ITEC 420 - Final Exam Material - Fall 2011

Updates: Last modified on

• 12/12/11: Update for Fall 2011

Material

• Topics from Exam 1 and 2

• Material from chapters 3, 4, 5, and 7, as specified

• Pumping lemma:
• For Regular Languages: State the PL and describe what it is used to prove.
• For Context Free Languages: State the PL and describe what it is used to prove.
• For Regular Languages: Write a proof using the PL.


• Likely points distribution: about 225 points total with about 70 points Chapter 1, 55 points for Chapter 2, and 100 points for 3, 4, 5, and 7 [roughly: 40 on 3, 40 on 4 and 5, 20 on 7].

Chapter 3 - Turing Machines

• Describe the specifications of a Turing Machine (TM), particularly as opposed to other machines.

• Give the formal definition of a TM.

• Use configurations to trace a TM computation.

• Define the terms Turing recognizable and Turing decidable (aka decidable).

• Write simple TM algorithms.

• Draw a TM to implement an algorithm.

• Describe the relationship among single tape deterministic TM, multi-tape deterministic TM, and nondeterministic TM.

• Describe how to simulate a multitape Turing machine with a single-tape machine. [not on fall 2011 exam]

• Describe how to simulate a non-deterministic Turing machine with deterministic Turing machine. [not on fall 2011 exam]

• Describe the Church-Turing Thesis and its relevance.

Chapter 4 - Decidability

• Define the following languages and state whether each is decidable and/or recognizable:

  • A_DFA
  • A_NFA
  • A_REX
  • E_DFA
  • EQ_DFA
  • A_CFG
  • E_CFG
  • EQ_CFG
  • Any CFL
  • A_TM
  • A_TM^c (ie A _TM bar)

Chapter 4 - Halting Problem

• Define the term countable. [not on fall 2011 exam]

• Use a diagonalization argument to prove that the real numbers are not countable. [not on fall 2011 exam]

• Outline the proof that A_TM is not decidable.

• Show the correspondence among language classes: all, context free, decidable, recognizable, regular.

• Give example languages that characterize each of the classes above (ie that are in a class and not in the next smaller class)


• Describe the relationship between the number of Turing Machines and the number of languages.


• Describe the connection between a language and its complement with respect to Turing Decidability and Turing Recognizability.

Chapter 5 - Undecidable Problems from Language Theory

• Define the following languages and state whether each is decidable and/or recognizable:

  • HALT_TM
  • E_TM
  • EQ_TM


• Define the term reduce and describe how reduction applies to proving that a language is not recognizable.

• Outline a proof that HALT_TM is not decidable.

Chapter 7 - P and NP

• Define the class P.

• Explain how to show that a problem such as PATH is in P.

• Define the class NP

• State the Cook-Levin Theorem (either version). [not on fall 2011 exam]

• Define the term NP-Complete.

• Explain the term reduce and how it applies to NP-Complete problems.

• Explain the significance of NP-Complete languages.

• Explain the significance of finding a polynomial time solution to an NP-Complete problem.

• Explain how one NP-Complete problem can be used to show that another problem is NP-complete

• Show that P = NP!

Not on Fall 2011 Exam

• Describe Hilbert's Problems.

• Explain the term verifier.

• Define SAT .

• Define 3SAT .

• Show that problems such as HAMPATH, RELPRIME, and COMPOSITES are in NP.

• Explain the relationship between the two versions of the Cook-Levin Theorem.