CS6014 - Advanced Theory of Computation

Meetings  

• Theme 1 : Turing Computability - 24 meetings

  • Meeting 01 : Mon, Jul 30, 08:00 am-08:50 am

  Administrative Announcements. Overview of the course. Discussion of the course requirements.<br> Hilbert's program, different attempts to capture the notion of an effective procedure, Church-Turing thesis.

  References	Lecture 28 of [K1 Book]
  Exercises
  Reading	<ul><li><a href="http://plato.stanford.edu/entries/church-turing/">Church-Turing Thesis</a></li> <li><a href="http://plato.stanford.edu/entries/hilbert-program/">Hilbert's Program</a>.</li> <li><a href="http://www.cse.uconn.edu/~dqg/papers/cie05.pdf">Church-Turing thesis - Breaking the Myth</a> - Read Critically.</li></ul>

  Administrative Announcements. Overview of the course. Discussion of the course requirements.
  Hilbert's program, different attempts to capture the notion of an effective procedure, Church-Turing thesis.
  

  References	:	Lecture 28 of [K1 Book]
  Reading	:	Church-Turing Thesis Hilbert's Program. Church-Turing thesis - Breaking the Myth - Read Critically.

  • Meeting 02 : Tue, Jul 31, 12:00 pm-12:50 pm

  Arriving at the Turing model - observations about computation - boundedness, locality and determinism. From the DFA to the Turing machines. The triplet definition.

  References	Lecture [28] in [K1 Book]
  Exercises
  Reading	<a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6014/Turing_Paper_1936.pdf">Turing's Paper published in 1936</a>.

  Arriving at the Turing model - observations about computation - boundedness, locality and determinism. From the DFA to the Turing machines. The triplet definition.
  

  References	:	Lecture [28] in [K1 Book]
  Reading	:	Turing's Paper published in 1936.

  • Meeting 03 : Thu, Aug 02, 11:00 am-11:50 am

  An intermediate model - Machines where the tape extends only up to the input lengths - Discussion on the term "finite". Mathematical notion of configurations.

  References	Lecture 28 in [K1 Book]
  Exercises
  Reading

  An intermediate model - Machines where the tape extends only up to the input lengths - Discussion on the term "finite". Mathematical notion of configurations.
  

  References

  :

  Lecture 28 in [K1 Book]

  • Meeting 04 : Fri, Aug 03, 10:00 am-10:50 am

  Configurations and moves. Language accepted by TMs, Decidability and Semi-decidability, Equivalent models to TMs - two-way infinite TMs.

  References	Lecture 28, 29, and 30 in [K1 Book]
  Exercises
  Reading

  Configurations and moves. Language accepted by TMs, Decidability and Semi-decidability, Equivalent models to TMs - two-way infinite TMs.
  

  References

  :

  Lecture 28, 29, and 30 in [K1 Book]

  • Meeting 05 : Mon, Aug 06, 08:00 am-08:50 am

  Tape reduction. Encodings of Turing machines. Universal Turing Machines.

  References
  Exercises
  Reading

  Tape reduction. Encodings of Turing machines. Universal Turing Machines.
  

  References

  :

  None

  • Meeting 06 : Tue, Aug 07, 09:00 am-09:50 am

  Undecidability of the Halting Problem (HP). Undecidability of the MP - Membership Problem, Notion of Reductions.

  References	Lecture 31 & 32 in [K1] book.
  Exercises	Can we use a total TM that solves HP problem in order to design a total TM for solving the MP problem? (Note that we showed it in the other way round in class).
  Reading	<ul> <li><a href="http://ebiquity.umbc.edu/blogger/2008/01/19/how-dr-suess-would-prove-the-halting-problem-undecidable/">Scooping the Loop Snooper</a> - a rhyming proof of the undecidability of the halting problem.</li> </ul>

  Undecidability of the Halting Problem (HP). Undecidability of the MP - Membership Problem, Notion of Reductions.
  

  References	:	Lecture 31 & 32 in [K1] book.
  Exercises	:	Can we use a total TM that solves HP problem in order to design a total TM for solving the MP problem? (Note that we showed it in the other way round in class).
  Reading	:	Scooping the Loop Snooper - a rhyming proof of the undecidability of the halting problem.

