb ¬A)
CPL theorem
6. Ka ( Kb ¬A  Kb B)  Ka (¬Kb B  ¬Kb ¬A)
Rule R 5
7. Ka (¬Kb B  ¬Kb ¬A)
MP 4,6
8. Ka (¬Kb B  ¬Kb ¬A)  ( Ka ¬Kb B  Ka ¬Kb ¬A) Distributivity
9. K a ¬Kb B  K a ¬Kb ¬A
MP 7,8
1.
Dist.: K a ( X  Y )  ( K a X  K aY )
Truth: Ka X  X
X Y
Rule R:
K a X 49K aY
Proof (cont’d)
9. K a ¬Kb B  K a ¬Kb ¬A
MP 7,8
It means that A knows that if B doesn’t knows
whether his forehead is muddy then A knows
that it is possible in B’s knowledge that A’s
Remember that: [i]A <i>A
Dist.: K a ( X  Y )  ( K a X  K aY )
Truth: Ka X  X
X Y
Rule R:
K a X 50K aY
Proof (cont’d)
9. K a ¬Kb B  K a ¬Kb ¬A
10. K a ¬Kb B
11. K a ¬Kb ¬A
MP 7,8
[b doesn’t know]
MP 9,10
It means that A knows that it is possible in B’s
knowledge that A’s forehead is muddy!
Dist.: K a ( X  Y )  ( K a X  K aY )
Truth: Ka X  X
X Y
Rule R:
K a X 51K aY
Proof (cont’d)
Ka ¬Kb B  K a ¬Kb ¬A
K a ¬Kb B
K a ¬Kb ¬A
Ka (¬Kb ¬A  Kb A)
13. Ka (¬Kb ¬A  Kb A)  ( Ka ¬Kb ¬A  Ka Kb A)
14. K a ¬Kb ¬A  K a Kb A
15. K a Kb A
9.
10.
11.
12.
MP 7,8
[b doesn’t know]
MP 9,10
[b sees a]
Distribution
MP 12,13
MP 11,14
It means that A knows that B knows A’s forehead
is muddy!
Dist.: K a ( X  Y )  ( K a X  K aY )
Truth: Ka X  X
X Y
Rule R:
K a X 52K aY
Proof (cont’d)
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Ka ¬Kb B  K a ¬Kb ¬A
K a ¬Kb B
K a ¬Kb ¬A
Ka (¬Kb ¬A  Kb A)
Ka (¬Kb ¬A  Kb A)  ( Ka ¬Kb ¬A  Ka Kb A)
K a ¬Kb ¬A  K a Kb A
K a Kb A
Kb A  A
K a Kb A  K a A
Ka A
Dist.: K a ( X  Y )  ( K a X  K aY )
Truth: Ka X  X
MP 7,8
[b doesn’t know]
MP 9,10
[b sees a]
Distribution
MP 12,13
MP 11,14
Truth
Rule R 16
MP 15,17
X Y
Rule R:
K a X 53K aY
[Vaughan Pratt 1974]
54
Dynamic Logic
 We will concentrate on:
Propositional Dynamic Logic
(PDL)
[Fischer & Lander 1977]
55
What is Dynamic Logic?
 Program verification ensures that a program is
correct, meaning that any possible input/
output combination is expected based on the
specifications of the program.
 A modal logic, called dynamic logic, was
developed to verify programs.
56
PDL Syntax
 Let ={p1, p2, p3, … } – a non-empty set of
propositions.
 An ‘atomic’ program is a smallest basic
program, meaning it does not consist of other
programs.
 Let ={a1, a2, a3, … } – a non-empty set of
atomic programs.
57
PDL Formulas
 Formulas:
 If p, then p is a formula.
 If  and  are formulas, then , , ,
,  are formulas.
 If  is a formula and  is a program, then
[], <> are formulas.
58
PDL Programs
 Programs:
 If a, then a is a program.
 If  and  are programs, then ;(sequential
composition), (nondeterministic choice),
and *(iteration) are programs.
 If
 is a formula, then ? (test) is a program.
 Operators precedence:
 Unary operators.
 The operator ‘;’, and the  operator .
 Classical Logic operators.
59
Program Operators Interpretation
 ;: means “do  and then ”.
 : means “do either  or (non-
deterministically)”.
 *: means “repeat  some finite number of
times”.
 ?: means “test : continue if  is true,
otherwise ‘fail’”.
60
Modal Operators Interpretation
 [] means that “if  terminates, then 
holds” or in other terms “after every
terminating execution of ,  holds”.
 <> means that “there is an execution of 
that terminates with  as true”.
61
Programming Statements
 We can write some classical programming
statements, such as loop constructs, using
PDL program operators:
 ‘if  then  else ’ =def (?;)(?;)
 ‘while  do ’ =def (?;)*;?
 ‘repeat  until ’ =def ;(?;)*;?
62
PDL Semantics
 A modal structure is M=(W,{Ra|a},V).
 W is a set of program states.
 Ra is one or more binary relation(s) that
determines which states are accessible from
any state in W.
 V is a function from {W} to {true, false}.
63
Accessibility Relations
 We consider (w,w’)Ra as the case that w is
the initial state of program a and w’ is an
ending state.
 Developed accessibility relations:

