CS6014 - Advanced Theory of Computation

Meetings  

• Theme 1 : Turing Computability - 21 meetings

  • Meeting 01 : Thu, Jul 31, 11:00 am-11:50 am

  Introduction. Administrative Announcements.

  References
  Exercises
  Reading

  Introduction. Administrative Announcements.
  

  References

  :

  None

  • Meeting 02 : Fri, Aug 01, 10:00 am-10:50 am

  Model of Computation. Post systems. Mu-calculus. Lambda-calculus.

  References
  Exercises
  Reading

  Model of Computation. Post systems. Mu-calculus. Lambda-calculus.
  

  References

  :

  None

  • Meeting 03 : Mon, Aug 04, 08:00 am-08:50 am

  Turing Machine Model. Church-Turing Thesis. Examples. Algorithms vs Turing Machines. Total Turing Machines.

  References
  Exercises
  Reading

  Turing Machine Model. Church-Turing Thesis. Examples. Algorithms vs Turing Machines. Total Turing Machines.
  

  References

  :

  None

  • Meeting 04 : Tue, Aug 05, 12:00 pm-12:50 pm

  Encoding Turing machines as strings. Existential argument for languages that are not even semi-decidable.

  References	Lecture 31 in [K1] and Class Notes.
  Exercises
  Reading

  Encoding Turing machines as strings. Existential argument for languages that are not even semi-decidable.
  

  References

  :

  Lecture 31 in [K1] and Class Notes.

  • Meeting 05 : Thu, Aug 07, 11:00 am-11:50 am

  One Turing machine as input to another. The Membership Problem. Universal Turing Machines. Membership problem is semi-decidable but not decidable. The proof using diagonalization.

  References	& Lecture 31 in [K1]
  Exercises
  Reading

  One Turing machine as input to another. The Membership Problem. Universal Turing Machines. Membership problem is semi-decidable but not decidable. The proof using diagonalization.
  

  References

  :

  & Lecture 31 in [K1]

  • Meeting 06 : Fri, Aug 08, 10:00 am-10:50 am

  Formalization of Reductions. Equivalence of halting and membership problems. A potentially easy variant of MP shown to be as hard as MP itself.

  References	Lecture 33 in [K1]
  Exercises
  Reading

  Formalization of Reductions. Equivalence of halting and membership problems. A potentially easy variant of MP shown to be as hard as MP itself.
  

  References

  :

  Lecture 33 in [K1]

  • Meeting 07 : Mon, Aug 11, 08:00 am-08:50 am

  MP_101, REG, CFL, DEC - a lot of undecidable problems through a master reduction.

  References	Lecture 32 in [K1]
  Exercises
  Reading

  MP_101, REG, CFL, DEC - a lot of undecidable problems through a master reduction.
  

  References

  :

  Lecture 32 in [K1]

  • Meeting 08 : Tue, Aug 12, 12:00 pm-12:50 pm

  An abstraction. Rice's 1st theorem. Completeness. Hardness.

  References	Lecture 34 in [K1]
  Exercises
  Reading

  An abstraction. Rice's 1st theorem. Completeness. Hardness.
  

  References

  :

  Lecture 34 in [K1]

  • Meeting 09 : Thu, Aug 14, 11:00 am-11:50 am

  More properties of reductions. Completeness and Hardness. SD-complete sets.

  References	Class Notes.
  Exercises
  Reading

  More properties of reductions. Completeness and Hardness. SD-complete sets.
  

  References

  :

  Class Notes.

  • Meeting 10 : Mon, Aug 18, 08:00 am-08:50 am

  Complements of Semi-decidable languages. Characterization of decidable languages. The story of FIN. Not even semi-decidable, not even co-semidecidable.

  References
  Exercises
  Reading

  Complements of Semi-decidable languages. Characterization of decidable languages. The story of FIN. Not even semi-decidable, not even co-semidecidable.
  