  • Meeting 07 : Thu, Aug 09, 11:00 am-11:50 am

  Does knowing the string in advance help? Another example of a reduction. Extending it to show the undecidability of regularity checking of the language of a given Turing machine.

  References
  Exercises
  Reading

  Does knowing the string in advance help? Another example of a reduction. Extending it to show the undecidability of regularity checking of the language of a given Turing machine.
  

  References

  :

  None

  • Meeting 08 : Fri, Aug 10, 10:00 am-10:50 am

  Showing undecidability of the following tasks. Testing Regularity, Testing Context-freeness, Testing decidability of the languages of the given TM.

  References
  Exercises
  Reading

  Showing undecidability of the following tasks. Testing Regularity, Testing Context-freeness, Testing decidability of the languages of the given TM.
  

  References

  :

  None

  • Meeting 09 : Tue, Aug 14, 12:00 pm-12:50 pm

  Rice's theorem. Infinitely many undecidable sets Rice's theorem I - All nontrivial properties of semidecidable sets are undecidable.

  References
  Exercises
  Reading

  Rice's theorem. Infinitely many undecidable sets Rice's theorem I - All nontrivial properties of semidecidable sets are undecidable.
  

  References

  :

  None

  • Meeting 10 : Thu, Aug 16, 11:00 am-11:50 am

  Complementation. L is decidable if and only of L and L-complement are semi-decidable. Applications to Halting Problem. Landscape of undecidable sets, Semidecidable and Co-semidecidable sets. Rice's Second theorem- statement and applications.

  References
  Exercises
  Reading

  Complementation. L is decidable if and only of L and L-complement are semi-decidable. Applications to Halting Problem. Landscape of undecidable sets, Semidecidable and Co-semidecidable sets. Rice's Second theorem- statement and applications.
  

  References

  :

  None

  • Meeting 11 : Tue, Aug 21, 12:00 pm-12:50 pm

  Rice's theorem II - No non-monotone property of SD sets is semi-decidable. Proof and applications to REG and its complement. Application to FIN, but inability to apply to its complement. A direct reduction from HP complement to FIN complement. Hence FIN is neither in SD nor in CoSD. REG and FIN are outside the landscape. Hints on relative computation.

  References	The lecture was based on the presentation in Kozen Textbook. But the material is distributed over different lectures.
  Exercises
  Reading

  Rice's theorem II - No non-monotone property of SD sets is semi-decidable. Proof and applications to REG and its complement. Application to FIN, but inability to apply to its complement. A direct reduction from HP complement to FIN complement. Hence FIN is neither in SD nor in CoSD. REG and FIN are outside the landscape. Hints on relative computation.
  

  References

  :

  The lecture was based on the presentation in Kozen Textbook. But the material is distributed over different lectures.

  • Meeting 12 : Thu, Aug 23, 11:00 am-11:50 am

  Relative Computation. Oracle Turing Machines. Relative Decidability and Turing Reductions. Relative Semi-decidability. Comparison with many-one reductions.

  References
  Exercises
  Reading

  Relative Computation. Oracle Turing Machines. Relative Decidability and Turing Reductions. Relative Semi-decidability. Comparison with many-one reductions.
  

  References

  :

  None

  • Meeting 13 : Mon, Aug 27, 08:00 am-08:50 am

  Landscape outside semi-decidables : Arithmetic hierarchy of languages. Sigma, Pi and Delta. The structure of the hierarchy.

  References	Supplementary lecture J, Page 276 of [K1].
  Exercises
  Reading

  Landscape outside semi-decidables : Arithmetic hierarchy of languages. Sigma, Pi and Delta. The structure of the hierarchy.
  

  References

  :

  Supplementary lecture J, Page 276 of [K1].

  • Meeting 14 : Tue, Aug 28, 12:00 pm-12:50 pm

  Hardness and Completeness of languages. Halting Problem and Membership Problem are complete for the class of semi-decidable languages. Canonical complete problem. Completeness in the higher levels of Arithmetic Hierarchy. MP_k

  References	Supplementary lecture J, Page 276 of [K1].
  Exercises
  Reading

  Hardness and Completeness of languages. Halting Problem and Membership Problem are complete for the class of semi-decidable languages. Canonical complete problem. Completeness in the higher levels of Arithmetic Hierarchy. MP_k
  

  References

  :

  Supplementary lecture J, Page 276 of [K1].

