:
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.
CHAITIN'S TOYLISP ON A CONNEX MEMORY MACHINE 1
Politechnical University of Bucharest, Department of Electronics
Introduction to the Theory of Computation, Michael Sipser
Chapter 0: Introduction
Automata, Computability and Complexity: They are linked by the question: o What are the fundamental capabilities and limitations of computers? The theories of computability
Design & Analysis of Algorithms  HW 4
Submission date: 22.01.2009 14:00 1. A truck owner is performing a delivery from Haifa to Eilat. The truck has three containers in the front, middle and back of the truck. The sizes and of the containers (in cubic me
The University of Nottingham
School of Computer Science and IT
LECTURE 5 TURING MACHINES PART 1
Overview
The Turing Machine Model Examples of Turing Machine
The Turing Machine Model
The Turing machine (TM) is a simple and powerful mathematical model of c
CS5371 Theory of Computation
Homework 4 (Suggested Solution) 1. Ans. Suppose on the contrary that T is decidable. Let R be its decider. Then, the following TM Q is a decider for AT M : Q = On input M , w , 1. Construct a TM M as follows: M = On input x, 1
CS5371 Theory of Computation
Homework 4 Due: 3:20 pm, December 21, 2007 (before class) 1. (20%) Let T = cfw_ M  M is a TM that accepts wR whenever it accepts w. Show that T is undecidable. 2. (15%) In the silly Post Correspondence Problem, SPCP, in each
CS5371 Theory of Computation
Homework 3 (Suggested Solution) 1. Ans. The language cfw_0n 1n 2n  n 1 is not contextfree, so that it cannot be recognized by a 1PDA. However, we can easily design a 2PDA to recognize this language as follows (Let S1 and S
CS5371 Theory of Computation
Homework 3 (Solution) 1. Show that singletape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages. Answer: Let M = (Q, , , q0 , qaccept , qreject ) be a singletape TM
CS5371 Theory of Computation
Homework 3 Due: 2:10 pm, December 4, 2007 (before class) 1. Let k PDA be a pushdown automaton that has k stacks. Thus a 0PDA is an NFA and a 1PDA is a conventional PDA. We already know that 1PDAs are more powerful than 0P
Department of Computer Science COMPSCI 350 Assignment 2 Due: May 15
1. Show that ALLDFA = cfw_ B  B is DFA and L(B ) = is decidable. [10 marks] Solution: The following Turing machine decides ALLDFA : M = on input B where B is a DFA: 1. Let C be the DFA
Recitation 6
John Chilton
July 18, 2007
John Chilton
Recitation 6
At the bottom of the proof for 5.13, write down one specic thing that you were the most unclear about while doing homework 5. Try to be as specic as possible.
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 =
3515ICT Theory of Computation Tutorial problems: Turing machines, decidability and undecidability 1. Give implementation level and formal descriptions of a Turing machine to recognise the language L = cfw_ an bn cn  n 0 . You may assume either a single
CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
Homework 10 Solutions
1. If A m B and B is a regular language, does that imply that A is a regular language? Answer: No. For example, dene the languages A = cfw_ 0n 1n  n 0 and B = cfw_1,
CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
Homework 9 Solutions
1. Let B be the set of all innite sequences over cfw_0, 1. Show that B is uncountable, using a proof by diagonalization. Answer: Each element in B is an innite sequence
TWCd )
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
Homework 3CSCI6339 Fall 2009
Due on November 18, 2008 (Wednesday)
Type your solution with Microsoft word. 1. Show that T=cfw_(i,j,k) i, j, k are all integers is countable.
2. Show that there is a correspondence between the two intervals (0,1) and [0,1].
ECS 120: Intoduction to the Theory of Computation Homework 8
Due: Friday, May 30, 2003 at the beginning of class
Problem 1. Classify the following languages as decidable, recognizable (but not decidable), corecognizable (but not decidable), or neither re
CS 154  Introduction to Automata and Complexity Theory
Spring Quarter, 2000 Assignment 6  Due date: Wednesday, 5 24 00
ALERT: This is your last chance to use the late credit for homeworks. If you have been wise and
have saved up your credit, now is the
Applied Computer Science II Winter semester 2001/2002 Exercise sheet 5
Institut fr Informatik, u AlbertLudwigsUniversitt Freiburg. a Date: 21.11.2001 Deadline: 28.11.2001 (before the lecture)
1. (4 points) Examine the formal denition of a Turing machine
CpSc 421
Homework 9
Solution
Attempt any three of the six problems below. The homework is graded on a scale of 100 points, even though you can attempt fewer or more points than that. Your recorded grade will be the total score on the problems that you att
Selected exercises with solutions on Computability theory in the eld of the
Theory of computation
Part 8
Amir Semmo
Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam
October 6, 2008
Exercise
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
Selected exercises with solutions on Computability theory in the eld of the
Theory of computation
Part 6
Amir Semmo
Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam
October 6, 2008
Exercise