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)

• Exercitii simple cu MT
• Exercitii cu MTU si conceptul de simulare

Von Newmann Model

• Algoritmi

• Ce este o problema de decizie?
• Problema si rezolvare;
• Lista de probleme care (pot/nu pot) fi rezolvate;
• Reduceri
• Further work: coRE, complement, dovetailing

• Implementari care sa ilustreze faptul ca constanta conteaza/nuconteaza
• Implementari care sa ilustreze faptul ca log n » n » n2 »> n3
• Grafice (in ?!)
• Exercitii clasice

• Cativa algoritmi si recurentele lor
• Exercitii clasice

• Implementare care sa ilustreze exponentiala (backtracking); legatura cu MTN.
• Reduceri pentru SAT solvere
• Exercitii clasice cu reduceri

• Conceptul de operator vs cel de functie (exercitiu in C, exercitiu in Haskell, pe Liste)
• Exercitii clasice