  • Meeting 15 : Thu, Aug 30, 11:00 am-11:50 am

  Quantified Predicate characterization of the Halting Problem. Extending it to all semi-decidable languages. Further extending it to arithmetic Hierarchy.

  References
  Exercises
  Reading

  Quantified Predicate characterization of the Halting Problem. Extending it to all semi-decidable languages. Further extending it to arithmetic Hierarchy.
  

  References

  :

  None

  • Meeting 16 : Fri, Aug 31, 10:00 am-10:50 am

  Positioning the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy. FIN is complete for the second level of the hierarchy.

  References	Page 278-280 of [K1] Book.
  Exercises
  Reading

  Positioning the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy. FIN is complete for the second level of the hierarchy.
  

  References

  :

  Page 278-280 of [K1] Book.

  • Meeting 17 : Sat, Sep 01, 11:00 am-12:45 pm

  Proof of the quantifier characterization of k-th level of arithmetic hierarchy.

  References	Notes sent to the course mailing list.
  Exercises
  Reading

  Proof of the quantifier characterization of k-th level of arithmetic hierarchy.
  

  References

  :

  Notes sent to the course mailing list.

  • Meeting 18 : Mon, Sep 03, 08:00 am-08:50 am

  Post's Theorem : There is an semi-decidable but undecidable language which is not complete for the class of semi-decidable languages. Simple and productive sets. Simple sets cannot be co-productive. The self-halting set(K) is co-productive.

  References	Notes scribed by Dinesh Krishnamoorthy has been sent to the course mailing list.
  Exercises	Show that K is hard for semi-decidable sets.
  Reading

  Post's Theorem : There is an semi-decidable but undecidable language which is not complete for the class of semi-decidable languages. Simple and productive sets. Simple sets cannot be co-productive. The self-halting set(K) is co-productive.
  

  References	:	Notes scribed by Dinesh Krishnamoorthy has been sent to the course mailing list.
  Exercises	:	Show that K is hard for semi-decidable sets.

  • Meeting 19 : Tue, Sep 04, 12:00 pm-12:50 pm

  Every SD-complete set is co-productive. Post's construction of Simple sets.

  References	Notes scribed by Dinesh Krishnamoorthy has been sent to the course mailing list.
  Exercises
  Reading

  Every SD-complete set is co-productive. Post's construction of Simple sets.
  

  References

  :

  Notes scribed by Dinesh Krishnamoorthy has been sent to the course mailing list.

  • Meeting 20 : Fri, Sep 07, 10:00 am-10:50 am

  Freidberg-Muchnik theorem. Low sets. Construction of low simple sets. Finite injury priority argument - the strategy.

  References	From Kozen2 Texbook.
  Exercises
  Reading	<a href="https://www1.maths.leeds.ac.uk/~pmt6sbc/3163/FMspecial.pdf">Notes for a variant of the theorem</a>

  Freidberg-Muchnik theorem. Low sets. Construction of low simple sets. Finite injury priority argument - the strategy.
  

  References	:	From Kozen2 Texbook.
  Reading	:	Notes for a variant of the theorem

  • Meeting 21 : Mon, Sep 10, 12:00 pm-01:00 pm

  Detailed Construction of Finite Simple Sets - Details of Finite injury priority argument.

  References	Lecture 38 in Kozen2 Book.
  Exercises
  Reading

  Detailed Construction of Finite Simple Sets - Details of Finite injury priority argument.
  

  References

  :

  Lecture 38 in Kozen2 Book.

  • Meeting 22 : Tue, Sep 11, 12:00 pm-12:50 pm

  Language of Numbers, Peano's Arithmetic. Godel's First Incompleteness theorem. Theorems and True statements - interpretation of the theorem.

  References	Lecture 38 of Kozen1 Book.
  Exercises
  Reading

  Language of Numbers, Peano's Arithmetic. Godel's First Incompleteness theorem. Theorems and True statements - interpretation of the theorem.
  

  References

  :

  Lecture 38 of Kozen1 Book.

  • Meeting 23 : Thu, Sep 13, 11:00 am-11:50 am

  A proof of incompleteness using computability.

  References	Lecture 38 of Kozen1 Book.
  Exercises
  Reading

  A proof of incompleteness using computability.
  

  References

  :

  Lecture 38 of Kozen1 Book.

