CS2200 - Languages, Machines and Computation

Meetings  

• Theme 1 : Finite Automata & Regular Languages - 17 meetings

  • Meeting 01 : Tue, Jan 03, 10:00 pm-10:50 pm

  Introduction. Outline of the course. Administrative details. The science vs engineering perspectives towards computation. The theory to aim for.

  References
  Exercises
  Reading

  Introduction. Outline of the course. Administrative details. The science vs engineering perspectives towards computation. The theory to aim for.
  

  References

  :

  None

  • Meeting 02 : Wed, Jan 04, 09:00 am-09:50 am

  Engineering side : Need of system modeling and verification. Intel FDIV bug. Push-button system as a simple system to study. The notion of a state.

  References	Section 1.1 of HMU textbook.
  Exercises	Correctness Proof strategy for Push Button Switch
  Reading	<a href="http://www.trnicely.net/pentbug/pentbug.html">Pentium FDIV flaw</a>

  Engineering side : Need of system modeling and verification. Intel FDIV bug. Push-button system as a simple system to study. The notion of a state.
  

  References	:	Section 1.1 of HMU textbook.
  Exercises	:	Correctness Proof strategy for Push Button Switch
  Reading	:	Pentium FDIV flaw

  • Meeting 03 : Thu, Jan 05, 01:00 pm-01:50 pm

  Science side : Notion of computation. Compass-straight edge construction from Ancient greeks. Computation as problem solving. View of the push button switch as a problem solver. Coffee-Tea machine as an additional example.

  References	Section 1.1-1.4 of HMU textbook.
  Exercises	(semi)formally express : what problem is solved by the Coffee-Tea machine?
  Reading	<a href="http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions">Compass and Straight-edge constructions</a> - Wikipedia Link.

  Science side : Notion of computation. Compass-straight edge construction from Ancient greeks. Computation as problem solving. View of the push button switch as a problem solver. Coffee-Tea machine as an additional example.
  

  References	:	Section 1.1-1.4 of HMU textbook.
  Exercises	:	(semi)formally express : what problem is solved by the Coffee-Tea machine?
  Reading	:	Compass and Straight-edge constructions - Wikipedia Link.

  • Meeting 04 : Tue, Jan 10, 10:00 pm-10:50 pm

  What language does Push-button switch know? How do we express languages formally? The language for languages : Alphabets, Strings, Sigma*, Languages. Languages vs Problems.

  References	Section 1.4 and 2.2 of HMU textbook. Kozen Book Lecture 3.
  Exercises
  Reading

  What language does Push-button switch know? How do we express languages formally? The language for languages : Alphabets, Strings, Sigma*, Languages. Languages vs Problems.
  

  References

  :

  Section 1.4 and 2.2 of HMU textbook. Kozen Book Lecture 3.

  • Meeting 05 : Wed, Jan 11, 09:00 am-09:50 am

  Modeling the push-button switch, Finite states, Finite alphabet, start state, transitions. Need of final states for the theory. Language accepted by a machine (informally).

  References	Section 1.5 of HMU textbook.
  Exercises
  Reading

  Modeling the push-button switch, Finite states, Finite alphabet, start state, transitions. Need of final states for the theory. Language accepted by a machine (informally).
  

  References

  :

  Section 1.5 of HMU textbook.

  • Meeting 06 : Thu, Jan 12, 01:00 pm-01:50 pm

  Extended transition functions, Language accepted by a finite state automaton. A formal proof that the push-button switch is indeed correct by design. The decimal-mod-3 machine.

  References	HMU Book Section 2.2.4, Kozen Book Lecture 3 and 4.
  Exercises
  Reading

  Extended transition functions, Language accepted by a finite state automaton. A formal proof that the push-button switch is indeed correct by design. The decimal-mod-3 machine.
  

  References

  :

  HMU Book Section 2.2.4, Kozen Book Lecture 3 and 4.

  • Meeting 07 : Mon, Jan 16, 11:00 am-11:50 am

  Proof of the correctness of decimal-mod3 machine. Regular Languages.<br> Operations on languages. Closure of Regular Languages under complementation.

  References	Lecture 4 in Kozen's Book.
  Exercises	Attempt on the formal proof that Regular languages are closed under complementation.
  Reading

  Proof of the correctness of decimal-mod3 machine. Regular Languages.
  Operations on languages. Closure of Regular Languages under complementation.
  

  References	:	Lecture 4 in Kozen's Book.
  Exercises	:	Attempt on the formal proof that Regular languages are closed under complementation.

