Laborator AA

Key concepts:

1. computation (the output of the tape) and acceptance (acceptare);
2. mechanical description of an algorithm

1. (Answers online - discussion) A Turing Machine consists of:
   • an alphabet $ \Sigma$
   • a set of states $ K$
   • an initial state $ q_0$
   • a transition function $ \delta : K \times \Sigma \rightarrow K \times \Sigma \times \{L,H,R\}$
   • a set of final states $ F \subseteq K$

Which of the following components of an assembly language would best correspond to the above? $ K,\Sigma, \delta, q_0, F$

• the processor
• the memory
• registers
• assembly instructions

1. (Answers online) What does the following TM do? (bitwise complement)
2. (Answers online) Write a TM which accepts only if the input is a binary encoding of a even natural number.
3. (Answers online) Write a TM which adds 5 to a number encoded in binary on the tape. The machine will always accept.
4. (Answers online) Check if a symbol is present on the tape.
5. (Discussion) How would the following algorithm be represented as a Turing Machine:

Algorithm(vector V, integer M) {
   integer s = 0
   for-each x in V
      s += x
   if (x > 1000)
   then  return 1
   else  return 0
}

Helpful questions:

• how should the tape be organised?
• when should the machine accept?
• how would foreach x in V be implemented?
• how would s += x be implemented?
• how would if (x > 1000) then … else … be implemented ?

Homework:

• Write a TM which verifies if a string has the same number of ones and zeroes. Give hints - live (what should the machine do?)
• write a TM which accepts a given regular expression
• write a TM which reverses a given binary string (always accepts)

Key concepts:

• acceptance vs decision
• complement of a problem.

• Can the following problem be accepted by a TM? (f(x) = 0)
  • What is the complement of this problem?
• Can a problem be accepted by two different TMs? Can a TM accept two different problems?
• If a problem is accepted by some TM, can its complement also be accepted?
• If a problem is decided by some TM, can its complement also be decided?
• Write a TM which accepts is-odd problem but which does not decide it.
• Which of the following problems you think can be accepted and which can be decided? Use pseudocode instead of writing a TM.
  • a) V Undecidable example 1
  • b) V Hilbert undecidable
  • c) V Wang Tile
  • e) k-color
  • f) Linear Integer Programming

Key concepts:

• simulation

Exercises:

• The Von Newmann architecture - explained.
• Which of the components of Von Newmann arch. corresponds best to the TM?
• Write a TM pseudocode which verifies if a word is the proper encoding of a TM.

Discussion on the pseudocode.

• Write a TM pseudocode which accepts if there exists a word which is accepted by a given TM in k steps.

Discussion on the pseudocode

• Which of the following is a suitable pseudocode for a TM:

Algoritm(M,w){
   if size(w) > 10
     then if M halts for w in k steps
          accept.
} 

Algoritm(M1,M2,w){
   k = 0
   while true
       if M1(w) has the same behaviour as M2(w) after k steps
          then accept
       else k = k + 1
} 

Algorithm(M,A) {
   // A is a finite set of words
   for each w in A
       if M(w) halts  //undecidable! Pseudocode is ok, but this machine may not terminate
          then accept
}

Algorithm(M,w) {
   build the machine M' such that M(x) accepts iff M'(x) does not accept, for all words x
   if M'(w) in 1000 steps
      accept
}

Algorithm(M1,M2) {
    if M1 always halts then     //we know of no procedure, terminating or not, which can achieve this. This is not a proper TM/algorithm.
       if M2 always halts then 
          accept
}

• Write the problem which is accepted by each of the above machines.
• Write a TM pseudocode which accepts if a given word is accepted by two given TMs. Explain the dovetailing technique.

Homework:

• Write a TM pseudocode which accepts if there exists a word which is accepted by two given TMs.
• Write a TM pseudocode which accepts if there exists a TM which accepts a given word.
• Write a TM pseudocode which accepts if a given TM accepts some word of a given finite set A.
• Write a TM pseudocode which accepts if a given TM accepts all words of a given finite set A.

Von Newmann Model

Key concept:

• reduction
• proving a problem is not in R
• proving a problem is not in RE

• (Homework) Implement mergesort and insertionsort in python. Use a large dataset (provided by us) to test your implementation. Plot the execution times together with the functions $ n^2$ and $ n\cdot \log{n}$ using gnuplot. What do you observe? Adjust the constants for the previous functions so that the rate of growth can be better observed.
• Exercitii clasice

• Cativa algoritmi si recurentele lor
  • Merge-sort,
  • Quick-sort (curs)
  • Exemplul cu sqrt(n) al lui Sebi.
• Exercitii clasice

• Classical exercises

• Implement a SAT solver which encodes formulae as matrices and iterates over interpretations treating them as binary counters. Plot execution times.
• Implement a better SAT solver which uses BDDs to encode a formula. The variable ordering is known in advance. Plot execution times.
• Implement a k-Vertex-Cover solver using a reduction from SAT, and any of the above solvers.
• Exercitii clasice cu choice si reduceri

• Conceptul de operator vs cel de functie (exercitiu in C, exercitiu in Haskell, pe Liste)
• (Homework) Implementare LinkedList si ArrayList in Python, impreuna cu operatii. Implementare Haskell a operatiilor, dupa o discutie la curs despre acestea.
• Exercitii clasice