CS6014 - Computability and Complexity

Meetings  

• Theme 1 : Computability Theory - 29 meetings

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

  Brief introduction to the course and the contents. Administrative Announcements. Grading policies.<br> Languages vs Algorithmic Problems. Automata theory vs algorithms courses. Historical aspects of the idea of computation. Ruler and Compass problem. Hilbert's problem. Need of understanding computation in very formal terms.

  References
  Exercises
  Reading

  Brief introduction to the course and the contents. Administrative Announcements. Grading policies.
  Languages vs Algorithmic Problems. Automata theory vs algorithms courses. Historical aspects of the idea of computation. Ruler and Compass problem. Hilbert's problem. Need of understanding computation in very formal terms.
  

  References

  :

  None

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

  Problem of Searching the Treasure on the number line. Algorithms. Drawbacks. Diophantine problem.<br><br> Formal model of computation. Post Systems, Mu-calculus, Lambda calculus. Turing machines. Formal Model of Turing machines.

  References
  Exercises
  Reading

  Problem of Searching the Treasure on the number line. Algorithms. Drawbacks. Diophantine problem.
  
  Formal model of computation. Post Systems, Mu-calculus, Lambda calculus. Turing machines. Formal Model of Turing machines.
  

  References

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  None

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

  Configurations of TM model. Acceptance of a string by Turing machines. Decidable, semi-decidable and undecidable. Robustness of the Turing machine model. Tape reduction theorem.

  References
  Exercises
  Reading

  Configurations of TM model. Acceptance of a string by Turing machines. Decidable, semi-decidable and undecidable. Robustness of the Turing machine model. Tape reduction theorem.
  

  References

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  None

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

  Encoding Turing machines as strings. Enumeration of Turing machines. Cantor's diagonalization argument for showing that R is uncountably infinite.

  References
  Exercises
  Reading

  Encoding Turing machines as strings. Enumeration of Turing machines. Cantor's diagonalization argument for showing that R is uncountably infinite.
  

  References

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  None

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

  First undecidable task. The proof that "entry filling machines" for the matrix defined, cannot exist. Discussions related to the implications.

  References
  Exercises
  Reading

  First undecidable task. The proof that "entry filling machines" for the matrix defined, cannot exist. Discussions related to the implications.
  

  References

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  None

  • Meeting 06 : Tue, Aug 07, 12:00 pm-12:50 pm

  The halting problem. The membership problem. The undecidability of both languages. Relationship among that two languages. Two ways of observing the relation. Comparing the two methods.

  References
  Exercises
  Reading

  The halting problem. The membership problem. The undecidability of both languages. Relationship among that two languages. Two ways of observing the relation. Comparing the two methods.
  

  References

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  None

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

  Formally capturing the relation between MP and HP. Definition of many-one reductions. Other natural forms of reductions.

  References
  Exercises
  Reading

  Formally capturing the relation between MP and HP. Definition of many-one reductions. Other natural forms of reductions.
  

  References

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  None

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

  Does knowing the input in advance help? MP-101 problem. Proof that MP-101 is undecidable. MP-101 is equivalent to MP. Transitivity of reductions. Observations about the reduction. Showing that FULL, and EMPTY are undecidable.

  References
  Exercises
  Reading

  Does knowing the input in advance help? MP-101 problem. Proof that MP-101 is undecidable. MP-101 is equivalent to MP. Transitivity of reductions. Observations about the reduction. Showing that FULL, and EMPTY are undecidable.
  

  References

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  None

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

  Other languages like EMPTY and FULL.<br> INF, REG, CFL, DEC. Undecidability of INF. Reduction from MP to INF. Reduction from MP to NREG. Reduction from MP to REG. Reduction from MP to NCFL. Reduction from MP to NDEC. Observations about the reductions.

  References
  Exercises
  Reading

  Other languages like EMPTY and FULL.
  INF, REG, CFL, DEC. Undecidability of INF. Reduction from MP to INF. Reduction from MP to NREG. Reduction from MP to REG. Reduction from MP to NCFL. Reduction from MP to NDEC. Observations about the reductions.
  

