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 : Computability Theory**- 20 meetings- Meeting 01 : Mon, Aug 01, 08:00 am-08:50 am
- Meeting 02 : Tue, Aug 02, 12:00 pm-12:50 pm
- Meeting 03 : Thu, Aug 04, 11:00 am-11:50 am
- Meeting 04 : Fri, Aug 05, 10:00 am-10:50 am
- Meeting 05 : Mon, Aug 08, 08:00 am-08:50 am
- Meeting 06 : Tue, Aug 09, 12:00 pm-12:50 pm
- Meeting 07 : Thu, Aug 11, 11:00 am-11:50 am
- Meeting 08 : Fri, Aug 12, 10:00 am-10:50 am
- Meeting 09 : Tue, Aug 16, 12:00 pm-12:50 pm
- Meeting 10 : Thu, Aug 18, 11:00 am-11:50 am
- Meeting 11 : Fri, Aug 19, 10:00 am-10:50 am
- Meeting 12 : Mon, Aug 22, 08:00 am-08:50 am
- Meeting 13 : Tue, Aug 23, 12:00 pm-12:50 pm
- Meeting 14 : Thu, Aug 25, 11:00 am-11:50 am
- Meeting 15 : Fri, Aug 26, 10:00 am-10:50 am
- Meeting 16 : Mon, Aug 29, 08:00 am-08:50 am
- Meeting 17 : Tue, Aug 30, 12:00 pm-12:50 pm
- Meeting 18 : Thu, Sep 01, 11:00 am-11:50 am
- Meeting 19 : Fri, Sep 02, 10:00 am-10:50 am
- Meeting 20 : Tue, Sep 06, 12:00 pm-12:50 pm

Administrative announcements. The overview of the course. Historical perspective. Models of computation. Algorithms vs Turing machines. Languages vs Computational Problems. References Lectuer 28 in [K1 Book]. Exercises Reading Administrative announcements. The overview of the course. Historical perspective. Models of computation. Algorithms vs Turing machines. Languages vs Computational Problems.References : Lectuer 28 in [K1 Book]. The Turing machine model in formal terms. Acceptance. Configurations. Examples. Tape reduction in Turing machines. References Lectuer 28,29,30 in [K1 Book]. Exercises Reading The Turing machine model in formal terms. Acceptance. Configurations. Examples. Tape reduction in Turing machines.References : Lectuer 28,29,30 in [K1 Book]. Two computational problems. Total Turing machines. Decidability vs Semi-decidability. References Lecture 29,30 in [K1 Book] Exercises Reading Two computational problems. Total Turing machines. Decidability vs Semi-decidability.References : Lecture 29,30 in [K1 Book] Encoding of Turing machines. Two consequences. Universal Turing machines, and Diagonalization. A proof that the halting Problem is undecidable. Variants of diagonalization. References Lecture 31 in [K1 Book] Exercises Reading Encoding of Turing machines. Two consequences. Universal Turing machines, and Diagonalization. A proof that the halting Problem is undecidable. Variants of diagonalization.References : Lecture 31 in [K1 Book] HP-42 is also undecidable. Notion of reductions. MP vs HP problem. References Lecture 32-33 in [K1 Book] Exercises Reading HP-42 is also undecidable. Notion of reductions. MP vs HP problem.References : Lecture 32-33 in [K1 Book] More undecidable problems - EMPTY, TOTAL, REG, CFL, DEC. Abstracting the proof idea. Statement of Rice-Myhill-Shapira theorem. References Lecture 34 in [K1 Book] Exercises Reading More undecidable problems - EMPTY, TOTAL, REG, CFL, DEC. Abstracting the proof idea. Statement of Rice-Myhill-Shapira theorem.References : Lecture 34 in [K1 Book] Proof of Rice-Myhill-Shapira theorem. Example applications. REG, DEC, EMPTY, FIN, CFL. References Lecture 34 in [K1 Book] Exercises Reading Proof of Rice-Myhill-Shapira theorem. Example applications. REG, DEC, EMPTY, FIN, CFL.References : Lecture 34 in [K1 Book] Beyond semi-decidables The class Co-SD. D = SD intersection Co-SD. FIN is not in SD and FIN is not in CoSD. References Lecture 34 in [K1 Book] Exercises Reading Beyond semi-decidables The class Co-SD. D = SD intersection Co-SD. FIN is not in SD and FIN is not in CoSD.References : Lecture 34 in [K1 Book] SD-completeness of MP. Beyond SD an Co-SD. Rice's Second Theorem about monotone properties of Semi-decidable languages. References Supplementary Lecture J in [K1 Book] Exercises Reading SD-completeness of MP. Beyond SD an Co-SD. Rice's Second Theorem about monotone properties of Semi-decidable languages.References : Supplementary Lecture J in [K1 Book] What languages become decidable if HP is decidable? The example beyond SD and CoSD. MPXOR. Oracle Turing machines. Relative Computability. Turing reductions. References Exercises Reading What languages become decidable if HP is decidable? The example beyond SD and CoSD. MPXOR. Oracle Turing machines. Relative Computability. Turing reductions.References : None Languages Semi-decidable in HP. Relativizing proofs. The Arithmetic Hierarchy. Diagonalization proof relativizes. Arithmetic hierarchy is strict. References Exercises Reading Languages Semi-decidable in HP. Relativizing proofs. The Arithmetic Hierarchy. Diagonalization