Meetings

Click on the theme item for the meeting plan for that theme.Click on the meeting item for references, exercises, and additional reading related to it.

**Theme 1 : Turing Computability**- 21 meetings- Meeting 01 : Thu, Jul 31, 11:00 am-11:50 am
- Meeting 02 : Fri, Aug 01, 10:00 am-10:50 am
- Meeting 03 : Mon, Aug 04, 08:00 am-08:50 am
- Meeting 04 : Tue, Aug 05, 12:00 pm-12:50 pm
- Meeting 05 : Thu, Aug 07, 11:00 am-11:50 am
- Meeting 06 : Fri, Aug 08, 10:00 am-10:50 am
- Meeting 07 : Mon, Aug 11, 08:00 am-08:50 am
- Meeting 08 : Tue, Aug 12, 12:00 pm-12:50 pm
- Meeting 09 : Thu, Aug 14, 11:00 am-11:50 am
- Meeting 10 : Mon, Aug 18, 08:00 am-08:50 am
- Meeting 11 : Tue, Aug 19, 12:00 pm-12:50 pm
- Meeting 12 : Thu, Aug 21, 11:00 am-11:50 am
- Meeting 13 : Fri, Aug 22, 10:00 am-10:50 am
- Meeting 14 : Mon, Aug 25, 08:00 am-08:50 am
- Meeting 15 : Tue, Aug 26, 12:00 pm-12:50 pm
- Meeting 16 : Thu, Aug 28, 11:00 am-11:50 am
- Meeting 17 : Tue, Sep 02, 12:00 pm-12:50 pm
- Notes sent to the mailing list
- Meeting 18 : Thu, Sep 04, 11:00 am-11:50 am
- Notes sent to the mailing list
- Meeting 19 : Fri, Sep 05, 10:00 am-10:50 am
- Meeting 20 : Mon, Sep 08, 12:00 pm-12:50 pm
- Meeting 21 : Tue, Sep 09, 12:00 pm-12:50 pm

Introduction. Administrative Announcements. References Exercises Reading Introduction. Administrative Announcements.References : None Model of Computation. Post systems. Mu-calculus. Lambda-calculus. References Exercises Reading Model of Computation. Post systems. Mu-calculus. Lambda-calculus.References : None 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 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. 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] 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] 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] 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] 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. 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 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 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 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 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 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 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. 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 : 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 : 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. 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). 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).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
- [Kozen1] Lecture 39 and first part of Lecture 38
- [Kozen2] Supplementary Lecture J, Page 231-234
- 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.
- Meeting 23 : Fri, Sep 12, 10:00 am-10:50 am
- Meeting 24 : Mon, Sep 15, 08:00 am-08:50 am
- Meeting 25 : Tue, Sep 16, 12:00 pm-12:50 pm
- Meeting 26 : Thu, Sep 18, 11:00 am-11:50 am
- Meeting 27 : Fri, Sep 19, 10:00 am-10:50 am
- Notes by Katz
- Meeting 28 : Mon, Sep 22, 08:00 am-08:50 am
- Meeting 29 : Tue, Sep 23, 12:00 pm-12:50 pm
- Meeting 30 : Thu, Sep 25, 11:00 am-11:50 am
- Meeting 31 : Fri, Sep 26, 10:00 am-10:50 am
- Meeting 32 : Mon, Sep 29, 10:00 am-10:50 am
- Meeting 33 : Mon, Sep 29, 05:00 pm-06:30 pm
- Meeting 34 : Tue, Sep 30, 12:00 pm-12:50 pm
- Meeting 35 : Tue, Oct 07, 12:00 pm-12:50 pm
- Meeting 36 : Thu, Oct 09, 11:00 am-11:50 am
- Meeting 37 : Fri, Oct 10, 10:00 am-10:50 am
- Meeting 38 : Sun, Oct 12, 11:30 am-01:00 pm
- Meeting 39 : Mon, Oct 13, 08:00 am-08:50 am
- Meeting 40 : Tue, Oct 14, 12:00 pm-12:50 pm
- Meeting 41 : Thu, Oct 16, 11:00 am-11:50 am
- ATIME(t(n)) contained in DSPACE(t(n))
- NSPACE(s(n)) contained in ATIME(s(n)^2)
- Meeting 42 : Fri, Oct 17, 10:00 am-10:50 am
- DTIME(t(n)) contained in ASPACE(log t(n))
- ASPACE(s(n)) contained in DTIME(2^O(s(n))
- Meeting 43 : Mon, Oct 20, 08:00 am-08:50 am
- Meeting 44 : Tue, Oct 21, 12:00 pm-12:50 pm
- Meeting 45 : Thu, Oct 23, 11:00 am-11:50 am
- Meeting 46 : Fri, Oct 24, 10:00 am-10:50 am
- Meeting 47 : Mon, Oct 27, 08:00 am-08:50 am
- Meeting 48 : Tue, Oct 28, 12:00 pm-12:50 pm
- Meeting 49 : Thu, Oct 30, 11:00 am-11:50 am
- Meeting 50 : Fri, Oct 31, 10:00 am-10:50 am
- Meeting 51 : Sun, Nov 02, 10:15 am-12:15 pm
- Meeting 52 : Mon, Nov 03, 08:00 am-08:50 am
- Meeting 53 : Tue, Nov 04, 10:00 am-12:00 pm
- Meeting 54 : Thu, Nov 13, 11:00 am-12:00 pm
- Meeting 55 : Thu, Nov 13, 05:30 pm-06:30 am
- Meeting 56 : Fri, Nov 14, 10:00 am-11:00 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 : Exercises : Reading : For a proof which does not use the computability language, see [Kozen1] Supplementary Lecture K. 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. 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 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 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. 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 : Reading : 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. 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 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 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 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 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 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. 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) 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) 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) 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 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. 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 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.References : We followed roughly the notes here. <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 References : We followed roughly the notes here. 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. 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 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 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 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. 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 : 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. 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 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.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. 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 BPP is in Sigma^2. References Exercises Reading BPP is in Sigma^2.References : None 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 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
- Meeting 58 : Sun, Oct 12, 09:30 am-11:15 am
- Meeting 59 : Wed, Nov 19, 09:00 am-12:00 pm

Quiz 1 References Exercises Reading Quiz 1References : None Quiz 2 References Exercises Reading Quiz 2References : None Endsem Exam References Exercises Reading Endsem ExamReferences : None