  References

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  None

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

  Rice-Myhill-Shapiro Theorem - I. Every nontrivial property of semi-decidable languages is undecidable. Example applications.

  References
  Exercises
  Reading

  Rice-Myhill-Shapiro Theorem - I. Every nontrivial property of semi-decidable languages is undecidable. Example applications.
  

  References

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  None

  • Meeting 11 : Fri, Aug 17, 10:00 am-10:50 am

  Importance of the study of the class of semi-decidable languages. Closure properties of D and SD. Characterization of decidable languages and semi-decidable languages. Examples of languages which are not even semi-decidable. Showing that FIN is not even semi-decidable.

  References
  Exercises
  Reading

  Importance of the study of the class of semi-decidable languages. Closure properties of D and SD. Characterization of decidable languages and semi-decidable languages. Examples of languages which are not even semi-decidable. Showing that FIN is not even semi-decidable.
  

  References

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  None

  • Meeting 12 : Mon, Aug 20, 08:00 am-08:50 am

  Rice-Myhill-Shapiro Theorem - II. Every non-monotone property of semi-decidable languages is outside SD. Proof and Applications.

  References
  Exercises
  Reading

  Rice-Myhill-Shapiro Theorem - II. Every non-monotone property of semi-decidable languages is outside SD. Proof and Applications.
  

  References

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  None

  • Meeting 13 : Tue, Aug 21, 12:00 pm-12:50 pm

  Enumerability and semi-decidability. A characterization.

  References
  Exercises
  Reading

  Enumerability and semi-decidability. A characterization.
  

  References

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  None

  • Meeting 14 : Thu, Aug 23, 11:00 am-11:50 am

  An application : Godel's Incompleteness Theorem. Set up and the statement of the theorem.

  References
  Exercises
  Reading

  An application : Godel's Incompleteness Theorem. Set up and the statement of the theorem.
  

  References

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  None

  • Meeting 15 : Fri, Aug 24, 10:00 am-10:50 am

  Proof of Godel's incompleteness theorem. Showing the Th(PA) is semi-decidable and T(N) is not even semidecidable. Reducing MP-complement to T(N). Validity of a computation history can be expressed as a FO formula VALCOMP(a) over the natural numbers.

  References	[K1] Lecture on Proof of completeness theorem.
  Exercises
  Reading

  Proof of Godel's incompleteness theorem. Showing the Th(PA) is semi-decidable and T(N) is not even semidecidable. Reducing MP-complement to T(N). Validity of a computation history can be expressed as a FO formula VALCOMP(a) over the natural numbers.
  

  References

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  [K1] Lecture on Proof of completeness theorem.

  • Meeting 16 : Mon, Aug 27, 08:00 am-08:50 am

  Hardness and Completeness in SD. MP is SD-complete. Other SD-complete languages. SD-complete languages are not decidable.

  References	Class Notes
  Exercises
  Reading

  Hardness and Completeness in SD. MP is SD-complete. Other SD-complete languages. SD-complete languages are not decidable.
  

  References

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  Class Notes

  • Meeting 17 : Tue, Aug 28, 12:00 pm-12:50 pm

  Rich world outside SD. The MPXOR language. Oracle Turing machines. The class Sigma-2 and Delta-2. The class Pi-2.

  References	Class Notes
  Exercises
  Reading

  Rich world outside SD. The MPXOR language. Oracle Turing machines. The class Sigma-2 and Delta-2. The class Pi-2.
  

  References

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  Class Notes

  • Meeting 18 : Thu, Aug 30, 11:00 am-11:50 am

  The arithmetic hierarchy. Proof that Delta-2 = Sigma2 intersection Pi-2.

  References	Class Notes
  Exercises
  Reading

  The arithmetic hierarchy. Proof that Delta-2 = Sigma2 intersection Pi-2.
  