proof relativizes. Arithmetic hierarchy is strict.References : None Quantified Predicate characterization of the Membership Problem. Extending it to all semi-decidable languages. Quantifier characterization of AH. Placing the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy. References Exercises Reading Quantified Predicate characterization of the Membership Problem. Extending it to all semi-decidable languages. Quantifier characterization of AH. Placing the languages FIN, EMPTY, REG, CFL, DEC in the hierarchy.References : None FIN is complete for the second level(Sigma-2) of the hierarchy. References Exercises Reading FIN is complete for the second level(Sigma-2) of the hierarchy.References : None Proof of the Quantifier characterization. References Notes sent to the mailing list. Exercises Reading Proof of the Quantifier characterization.References : Notes sent to the mailing list. Proof of the Quantifier Characterization of Arithmetic Hierarchy. References Exercises Reading Proof of the Quantifier Characterization of Arithmetic Hierarchy.References : None Peano's Axioms, Godels Incompleteness Theorem. Computability based non-constructive proof. References Exercises Reading Peano's Axioms, Godels Incompleteness Theorem. Computability based non-constructive proof.References : None Godel's Incompleteness Theorem - details of the proof. References Exercises Reading Godel's Incompleteness Theorem - details of the proof.References : None Post's Theorem : There is an semi-decidable but undecidable language which is not complete for the class of semi-decidable languages. Simple sets cannot be SD-complete. Post's construction of Simple sets. References Exercises Reading Post's Theorem : There is an semi-decidable but undecidable language which is not complete for the class of semi-decidable languages. Simple sets cannot be SD-complete. Post's construction of Simple sets.References : None Productive Sets. Simple sets cannot be co-productive. The self-halting set(K) is co-productive. References Exercises Reading Productive Sets. Simple sets cannot be co-productive. The self-halting set(K) is co-productive.References : None Every SD-complete set is co-productive. References Exercises Reading Every SD-complete set is co-productive.References : None **Theme 2 : Resource Bounded Complexity Theory**- 35 meetings- Meeting 21 : Thu, Sep 08, 11:00 am-11:50 am
- Meeting 22 : Fri, Sep 09, 10:00 am-10:50 am
- Meeting 23 : Thu, Sep 15, 11:00 am-11:50 am
- Meeting 24 : Fri, Sep 16, 10:00 am-10:50 am
- Meeting 25 : Mon, Sep 19, 08:00 am-08:50 am
- Meeting 26 : Tue, Sep 20, 12:00 pm-12:50 pm
- Meeting 27 : Thu, Sep 22, 11:00 am-11:50 am
- Meeting 28 : Fri, Sep 23, 10:00 am-10:50 am
- Meeting 29 : Mon, Sep 26, 08:00 am-08:50 am
- Meeting 30 : Tue, Sep 27, 12:00 pm-12:50 pm
- Meeting 31 : Thu, Sep 29, 11:00 am-11:50 am
- Meeting 32 : Fri, Sep 30, 10:00 am-10:50 am
- Meeting 33 : Mon, Oct 03, 08:00 am-08:50 am
- Meeting 34 : Tue, Oct 04, 12:00 pm-12:50 pm
- Meeting 35 : Thu, Oct 06, 11:00 am-11:50 am
- Meeting 36 : Fri, Oct 07, 10:00 am-10:50 am
- Meeting 37 : Thu, Oct 13, 11:00 am-11:50 am
- Meeting 38 : Fri, Oct 14, 10:00 am-10:50 am
- Meeting 39 : Tue, Oct 18, 12:00 pm-12:50 pm
- Meeting 40 : Thu, Oct 20, 11:00 am-11:50 am
- Meeting 41 : Fri, Oct 21, 10:00 am-10:50 am
- Meeting 42 : Mon, Oct 24, 08:00 am-08:50 am
- Meeting 43 : Tue, Oct 25, 12:00 pm-12:50 pm
- Meeting 44 : Thu, Oct 27, 11:00 am-11:50 am
- Meeting 45 : Fri, Oct 28, 10:00 am-10:50 am
- Meeting 46 : Mon, Oct 31, 08:00 am-08:50 am
- Meeting 47 : Tue, Nov 01, 12:00 pm-12:50 pm
- Meeting 48 : Wed, Nov 02, 05:00 pm-06:00 pm
- Meeting 49 : Thu, Nov 03, 11:00 am-11:50 am
- Meeting 50 : Fri, Nov 04, 10:00 am-10:50 am
- Meeting 51 : Sat, Nov 05, 12:30 pm-01:30 pm
- Meeting 52 : Mon, Nov 07, 08:00 am-08:50 am
- Meeting 53 : Tue, Nov 08, 12:00 pm-12:50 pm
- Meeting 54 : Thu, Nov 10, 11:00 am-11:50 am
- Meeting 55 : Fri, Nov 11, 10:00 am-10:50 am