  • Meeting 08 : Tue, Jan 17, 10:00 am-10:50 am

  Product construction and the formal proof of correctness. Regular Languages are closed under complementation, intersection. DeMorgan's Law for union case.

  References	Lecture 4 in Kozen's Book.
  Exercises	Formally prove that regular languages are closed under union using the product construction (without using the DeMorgan's law).
  Reading

  Product construction and the formal proof of correctness. Regular Languages are closed under complementation, intersection. DeMorgan's Law for union case.
  

  References	:	Lecture 4 in Kozen's Book.
  Exercises	:	Formally prove that regular languages are closed under union using the product construction (without using the DeMorgan's law).

  • Meeting 09 : Mon, Jan 23, 11:00 am-11:50 am

  Are all languages regular? Question 5 in the Tutorial -1. Developing a proof strategy. A complete proof that the language is not regular.

  References	Lecture 11 in Kozen, Section 4.1 in HMU.
  Exercises
  Reading

  Are all languages regular? Question 5 in the Tutorial -1. Developing a proof strategy. A complete proof that the language is not regular.
  

  References

  :

  Lecture 11 in Kozen, Section 4.1 in HMU.

  • Meeting 10 : Tue, Jan 24, 10:00 am-10:50 am

  Another complete proof : a^nb^n is not regular. Another one, d^ma^nb^n is not regular. Observations from the proofs. Abstraction of the proof strategies as a Lemma. Pumping Lemma for Regular Languages. Proof as a game with the demon.

  References	Lecture 11 in Kozen's Book. <br> Chapter 4 in HMU Book.
  Exercises	We wrote pumping lemma as implied by the proofs that we did in the class. Write a complete proof of the pumping lemma combining the proofs of the claims that we have done in the class.
  Reading	A natural question about which you could read and think : Is the converse of pumping Lemma true?. If the RHS is true for a language A, does it imply that the language A is regular? <br> If yes, how will we construct the automaton just from the fact that it satisfies the RHS? <br> If no, we have to show an example of a language A satisfying the RHS of the pumping lemma . Aha, but then we need a method of showing that some language is not regular without using the pumping lemma !.

  Another complete proof : a^nb^n is not regular. Another one, d^ma^nb^n is not regular. Observations from the proofs. Abstraction of the proof strategies as a Lemma. Pumping Lemma for Regular Languages. Proof as a game with the demon.
  

  References	:	Lecture 11 in Kozen's Book. Chapter 4 in HMU Book.
  Exercises	:	We wrote pumping lemma as implied by the proofs that we did in the class. Write a complete proof of the pumping lemma combining the proofs of the claims that we have done in the class.
  Reading	:	A natural question about which you could read and think : Is the converse of pumping Lemma true?. If the RHS is true for a language A, does it imply that the language A is regular? If yes, how will we construct the automaton just from the fact that it satisfies the RHS? If no, we have to show an example of a language A satisfying the RHS of the pumping lemma . Aha, but then we need a method of showing that some language is not regular without using the pumping lemma !.

  • Meeting 11 : Wed, Jan 25, 01:00 pm-01:50 pm

  Myhill-Nerode Relations. ( <a href="CS2200/myhill-nerode.pdf">Notes</a> ).

  References	Chapter on Myhill-Nerode relations in Kozen's Book.
  Exercises	We showed in class a right congruence property for the Myhill-Nerode relations defined from an automata. How about a left-congruence relation? Try to define this notion, and argue.
  Reading

  Myhill-Nerode Relations. ( Notes ).
  

  References	:	Chapter on Myhill-Nerode relations in Kozen's Book.
  Exercises	:	We showed in class a right congruence property for the Myhill-Nerode relations defined from an automata. How about a left-congruence relation? Try to define this notion, and argue.

  • Meeting 12 : Mon, Jan 30, 11:00 am-11:50 am

  Myhill-Nerode theorem. Forward direction. Showing non-regularity of languages using Myhill-Nerode theorem.

  References	The notes sent to the course mailing list
  Exercises	Try showing that { a^p : p is a prime } is not regular using MN theorem.
  Reading

  Myhill-Nerode theorem. Forward direction. Showing non-regularity of languages using Myhill-Nerode theorem.
  

  References	:	The notes sent to the course mailing list
  Exercises	:	Try showing that { a^p : p is a prime } is not regular using MN theorem.

  • Meeting 13 : Tue, Jan 31, 10:00 am-10:50 am

  Proof of reverse direction in Myhill-Nerode Theorem. Showing lower bounds on the number of states of a finite automata.