  • Meeting 24 : Fri, Sep 14, 10:00 am-10:50 am

  Kolmogorov Complexity : Generating statements that, with high probability, are true and unprovable.

  References	<a href="theory.stanford.edu/~trevisan/cs154-12/notek.pdf">Note from a lecture by Luca Trevisan and Ryan Williams</a>.
  Exercises
  Reading

  Kolmogorov Complexity : Generating statements that, with high probability, are true and unprovable.
  

  References

  :

  Note from a lecture by Luca Trevisan and Ryan Williams.

• Theme 2 : Basic Time & Space Complexity - 33 meetings

  • Meeting 25 : Mon, Sep 17, 08:00 am-08:50 am

  Resources of computation, Blum's axioms, Examples and non-examples of resources.

  References	Notes from class.
  Exercises
  Reading

  Resources of computation, Blum's axioms, Examples and non-examples of resources.
  

  References

  :

  Notes from class.

  • Meeting 26 : Fri, Sep 21, 10:00 am-10:50 am

  The landscape in the decidable world with respect to resources of computation. Language outside the g-decidables. Measuring Resources in terms of input length, DTIME(t(n)), DSPACE(s(n)).

  References	Prop 1.12 and [1.13] in [DK].
  Exercises
  Reading

  The landscape in the decidable world with respect to resources of computation. Language outside the g-decidables. Measuring Resources in terms of input length, DTIME(t(n)), DSPACE(s(n)).
  

  References

  :

  Prop 1.12 and [1.13] in [DK].

  • Meeting 27 : Mon, Sep 24, 08:00 am-08:50 am

  Tape Compression, Linear Speed up theorems, Tape reduction theorem.

  References	Prop 1.12 and [1.13] in [DK].
  Exercises
  Reading

  Tape Compression, Linear Speed up theorems, Tape reduction theorem.
  

  References

  :

  Prop 1.12 and [1.13] in [DK].

  • Meeting 28 : Tue, Sep 25, 12:00 pm-12:50 pm

  Time Hierarchy Theorem. Hennie-Stearns Tape reduction theorem (statement). Limitations of 1-tape Turing machines. Crossing Sequences.

  References	Notes from class.
  Exercises
  Reading

  Time Hierarchy Theorem. Hennie-Stearns Tape reduction theorem (statement). Limitations of 1-tape Turing machines. Crossing Sequences.
  

  References

  :

  Notes from class.

  • Meeting 29 : Wed, Sep 26, 04:45 pm-05:35 pm

  Crossing sequence arguments - Quadratic time lower bounds for Palindrome language. Any one tape Turing machine running in time o(n log n) can accept only a regular language.

  References	<a href" www.cs.umd.edu/~jkatz/complexity/f05/lower_bounds.pdf">Notes</a> by Katz.
  Exercises
  Reading

  Crossing sequence arguments - Quadratic time lower bounds for Palindrome language. Any one tape Turing machine running in time o(n log n) can accept only a regular language.
  

  References

  :

  Notes by Katz.

  • Meeting 30 : Thu, Sep 27, 11:00 am-11:50 am

  What is the notion of time efficient algorithms. Discussion of efficiency. Composibility. The definition of the class P.

  References	Notes from class.
  Exercises
  Reading

  What is the notion of time efficient algorithms. Discussion of efficiency. Composibility. The definition of the class P.
  

  References

  :

  Notes from class.

  • Meeting 31 : Fri, Sep 28, 11:00 am-11:50 am

  The class EXP, E. Space complexity. Space hierarchy theorem. Classes L.

  References	Notes from class.
  Exercises
  Reading

  The class EXP, E. Space complexity. Space hierarchy theorem. Classes L.
  

  References

  :

  Notes from class.

  • Meeting 32 : Wed, Oct 03, 04:45 pm-05:35 pm

  CLIQUE and REACH - two computational problems. EXP time algorithm for CLIQUE, PSPACE algorithm for CLIQUE. Poly time algorithm for REACH, log-space algorithm for TREE-REACH.

  References	Notes from class.
  Exercises
  Reading

  CLIQUE and REACH - two computational problems. EXP time algorithm for CLIQUE, PSPACE algorithm for CLIQUE. Poly time algorithm for REACH, log-space algorithm for TREE-REACH.
  

  References

  :

  Notes from class.