  References

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  Class Notes

  • Meeting 19 : Fri, Aug 31, 10:00 am-10:50 am

  Relatizing the completeness proofs. MP^MP is Sigma-2 complete. The third level of the hierarchy.

  References	Class Notes
  Exercises
  Reading

  Relatizing the completeness proofs. MP^MP is Sigma-2 complete. The third level of the hierarchy.
  

  References

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  Class Notes

  • Meeting 20 : Tue, Sep 04, 12:00 pm-12:50 pm

  Quantifier Characterization of Arithmetic Hierarchy. The base case for SD. Applications to place languages in the AH. Placing REG, FIN, etc in AH.

  References	Class Notes
  Exercises
  Reading

  Quantifier Characterization of Arithmetic Hierarchy. The base case for SD. Applications to place languages in the AH. Placing REG, FIN, etc in AH.
  

  References

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  Class Notes

  • Meeting 21 : Thu, Sep 06, 11:00 am-11:50 am

  Placing MP^MP in Sigma_2 via the characterizations. Taking clues from the proof to lead to the proof of the characterization itself.

  References	Class Notes
  Exercises
  Reading

  Placing MP^MP in Sigma_2 via the characterizations. Taking clues from the proof to lead to the proof of the characterization itself.
  

  References

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  Class Notes

  • Meeting 22 : Fri, Sep 07, 10:00 am-10:50 am

  Proof of Quantifier Characterization of Arithmetic Hierarchy.

  References	Class Notes
  Exercises
  Reading

  Proof of Quantifier Characterization of Arithmetic Hierarchy.
  

  References

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  Class Notes

  • Meeting 23 : Mon, Sep 10, 08:00 am-08:50 am

  FIN is Sigma-2 complete. SD-intermediate languages. Proof strategy and discussions.

  References	[K1] Textbook. Lecture on Arithmetic Hierarchy.
  Exercises
  Reading

  FIN is Sigma-2 complete. SD-intermediate languages. Proof strategy and discussions.
  

  References

  :

  [K1] Textbook. Lecture on Arithmetic Hierarchy.

  • Meeting 24 : Tue, Sep 11, 12:00 pm-12:50 pm

  Posts theorem. Simple sets. Arriving at the definition of simple sets from the requirements. From simple sets Construction of simple sets. The algorithm and the proof idea.

  References	Lecture 37 in [K2]
  Exercises
  Reading

  Posts theorem. Simple sets. Arriving at the definition of simple sets from the requirements. From simple sets Construction of simple sets. The algorithm and the proof idea.
  

  References

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  Lecture 37 in [K2]

  • Meeting 25 : Fri, Sep 14, 10:00 am-10:50 am

  Productive sets. Complements of Simple sets cannot be productive.

  References	Lecture 37 in [K2]
  Exercises
  Reading

  Productive sets. Complements of Simple sets cannot be productive.
  

  References

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  Lecture 37 in [K2]

  • Meeting 26 : Mon, Sep 17, 08:00 am-08:50 am

  Complements of SD-complete sets are productive. Relativizing the Post's theorem.

  References	Lecture 37 in [K2]
  Exercises
  Reading

  Complements of SD-complete sets are productive. Relativizing the Post's theorem.
  

  References

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  Lecture 37 in [K2]

  • Meeting 27 : Tue, Sep 18, 12:00 pm-12:50 pm

  Freidburg-Muchnik Theorem. Formulating the proof strategy. Low simple sets.

  References	Lecture 38 in [K2]
  Exercises
  Reading

  Freidburg-Muchnik Theorem. Formulating the proof strategy. Low simple sets.
  

  References

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  Lecture 38 in [K2]

  • Meeting 28 : Thu, Sep 20, 11:00 am-11:50 am

  Construction of the low simple sets. Positive and Negative conditions.

  References	Lecture 38 in [K2]
  Exercises
  Reading

  Construction of the low simple sets. Positive and Negative conditions.
  

