Lectures

Following is an outline of the lectures given along with references and links to additional reading materials.

\( \def\SAT{\mathsf{SAT}} \def\USAT{\mathsf{USAT}} \def\DTIME{\mathsf{DTIME}} \def\DTIME{\mathsf{DTIME}} \def\NTIME{\mathsf{NTIME}} \def\DSPACE{\mathsf{DSPACE}} \def\NSPACE{\mathsf{NSPACE}} \def\poly{\mathsf{poly}} \def\P{\mathsf{P}} \def\NP{\mathsf{NP}} \def\co{\mathsf{co}} \def\coNP{\mathsf{coNP}} \def\L{\mathsf{L}} \def\NL{\mathsf{NL}} \def\PSPACE{\mathsf{PSPACE}} \def\NPSPACE{\mathsf{NPSPACE}} \def\EXP{\mathsf{EXP}} \def\E{\mathsf{E}} \def\NEXP{\mathsf{NEXP}} \def\RP{\mathsf{RP}} \def\coRP{\mathsf{coRP}} \def\ZPP{\mathsf{ZPP}} \def\PP{\mathsf{PP}} \def\FP{\mathsf{FP}} \def\sharpP{\#\mathsf{P}} \def\BPP{\mathsf{BPP}} \def\BP{\mathsf{BP}} \def\AC{\mathsf{AC}} \def\NC{\mathsf{NC}} \def\SC{\mathsf{SC}} \def\RL{\mathsf{RL}} \def\BPL{\mathsf{BPL}} \def\ZPL{\mathsf{ZPL}} \def\PAL{\rm PAL} \def\HP{\rm HP} \def\FIN{\rm FIN} \def\REG{\rm REG} \def\TOTAL{\rm TOTAL} \def\COF{\rm COF} \def\D{\mathcal{D}} \def\SD{\mathcal{SD}} \def\MP{\rm MP} \)

Lec 1 (17 Jan, Tue) : Course plan and administrative information. Computation, Turing machines, Church-Turing thesis.

^{Reading : Kozen (A&C) Lec 28 to 30}

Lec 2 (18 Jan, Wed) : Turing machines - definitions and variants. Review of undecidability. Overview of the proof via Cantor's diagonalisation.

^{Reading : Kozen (A&C) Lec 31, Notes on Turing machines}

Lec 3 (19 Jan, Thu) : Proof of undecidability of Halting problem ($\HP$) via diagonalisation. Undecidability of Membership problem ($\MP$). Relation between undecidable problems and motivation for reductions.

^{Reading : Kozen (A&C) Lec 33}

Lec 4 (24 Jan, Tue) : Reducing $\HP$ to $\MP$. Proof details. $\HP$ where the string is fixed reduces to $\HP$.

^{Reading : Kozen (A&C) Lec 33}

Lec 5 (25 Jan, Wed) : (Shorter lecture) $\HP$ reduces to $\HP$ where the string is fixed. Decidable or not ? More examples.

^{Reading : Kozen (A&C) Lec 32}

Lec () (26 Jan, Thu) : Holiday due to Republic Day

Lec 6 (31 Jan, Tue) : Many one reductions. $\overline{\HP}$ is not recursively enumerable. Argued this by showing $L$ is recursive if and only if $L$, $\overline{L}$ are RE. Showing $\HP \not \le_m \overline{\HP}$. Using reductions to show non-semi decidability of languages.

^{Reading : Kozen (A&C) Lec 33}

Lec 7 (01 Feb, Wed) : $\HP$ reduces to $\REG$. $\FIN$ as well as $\overline{\FIN}$ are not in RE.

^{Reading : Kozen (A&C) Lec 33, 34}

Lec 8 (02 Feb, Thu) : Completed the proof. Properties of semi-decidable languages. Rice's theorem.

^{Reading : Kozen (A&C) Lec 33, 34}

Lec 9 (07 Feb, Tue) : Examples and non-examples of non-trivial properties of semi-decidable languages. Proof of Rice's theorem. Property being monotone.

^{Reading : Kozen (A&C) Lec 34}

Lec 10 (08 Feb, Wed) : Rice's theorem for non-monotone properties. Machines for RE sets - Enumeration Turing machines.