  References

  :

  None

  • Meeting 11 : Tue, Aug 19, 12:00 pm-12:50 pm

  Rice's Second theorem. Applications to REG, CFL. Relative Computation. Oracle Turing machines.

  References
  Exercises
  Reading

  Rice's Second theorem. Applications to REG, CFL. Relative Computation. Oracle Turing machines.
  

  References

  :

  None

  • Meeting 12 : Thu, Aug 21, 11:00 am-11:50 am

  Semi-deciding FIN with oracle access to MP. Two formalizations of Oracle Turing Machines. Language accepted by Oracle Turing machines equipped with an oracle.

  References
  Exercises
  Reading

  Semi-deciding FIN with oracle access to MP. Two formalizations of Oracle Turing Machines. Language accepted by Oracle Turing machines equipped with an oracle.
  

  References

  :

  None

  • Meeting 13 : Fri, Aug 22, 10:00 am-10:50 am

  Decidability and Semidecidability with respect to an Oracle Language. Arithmetic Hierarchy. Turing versus Many-one reductions. The class Delta_2. Proof that Delta_2 strictly contains SD and Co-SD. MPXOR language is the example.

  References
  Exercises
  Reading

  Decidability and Semidecidability with respect to an Oracle Language. Arithmetic Hierarchy. Turing versus Many-one reductions. The class Delta_2. Proof that Delta_2 strictly contains SD and Co-SD. MPXOR language is the example.
  

  References

  :

  None

  • Meeting 14 : Mon, Aug 25, 08:00 am-08:50 am

  Relativizing proofs. Arithmetic hierarchy is strict. Quantified Predicate characterization of the Membership Problem. Extending it to all semi-decidable languages.

  References
  Exercises
  Reading

  Relativizing proofs. Arithmetic hierarchy is strict. Quantified Predicate characterization of the Membership Problem. Extending it to all semi-decidable languages.
  

  References

  :

  None

  • Meeting 15 : Tue, Aug 26, 12:00 pm-12:50 pm

  Further extending it to arithmetic Hierarchy (statement). Positioning the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy.

  References
  Exercises
  Reading

  Further extending it to arithmetic Hierarchy (statement). Positioning the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy.
  

  References

  :

  None

  • Meeting 16 : Thu, Aug 28, 11:00 am-11:50 am

  FIN is complete for the second level(Sigma-2) of the hierarchy. TOTAL is complete for Pi-2.

  References	Reference sent to the course emailing list.
  Exercises
  Reading

  FIN is complete for the second level(Sigma-2) of the hierarchy. TOTAL is complete for Pi-2.
  

  References

  :

  Reference sent to the course emailing list.

  • Meeting 17 : Tue, Sep 02, 12:00 pm-12:50 pm

  Proof of the Quantifier Characterization of Arithmetic Hierarchy.

  References	<ul> <li><a href="">Notes</a> sent to the mailing list</li> </ul>
  Exercises
  Reading

  Proof of the Quantifier Characterization of Arithmetic Hierarchy.
  

  References

  :

  Notes sent to the mailing list

  • Meeting 18 : Thu, Sep 04, 11:00 am-11:50 am

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

  References	<ul> <li><a href="">Notes</a> sent to the mailing list</li> </ul>
  Exercises
  Reading	Is there a simpler way to show this than constructing "simple" sets? This was a question raised in class. Myhill, in 1952 showed this on the negative by arguing that, any "intermediate set" should be many-one equivalent to a simple set. Read about it <a href="">here</a>.

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

  References	:	Notes sent to the mailing list
  Reading	:	Is there a simpler way to show this than constructing "simple" sets? This was a question raised in class. Myhill, in 1952 showed this on the negative by arguing that, any "intermediate set" should be many-one equivalent to a simple set. Read about it here.

  • Meeting 19 : Fri, Sep 05, 10:00 am-10:50 am

  Simple and productive sets. Simple sets cannot be SD-complete. Post's construction of Simple sets.