  References

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  Lecture 38 in [K2]

  • Meeting 29 : Tue, Sep 25, 12:00 pm-12:50 pm

  Details of Freidburg-Muchnik Theorem. Finite Injury arguments to satisfy the negative conditions.

  References	Lecture 38 in [K2]
  Exercises
  Reading

  Details of Freidburg-Muchnik Theorem. Finite Injury arguments to satisfy the negative conditions.
  

  References

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  Lecture 38 in [K2]

• Theme 2 : Time & Space Complexity Theory - 28 meetings

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

  Complexity Measures. Blum's Axiom. Examples and Non-examples.

  References
  Exercises
  Reading

  Complexity Measures. Blum's Axiom. Examples and Non-examples.
  

  References

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  None

  • Meeting 31 : Fri, Sep 28, 10:00 am-10:50 am

  More Examples and Non-examples of resources. Space, ITIME. The class of g-decidables of a computable function g. Complexity classes. Two questions.

  References	Class Notes
  Exercises
  Reading	Supplementary Lecture J in [K2]

  More Examples and Non-examples of resources. Space, ITIME. The class of g-decidables of a computable function g. Complexity classes. Two questions.
  

  References	:	Class Notes
  Reading	:	Supplementary Lecture J in [K2]

  • Meeting 32 : Mon, Oct 01, 08:00 am-08:50 am

  Proof of the Borodin's Gap Theorem.

  References	Lecture 32 in [K2]
  Exercises
  Reading	Supplementary Lecture J in [K2]

  Proof of the Borodin's Gap Theorem.
  

  References	:	Lecture 32 in [K2]
  Reading	:	Supplementary Lecture J in [K2]

  • Meeting 33 : Tue, Oct 02, 12:00 pm-12:50 pm

  Linear Speed up Theorem.

  References	[DK] Page 21
  Exercises
  Reading	[DK] Pages 20-30

  Linear Speed up Theorem.
  

  References	:	[DK] Page 21
  Reading	:	[DK] Pages 20-30

  • Meeting 34 : Thu, Oct 04, 11:00 am-11:50 am

  Time Hierarchy Theorem. Attempt 1. Failures. Overheads in UTM simulations. Importance of the tape reduction.

  References	Class Notes
  Exercises
  Reading	[DK] Pages 20-30

  Time Hierarchy Theorem. Attempt 1. Failures. Overheads in UTM simulations. Importance of the tape reduction.
  

  References	:	Class Notes
  Reading	:	[DK] Pages 20-30

  • Meeting 35 : Fri, Oct 05, 10:00 am-10:50 am

  Tape reduction theorem. Quadratic loss in time. Improved one due to Hennie-Stearns.

  References	Class notes <br> [DK] Page 24.
  Exercises
  Reading

  Tape reduction theorem. Quadratic loss in time. Improved one due to Hennie-Stearns.
  

  References

  :

  Class notes [DK] Page 24.

  • Meeting 36 : Mon, Oct 08, 08:00 am-08:50 am

  Optimality of the k-tape to 1-tape reduction. PAL languages requires n^2 time on 1-tape TMs but has a linear time 2-tape TM. Proof strategy. Crossing Sequences.

  References	Class notes. [K2] Lecture 1
  Exercises
  Reading

  Optimality of the k-tape to 1-tape reduction. PAL languages requires n^2 time on 1-tape TMs but has a linear time 2-tape TM. Proof strategy. Crossing Sequences.
  

  References

  :

  Class notes. [K2] Lecture 1

  • Meeting 37 : Tue, Oct 09, 12:00 pm-12:50 pm

  Non-intersecting Crossing Sequence sets. Quadratic Lower Bounds for 1-tape Turing machines for PAL. Comments on Time-space trade offs. <br><br> Notion of efficiency in terms of time. Edmond's proposition. The class P. Union Theorem.

  References	[K2] Lecture 1
  Exercises	Prove a time space tradeoff for 1-tape Turing machines.
  Reading	Supplementary Lecture J in [K2]

  Non-intersecting Crossing Sequence sets. Quadratic Lower Bounds for 1-tape Turing machines for PAL. Comments on Time-space trade offs.
  