  • Meeting 33 : Thu, Oct 04, 11:00 am-11:50 am

  EXACTCLIQUE problem vs CLIQUE problem. Efficient verifiability property for membership in CLIQUE. Non-determinism, definition of time and space.

  References	Notes from class.
  Exercises
  Reading

  EXACTCLIQUE problem vs CLIQUE problem. Efficient verifiability property for membership in CLIQUE. Non-determinism, definition of time and space.
  

  References

  :

  Notes from class.

  • Meeting 34 : Fri, Oct 05, 10:00 am-10:50 am

  Simulation and containments. NP is contained in EXP and is in PSPACE. Guess+Verify characterisation of NP.

  References	Theorem 2.1 ok [DK]
  Exercises
  Reading

  Simulation and containments. NP is contained in EXP and is in PSPACE. Guess+Verify characterisation of NP.
  

  References

  :

  Theorem 2.1 ok [DK]

  • Meeting 35 : Mon, Oct 08, 08:00 am-08:50 am

  Padding Arguments : P = NP implies EXP = NEXP. <br> Input length and complexity - Eg : PRIMES. Complement of PRIMES is in NP. The class CoNP. Relationship between NP, CoNP, P and PSAPCE.

  References
  Exercises
  Reading	Theorem 1.27 in [DK]

  Padding Arguments : P = NP implies EXP = NEXP.
  Input length and complexity - Eg : PRIMES. Complement of PRIMES is in NP. The class CoNP. Relationship between NP, CoNP, P and PSAPCE.
  

  References	:	None
  Reading	:	Theorem 1.27 in [DK]

  • Meeting 36 : Tue, Oct 09, 12:00 pm-12:50 pm

  Discussion on P vs NP problem. Nondeterministic space. Simulation using deterministic algorithms. Analysis of time. Analysis of space. Savitch's theorem statement.

  References
  Exercises
  Reading

  Discussion on P vs NP problem. Nondeterministic space. Simulation using deterministic algorithms. Analysis of time. Analysis of space. Savitch's theorem statement.
  

  References

  :

  None

  • Meeting 37 : Thu, Oct 11, 11:00 am-11:50 am

  Proof of Savitch's theorem. Review of complexity classes and containments so far.

  References	Theorem 1.3 in [DK]
  Exercises
  Reading

  Proof of Savitch's theorem. Review of complexity classes and containments so far.
  

  References

  :

  Theorem 1.3 in [DK]

  • Meeting 38 : Fri, Oct 12, 10:00 am-10:50 am

  INDSET problem, NP algorithm, connection between CLIQUE and INDSET. Reductions, Completeness, Bounded Halting Problem is NP-complete.

  References	Section 2.1, 2.2 and 2.3 of [DK]
  Exercises
  Reading

  INDSET problem, NP algorithm, connection between CLIQUE and INDSET. Reductions, Completeness, Bounded Halting Problem is NP-complete.
  

  References

  :

  Section 2.1, 2.2 and 2.3 of [DK]

  • Meeting 39 : Sat, Oct 13, 10:45 am-12:00 pm

  Composability of log-space reductions. NL-completeness of REACH under log-space reductions.

  References	Refer to class notes.
  Exercises
  Reading

  Composability of log-space reductions. NL-completeness of REACH under log-space reductions.
  

  References

  :

  Refer to class notes.

  • Meeting 40 : Mon, Oct 15, 08:00 am-08:50 am

  CLIQUE, INDSET, VC, GI, SAT - Five problems in NP. Structure and reductions.

  References	Section 2.1, 2.2 and 2.3 of [DK]
  Exercises
  Reading

  CLIQUE, INDSET, VC, GI, SAT - Five problems in NP. Structure and reductions.
  

  References

  :

  Section 2.1, 2.2 and 2.3 of [DK]

  • Meeting 41 : Tue, Oct 16, 12:00 pm-12:50 pm

  Cook-Levin theorem. Statement, reasons and implications.

  References	Section 2.1, 2.2 and 2.3 of [DK]
  Exercises
  Reading

  Cook-Levin theorem. Statement, reasons and implications.
  

  References

  :

  Section 2.1, 2.2 and 2.3 of [DK]

  • Meeting 42 : Wed, Oct 17, 12:00pm-12:50pm

  Details of Cook-Levin theorem. Reduction to CNF SAT, reduction to 3-SAT. boundaries for k-SAT.