  References	The notes sent to the course mailing list
  Exercises
  Reading

  Proof of reverse direction in Myhill-Nerode Theorem. Showing lower bounds on the number of states of a finite automata.
  

  References

  :

  The notes sent to the course mailing list

  • Meeting 14 : Wed, Feb 01, 09:00 am-09:50 am

  Myhill-Nerode relations, refinement, the coarsest Myhill-Nerode relation.

  References	The notes sent to the course mailing list
  Exercises
  Reading

  Myhill-Nerode relations, refinement, the coarsest Myhill-Nerode relation.
  

  References

  :

  The notes sent to the course mailing list

  • Meeting 15 : Thu, Feb 02, 01:00 pm-02:00 pm

  Coarsest MN relation, Connections to minimization, Examples of Minimization Process. Need of an algorithm.

  References	Notes sent to the course mailing list.
  Exercises
  Reading

  Coarsest MN relation, Connections to minimization, Examples of Minimization Process. Need of an algorithm.
  

  References

  :

  Notes sent to the course mailing list.

  • Meeting 16 : Mon, Feb 06, 11:00 am-11:50 am

  A criterion for merging of states. Marking Algorithm, Examples

  References	Kozen Book - Lecture 13, last section (for the criterion). Lecture 14, for the algorithm and the examples.
  Exercises
  Reading

  A criterion for merging of states. Marking Algorithm, Examples
  

  References

  :

  Kozen Book - Lecture 13, last section (for the criterion). Lecture 14, for the algorithm and the examples.

  • Meeting 17 : Tue, Feb 07, 10:00 am-10:50 am

  Proof of correctness of Marking Algorithm for Minimization of Automaton. An introduction to non-determinism.

  References	Induction Proof done in class.
  Exercises
  Reading

  Proof of correctness of Marking Algorithm for Minimization of Automaton. An introduction to non-determinism.
  

  References

  :

  Induction Proof done in class.

• Theme 2 : Non-determinism & Two-way Access - 8 meetings

  • Meeting 18 : Wed, Feb 08, 09:00 am-09:50 am

  An introduction to non-determinism. Non-deterministic Finite Automaton, Formal model, extended transition function.

  References	Kozen's Book - Lecture 5 first two sections, Lecture 6 first two sections.
  Exercises
  Reading

  An introduction to non-determinism. Non-deterministic Finite Automaton, Formal model, extended transition function.
  

  References

  :

  Kozen's Book - Lecture 5 first two sections, Lecture 6 first two sections.

  • Meeting 19 : Mon, Feb 13, 11:00 am-11:50 am

  Example NFAs, Subset construction, Formal Proof of correctness. Epsilon Transitions

  References	Lecture 5,6 of Kozen Book.
  Exercises
  Reading

  Example NFAs, Subset construction, Formal Proof of correctness. Epsilon Transitions
  

  References

  :

  Lecture 5,6 of Kozen Book.

  • Meeting 20 : Tue, Feb 14, 10:00 pm-10:50 pm

  Epsilon Transitions. Example constructions

  References	Lecture 5 and 6 in Kozen Textbook
  Exercises	Try to prove that if A, B, C are regular languages, then the concatenation language ABC is also regular. The definition is same as what we used in class.
  Reading

  Epsilon Transitions. Example constructions
  

  References	:	Lecture 5 and 6 in Kozen Textbook
  Exercises	:	Try to prove that if A, B, C are regular languages, then the concatenation language ABC is also regular. The definition is same as what we used in class.

  • Meeting 21 : Mon, Feb 20, 11:00 am-11:50 am

  *-closure of regular languages. Regular Expressions, Constructing automaton from regular expressions. Simplification of regular expressions.

  References	Page 37-39 in Kozen's Book.
  Exercises	Construct the equivalent automaton for regular expression (a+b)*+(a*b^2)*.
  Reading

  *-closure of regular languages. Regular Expressions, Constructing automaton from regular expressions. Simplification of regular expressions.
  

  References	:	Page 37-39 in Kozen's Book.
  Exercises	:	Construct the equivalent automaton for regular expression (a+b)*+(a*b^2)*.

  • Meeting 22 : Tue, Feb 21, 10:00 pm-10:50 pm

  Constructing Regular expressions from Automaton - the recursive construction. Simplification of regular expressions.

  References	Lecture 9 in Kozen's Book.
  Exercises	Try to simplify the regular expression that we obtained for the example automaton in class.
  Reading

  Constructing Regular expressions from Automaton - the recursive construction. Simplification of regular expressions.
  

  References	:	Lecture 9 in Kozen's Book.
  Exercises	:	Try to simplify the regular expression that we obtained for the example automaton in class.

