CS6014 - Computability and Complexity

Meetings  

• Theme 1 : Basic Computability - 25 meetings

  • Meeting 01 : Mon, Jul 28, 08:00 am-08:50 am

  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 program. Formal model of computation. Post Systems, Mu-calculus, Lambda calculus. Turing machines. Church-Turing thesis.

  References	Lecture 28 in [K1]
  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 program. Formal model of computation. Post Systems, Mu-calculus, Lambda calculus. Turing machines. Church-Turing thesis.
  

  References

  :

  Lecture 28 in [K1]

  • Meeting 02 : Tue, Jul 29, 12:00 pm-12:50 pm

  Formal Model of Turing machines. Nuances of the definition. Configurations of a Turing Machine.

  References	Lecture 28 in [K1]
  Exercises
  Reading

  Formal Model of Turing machines. Nuances of the definition. Configurations of a Turing Machine.
  

  References

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  Lecture 28 in [K1]

  • Meeting 03 : Thu, Jul 31, 11:00 am-11:50 am

  Language accepted by Turing machines. Total Turing machines. Algorithms vs Turing machines. Drawbacks. Diophantine problem. Is there a computational task that total Turing machines cannot do? Two observations about TM model (1) encoding a TM as a string (2) Universal TM.

  References	Lecture 28, 29, 30 in [K1]
  Exercises
  Reading

  Language accepted by Turing machines. Total Turing machines. Algorithms vs Turing machines. Drawbacks. Diophantine problem. Is there a computational task that total Turing machines cannot do? Two observations about TM model (1) encoding a TM as a string (2) Universal TM.
  

  References

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  Lecture 28, 29, 30 in [K1]

  • Meeting 04 : Fri, Aug 01, 10:00 am-10:50 am

  Review of Cantor's diagonalization and infinities. The infinite matrix form. Entry filling task. Impossibility proof.

  References	Lecture 31 in [K1]
  Exercises
  Reading

  Review of Cantor's diagonalization and infinities. The infinite matrix form. Entry filling task. Impossibility proof.
  

  References

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  Lecture 31 in [K1]

  • Meeting 05 : Mon, Aug 04, 10:00 am-10:50 am

  Undecidability of HP. The language MP and undecidability through mimicking the diagonalization.

  References	Lecture 31 in [K1]
  Exercises
  Reading

  Undecidability of HP. The language MP and undecidability through mimicking the diagonalization.
  

  References

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  Lecture 31 in [K1]

  • Meeting 06 : Tue, Aug 05, 12:00 pm-12:50 pm

  MP vs HP. Building HP-solver using MP-solver. Building MP-solver using HP-solver. Two different methods. Reductions and many-one reductions.

  References	Lecture 33 in [K1]
  Exercises
  Reading

  MP vs HP. Building HP-solver using MP-solver. Building MP-solver using HP-solver. Two different methods. Reductions and many-one reductions.
  

  References

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  Lecture 33 in [K1]

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

  Recap of reductions. A and A-bar are semidecidable if and only if A is decidable. The complement of MP is outside SD. The CoSD class. SD intersection CoSD is exactly the decidables. MP cannot many-one reduce to MP-har. <br><br> MP_42 language. Undecidability of MP_42. The generic nature of the reduction.

  References	Lecture 33 in [K1]
  Exercises
  Reading

  Recap of reductions. A and A-bar are semidecidable if and only if A is decidable. The complement of MP is outside SD. The CoSD class. SD intersection CoSD is exactly the decidables. MP cannot many-one reduce to MP-har.
  
  MP_42 language. Undecidability of MP_42. The generic nature of the reduction.
  

  References

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  Lecture 33 in [K1]

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

  Demonstrating the generality of the idea. FULL, REG, CFL, DEC, FIN are all undecidable. MP reduces to REG, MP reduces to REG-bar. The first language outside the SD and CoSD.