  References	Section 2.1, 2.2 and 2.3 of [DK]
  Exercises
  Reading

  Details of Cook-Levin theorem. Reduction to CNF SAT, reduction to 3-SAT. boundaries for k-SAT.
  

  References

  :

  Section 2.1, 2.2 and 2.3 of [DK]

  • Meeting 43 : Fri, Oct 19, 10:00 am-10:50 am

  3SAT to VC, Chain of reductions. Ladner's theorem (statement).

  References	Section 2.1, 2.2 and 2.3 of [DK]
  Exercises
  Reading

  3SAT to VC, Chain of reductions. Ladner's theorem (statement).
  

  References

  :

  Section 2.1, 2.2 and 2.3 of [DK]

  • Meeting 44 : Mon, Oct 22, 08:00 am-08:50 am

  Ladner's theorem introduction and proof idea

  References	Lecture notes sent to the group.<br> Section 3.3 of [AB]
  Exercises
  Reading

  Ladner's theorem introduction and proof idea
  

  References

  :

  Lecture notes sent to the group. Section 3.3 of [AB]

  • Meeting 45 : Tue, Oct 23, 12:00 pm-12:50 pm

  Impagliazzo's Proof of Ladner's Theorem.

  References	Lecture notes sent to the group.<br> Section 3.3 of [AB]
  Exercises
  Reading

  Impagliazzo's Proof of Ladner's Theorem.
  

  References

  :

  Lecture notes sent to the group. Section 3.3 of [AB]

  • Meeting 46 : Thu, Oct 25, 11:00 am-11:50 am

  Turing reductions, Examples, Oracle Machines, Diagonalization hits back : Relativisation, Baker-Gill-Solovoy theorem.

  References	Section 3.4 in [AB]
  Exercises
  Reading

  Turing reductions, Examples, Oracle Machines, Diagonalization hits back : Relativisation, Baker-Gill-Solovoy theorem.
  

  References

  :

  Section 3.4 in [AB]

  • Meeting 47 : Fri, Oct 26, 10:00 am-10:50 am

  Details of the proof of BGS Theorem. Implications and discussion.

  References	Section 3.4 in [AB]
  Exercises
  Reading

  Details of the proof of BGS Theorem. Implications and discussion.
  

  References

  :

  Section 3.4 in [AB]

  • Meeting 48 : Mon, Oct 29, 08:00 am-08:50 am

  Three generalizations of NP. <br> 1) Based on oracle access. Definition of Polynomial Hierarchy, PSPACE upper bound. <br> 2) Characterization of PH in terms of Quantifiers. Examples.<br> EXACT-CLIQUE, and MIN-CKT problems as examples.

  References	Section 3.1, 3.2 and 3.4 in [DK]
  Exercises
  Reading

  Three generalizations of NP.
  1) Based on oracle access. Definition of Polynomial Hierarchy, PSPACE upper bound.
  2) Characterization of PH in terms of Quantifiers. Examples.
  EXACT-CLIQUE, and MIN-CKT problems as examples.
  

  References

  :

  Section 3.1, 3.2 and 3.4 in [DK]

  • Meeting 49 : Tue, Oct 30, 12:00 pm-12:50 pm

  3) Alternating Turing machines. Definition, Time and Space. Alternating Turing machine for EXACT-CLIQUE problem. ATIME, ASPACE. Simulations of Alternating Turing machines with Deterministic Machines.

  References	Section 3.4 in [DK]
  Exercises
  Reading

  3) Alternating Turing machines. Definition, Time and Space. Alternating Turing machine for EXACT-CLIQUE problem. ATIME, ASPACE. Simulations of Alternating Turing machines with Deterministic Machines.
  

  References

  :

  Section 3.4 in [DK]

  • Meeting 50 : Thu, Nov 01, 11:00 am-11:50 am

  AP = PSPACE, AL = P, APSPACE = EXP, Completeness for PH.

  References	Section 3.4 in [DK]
  Exercises
  Reading

  AP = PSPACE, AL = P, APSPACE = EXP, Completeness for PH.
  

  References

  :

  Section 3.4 in [DK]

  • Meeting 51 : Mon, Nov 05, 08:00 am-08:50 am

  PSPACE completeness of QBF, Geography.