  • Meeting 23 : Wed, Feb 22, 09:00 am-09:50 am

  Simplification of regular expressions. <br>Introduction to 2DFAs. Informal view. 2-way access to the tape. The properties that the transition function should satisfy to formally capture the model.

  References	Lecture 17 of Kozen's textbook.
  Exercises	Try designing a 2DFA for the language: <br> A = {x : the number of as in x is divisible by 5 and the number bs in x is divisible by 7} <br> But note that this language is regular (closure under intersection !!). <br> Are there languages where this could actually help?
  Reading

  Simplification of regular expressions.
  Introduction to 2DFAs. Informal view. 2-way access to the tape. The properties that the transition function should satisfy to formally capture the model.
  

  References	:	Lecture 17 of Kozen's textbook.
  Exercises	:	Try designing a 2DFA for the language: A = {x : the number of as in x is divisible by 5 and the number bs in x is divisible by 7} But note that this language is regular (closure under intersection !!). Are there languages where this could actually help?

  • Meeting 24 : Thu, Feb 23, 01:00 pm-01:50 pm

  Two-way access to the input tape. Model, Configurations, Acceptance, Language computed.

  References	Lecture 17-18 of Kozen's textbook.
  Exercises	Write down the transition table for the 2DFA that we described informally. Solve Example 17.2 completely.
  Reading

  Two-way access to the input tape. Model, Configurations, Acceptance, Language computed.
  

  References	:	Lecture 17-18 of Kozen's textbook.
  Exercises	:	Write down the transition table for the 2DFA that we described informally. Solve Example 17.2 completely.

  • Meeting 25 : Mon, Feb 27, 11:00 am-11:50 am

  2DFAs can accept only regular sets. <br> A proof using Myhill-Nerode theorem.

  References	Lecture 17 of Kozen's textbook.
  Exercises	<ul> <li>A nontrivial application of 2DFA. Try the following: <br> Let A be a regular language. Consider the set B = { x : x^|x| is in A }. Prove that B is regular. <br> Hint : Designing 2DFA for accepting might be easier, and is equivalent to designing an FA as we proved in the class.</li> <li>What about non-deterministic 2way DFAs? Can they accept non-regular sets? <br> Hint : try the subset construction.</li> </ul>
  Reading	An alternative proof of the fact that 2DFAs can accept only regular sets. One can design an NFA directly from the 2DFA (and since NFAs accept only regular sets this will imply that 2DFAs can accept only regular sets). This is presented as an exercise (giving the individual steps) in Kozen : Misc.Ex.61 (Page 331).

  2DFAs can accept only regular sets.
  A proof using Myhill-Nerode theorem.
  

  References	:	Lecture 17 of Kozen's textbook.
  Exercises	:	A nontrivial application of 2DFA. Try the following: Let A be a regular language. Consider the set B = { x : x^|x| is in A }. Prove that B is regular. Hint : Designing 2DFA for accepting might be easier, and is equivalent to designing an FA as we proved in the class. What about non-deterministic 2way DFAs? Can they accept non-regular sets? Hint : try the subset construction.
  Reading	:	An alternative proof of the fact that 2DFAs can accept only regular sets. One can design an NFA directly from the 2DFA (and since NFAs accept only regular sets this will imply that 2DFAs can accept only regular sets). This is presented as an exercise (giving the individual steps) in Kozen : Misc.Ex.61 (Page 331).

• Theme 3 : Grammars & Context Free Languages - 17 meetings

  • Meeting 26 : Tue, Feb 28, 10:00 pm-10:50 pm

  Language centric view, Grammars, Formal definition, derivation of strings. <br> Grammar for {a^nb^n : n >= 0} language.

  References	Lecture 19 of Kozen's Textbook.
  Exercises	Complete the induction proof that we outlined in class. For the grammar that we defined show that L(G) = {a^nb^n : n >= 0}.
  Reading	Lecture 20 of Kozen's textbook.

  Language centric view, Grammars, Formal definition, derivation of strings.
  Grammar for {a^nb^n : n >= 0} language.
  

  References	:	Lecture 19 of Kozen's Textbook.
  Exercises	:	Complete the induction proof that we outlined in class. For the grammar that we defined show that L(G) = {a^nb^n : n >= 0}.
  Reading	:	Lecture 20 of Kozen's textbook.

  • Meeting 27 : Wed, Feb 29, 09:00 am-09:50 am

  Context free Grammars. PAREN language. <br> Formal definition and Proof of acceptance by induction.