  References	Lecture 34 in [K1]
  Exercises
  Reading

  Demonstrating the generality of the idea. FULL, REG, CFL, DEC, FIN are all undecidable. MP reduces to REG, MP reduces to REG-bar. The first language outside the SD and CoSD.
  

  References

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  Lecture 34 in [K1]

  • Meeting 09 : Mon, Aug 11, 08:00 am-08:50 am

  Rice's first theorem. Applications. Statement of Rice's second theorem.

  References	Lecture 34 in [K1]
  Exercises
  Reading

  Rice's first theorem. Applications. Statement of Rice's second theorem.
  

  References

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  Lecture 34 in [K1]

  • Meeting 10 : Tue, Aug 12, 12:00 pm-12:50 pm

  Rice's Second Theorem and Proof. Example Applications. Cases where it does not apply. Other reduction techniques.

  References	Lecture 34 in [K1]
  Exercises
  Reading

  Rice's Second Theorem and Proof. Example Applications. Cases where it does not apply. Other reduction techniques.
  

  References

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  Lecture 34 in [K1]

  • Meeting 11 : Thu, Aug 14, 12:00 pm-12:50 pm

  PA Axioms. Incompleteness theorem. Godel's Theorem statement. Computability based proof overview. Th(N) is semidecidable.

  References	Lecture 38 in [K1]
  Exercises
  Reading

  PA Axioms. Incompleteness theorem. Godel's Theorem statement. Computability based proof overview. Th(N) is semidecidable.
  

  References

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  Lecture 38 in [K1]

  • Meeting 12 : Mon, Aug 18, 08:00 am-08:50 am

  Buidling towards showing T(N) is not semidecidable. Toolkit for the reduction. Expressions of properties of strings using numbers. Bit extraction predicate. Configuration seuqence. The VALCOM predicate. Predicates for start configuration and final configuration.

  References	Lecture 39 in [K1]
  Exercises
  Reading

  Buidling towards showing T(N) is not semidecidable. Toolkit for the reduction. Expressions of properties of strings using numbers. Bit extraction predicate. Configuration seuqence. The VALCOM predicate. Predicates for start configuration and final configuration.
  

  References

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  Lecture 39 in [K1]

  • Meeting 13 : Tue, Aug 19, 12:00 pm-12:50 pm

  Details of the reduction from MP-bar to T(N). Writing down the "valid move" predicate buiding on digit extraction. Completing the proof. Reflections on the incompleteness theorem.

  References	Lecture 39 in [K1]
  Exercises
  Reading

  Details of the reduction from MP-bar to T(N). Writing down the "valid move" predicate buiding on digit extraction. Completing the proof. Reflections on the incompleteness theorem.
  

  References

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  Lecture 39 in [K1]

  • Meeting 14 : Thu, Aug 21, 11:00 am-11:50 am

  SD-hardness and SD-completeness. Proof that MP is SD-complete. HP is SD-complete via transitivity and a direct proof. SDC cannot contain a decidable language. Question - does SDC and decidables cover the set of semidecidable language? Is there a gap in the landscape?

  References	No reference in [K1]. Class notes.
  Exercises
  Reading

  SD-hardness and SD-completeness. Proof that MP is SD-complete. HP is SD-complete via transitivity and a direct proof. SDC cannot contain a decidable language. Question - does SDC and decidables cover the set of semidecidable language? Is there a gap in the landscape?
  

  References

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  No reference in [K1]. Class notes.

  • Meeting 15 : Fri, Aug 22, 10:00 am-10:50 am

  Gaps in the landscape. Post's theorem. There is a language which is semidecidable, but is neither decidable nor SD-complete. Definition of simple sets. Simple sets are semidecidable and are undecidable. Construction of explicit simple sets by stating the enumeration TM. Proof of simplicity.

  References	Class notes shared. Lecture 37 in [K2]
  Exercises
  Reading

  Gaps in the landscape. Post's theorem. There is a language which is semidecidable, but is neither decidable nor SD-complete. Definition of simple sets. Simple sets are semidecidable and are undecidable. Construction of explicit simple sets by stating the enumeration TM. Proof of simplicity.
  