  References
  Exercises
  Reading	Is there a simpler way to show this than constructing "simple" sets? This was a question raised in class. Dekker, in 1952 showed this on the negative by arguing that, any "intermediate set" should be Turing equivalent to a simple set. Read about it <a href="http://www.math.wisc.edu/~miller/old/m773-07/cmpthy.pdf">here</a> (Dekker's Deficieny Set, Section 17, Page 37/137).

  Simple and productive sets. Simple sets cannot be SD-complete. Post's construction of Simple sets.
  

  References	:	None
  Reading	:	Is there a simpler way to show this than constructing "simple" sets? This was a question raised in class. Dekker, in 1952 showed this on the negative by arguing that, any "intermediate set" should be Turing equivalent to a simple set. Read about it here (Dekker's Deficieny Set, Section 17, Page 37/137).

  • Meeting 20 : Mon, Sep 08, 12:00 pm-12:50 pm

  Every SD-complete set is co-productive. Freidberg-Muchnik theorem. Low sets. Construction of low simple sets. Finite injury priority argument - the strategy.

  References	Notes sent to the emailing list. Lectures 37-38 in [Kozen2] Book.
  Exercises
  Reading	<i>Are we doing more than what is required? Does co-productivity characterize SD-complete sets?</i> <br>Yes, they do. Myhill showed in the 50s, that if a set is productive then its complement must be SD-complete. Known as "Myhill's characterization of creative sets" can be read about in detail in <a href="http://www.math.wisc.edu/~miller/old/m773-07/cmpthy.pdf">Miller's Notes</a> (Page 33, Section 14).

  Every SD-complete set is co-productive. Freidberg-Muchnik theorem. Low sets. Construction of low simple sets. Finite injury priority argument - the strategy.
  

  References	:	Notes sent to the emailing list. Lectures 37-38 in [Kozen2] Book.
  Reading	:	Are we doing more than what is required? Does co-productivity characterize SD-complete sets? Yes, they do. Myhill showed in the 50s, that if a set is productive then its complement must be SD-complete. Known as "Myhill's characterization of creative sets" can be read about in detail in Miller's Notes (Page 33, Section 14).

  • Meeting 21 : Tue, Sep 09, 12:00 pm-12:50 pm

  Details of Finite Injury Argument. Construction of low simple set - the proof.

  References	[Kozen2] Lecture 38, Pages 253-256.
  Exercises
  Reading

  Details of Finite Injury Argument. Construction of low simple set - the proof.
  

  References

  :

  [Kozen2] Lecture 38, Pages 253-256.

• Theme 2 : Resource-bounded Complexity - 35 meetings

  • Meeting 22 : Thu, Sep 11, 11:00 am-11:50 am

  Half Lecture : Outline of the proof of Godel's Incompleteness Theorem using Computability.<br> World of Decidable Languages. Notion of a Computational Resource. Blum's Axioms.

  References	<ul> <li>[Kozen1] Lecture 39 and first part of Lecture 38</li> <li>[Kozen2] Supplementary Lecture J, Page 231-234</li> </ul>
  Exercises	<ul> <li>Think about alternatives for the axioms and attempt to compare them with Blum's axioms.</li> <li>Check if Space and Ink we defined in class are resources.</li> </ul>
  Reading	For a proof which does not use the computability language, see [Kozen1] Supplementary Lecture K.

  Half Lecture : Outline of the proof of Godel's Incompleteness Theorem using Computability.
  World of Decidable Languages. Notion of a Computational Resource. Blum's Axioms.
  

  References	:	[Kozen1] Lecture 39 and first part of Lecture 38 [Kozen2] Supplementary Lecture J, Page 231-234
  Exercises	:	Think about alternatives for the axioms and attempt to compare them with Blum's axioms. Check if Space and Ink we defined in class are resources.
  Reading	:	For a proof which does not use the computability language, see [Kozen1] Supplementary Lecture K.