  References	Lecture 20 of Kozen's Textbook.
  Exercises	Reading assignment. Write the induction proof on your own for the correctness of the grammar for PAREN. Work out the details from the textbook (lecture 20). <br> Write a grammar for the set of palindromes over {a,b}^*
  Reading

  Context free Grammars. PAREN language.
  Formal definition and Proof of acceptance by induction.
  

  References	:	Lecture 20 of Kozen's Textbook.
  Exercises	:	Reading assignment. Write the induction proof on your own for the correctness of the grammar for PAREN. Work out the details from the textbook (lecture 20). Write a grammar for the set of palindromes over {a,b}^*

  • Meeting 28 : Thu, Mar 01, 01:00 pm-01:50 pm

  All regular languages are context-free. Right-Linear Grammars. <br> Normal forms.

  References	Class notes.<br> Lecture 21 of Kozen's Book for CNF.
  Exercises	Show that a language is right-linear CFL if and only if it is regular. We showed only one direction which itself was sufficient to argue that all regular langauges are context-free. Show that we can design an automaton from a right linear grammar, accepting the same language.
  Reading	Read about Left Linear Grammars.

  All regular languages are context-free. Right-Linear Grammars.
  Normal forms.
  

  References	:	Class notes. Lecture 21 of Kozen's Book for CNF.
  Exercises	:	Show that a language is right-linear CFL if and only if it is regular. We showed only one direction which itself was sufficient to argue that all regular langauges are context-free. Show that we can design an automaton from a right linear grammar, accepting the same language.
  Reading	:	Read about Left Linear Grammars.

  • Meeting 29 : Mon, Mar 05, 11:00 am-11:50 am

  Conversion of an arbitrary CFG to CNF. Removal of epsilon and unit productions. Final conversion to a CNF. An argument to why the modifications does not change the language (except losing out the epsilon string).

  References	Lecture 21 in Kozen's Textbook. <br> Additional reference is <a href"http://www.cs.nyu.edu/courses/fall07/V22.0453-001/cnf.pdf">here</a>.
  Exercises	Write down the complete formal proof that our modification does not change the language generated by the grammar.See Lecture 21 in Kozen's textbook for details.
  Reading

  Conversion of an arbitrary CFG to CNF. Removal of epsilon and unit productions. Final conversion to a CNF. An argument to why the modifications does not change the language (except losing out the epsilon string).
  

  References	:	Lecture 21 in Kozen's Textbook. Additional reference is here.
  Exercises	:	Write down the complete formal proof that our modification does not change the language generated by the grammar.See Lecture 21 in Kozen's textbook for details.

  • Meeting 30 : Tue, Mar 06, 10:00 pm-10:50 pm

  Pumping Lemma for Context Free Languages.

  References	Lecture 22, Kozen's Textbook.
  Exercises	Try to prove that L = {a^nb^na^n} is not a CFL by applying the pumping lemma. You need to come up with a winning strategy against the Demon.
  Reading

  Pumping Lemma for Context Free Languages.
  

  References	:	Lecture 22, Kozen's Textbook.
  Exercises	:	Try to prove that L = {a^nb^na^n} is not a CFL by applying the pumping lemma. You need to come up with a winning strategy against the Demon.

  • Meeting 31 : Wed, Mar 07, 09:00 am-09:50 am

  Example applications of Pumping Lemma. <br> Closure Properties : CFLs are closed under union, Kleene closure, concatenation. CFLs are not closed under intersection and complementation.

  References	Last part of Lecture 22, Page 155 of Kozen's Textbook.<br> Page 195-196 of Kozen's textbook.
  Exercises	We argued that CFLs are not closed under intersection. That is, we showed that there are two CFLs whose intersection is not a CFL. Notice that this does not say that intersection of two CFLs is never a CFL. Give examples of two CFLs whose intersection is actually a CFL.
  Reading	We stated Parikh's theorem (by Rohit Parikh, 1966) in class. Every CFL over a singleton alphabet is regular. That is, it is not just a coincidence that all the examples of such non-regular languages (that we tried out in class) happened to be non-context-free also. <br> For the motivated readers : here is very good reference for Parikh's theorem. Supplementary lecture H, Page 201, Kozen's textbook.<br> Here is another : <a href="http://www8.cs.umu.se/kurser/TDBC92/VT06/final/3.pdf">An exposition of Parikh's theorem</a>.

  Example applications of Pumping Lemma.
  Closure Properties : CFLs are closed under union, Kleene closure, concatenation. CFLs are not closed under intersection and complementation.
  