  References

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  Class notes shared. Lecture 37 in [K2]

  • Meeting 16 : Mon, Aug 25, 08:00 am-08:50 am

  Quiz 1

  References
  Exercises
  Reading

  Quiz 1
  

  References

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  None

  • Meeting 17 : Tue, Aug 26, 12:00 pm-12:50 pm

  Productive Sets. Simple sets cannot be co-productive. All SD-complete sets are co-productive. Hence simple sets cannot be SD-complete.

  References	Class notes shared. Lecture 37 in [K2]
  Exercises
  Reading

  Productive Sets. Simple sets cannot be co-productive. All SD-complete sets are co-productive. Hence simple sets cannot be SD-complete.
  

  References

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  Class notes shared. Lecture 37 in [K2]

  • Meeting 18 : Thu, Aug 28, 11:00 am-11:50 am

  Oracle Turing Machines. Relative Computation. Arithmetic Heirarchy. The arithmetic hierarchy is strict. Relativisation of the diagonalization proof.

  References	Lecture Notes
  Exercises
  Reading

  Oracle Turing Machines. Relative Computation. Arithmetic Heirarchy. The arithmetic hierarchy is strict. Relativisation of the diagonalization proof.
  

  References

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  Lecture Notes

  • Meeting 19 : Fri, Aug 29, 11:00 am-11:50 am

  Relativising the previous arguments. <br> Delta_k = Sigma_k cap Pi_k. <br>

  References	Lecture notes
  Exercises
  Reading

  Relativising the previous arguments.
  Delta_k = Sigma_k cap Pi_k.
  
  

  References

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  Lecture notes

  • Meeting 20 : Mon, Sep 01, 08:00 am-08:50 am

  MP^A is Sigma_k-complete.

  References
  Exercises
  Reading

  MP^A is Sigma_k-complete.
  

  References

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  None

  • Meeting 21 : Tue, Sep 02, 12:00 pm-12:50 pm

  Characterization of Arithmetic Hierarchy. Corollaries and quanfier alternation characterization for Sigma_k, Pi_k. FIN is in Sigma_2.

  References	[K2] and lecture notes
  Exercises
  Reading

  Characterization of Arithmetic Hierarchy. Corollaries and quanfier alternation characterization for Sigma_k, Pi_k. FIN is in Sigma_2.
  

  References

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  [K2] and lecture notes

  • Meeting 22 : Thu, Sep 04, 11:00 am-11:50 am

  REG. DEC, CFL is in Sigma_3.<br> COF in Sigma_3

  References	[K2] and lecture notes
  Exercises
  Reading

  REG. DEC, CFL is in Sigma_3.
  COF in Sigma_3
  

  References

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  [K2] and lecture notes

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

  FIN is Sigma_2-hard. Relativising this proof to show that FIN^HP is Sigma_3-hard.

  References	Lecture 36 in [K2]
  Exercises
  Reading

  FIN is Sigma_2-hard. Relativising this proof to show that FIN^HP is Sigma_3-hard.
  

  References

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  Lecture 36 in [K2]

  • Meeting 24 : Tue, Sep 09, 12:00 pm-12:50 pm

  Proof that FIN^HP is in Sigma_3. Reduction from FIN^HP to COF. Technique of accepting the non-accepting computation histories.

  References	Lecture 36 in [K2]
  Exercises
  Reading

  Proof that FIN^HP is in Sigma_3. Reduction from FIN^HP to COF. Technique of accepting the non-accepting computation histories.
  

  References

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  Lecture 36 in [K2]

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

  Proof of the Characterization of the Arithmetic Hierarchy.

  References	Notes shared.
  Exercises
  Reading

  Proof of the Characterization of the Arithmetic Hierarchy.
  

  References

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  Notes shared.

