:
91
Neighbor Joining Algorithm
A
B
C
D
A
B
C
D

17
21
27

12
18

14

Neighbor Joining Algorithm
A
B
C
D
A
B
C
D
i
ui

17
21
27
A
(17+21+27)/2=32.5

12
18
B
(17+12+18)/2=23.5

14
C
(21+12+14)/2=23.5

D
(27+18+14)/2=29.5
Neighbor Joining
2421
3230
:
91
A (0,0) = A dX (0,0) = A dY (0,0) = 0
A (i , 0) = A dX (i , 0) = (i 1) , A dX (0, j) =
A (0, j) = A dY (0, j) = ( j 1) , A dY (i ,0) =
( 21
v=v1.vn M(v,w) w=w1wm
) (i,j vi
Incomputable Languages
Zeph Grunschlag
1
Announcements
HW Due Tuesday
2
Agenda
Incomputable Languages Existence Proof Explicit undecidable language: ATM Unrecognizable Language Neither recognizable nor corecognizable Example of a "real world" unsolvable
Journal of Universal Computer Science, vol. 2, no. 5 (1996), 410426 submitted: 13/5/96, accepted: 13/5/96, appeared: 28/5/96 Springer Pub. Co.
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John Chilton
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John Chilton
Recitation 6
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John Chilton
Recitation 6
LEFT
1 Post Correspondence Problem (PCP): Given a nite set of ordered pairs (x1 , y1 ), . . . , (xn , yn ) of strings over , determine whether there is a nite sequence of integers (i1 , i2 , . . . , im ), with each ij cfw_1, . . . , n, such that xi1 xi2 xim =
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V x X ` Xx Y)p$rYYf0TY) w p)x ssm Y0f)Yb)f w ) X y ot  X ` V x X ` Xx rp0)Yb$uprT) w p)qx pupm ! X y ot y a V f a xx x ` YdYYprrf0T Xx V V y f a X V a x `x x y X V y Xx  f a xx x ` Xx r)p4&YdfTY0)ep6rbr)f!T6fF0fpT)srYDcfw_)srr)b)0T) )p!W600&pYW
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Selected exercises with solutions on Computability theory in the eld of the
Theory of computation
Part 7
Amir Semmo
Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam
October 6, 2008
Exercise