^{Reading : Kozen (A&C) Lec 34}

Lec 11 (09 Feb, Thu) : Languages accepted by Enumeration TMs is precisely RE sets. Notion of hardness and completeness for semi-decidable languages. $\MP$ is SD-complete.

^{Reading : Kozen (A&C) Lec 30 and Supplementary lecture J}

Lec 12 (14 Feb, Tue) : $\HP$ is SD-complete. Expressing languages HP, HP101, EMPTY, ALL and FIN in terms of quantifiers. Quantifier characterisation of SD and co-SD. EMPTY is in co-SD.

^{Reading : Kozen (A&C) Supplementary lecture J}

Lec 13 (15 Feb, Wed) : Proof of Quantifier characterisation. Relative computation. Oracle Turing machines. Examples.

^{Reading : Kozen (A&C) Supplementary lecture J}

Lec 14 (16 Feb, Thu) : A total oracle TM for $\HP$ with $\MP$ as an oracle. Oracle TM for FIN with $\HP$ as an oracle. Notion of decidable and semi-decidable sets with respect to oracles. Towards defining arithmetic hierarchy.

^{Reading : Kozen (A&C) Supplementary lecture J, Kozen (ToC) Lec 35}

Lec 15 (21 Feb, Tue) : Relativized language classes - $\D^A$ and $\SD^A$ for a language $A$. Basic containments when $A$ is recursive. Diagonalization to show strictness of containment when $A=\HP$.

^{Reading : Kozen (ToC) Lec 35, Lec 36}

Lec 16 (22 Feb, Wed) : Arithmetic hierachy - classes $\Sigma_i$, $\Pi_i$ and $\Delta_i$. Basic containments, statement of quantifier characterization.

^{Reading : Kozen (ToC) Lec 35, Lec 36}

Lec 17 (23 Feb, Thu) : Completeness in Arithmetic hierarchy. Proof of quantifier characterization - overview.

^{Reading : Kozen (ToC) Lec 35, Lec 36, Class notes}

Lec 18 (28 Feb, Tue) : Proof of quantifier characterization - argument for base case using computation histories.

^{Reading : Kozen (ToC) Lec 35, Lec 36, Class notes}

Lec 19 (01 Mar, Wed) : Completed the proof of characterization. Used the characterization to show that $\FIN \in \Sigma_2, \TOTAL \in \Pi_2$ and $\COF \in \Sigma_3$.

^{Reading : Kozen (ToC) Lec 35, Lec 36, Class notes}

Lec 20 (02 Mar, Thu) : Showed that $\FIN$ is $\Sigma_2$-complete under many one reductions. $\FIN^{\HP}$ is in $\Sigma_3$.

^{Reading : Kozen (ToC) Lec 35, Lec 36}

Lec 21 (07 Mar, Tue) : Showed $\FIN^{\HP}$ is $\Sigma_3$-complete. Generalization to $\FIN^A$ where $A$ is $\Sigma_n$-complete.

^{Reading : Kozen (ToC) Lec 35, Lec 36}

Lec () (08 Mar, Wed) : Holiday due to Holi

Lec 22 (09 Mar, Thu) : $\COF$ is $\Sigma_3$-complete via reduction.

^{Reading : Kozen (ToC) Lec 36}

Lec 23 (14 Mar, Tue) : Many-one and Turing degrees. Post's problem - simple and productive sets. Simple sets are undecidable. Complement of productive sets are not simple.

^{Reading : Kozen (ToC) Lec 37}

Lec 24 (15 Mar, Wed) : Complement of SD-hard sets are productive.

^{Reading : Kozen (ToC) Lec 37 and class notes}

Lec 25 (16 Mar, Thu) : Simple sets exists - construction and correctness. Post's problem - many one and Turing equivalence classes. Friedberg-Muchnick theorem.

^{Reading : Kozen (ToC) Lec 37 and class notes}

Lec 26 (21 Mar, Tue) : Kolomogrov Complexity. Counting arguments. Random strings have high Kolomogrov Complexity.

^{Reading : Class notes}

Lec 27 (22 Mar, Wed) : Undecidability and Kolomogrov Complexity. Resource bounded computation - models, paradigms and resources. Notion of resource.