  • Meeting 23 : Fri, Sep 12, 10:00 am-10:50 am

  Blum's Axioms. Examples and Non-examples of resources. Aiming to study g-decidable sets where g is a computable function. Plateau's in Hierarchy - Borodin's Gap Theorem.

  References	[Kozen2] Supplementary Lecture J, Page 231-234
  Exercises	Write down formal proofs on whether the number of head turns, number of distinct states visited are resources of computations.
  Reading

  Blum's Axioms. Examples and Non-examples of resources. Aiming to study g-decidable sets where g is a computable function. Plateau's in Hierarchy - Borodin's Gap Theorem.
  

  References	:	[Kozen2] Supplementary Lecture J, Page 231-234
  Exercises	:	Write down formal proofs on whether the number of head turns, number of distinct states visited are resources of computations.

  • Meeting 24 : Mon, Sep 15, 08:00 am-08:50 am

  Proof of Borodin's Gap Theorem. Blum's Speedup Theorem. Outline of the proof.

  References	[Kozen2] Lecture 32, Page 216-219
  Exercises
  Reading	[Kozen2] Supplementary Lecture J, Page 231-234

  Proof of Borodin's Gap Theorem. Blum's Speedup Theorem. Outline of the proof.
  

  References	:	[Kozen2] Lecture 32, Page 216-219
  Reading	:	[Kozen2] Supplementary Lecture J, Page 231-234

  • Meeting 25 : Tue, Sep 16, 12:00 pm-12:50 pm

  Construction of Blum's Speedup Set.

  References	[Kozen2] Lecture 32, Page 216-219
  Exercises
  Reading	[Kozen2] Supplementary Lecture J, Page 231-234

  Construction of Blum's Speedup Set.
  

  References	:	[Kozen2] Lecture 32, Page 216-219
  Reading	:	[Kozen2] Supplementary Lecture J, Page 231-234

  • Meeting 26 : Thu, Sep 18, 11:00 am-11:50 am

  Linear Speedup Theorem. Tape Compression Theorem. Time and Space Complexity Classes.

  References	[DK] Pages 19-21.
  Exercises
  Reading

  Linear Speedup Theorem. Tape Compression Theorem. Time and Space Complexity Classes.
  

  References

  :

  [DK] Pages 19-21.

  • Meeting 27 : Fri, Sep 19, 10:00 am-10:50 am

  Hartmanis-Stearns Tape reduction Bound. Optimality of this via Crossing Sequence Arguments. Hennie-Stearns Improved Tape reduction theorem for 2 tapes(statement).

  References	<ul> <li><a href="http://www.cs.umd.edu/~jkatz/complexity/f05/lower_bounds.pdf">Notes</a> by Katz</li> </ul>
  Exercises
  Reading	<ul> <li><a href="http://www.dcs.fmph.uniba.sk/~fduris/VKTI/2tape.pdf">Hennie-Stearns Tape Reduction</a></li> </ul>

  Hartmanis-Stearns Tape reduction Bound. Optimality of this via Crossing Sequence Arguments. Hennie-Stearns Improved Tape reduction theorem for 2 tapes(statement).
  

  References	:	Notes by Katz
  Reading	:	Hennie-Stearns Tape Reduction

  • Meeting 28 : Mon, Sep 22, 08:00 am-08:50 am

  Tape Alphabet Reduction. Time Hierarchy Theorem. Space Hierarchy Theorem.

  References	Follow Class Notes. Refer to <a href="http://www.eecs.berkeley.edu/~luca/cs172/noteh.pdf">Luca Trevisan's notes</a> for slightly different version of the proof.
  Exercises
  Reading

  Tape Alphabet Reduction. Time Hierarchy Theorem. Space Hierarchy Theorem.
  

  References

  :

  Follow Class Notes. Refer to Luca Trevisan's notes for slightly different version of the proof.