Shorter Lecture : Blum's Axioms. Examples and Non-examples of resources. References Exercises Reading Shorter Lecture : Blum's Axioms. Examples and Non-examples of resources.References : None More Examples. Complexity classes based on a resource bound. References Exercises Reading More Examples. Complexity classes based on a resource bound.References : None Borodin's Gap Theorem. References Exercises Reading Borodin's Gap Theorem.References : None Shorter Lecture : Tape reduction and time and space. Tape Compression Theorem. References Exercises Reading Shorter Lecture : Tape reduction and time and space. Tape Compression Theorem.References : None Linear Speedup Theorem. Constructibility of functions (time and space). Time Hierarchy Theorem. Space Hierarchy Theorem. The ideas of the proofs. References Exercises Reading Linear Speedup Theorem. Constructibility of functions (time and space). Time Hierarchy Theorem. Space Hierarchy Theorem. The ideas of the proofs.References : None Detailed proof of the hierarchy theorem. Importance of the tape reduction. Hennie-Stearns improved Tape reduction theorem for 2 tapes. Optimality of the Hartmani-Stearns Tape reduction. References Exercises Reading Detailed proof of the hierarchy theorem. Importance of the tape reduction. Hennie-Stearns improved Tape reduction theorem for 2 tapes. Optimality of the Hartmani-Stearns Tape reduction.References : None Optimality of k-tape to 1-tape reduction via Crossing Sequence Arguments. An application to show that TMs using space less than loglog n accept only regular languages. References Exercises Reading Optimality of k-tape to 1-tape reduction via Crossing Sequence Arguments. An application to show that TMs using space less than loglog n accept only regular languages.References : None Complexity classes. 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 Complexity classes. 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 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 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. 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 CIRCUIT SAT. From CIRCUIT SAT to 3CNFSAT. References Exercises Reading Details of Cook-Levin Theorem. Reductions to CIRCUIT SAT. From CIRCUIT SAT to 3CNFSAT.References : None Reduction from SAT to VC. References Exercises Reading Reduction from SAT to VC.References : None 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. 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.References : None NP-Intermediate languages? Ladner's Theorem. Padding Arguments : P = NP implies EXP = NEXP. Separating E from PSPACE. Discussion on P vs NP problem. References Exercises Reading NP-Intermediate languages? Ladner's Theorem. Padding Arguments : P = NP implies EXP = NEXP. Separating E from PSPACE. Discussion on P vs NP problem.References : None Impagliazzo's proof of Ladner's theorem. References Exercises Reading Impagliazzo's proof of Ladner's theorem.References : None Impagliazzo's proof of Ladner's theorem. References Exercises Reading Impagliazzo's proof of Ladner's theorem.References : None Relativization. Baker-Gill-Solovoy theorem. References Exercises Reading Relativization. Baker-Gill-Solovoy theorem.References : None References Exercises Reading References : None References Exercises Reading References : None References Exercises Reading References : None References Exercises Reading References : None References Exercises Reading References : None Structure of NP-complete sets. Can a sparse set be NP-complete? Berman-Hartmanis Conjecture. Mahaney's theorem (Statement). References Exercises Reading Structure of NP-complete sets. Can a sparse set be NP-complete? Berman-Hartmanis Conjecture. Mahaney's theorem (Statement).References : None Mahaney's Theorem. A sparse set cannot be NP-complete unless P=NP. References Exercises Reading Mahaney's Theorem. A sparse set cannot be NP-complete unless P=NP.References : None Space Complexity. Complexity of Reductions. Composition of Logspace reductions. Savitch's Theorem. References Exercises Reading Space Complexity. Complexity of Reductions. Composition of Logspace reductions. Savitch's Theorem.References : None Immerman-Szelepsinyi Theorem. Inductive Counting. References Exercises Reading Immerman-Szelepsinyi Theorem. Inductive Counting.References : None QBF is PSPACE-complete. References Exercises Reading QBF is PSPACE-complete.References : None CVP is P-complete. Story of Satisfiability 2SAT is in NP. References Exercises Reading CVP is P-complete. Story of Satisfiability 2SAT is in NP.References : None 2SAT is NP-hard, HornSAT is P-complete. Story of Reachability. Randomized Algorithms. References Exercises Reading 2SAT is NP-hard, HornSAT is P-complete. Story of Reachability. Randomized Algorithms.References : None 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 56 : Tue, Sep 13, 10:00 am-11:15 am
- Meeting 57 : Mon, Oct 17, 08:00 am-08:50 am
- Meeting 58 : Tue, Nov 15, 12:00 pm-12:50 pm

To Be Announced References Exercises Reading To Be AnnouncedReferences : None Quiz 1 (Syllabus : Lectures 1-21 References Exercises Reading Quiz 1 (Syllabus : Lectures 1-21References : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None