• Theme 2 : Time Complexity - 7 meetings

  • Meeting 26 : Fri, Sep 12, 08:00 am-09:00 am

  Blum's axioms for computational resources. Examples.

  References	Supplemetary lecture J in [K2]
  Exercises
  Reading

  Blum's axioms for computational resources. Examples.
  

  References

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  Supplemetary lecture J in [K2]

  • Meeting 27 : Mon, Sep 15, 08:00 am-09:00 am

  Defining a complexity class. How do we capture asymptotics. Decision to use input length. Linear Speedup Theorem, Tape Compression Theorem

  References	For linear speed up theorem and tape compression theorem, see Proposition 1.12, 1.13 (page 20-21)
  Exercises
  Reading

  Defining a complexity class. How do we capture asymptotics. Decision to use input length. Linear Speedup Theorem, Tape Compression Theorem
  

  References

  :

  For linear speed up theorem and tape compression theorem, see Proposition 1.12, 1.13 (page 20-21)

  • Meeting 28 : Tue, Sep 16, 12:00 pm-12:50 pm

  Proof of Borodin Gap Theorem. Statement of Blum's Speedup Theorem.

  References	Lecture 32, Theorem 32.1 in [K2]
  Exercises
  Reading

  Proof of Borodin Gap Theorem. Statement of Blum's Speedup Theorem.
  

  References

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  Lecture 32, Theorem 32.1 in [K2]

  • Meeting 29 : Thu, Sep 18, 11:00 am-11:50 am

  Tape reduction theorem for time and space. Statement of Hierarchy theorems. Henny-Stearns tape reduction (statement).

  References	Theorem 1.18 in [DK]
  Exercises
  Reading

  Tape reduction theorem for time and space. Statement of Hierarchy theorems. Henny-Stearns tape reduction (statement).
  

  References

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  Theorem 1.18 in [DK]

  • Meeting 30 : Fri, Sep 19, 10:00 am-10:50 am

  Space Hierarchy Theorem and the proof.

  References	Theorem 1.22 in [DK]
  Exercises
  Reading

  Space Hierarchy Theorem and the proof.
  

  References

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  Theorem 1.22 in [DK]

  • Meeting 31 : Mon, Sep 22, 08:00 am-08:50 am

  Time hierarchy theorem and its proof.<br> Optimality of tape reduction. Crossing sequence arguments.

  References	<a href="https://lucatrevisan.github.io/teaching/cs172-15/noteh.pdf">Notes on Time Hierarchy Theorems</a> - (Late) Luca Trevisan.<br> <a href="https://www.cs.umd.edu/~jkatz/complexity/f05/lower_bounds.pdf">Lower Bounds via Crossing Sequence Arguments</a> - Jonathan Katz
  Exercises
  Reading

  Time hierarchy theorem and its proof.
  Optimality of tape reduction. Crossing sequence arguments.
  

  References

  :

  Notes on Time Hierarchy Theorems - (Late) Luca Trevisan. Lower Bounds via Crossing Sequence Arguments - Jonathan Katz

  • Meeting 32 : Tue, Sep 23, 12:00 pm-12:50 pm

  Finishing the crossing sequence arguments. Defining the notion of efficiency. Growth requirements for the bound and composibility requirements for the class. Union theorem. Definition of the class P.

  References	<a href="https://www.cs.umd.edu/~jkatz/complexity/f05/lower_bounds.pdf">Lower Bounds via Crossing Sequence Arguments</a> - Jonathan Katz
  Exercises
  Reading	Union theorem proof - page 234 as theorem J.3 in [K2].

  Finishing the crossing sequence arguments. Defining the notion of efficiency. Growth requirements for the bound and composibility requirements for the class. Union theorem. Definition of the class P.
  

  References	:	Lower Bounds via Crossing Sequence Arguments - Jonathan Katz
  Reading	:	Union theorem proof - page 234 as theorem J.3 in [K2].