  • Meeting 29 : Tue, Sep 23, 12:00 pm-12:50 pm

  Notion of Efficiency. Edmond's proposition. Union Theorem. Complexity classes P, E and EXP. Space Complexity classes L, PSPACE, and EXPSPACE. From time bound to space bound and from a space bound to a time bound.

  References
  Exercises
  Reading

  Notion of Efficiency. Edmond's proposition. Union Theorem. Complexity classes P, E and EXP. Space Complexity classes L, PSPACE, and EXPSPACE. From time bound to space bound and from a space bound to a time bound.
  

  References

  :

  None

  • Meeting 30 : Thu, Sep 25, 11:00 am-11:50 am

  CLIQUE, EXACTCLIQUE, REACH, GI - four computational problems. EXP time algorithm for all of them. PSPACE algorithms for all of them. Poly time algorithm for REACH. EXACTCLIQUE problem vs CLIQUE problem. Efficient verifiability property for membership in CLIQUE.

  References
  Exercises
  Reading

  CLIQUE, EXACTCLIQUE, REACH, GI - four computational problems. EXP time algorithm for all of them. PSPACE algorithms for all of them. Poly time algorithm for REACH. EXACTCLIQUE problem vs CLIQUE problem. Efficient verifiability property for membership in CLIQUE.
  

  References

  :

  None

  • Meeting 31 : Fri, Sep 26, 10:00 am-10:50 am

  Non-determinism, definition of time and space. The Simulation and Containments.

  References
  Exercises
  Reading

  Non-determinism, definition of time and space. The Simulation and Containments.
  

  References

  :

  None

  • Meeting 32 : Mon, Sep 29, 10:00 am-10:50 am

  Padding Arguments : P = NP implies EXP = NEXP. Separating E from PSPACE. Discussion on P vs NP problem. Quest for structure within NP. A structural observation about the NP algorithm for Clique. Can that be exploited? No, it seems to be there for every NP algorithm - quantifier characterization for NP.

  References
  Exercises
  Reading

  Padding Arguments : P = NP implies EXP = NEXP. Separating E from PSPACE. Discussion on P vs NP problem. Quest for structure within NP. A structural observation about the NP algorithm for Clique. Can that be exploited? No, it seems to be there for every NP algorithm - quantifier characterization for NP.
  

  References

  :

  None

  • Meeting 33 : Mon, Sep 29, 05:00 pm-06:30 pm

  Quantifier characterization of NP. <br> CLIQUE, INDSET, VC, GI, SAT - Five problems in NP. Quest for structural similarities between problems. CLIQUE and INDSET, INDSET and VC. Structure and reductions. Reductions, Completeness. Cook-Levin Theorem and proof outline.

  References
  Exercises
  Reading

  Quantifier characterization of NP.
  CLIQUE, INDSET, VC, GI, SAT - Five problems in NP. Quest for structural similarities between problems. CLIQUE and INDSET, INDSET and VC. Structure and reductions. Reductions, Completeness. Cook-Levin Theorem and proof outline.
  

  References

  :

  None

  • Meeting 34 : Tue, Sep 30, 12:00 pm-12:50 pm

  Details of Cook-Levin Theorem. Reductions to CNF SAT. Reductions to 3CNFSAT.

  References	Du-Ko Book, Page 47.
  Exercises
  Reading

  Details of Cook-Levin Theorem. Reductions to CNF SAT. Reductions to 3CNFSAT.
  

  References

  :

  Du-Ko Book, Page 47.

  • Meeting 35 : Tue, Oct 07, 12:00 pm-12:50 pm

  Reduction from SAT to VC. Status of Graph Isomorphism Problem. Intermediate languages? Ladner's Theorem.

  References	Notes sent to the mailing list (Ladner's theorem)
  Exercises
  Reading

  Reduction from SAT to VC. Status of Graph Isomorphism Problem. Intermediate languages? Ladner's Theorem.
  