  References
  Exercises
  Reading

  PSPACE completeness of QBF, Geography.
  

  References

  :

  None

  • Meeting 52 : Tue, Nov 06, 12:00 pm-12:50 pm

  Immerman-Szelepscenyi Theorem, NL = CoNL.

  References	<a href="http://www.karlin.mff.cuni.cz/~krajicek/is.pdf">Kozen2 Chapter</a>
  Exercises
  Reading

  Immerman-Szelepscenyi Theorem, NL = CoNL.
  

  References

  :

  Kozen2 Chapter

  • Meeting 53 : Wed, Nov 07, 04:45 pm-05:35 pm

  Reachability in Graphs with expander components. Eigen values, Expansion and Graph Products. Algebraic and combinatorial definition of expansion.

  References	Arora Barak Textbook Section 21.2.2
  Exercises
  Reading

  Reachability in Graphs with expander components. Eigen values, Expansion and Graph Products. Algebraic and combinatorial definition of expansion.
  

  References

  :

  Arora Barak Textbook Section 21.2.2

  • Meeting 54 : Thu, Nov 08, 11:00 am-11:50 am

  Significance of Second Largest Eigen Value. The reduction of the orthogonal component at every step. Random walk mixing well. An intuitive reason for a relation to "high connectivity".

  References	Arora Barak Textbook Section 21.1
  Exercises
  Reading

  Significance of Second Largest Eigen Value. The reduction of the orthogonal component at every step. Random walk mixing well. An intuitive reason for a relation to "high connectivity".
  

  References

  :

  Arora Barak Textbook Section 21.1

  • Meeting 55 : Fri, Nov 09, 10:00 am-10:50 am

  Algebraic Expansion implies Combinatorial Expansion. Algebraic Expanders do not have small balanced cuts.<br> Target : To do reachability, expanderize each component of the given graph. Methods of Expanderization - Powering. Effects on the second largest eigen value and the degree.

  References	Arora Barak Textbook Section 21.1, 21.2
  Exercises
  Reading

  Algebraic Expansion implies Combinatorial Expansion. Algebraic Expanders do not have small balanced cuts.
  Target : To do reachability, expanderize each component of the given graph. Methods of Expanderization - Powering. Effects on the second largest eigen value and the degree.
  

  References

  :

  Arora Barak Textbook Section 21.1, 21.2

  • Meeting 56 : Wed, Nov 14, 04:45 pm-05:35 pm

  Every graph has a small spectral Gap. Powering and Replacement Product. Eigen value change. Reingold's log-space algorithm for undirected reachability.

  References	Arora Barak Textbook Section 21.3 21.3 and 21.4
  Exercises
  Reading

  Every graph has a small spectral Gap. Powering and Replacement Product. Eigen value change. Reingold's log-space algorithm for undirected reachability.
  

  References

  :

  Arora Barak Textbook Section 21.3 21.3 and 21.4

  • Meeting 57 : Thu, Nov 15, 11:00 am-11:50 am

  Randomized algorithm for undirected connectivity. Success Probability Amplifications. Polynomial identity testing problem. Schwartz-Zippel Lemma, BPP. Error reduction, containments. Complexity classes BPP and RP. Containments <br> Conclusion.

  References	Arora Barak Textbook Section 21.1, 21.3 and 21.4
  Exercises
  Reading

  Randomized algorithm for undirected connectivity. Success Probability Amplifications. Polynomial identity testing problem. Schwartz-Zippel Lemma, BPP. Error reduction, containments. Complexity classes BPP and RP. Containments
  Conclusion.
  

  References

  :

  Arora Barak Textbook Section 21.1, 21.3 and 21.4

• Evaluation Meetings - 3 meetings

  • Meeting 58 : Sat, Sep 01, 09:00 am-10:30 am

  Quiz I (20%)

  References
  Exercises
  Reading

  Quiz I (20%)
  

  References

  :

  None

  • Meeting 59 : Sat, Oct 13, 09:00 am-10:30 am

  Quiz II (20%)

  References
  Exercises
  Reading

  Quiz II (20%)
  

  References

  :

  None

  • Meeting 60 : Mon, Nov 19, 09:00 am-12:00 pm

  End-semester Examination

  References
  Exercises
  Reading

  End-semester Examination
  

  References

  :

  None