  References	:	Last part of Lecture 22, Page 155 of Kozen's Textbook. Page 195-196 of Kozen's textbook.
  Exercises	:	We argued that CFLs are not closed under intersection. That is, we showed that there are two CFLs whose intersection is not a CFL. Notice that this does not say that intersection of two CFLs is never a CFL. Give examples of two CFLs whose intersection is actually a CFL.
  Reading	:	We stated Parikh's theorem (by Rohit Parikh, 1966) in class. Every CFL over a singleton alphabet is regular. That is, it is not just a coincidence that all the examples of such non-regular languages (that we tried out in class) happened to be non-context-free also. For the motivated readers : here is very good reference for Parikh's theorem. Supplementary lecture H, Page 201, Kozen's textbook. Here is another : An exposition of Parikh's theorem.

  • Meeting 32 : Mon, Mar 12, 11:00 am-11:50 am

  Machine Model - Nondeterministic Pushdown Automata - Machine accepting a^nb^n and PAREN.

  References	Lecture in Kozens Book.
  Exercises	We defined the PDAs formally as a tuple. We also described the machines for the languages that we described. Describe formally the tuple representation of the PDAs for a^nb^n and PAREN.
  Reading

  Machine Model - Nondeterministic Pushdown Automata - Machine accepting a^nb^n and PAREN.
  

  References	:	Lecture in Kozens Book.
  Exercises	:	We defined the PDAs formally as a tuple. We also described the machines for the languages that we described. Describe formally the tuple representation of the PDAs for a^nb^n and PAREN.

  • Meeting 33 : Tue, Mar 13, 10:00 pm-10:50 pm

  Configurations. Acceptance. Final state vs Empty stack.

  References	Page 159-161, Lecture 23 Kozen's textbook. We did Example 23.1 towards the end of the lecture.
  Exercises	Try out acceptance on empty stack vs final state for our running examples. {a^nb^n}, PAREN
  Reading	We described informally why the two acceptance conditions are essentially equivalent. A formal proof of this is presented in Page 164, Supplementary Lecture E, Kozen's Book. It is worth reading.

  Configurations. Acceptance. Final state vs Empty stack.
  

  References	:	Page 159-161, Lecture 23 Kozen's textbook. We did Example 23.1 towards the end of the lecture.
  Exercises	:	Try out acceptance on empty stack vs final state for our running examples. {a^nb^n}, PAREN
  Reading	:	We described informally why the two acceptance conditions are essentially equivalent. A formal proof of this is presented in Page 164, Supplementary Lecture E, Kozen's Book. It is worth reading.

  • Meeting 34 : Wed, Mar 14, 09:00 am-09:50 am

  From CFLs to PDAs. The Greibach normal form (GNF). Construction of PDA from the productions of the grammar in GNF.

  References	Lecture 24, Page 167-169, Kozen's Book.
  Exercises
  Reading

  From CFLs to PDAs. The Greibach normal form (GNF). Construction of PDA from the productions of the grammar in GNF.
  

  References

  :

  Lecture 24, Page 167-169, Kozen's Book.

  • Meeting 35 : Thu, Mar 15, 01:00 pm-01:50 pm

  Example run of the PDA simulating the leftmost derivation. Correctness claim. <br> Converting arbitrary CFGs into GNF form.

  References	<a href="http://www.iitg.ernet.in/gkd/ma513/oct/oct18/note.pdf">Notes</a>.<br> The above linked notes gives a better presentation of the conversion procedure than the textbook.
  Exercises
  Reading

  Example run of the PDA simulating the leftmost derivation. Correctness claim.
  Converting arbitrary CFGs into GNF form.
  

  References

  :

  Notes. The above linked notes gives a better presentation of the conversion procedure than the textbook.

  • Meeting 36 : Fri, Mar 16, 09:00 am-09:50 am

  <a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS2200/tutorial6.pdf">Tutorial 6</a>

  References
  Exercises
  Reading

  Tutorial 6
  

  References

  :

  None

  • Meeting 37 : Mon, Mar 19, 11:00 am-11:50 am

  Converting CFGs to GNF form. General procedure.

  References	<a href="http://www.iitg.ernet.in/gkd/ma513/oct/oct18/note.pdf">Notes</a>.
  Exercises
  Reading

  Converting CFGs to GNF form. General procedure.
  

  References

  :

  Notes.

  • Meeting 38 : Tue, Mar 20, 10:00 pm-10:50 pm

  An example conversion from CFG to GNF form.

  References
  Exercises
  Reading

  An example conversion from CFG to GNF form.
  