  References

  :

  Notes sent to the mailing list (Ladner's theorem)

  • Meeting 36 : Thu, Oct 09, 11:00 am-11:50 am

  Impagliazzo's proof of Ladner's theorem.

  References	Notes sent to the mailing list (Ladner's theorem)
  Exercises
  Reading

  Impagliazzo's proof of Ladner's theorem.
  

  References

  :

  Notes sent to the mailing list (Ladner's theorem)

  • Meeting 37 : Fri, Oct 10, 10:00 am-10:50 am

  Construction of the padding function in the proof of Ladner's Theorem. Complement of PRIMES is in NP. The class CoNP.

  References	Notes sent to the mailing list (Ladner's theorem)
  Exercises
  Reading

  Construction of the padding function in the proof of Ladner's Theorem. Complement of PRIMES is in NP. The class CoNP.
  

  References

  :

  Notes sent to the mailing list (Ladner's theorem)

  • Meeting 38 : Sun, Oct 12, 11:30 am-01:00 pm

  Relationship between NP, CoNP, P and PSAPCE. Two 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. Relativization. Baker-Gill-Solovoy theorem (statement).

  References
  Exercises
  Reading

  Relationship between NP, CoNP, P and PSAPCE. Two 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. Relativization. Baker-Gill-Solovoy theorem (statement).
  

  References

  :

  None

  • Meeting 39 : Mon, Oct 13, 08:00 am-08:50 am

  Proof of Baker-Gill-Solovoy Theorem.

  References	Chapter titled "diagonalization" - the second last section. Please check the above link. I am not mentioning page numbers because it is different in different versions. In the print version version, it is chapter 3, page 72, section 3.4.
  Exercises
  Reading

  Proof of Baker-Gill-Solovoy Theorem.
  

  References

  :

  Chapter titled "diagonalization" - the second last section. Please check the above link. I am not mentioning page numbers because it is different in different versions. In the print version version, it is chapter 3, page 72, section 3.4.

  • Meeting 40 : Tue, Oct 14, 12:00 pm-12:50 pm

  Nondeterministic space. Simulations. Abstraction of space bounded machines computation as a reachability problem. Savitch's theorem.

  References
  Exercises
  Reading

  Nondeterministic space. Simulations. Abstraction of space bounded machines computation as a reachability problem. Savitch's theorem.
  

  References

  :

  None

  • Meeting 41 : Thu, Oct 16, 11:00 am-11:50 am

  Alternating Turing Machines. Definition, Example. Simulations. <ul> <li>ATIME(t(n)) contained in DSPACE(t(n))</li> <li>NSPACE(s(n)) contained in ATIME(s(n)^2)</li> </ul> Conclusion : AP = PSPACE.

  References	We followed roughly the notes <a href="http://www.cs.yale.edu/homes/spielman/AdvComplexity/2000/lecture2.ps">here</a>.
  Exercises
  Reading

  Alternating Turing Machines. Definition, Example. Simulations.

  • ATIME(t(n)) contained in DSPACE(t(n))
  • NSPACE(s(n)) contained in ATIME(s(n)^2)

  Conclusion : AP = PSPACE.
  

  References

  :

  We followed roughly the notes here.

  • Meeting 42 : Fri, Oct 17, 10:00 am-10:50 am

  <ul> <li>DTIME(t(n)) contained in ASPACE(log t(n))</li> <li>ASPACE(s(n)) contained in DTIME(2^O(s(n))</li> </ul> Conclusion AL = P.

  References	We followed roughly the notes <a href="http://www.cs.yale.edu/homes/spielman/AdvComplexity/2000/lecture2.ps">here</a>.
  Exercises
  Reading

  • DTIME(t(n)) contained in ASPACE(log t(n))
  • ASPACE(s(n)) contained in DTIME(2^O(s(n))

  Conclusion AL = P.
  

  References

  :

  We followed roughly the notes here.