Rα ; β  def {( w, w ') | w '' s.t. wRα w '' w '' Rβ w '}

Rα  β  def Rα  Rβ

Rα* def {(u, v) | u0 ,..., un where n  0, u  u0 , v  un
s.t. (ui , ui 1 )  Rα for 0  i  n  1}
 We will give the definition of R? after presenting the
definition of .
64
The Semantic Relation
 The relation
‘
between a pair (M,w) where
M is a model and w is a state, and a formula
A, is defined recursively as follows:
 M , w ‘ p, p  Φ  V ( p, w)  true

M , w ‘ pA Ù B  M , w ‘ A and M , w ‘ B
 Similar for the other classical logic connectors.
M , w ‘ [[α] A  x W if (w, x)  Rα then M , x ‘ A
 M,w‘ <
 α  A  x W s.t. (w, x)  Rα Ù M , x ‘ A

65
The R? relation
 R? =def {(u,u) | M,u  }
 For example, we can define the accessibility
relation for the while-do program (;)*;?:
 Rwhile  do  def {(u , v) | u0 ,..., un where n  0, u  u0 , v  un
s.t. M , ui ‘ φ, 0  i  n  1 and (ui , ui 1 )  Rα
for 0  i  n  1, and M , un ‘ φ}
66
Applications of PDL
 A correctness specification is a formal
description of how a program is to behave.
 A program is correct if its output meets the
correctness specification.
 PDL, and hence dynamic logic, is not well-
intermediary states. Other logics that do so
are process logic and temporal logic.
67
Applications of PDL
 PDL is better suited to reasoning about program
behavior with respect to only input and output
states.
 For example, the accessibility relation for a
program  only contains information about an
input and an output state, i.e., (w,w’)R means
that w’ is an output state when program  is run
with initial state w.
 Thus, a reasonable restriction for dynamic logic is
to only consider programs that halt (so its
correctness specifications are usually in the form
of input/output).
68
A PDL Example
 Let a,b be atomic programs.
 Let p be an atomic proposition.
 Suppose M=(W,Ra,Rb,V)
 W = {s,t,u,v}
 Ra = {(u,s),(v,t),(s,u),(t,v)}
 Rb = {(u,v),(v,u),(s,t),(t,s)}
s
b
a
p
u
t
a
b
V(p,u) = V(p,v) = true
v
70
s
A PDL Example
 Prove:
 Mp[(ab*a)*]p
b
a
p
u
t
a
b
v
 Proof:
 M,wp[(ab*a)*]p 
 (xW.(w,x)R(ab*a)*M,xp)  (M,wp)
 What is R(ab*a)*?
71
s
A PDL Example
 R(ab*a)*:
b
a
p
u
t
a
b
v
 Let’s build it from:
 Rb*={(u,u), (u,v), (v,u), (v,v), (s,s), (s,t),
(t,s), (t,t)}
 Rab*={(u,s), (u,t), (v,s), (v,t), (s,u), (s,v),
(t,u), (t,v)}
 Rab*a={(u,u), (u,v), (v,u), (v,v), (s,s), (s,t),
(t,s), (t,t)}
 R(ab*a)*={(u,u), (u,v), (v,u), (v,v), (s,s),
(s,t), (t,s), (t,t)}
72
s
A PDL Example
 M,wp[(ab*a)*]p 
b
a
p
u
t
a
b
v
(xW.(w,x)R(ab*a)*M,xp)  (M,wp)
 R(ab*a)*={(u,u), (u,v), (v,u), (v,v), (s,s), (s,t), (t,s),
(t,t)}
 If M,wp then w{u,v} and so:
 For each accessible state x from w (that are u
and v), M,xp.
 If M,wp then w{s,t} and so:
 There is an accessible state x from w (for
instance, s itself), such that M,xp.
 Thus, Mp[(ab*a)*]p.
73
s
A PDL Example
b
a
p
u
t
a
b
v
 Let:  = (aabb(abba)(aabb)*(abba))*
 M  []
 Proof:
 M,w [] 
 (xW.(w,x)RM,x)  (M,w)
 What is R?
74
s
A PDL Example
 R:
b
t
a
p
u
a
b
v
 Let’s build it from:
 Raa = {(u,u), (v,v), (s,s), (t,t)}
 Rbb = {(u,u), (v,v), (s,s), (t,t)}
 Rab = {(u,t), (v,s), (s,v), (t,u)}
 Rba = {(u,t), (v,s), (s,v), (t,u)}
 R = {(u,u), (v,v), (s,s), (t,t)}
The
identity
relation RI
  = (aabb(abba)(aabb)*(abba))*
75
s
A PDL Example
 M,w [] 
b
a
p
u
t
a
b
v
(xW.(w,x)RM,x)  (M,w)
 R = {(u,u), (v,v), (s,s), (t,t)}
 In conclusion,
 The only state accessible from w is w itself.
 And so, (xW.(w,x)RM,xp)  (M,wp)
 Thus, M [].
76
Summary
 Modal logic as an extension of classical logic
 Possible worlds semantics
 Logics and normal logics
 Axiomatic systems
 Extensions of multi-modal logic.
 First order modal logic
 Various Applications of modal logic- focus on
artificial intelligence and program verification
77
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