Download 10 - SFU Computing Science

Computability and Complexity
10-1
Gödel’s Incompleteness
Theorem
Computability and Complexity
Andrei Bulatov
Computability and Complexity
Proof Systems We Use
Axioms:
Proof rules:
Logic axioms AX1-AX4 + Non-Logic axioms
modus ponens
,  | 
10-2
Computability and Complexity
Axioms of Number Theory
NT1 :
NT2 :
NT3 :
NT4 :
NT5 :
NT6 :
X ( ( X )  0)
X Y (( ( X )   (Y ))  ( X  Y ))
X ( X  0)  (Y ( (Y )  X )))
X ( X  0  X )
X Y ( X   (Y )   ( X  Y ))
X ( X  0  0)
NT7 : X Y ( X   (Y )  X  Y  X )
NT8 : X ( X ^0   (0))
NT9 : X Y ( X ^  (Y )  ( X ^ Y )  X )
NT10 : X ( X   ( X ))
NT11 : X Y (( X  Y )  ( ( X )  Y ))
NT12 : X Y (( X  Y )  (Y  X ))
NT13 : X Y Z ((( X  Y )  (Y  Z ))  ( X  Z ))
NT14 : X Y Z T ((mod( X ,Y , Z )  mod( X ,Y , T ))  ( Z  T ))
10-3
Computability and Complexity
Some Theorems (High School Identities)
(1) X  Y  Y  X
(2) X  (Y  Z )  ( X  Y )  Z
(3)
X 1  X
(4)
X Y  Y  X
(5)
X  (Y  Z )  ( X  Y )  Z
(6)
X  (Y  Z )  ( X  Y )  ( X  Z )
(7) 1X  1
(8)
X1  X
( 9)
X Y Z  X Y  X Z
(10) ( X  Y ) Z  X Z  Y Z
(11) ( X )Y ) Z  X Y Z
10-4
Computability and Complexity
Good Proof Systems
Definition
A proof system with the set of non-logical axioms  is said to be
consistent if there is no formula, , such that
   and   
Theorem
NT1-NT14 is consistent.
10-5
Computability and Complexity
Good Proof Systems
Definition
A proof system with the set of non-logical axioms  is said to be
acceptable if  is acceptable
Theoremhood
Instance: A proof system with the set of non-logical axioms  and
a formula .
Question:    ?
The corresponding language is:
Theorem
LTheorem  {" " |    }
If  is acceptable, then LTheorem is acceptable.
10-6
Computability and Complexity
Proof Idea
Given a formula , let S1 , S 2 , be a list of all sequences
of formulas which end with .
S1 :
S2 :
S3 :

11
 21
 31

12
 22
 32

13
 23
 33




Perform 1st step of an acceptor for  11
Perform 2nd step of an acceptor for  11 and 1st step of an acceptor for 12
Perform 3rd step for  11 , 2nd step for 12 and 1st step for  21
…
10-7
Computability and Complexity
Proof Systems and Models
Let M be a model
Definition
A proof system  is sound for M, if every theorem of 
belongs to Th(M)
Theorem
NT1-NT14 is sound for N.
Definition
A proof system  is complete for M, if every sentence from
Th(M) is a theorem of 
10-8
Computability and Complexity
Gödel’s Incompleteness Theorem
Theorem
Any acceptable proof system for N is either inconsistent
or incomplete.
Corollary
Any acceptable proof system that is powerful enough (to reason
about N) is either inconsistent or incomplete.
10-9
Computability and Complexity
10-10
Proof Idea (we use)
Step 1:
Encode TM descriptions, configurations and computations using natural
numbers
Step 2:
Encode properties of TMs as properties of numbers representing them
Step 3:
Reducing the Halting problem show that Th(M) and its complement are
undecidable
Step 4:
Using the theorem about acceptability of Theoremhood and observing that
Th(M) is acceptable if and only if its complement is, conclude the theorem
Computability and Complexity
Proof Idea (Gödel used)
Step 1:
Encode variables, predicate and function symbols, quantifiers and first
order formulas using natural numbers
Step 2:
Encode properties of first order formulas (in the vocabulary of number
theory) as properties of numbers representing them
Step 3:
Construct a formula claiming “I am not a theorem in your proof system.”
Step 4:
Observe that if this formula is true (in N), then it is not a theorem in the
proof system and, therefore, the system is incomplete; if it is false, then
there is a false theorem, i.e. the proof system is not sound
10-11
Computability and Complexity
Computations as Natural Numbers
We design a computable function  that maps TM descriptions,
configurations and computations into N
We know how all these objects can be encoded into 01-strings.
 just outputs the number for which this string is the binary representation
Note that the converse function is also computable, because the ith bit of
the binary representation of a number n can be computed:
m
n
m  2   where m   i 1 
2
2 
Similarly, there is a first order formula (X) meaning “the ith bit of X is 1”:
( X )  Y ((( X  2  Y )  ( X  2  Y  1))  ( X  2  Y  1))
(this is for the last bit)
10-12
Computability and Complexity
10-13
Example
a|a|R
q0
Encoding:
b|b|RR
q1
( q0 ,  ), ( q1 , , R )
( q0 , a ), ( q0 , a, R )
( q0 , b), ( q1 , b, R )
q0  0, q1  00,
  0, a  00, b  000
R  0, L  00, S  000
0101001010110100101001011010001001000101
Configuration:
a, q0 , a, b
0011011001000
Computability and Complexity
We construct a formula that, given 3 numbers X, Y and Z, is true
if and only if the machine encoded X moves from the configuration
encoded Y into the configuration encoded Z
a, q0 , a, b
0011011001000


a, a, q0 , b
0010011011000

( X , Y , Z )  block of bits
( the correspondent pieces of Y and Z are a rule of X )
10-14
Computability and Complexity
Claim 1.
There is a first order formula (X,Y) which is true if and only if Y is a
computation of the TM encoded X
Claim 2.
There is a first order formula (X,Y,Z) which is true if and only if Y is a
computation of the TM encoded X on input Z
Claim 3.
There is a first order formula (X,Y) which is true if and only if Y is a
computation of the TM encoded X and Y ends in a final state
Claim 4.
There is a first order formula (X,Y) which is true if and only if the TM
encoded X halts on input Y
( X ,Y )  Z (( X , Z ,Y )  ( X , Z ))
10-15
Computability and Complexity
10-16
Finally, to reduce LHalting to Th(N), we define a mapping as follows:
"T ; w"  (e(T ), e( w))
Observe that
• This mapping is computable
• The obtained formula is a sentence
• This sentence is true if and only if T halts on w
QED