  • Meeting 43 : Mon, Oct 20, 08:00 am-08:50 am

  QBF is PSPACE-complete. Winning strategy in Games and PSPACE. Notion of reductions in space bounded settings.

  References	Arora Barak Chapter on Space Bounded Computation.
  Exercises
  Reading

  QBF is PSPACE-complete. Winning strategy in Games and PSPACE. Notion of reductions in space bounded settings.
  

  References

  :

  Arora Barak Chapter on Space Bounded Computation.

  • Meeting 44 : Tue, Oct 21, 12:00 pm-12:50 pm

  Log-space reductions. Composibility. NL-completeness. Reach is NL-complete. Immerman-Szelepscenyi Theorem, NL = CoNL.

  References
  Exercises
  Reading

  Log-space reductions. Composibility. NL-completeness. Reach is NL-complete. Immerman-Szelepscenyi Theorem, NL = CoNL.
  

  References

  :

  None

  • Meeting 45 : Thu, Oct 23, 11:00 am-11:50 am

  Immerman-Szelepscenyi Theorem. Inductive counting technique.

  References	<a href="http://blog.computationalcomplexity.org/2003/06/foundations-of-complexity-lesson-19.html">A Blog Entry</a>
  Exercises
  Reading

  Immerman-Szelepscenyi Theorem. Inductive counting technique.
  

  References

  :

  A Blog Entry

  • Meeting 46 : Fri, Oct 24, 10:00 am-10:50 am

  Details of Inductive Counting. Space complexity of reductions. NP-completeness of problems under logspace many-one reductions.

  References	<a href="http://blog.computationalcomplexity.org/2003/06/foundations-of-complexity-lesson-19.html">A Blog Entry</a>
  Exercises
  Reading

  Details of Inductive Counting. Space complexity of reductions. NP-completeness of problems under logspace many-one reductions.
  

  References

  :

  A Blog Entry

  • Meeting 47 : Mon, Oct 27, 08:00 am-08:50 am

  DAG-reach is NL-complete under Log-space reductions. Circuit Value Problem. P-completeness.

  References	Class Notes.
  Exercises
  Reading

  DAG-reach is NL-complete under Log-space reductions. Circuit Value Problem. P-completeness.
  

  References

  :

  Class Notes.

  • Meeting 48 : Tue, Oct 28, 12:00 pm-12:50 pm

  CVP is P-complete. Monotone-CVP. Horn SAT is P-complete.

  References	<ul> <li><a href="http://www.cs.cornell.edu/courses/CS6820/2012sp/Handouts/cvp.pdf">P-completeness of CVP</a></li> <li><a href="http://www.tcs.hut.fi/Studies/T-79.5103/2007AUT/tutorials/solutions_4.ps">Horn-SAT is P-complete</a></li> </ul>
  Exercises
  Reading

  CVP is P-complete. Monotone-CVP. Horn SAT is P-complete.
  

  References

  :

  P-completeness of CVP Horn-SAT is P-complete

  • Meeting 49 : Thu, Oct 30, 11:00 am-11:50 am

  Structure of NP-complete sets. Can a sparse set be NP-complete? Berman-Hartmanis Conjecture. A proof that Co-NP complete set cannot be sparse unless P=NP.

  References	<a href="http://www.villesalo.com/OtEoDoSNPHL.pdf">notes</a>
  Exercises
  Reading	Chapter 7, Du-Ko Book.

  Structure of NP-complete sets. Can a sparse set be NP-complete? Berman-Hartmanis Conjecture. A proof that Co-NP complete set cannot be sparse unless P=NP.
  

  References	:	notes
  Reading	:	Chapter 7, Du-Ko Book.

  • Meeting 50 : Fri, Oct 31, 10:00 am-10:50 am

  Mahaney's Theorem. A sparse set cannot be NP-complete unless P=NP.

  References	<a href="http://www.villesalo.com/OtEoDoSNPHL.pdf">notes</a>
  Exercises
  Reading

  Mahaney's Theorem. A sparse set cannot be NP-complete unless P=NP.
  

