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**- 24 meetings- Meeting 01 : Mon, Jul 30, 08:00 am-08:50 am
- 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
- Meeting 03 : Thu, Aug 02, 11:00 am-11:50 am
- Meeting 04 : Fri, Aug 03, 10:00 am-10:50 am
- Meeting 05 : Mon, Aug 06, 08:00 am-08:50 am
- Meeting 06 : Tue, Aug 07, 09:00 am-09:50 am
- 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
- Meeting 08 : Fri, Aug 10, 10:00 am-10:50 am
- Meeting 09 : Tue, Aug 14, 12:00 pm-12:50 pm
- Meeting 10 : Thu, Aug 16, 11:00 am-11:50 am
- Meeting 11 : Tue, Aug 21, 12:00 pm-12:50 pm
- Meeting 12 : Thu, Aug 23, 11:00 am-11:50 am
- Meeting 13 : Mon, Aug 27, 08:00 am-08:50 am
- Meeting 14 : Tue, Aug 28, 12:00 pm-12:50 pm
- Meeting 15 : Thu, Aug 30, 11:00 am-11:50 am
- Meeting 16 : Fri, Aug 31, 10:00 am-10:50 am
- Meeting 17 : Sat, Sep 01, 11:00 am-12:45 pm
- Meeting 18 : Mon, Sep 03, 08:00 am-08:50 am
- Meeting 19 : Tue, Sep 04, 12:00 pm-12:50 pm
- Meeting 20 : Fri, Sep 07, 10:00 am-10:50 am
- Meeting 21 : Mon, Sep 10, 12:00 pm-01:00 pm
- Meeting 22 : Tue, Sep 11, 12:00 pm-12:50 pm
- Meeting 23 : Thu, Sep 13, 11:00 am-11:50 am
- Meeting 24 : Fri, Sep 14, 10:00 am-10:50 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 : 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. 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] 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] Tape reduction. Encodings of Turing machines. Universal Turing Machines. References Exercises Reading Tape reduction. Encodings of Turing machines. Universal Turing Machines.References : None 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 : 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 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 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 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 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. 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 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]. 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_kReferences : Supplementary lecture J, Page 276 of [K1]. 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 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. 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. 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. 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. 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 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. 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. 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. 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.**Theme 2 : Basic Time & Space Complexity**- 33 meetings- Meeting 25 : Mon, Sep 17, 08:00 am-08:50 am
- Meeting 26 : Fri, Sep 21, 10:00 am-10:50 am
- Meeting 27 : Mon, Sep 24, 08:00 am-08:50 am
- Meeting 28 : Tue, Sep 25, 12:00 pm-12:50 pm
- Meeting 29 : Wed, Sep 26, 04:45 pm-05:35 pm
- Meeting 30 : Thu, Sep 27, 11:00 am-11:50 am
- Meeting 31 : Fri, Sep 28, 11:00 am-11:50 am
- Meeting 32 : Wed, Oct 03, 04:45 pm-05:35 pm
- Meeting 33 : Thu, Oct 04, 11:00 am-11:50 am
- Meeting 34 : Fri, Oct 05, 10:00 am-10:50 am
- Meeting 35 : Mon, Oct 08, 08:00 am-08:50 am
- Meeting 36 : Tue, Oct 09, 12:00 pm-12:50 pm
- Meeting 37 : Thu, Oct 11, 11:00 am-11:50 am
- Meeting 38 : Fri, Oct 12, 10:00 am-10:50 am
- Meeting 39 : Sat, Oct 13, 10:45 am-12:00 pm
- Meeting 40 : Mon, Oct 15, 08:00 am-08:50 am
- Meeting 41 : Tue, Oct 16, 12:00 pm-12:50 pm
- Meeting 42 : Wed, Oct 17, 12:00pm-12:50pm
- Meeting 43 : Fri, Oct 19, 10:00 am-10:50 am
- Meeting 44 : Mon, Oct 22, 08:00 am-08:50 am
- Meeting 45 : Tue, Oct 23, 12:00 pm-12:50 pm
- Meeting 46 : Thu, Oct 25, 11:00 am-11:50 am
- Meeting 47 : Fri, Oct 26, 10:00 am-10:50 am
- Meeting 48 : Mon, Oct 29, 08:00 am-08:50 am
- Meeting 49 : Tue, Oct 30, 12:00 pm-12:50 pm
- Meeting 50 : Thu, Nov 01, 11:00 am-11:50 am
- Meeting 51 : Mon, Nov 05, 08:00 am-08:50 am
- Meeting 52 : Tue, Nov 06, 12:00 pm-12:50 pm
- Meeting 53 : Wed, Nov 07, 04:45 pm-05:35 pm
- Meeting 54 : Thu, Nov 08, 11:00 am-11:50 am
- Meeting 55 : Fri, Nov 09, 10:00 am-10:50 am
- Meeting 56 : Wed, Nov 14, 04:45 pm-05:35 pm
- Meeting 57 : Thu, Nov 15, 11:00 am-11:50 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. 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]. 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]. 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. 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. 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. 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. 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. 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. 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] 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] 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 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] 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] 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. 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] 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] 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] 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] 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 ideaReferences : Lecture notes sent to the group.

Section 3.3 of [AB]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]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] 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] 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] 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] 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] PSPACE completeness of QBF, Geography. References Exercises Reading PSPACE completeness of QBF, Geography.References : None 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 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 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 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 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 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
- Meeting 59 : Sat, Oct 13, 09:00 am-10:30 am
- Meeting 60 : Mon, Nov 19, 09:00 am-12:00 pm

Quiz I (20%) References Exercises Reading Quiz I (20%)References : None Quiz II (20%) References Exercises Reading Quiz II (20%)References : None End-semester Examination References Exercises Reading End-semester ExaminationReferences : None