  References

  :

  None

  • Meeting 39 : Wed, Mar 21, 08:00 am-08:50 am

  Quiz 2

  References
  Exercises
  Reading

  Quiz 2
  

  References

  :

  None

  • Meeting 40 : Thu, Mar 22, 01:00 pm-01:30 pm

  Shorter Lecture : Simulating k-state NPDA using 1-state NPDA.

  References
  Exercises
  Reading

  Shorter Lecture : Simulating k-state NPDA using 1-state NPDA.
  

  References

  :

  None

  • Meeting 41 : Mon, Mar 26, 11:00 am-11:50 am

  Example : 3-state PDA to 1-state NPDA

  References
  Exercises
  Reading

  Example : 3-state PDA to 1-state NPDA
  

  References

  :

  None

  • Meeting 42 : Tue, Mar 27, 10:00 pm-10:50 pm

  Parsing and Membership Problems in CFLs. Cocke-Youger-Kasami algorithm.

  References	Lecture in Kozen's Book.
  Exercises
  Reading

  Parsing and Membership Problems in CFLs. Cocke-Youger-Kasami algorithm.
  

  References

  :

  Lecture in Kozen's Book.

• Theme 4 : Turing Machines & Computability - 13 meetings

  • Meeting 43 : Wed, Mar 28, 09:00 am-09:50 am

  Effective computability. Introduction to TMs, Historical perspectives. Church-Turing thesis.

  References	Lecture 28 of Kozen's Book.
  Exercises
  Reading

  Effective computability. Introduction to TMs, Historical perspectives. Church-Turing thesis.
  

  References

  :

  Lecture 28 of Kozen's Book.

  • Meeting 44 : Thu, Mar 29, 01:00 pm-01:50 pm

  Formal definition, configurations, and acceptance. Language Accepted, Example of TM accepting a non-context-free language.

  References	Lecture 28 and Lecture 29 of Kozen's Book.
  Exercises	Write down full set of transitions for the TM accepting a^nb^nc^n language.
  Reading

  Formal definition, configurations, and acceptance. Language Accepted, Example of TM accepting a non-context-free language.
  

  References	:	Lecture 28 and Lecture 29 of Kozen's Book.
  Exercises	:	Write down full set of transitions for the TM accepting a^nb^nc^n language.

  • Meeting 45 : Sat, Mar 31, 01:00 pm-01:50 pm

  Halting and Looping of Turing machine computations. Example Problems. Decidability and Semi-decidability.

  References
  Exercises
  Reading

  Halting and Looping of Turing machine computations. Example Problems. Decidability and Semi-decidability.
  

  References

  :

  None

  • Meeting 46 : Mon, Apr 02, 11:00 am-11:50 am

  Equivalent models - Multiple Tapes, Two-way infinite tapes, Two stacks, Multiple heads.

  References
  Exercises
  Reading

  Equivalent models - Multiple Tapes, Two-way infinite tapes, Two stacks, Multiple heads.
  

  References

  :

  None

  • Meeting 47 : Wed, Apr 04, 09:00 am-09:50 am

  <a href="./CS2200/tutorial7.pdf">Tutorial 7</a> (evaluated).

  References
  Exercises
  Reading

  Tutorial 7 (evaluated).
  

  References

  :

  None

  • Meeting 48 : Mon, Apr 09, 11:00 am-11:50 am

  Encoding Turing machines as strings, Universal Turing machine.

  References
  Exercises
  Reading

  Encoding Turing machines as strings, Universal Turing machine.
  

  References

  :

  None

  • Meeting 49 : Tue, Apr 10, 10:00 pm-10:50 pm

  Membership Problem(MP). Universal Turing machine semi-deciding MP. <br> Halting Problem (HP), A TM semi-deciding HP.

  References	Lecture 31 in Kozen's Book.
  Exercises	Try out the following thought : Suppose there is a total TM for the membership problem, can that be used as a "subroutine" to decide the halting problem? <br> We will formalize this in the next lecture. But it is worth attempting ahead of time.
  Reading	<a href="http://www14.informatik.tu-muenchen.de/konferenzen/Jass07/courses/1/Sadovnikov/Sadovnikov_Paper.pdf">Around the Hilbert's 10th problem</a>.

  Membership Problem(MP). Universal Turing machine semi-deciding MP.
  Halting Problem (HP), A TM semi-deciding HP.
  

  References	:	Lecture 31 in Kozen's Book.
  Exercises	:	Try out the following thought : Suppose there is a total TM for the membership problem, can that be used as a "subroutine" to decide the halting problem? We will formalize this in the next lecture. But it is worth attempting ahead of time.
  Reading	:	Around the Hilbert's 10th problem.