  References

  :

  notes

  • Meeting 51 : Sun, Nov 02, 10:15 am-12:15 pm

  Karp-Lipton Collapse Theorem. If a sparse set if NP-hard under Turing reductions, then PH collapses down to Sigma_2. <br> Randomized Algorithms and Machine Models. Yet another acceptance criterion for choice machines. BPP as counting classes. Containments and simulations.

  References	For the Karp-Lipton theorem - <a href="http://www.cs.rochester.edu/u/lane/=companion/self-reducible.pdf">Chapter on Self-reducibility</a>(Page 20-21).<br> For the second part refer to class-notes.
  Exercises
  Reading

  Karp-Lipton Collapse Theorem. If a sparse set if NP-hard under Turing reductions, then PH collapses down to Sigma_2.
  Randomized Algorithms and Machine Models. Yet another acceptance criterion for choice machines. BPP as counting classes. Containments and simulations.
  

  References

  :

  For the Karp-Lipton theorem - Chapter on Self-reducibility(Page 20-21). For the second part refer to class-notes.

  • Meeting 52 : Mon, Nov 03, 08:00 am-08:50 am

  Polynomial identity testing problem. Schwartz-Zippel Lemma. Class BPP. Success Probability Amplification.

  References	Class notes.
  Exercises
  Reading

  Polynomial identity testing problem. Schwartz-Zippel Lemma. Class BPP. Success Probability Amplification.
  

  References

  :

  Class notes.

  • Meeting 53 : Tue, Nov 04, 10:00 am-12:00 pm

  Two consequences of Success Probability Amplification. The class P/poly. BPP is in P/poly. Every language in P/poly Turing Reduces to sparse sets. If NP is contained in BPP then PH collapses. The class RP, CoRP and structural simulations.

  References
  Exercises
  Reading

  Two consequences of Success Probability Amplification. The class P/poly. BPP is in P/poly. Every language in P/poly Turing Reduces to sparse sets. If NP is contained in BPP then PH collapses. The class RP, CoRP and structural simulations.
  

  References

  :

  None

  • Meeting 54 : Thu, Nov 13, 11:00 am-12:00 pm

  BPP is in Sigma^2.

  References
  Exercises
  Reading

  BPP is in Sigma^2.
  

  References

  :

  None

  • Meeting 55 : Thu, Nov 13, 05:30 pm-06:30 am

  Randomized Logspace. Reachability in Undirected graphs. Significance of spectral gap. Every graph has a spectral gap. Algebraic Expanders. Combinatorial Expanders and equivalence. Reachability in Graphs with combinatorial expander components.

  References
  Exercises
  Reading

  Randomized Logspace. Reachability in Undirected graphs. Significance of spectral gap. Every graph has a spectral gap. Algebraic Expanders. Combinatorial Expanders and equivalence. Reachability in Graphs with combinatorial expander components.
  

  References

  :

  None

  • Meeting 56 : Fri, Nov 14, 10:00 am-11:00 am

  Graph Powering, Replacement product and their effects in the algebraic expansion. Details of Reingold's algorithm.

  References
  Exercises
  Reading

  Graph Powering, Replacement product and their effects in the algebraic expansion. Details of Reingold's algorithm.
  

  References

  :

  None

• Evaluation Meetings - 3 meetings

  • Meeting 57 : Mon, Sep 01, 08:00 am-08:50 am

  Quiz 1

  References
  Exercises
  Reading

  Quiz 1
  

  References

  :

  None

  • Meeting 58 : Sun, Oct 12, 09:30 am-11:15 am

  Quiz 2

  References
  Exercises
  Reading

  Quiz 2
  

  References

  :

  None

  • Meeting 59 : Wed, Nov 19, 09:00 am-12:00 pm

  Endsem Exam

  References
  Exercises
  Reading

  Endsem Exam
  

  References

  :

  None