  Notion of efficiency in terms of time. Edmond's proposition. The class P. Union Theorem.
  

  References	:	[K2] Lecture 1
  Exercises	:	Prove a time space tradeoff for 1-tape Turing machines.
  Reading	:	Supplementary Lecture J in [K2]

  • Meeting 38 : Thu, Oct 11, 11:00 am-11:50 am

  Classes EXP, E. Applying union theorem. Algorithmic Problems CLIQUE, VertexCover, IndSet. Trivial Algorithms and runtime bounds.

  References	Class Notes
  Exercises
  Reading	[DK] Chapter 1

  Classes EXP, E. Applying union theorem. Algorithmic Problems CLIQUE, VertexCover, IndSet. Trivial Algorithms and runtime bounds.
  

  References	:	Class Notes
  Reading	:	[DK] Chapter 1

  • Meeting 39 : Fri, Oct 12, 10:00 am-10:50 am

  GI, MinCKT, Exact Clique, Reachability. The special property of some of the languages - that they have efficient algorithms which correctly verify a "short certificate of membership".

  References	Class Notes
  Exercises
  Reading	[DK] Chapter 1

  GI, MinCKT, Exact Clique, Reachability. The special property of some of the languages - that they have efficient algorithms which correctly verify a "short certificate of membership".
  

  References	:	Class Notes
  Reading	:	[DK] Chapter 1

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

  Formal definition of the class NP. Definition of time in non-deterministic computation. Normal forms for polytime non-deterministic TMs.

  References
  Exercises
  Reading

  Formal definition of the class NP. Definition of time in non-deterministic computation. Normal forms for polytime non-deterministic TMs.
  

  References

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  None

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

  Characterization of NP using non-deterministic computation.

  References
  Exercises
  Reading

  Characterization of NP using non-deterministic computation.
  

  References

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  None

  • Meeting 42 : Thu, Oct 18, 11:00 am-11:50 am

  P vs NP problem. EXP vs NEXP problem. Padding Arguments.

  References
  Exercises
  Reading

  P vs NP problem. EXP vs NEXP problem. Padding Arguments.
  

  References

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  None

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

  Holiday.

  References
  Exercises
  Reading

  Holiday.
  

  References

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  None

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

  CLIQUE and INDSET, INDSET and VC. Structure and reductions. Reductions, Completeness. Cook-Levin Theorem and proof outline.

  References
  Exercises
  Reading

  CLIQUE and INDSET, INDSET and VC. Structure and reductions. Reductions, Completeness. Cook-Levin Theorem and proof outline.
  

  References

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  None

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

  Details of Cook-Levin Theorem. Reductions to CIRCUIT SAT.

  References
  Exercises
  Reading

  Details of Cook-Levin Theorem. Reductions to CIRCUIT SAT.
  

  References

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  None

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

  From CIRCUIT SAT to 3CNFSAT.k-SAT, 3-SAT and 2-SAT. Story of Satisfiability.

  References
  Exercises
  Reading

  From CIRCUIT SAT to 3CNFSAT.k-SAT, 3-SAT and 2-SAT. Story of Satisfiability.
  

  References

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  None

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

  Reduction from 3SAT to VC. Structure of NP-complete sets. Sparse sets. Berman-Hartmanis Conjecture.

  References
  Exercises
  Reading

  Reduction from 3SAT to VC. Structure of NP-complete sets. Sparse sets. Berman-Hartmanis Conjecture.
  

  References

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  None

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

  Mahaney's theorem : If a sparse set is NP-complete, then P = NP. LSAT as a step in the proof. Berman-Hartmanis Conjecture.

  References	<a href="www.cs.umd.edu/~jkatz/complexity/f05/lecture6.pdf">Notes by Jonathan Katz</a>.
  Exercises
  Reading

  Mahaney's theorem : If a sparse set is NP-complete, then P = NP. LSAT as a step in the proof. Berman-Hartmanis Conjecture.
  