  • Meeting 50 : Wed, Apr 11, 09:00 am-09:50 am

  Undecidability of Halting Problem. <br> Using total TM for MP to design a total TM for HP. <br> Using total TM for HP to design a total TM for MP.

  References	Last section in Lecture 31 - Kozen's Book.
  Exercises
  Reading	<a href="http://www14.informatik.tu-muenchen.de/konferenzen/Jass07/courses/1/Sadovnikov/Sadovnikov_Paper.pdf">Around the Hilbert's 10th problem</a>.

  Undecidability of Halting Problem.
  Using total TM for MP to design a total TM for HP.
  Using total TM for HP to design a total TM for MP.
  

  References	:	Last section in Lecture 31 - Kozen's Book.
  Reading	:	Around the Hilbert's 10th problem.

  • Meeting 51 : Thu, Apr 12, 01:00 pm-01:50 pm

  <a href="./CS2200/tutorial8.pdf">Tutorial 8 </a> (discussion based)

  References
  Exercises
  Reading

  Tutorial 8 (discussion based)
  

  References

  :

  None

  • Meeting 52 : Mon, Apr 16, 11:00 am-11:50 am

  The language L_101 (from Tutorial) - solution. Variants of the problem. Abstracting out the idea to define the formal notion of reductions between computational problems.

  References
  Exercises
  Reading

  The language L_101 (from Tutorial) - solution. Variants of the problem. Abstracting out the idea to define the formal notion of reductions between computational problems.
  

  References

  :

  None

  • Meeting 53 : Tue, Apr 17, 10:00 pm-10:50 pm

  More Examples in the language of the reductions. HP to MP and MP to HP. Techniques of proving undecidability.

  References
  Exercises
  Reading

  More Examples in the language of the reductions. HP to MP and MP to HP. Techniques of proving undecidability.
  

  References

  :

  None

  • Meeting 54 : Wed, Apr 18, 09:00 am-09:50 am

  Journey within the decidable world. The notion of efficient computation. Computation resource - Time and Space as examples. Properties we expect from the notion of efficient computation. The class P, NP, examples. The P vs NP problem.

  References	No reference. Please follow class notes.
  Exercises
  Reading

  Journey within the decidable world. The notion of efficient computation. Computation resource - Time and Space as examples. Properties we expect from the notion of efficient computation. The class P, NP, examples. The P vs NP problem.
  

  References

  :

  No reference. Please follow class notes.

  • Meeting 55 : Thu, Apr 19, 01:00 pm-02:00 pm

  1pm - 1:20pm - Teacher-Course Feedback. <br> 1:20pm - 2pm - <a href="./CS2200/tutorial9.pdf">Tutorial 9</a> (evaluated).

  References
  Exercises
  Reading

  1pm - 1:20pm - Teacher-Course Feedback.
  1:20pm - 2pm - Tutorial 9 (evaluated).
  

  References

  :

  None

• Theme 5 : Tutorials & Evaluation Meetings - 5 meetings

  • Meeting 56 : Wed, Jan 18, 09:00 am-09:50 am

  <a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS2200/tutorial1.pdf">Tutorial 1</a>.

  References
  Exercises	Attempt the questions that you could not attempt during the tutorial hour due to time constraints. Come back to us with questions.
  Reading

  Tutorial 1.
  

  References	:	None
  Exercises	:	Attempt the questions that you could not attempt during the tutorial hour due to time constraints. Come back to us with questions.

  • Meeting 57 : Wed, Jan 25, 04:00 pm-04:50 pm

  <a href="./CS2200/tutorial2.pdf">Tutorial 2</a>

  References	<a href="./CS2200/tutorial2.pdf">Tutorial 2</a>
  Exercises
  Reading

  Tutorial 2
  

  References

  :

  Tutorial 2

  • Meeting 58 : Thu, Feb 09, 01:00 pm-01:50 pm

  <a href="./CS2200/tutorial3.pdf">Tutorial 3</a> (evaluated).

  References
  Exercises
  Reading

  Tutorial 3 (evaluated).
  

  References

  :

  None

  • Meeting 59 : Wed, Feb 15, 08:00 am-08:50 am

  Quiz - I

  References
  Exercises
  Reading

  Quiz - I
  

  References

  :

  None

  • Meeting 60 : Fri, Mar 02, 09:00 am-09:50 pm

  <a href="./CS2200/tutorial4.pdf">Tutorial 4</a> (evaluated).

  References
  Exercises
  Reading

  Tutorial 4 (evaluated).
  

  References

  :

  None