  References

  :

  Notes by Jonathan Katz.

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

  NP-Intermediate languages? Ladner's Therem. Impagliazzo's Proof. Padded version of SAT (SAT_H). Controlled Padding. Properties of SAT_H with respect to properties of H function. Impagliazzo's design of the function H which is computable in polynomial time.

  References	Noted shared on Moodle.
  Exercises
  Reading

  NP-Intermediate languages? Ladner's Therem. Impagliazzo's Proof. Padded version of SAT (SAT_H). Controlled Padding. Properties of SAT_H with respect to properties of H function. Impagliazzo's design of the function H which is computable in polynomial time.
  

  References

  :

  Noted shared on Moodle.

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

  Details of Impagliazzo's proof of Ladner's theorem.

  References
  Exercises
  Reading

  Details of Impagliazzo's proof of Ladner's theorem.
  

  References

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  None

  • Meeting 51 : Fri, Nov 02, 10:00 am-10:50 am

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

  References
  Exercises
  Reading

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

  References

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  None

  • Meeting 52 : Mon, Nov 05, 08:00 am-08:50 am

  Baker-Gill-Solovoy Theorem

  References
  Exercises
  Reading

  Baker-Gill-Solovoy Theorem
  

  References

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  None

  • Meeting 53 : Tue, Nov 06, 12:00 pm-12:50 pm

  Space Complexity. Space hierarchy theorem. Space Complexity Classes. Deterministic Space to Deterministic Time.

  References
  Exercises
  Reading

  Space Complexity. Space hierarchy theorem. Space Complexity Classes. Deterministic Space to Deterministic Time.
  

  References

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  None

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

  Simulation Theorems. Non-deterministic Time to Deterministic Space. NP is contained in PSPACE. PH Is contained in PSPACE. QBF is in PSPACE. Space in non-deterministic Turing machines. Non-deterministic Space to Deterministic Time. Configuration Graphs.

  References
  Exercises
  Reading

  Simulation Theorems. Non-deterministic Time to Deterministic Space. NP is contained in PSPACE. PH Is contained in PSPACE. QBF is in PSPACE. Space in non-deterministic Turing machines. Non-deterministic Space to Deterministic Time. Configuration Graphs.
  

  References

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  None

  • Meeting 55 : Mon, Nov 12, 08:00 am-08:50 am

  Savitch's theorem. Determinism vs Non-determinism.

  References
  Exercises
  Reading

  Savitch's theorem. Determinism vs Non-determinism.
  

  References

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  None

  • Meeting 56 : Tue, Nov 13, 10:00 am-10:50 am

  Immerman-Szelepsinyi Theorem. Inductive Counting.

  References	Page 44 of <a href="http://pages.cs.wisc.edu/~jyc/810notes/book.pdf">Notes by Cai</a>.
  Exercises
  Reading

  Immerman-Szelepsinyi Theorem. Inductive Counting.
  

  References

  :

  Page 44 of Notes by Cai.

  • Meeting 57 : Tue, Nov 13, 12:00 pm-12:50 pm

  Reach is NL-complete. QBF is PSPACE-complete. CVP is P-complete.

  References
  Exercises
  Reading

  Reach is NL-complete. QBF is PSPACE-complete. CVP is P-complete.
  

  References

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  None

• Evaluation Meetings - 3 meetings

  • Meeting 58 : Mon, Sep 03, 08:00 am-08:50 am

  Quiz I

  References
  Exercises
  Reading

  Quiz I
  

  References

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  None

  • Meeting 59 : Sat, Oct 27, 10:00 am-11:30 am

  Quiz II

  References
  Exercises
  Reading

  Quiz II
  

  References

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  None

  • Meeting 60 : Wed, Nov 14, 01:30 pm-04:30 pm

  Endsemester Exam

  References
  Exercises
  Reading

  Endsemester Exam
  

  References

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  None