^{Reading : Class notes, Arora-Barak Chap 1 - 1.2, 1.3,}

Lec 28 (23 Mar, Thu) : Blum's axioms, Time and space satisfies Blum's axioms. Example of a non-resource. Input representation and its effect on resource. Issues to be addressed and the roadmap ahead.

^{Reading : Class notes, Kozen (ToC) Lecture J, Arora-Barak Chap 1, Preclude (Chap 1) and Epilogue (Chap 20) of the book Mathematics and Computation.}

Lec 29 (28 Mar, Tue) : $\ell(n)$-time decidable sets. Using diagonlization to obtain decidable sets outside $\ell(n)$-time decidable. Time and space constructibility.

^{Reading : Class notes, Arora-Barak, Chap. 1}

Lec 30 (29 Mar, Wed) : Tape reduction and alphabet reduction. Ther exists a language in $O(\ell^3(n)\log \ell(n))$ that is not $\ell(n)$-time decidable.

^{Reading : Arora-Barak, Chap. 1}

Lec 31 (30 Mar, Thu) : Tape compression theorem. Linear speed up theorem.

^{Reading : Arora-Barak, Chap. 1.}

Lec () (04 Apr, Tue) : Holiday due to recess break

Lec () (05 Apr, Wed) : Holiday due to recess break

Lec () (06 Apr, Thu) : Holiday due to recess break

Lec 32 (11 Apr, Tue) : Completed the proof of Linear speedup theorem.

^{Reading : (Arora-Barak) Chap. 1.}

Lec 33 (12 Apr, Wed) : Deterministic space hierarchy theorem

^{Reading : (Arora-Barak) Chap. 1., Kozen (ToC) Lect. 3}

Lec 34 (13 Apr, Thu) : Determinisitic time hierarchy - statement and proof. Hennie and Stearns - Simulating $\ell(n)$ time Turing machine in $O(\ell(n)\log\ell(n))$ time - overview.

^{Reading : (Arora-Barak) Chap. 1. Section 1.4.1 + Section 1.7}

Lec 35 (18 Apr, Tue) : Details of Hennie and Stearns simulation.

^{Reading : (Arora-Barak) Chap. 1. Section 1.4.1 + Section 1.7}

Lec 36 (19 Apr, Wed) : Limitations of one tape Turing machine - Crossing sequence arguments. Any one tape Turing machine deciding the palindrome language must take $\Omega(n^2)$ time.

^{Reading : Kozen (ToC) Lec. 1}

Lec 37 (20 Apr, Thu) : Basic classes $\DTIME$, $\DSPACE$, $\NTIME$, $\NSPACE$ and containments.

^{Reading : (Arora-Barak) Chap. 3, Section 3.1, Chap 4, Section 4.1, Kozen (ToC) Lect. 2}

Lec 38 (25 Apr, Tue) : Simulating non-determinisitic $s(n)$ space bounded Turing machine deterministically. Proof idea - Configuration graphs and its structure.

^{Reading : (Arora-Barak) Chap. 3, Section 3.1, Chap 4, Section 4.1, Kozen (ToC) Lect. 2}

Lec 39 (26 Apr, Wed) : Oblivious Turing machines. Every vertex in the configuration graph can be described in $O(s(n))$ space, edge between vertices can be detected using Boolean formula of size $O(s(n))$. Introduction to complexity classes $\P$, $\NP$, $\L$, $\NL$, $\PSPACE$, $\NPSPACE$. Composability of time and space.

^{Reading : (Arora-Barak) Chap 1, Section 6, Chap. 3, Section 3.1, Chap 4, Section 4.1, Kozen (ToC) Lect. 2}

Lec 40 (27 Apr, Thu) : Basic containments among the complexity classes and separations via hierarchy theorem. $\NP$ - choice machines, guess + verify view. The class $\coNP$. $PRIMES$ is in $\coNP$.

^{Reading : (Arora-Barak) Chap 3, Section 3.1, Chap 2}

Lec 41 (02 May, Tue) : Many-one polynomial reductions, $\NP$-completeness, bounded Halting problem. More examples. What is next and course wrap up.

^{Reading : (Arora-Barak) Chap 2, Section 2.1, 2.2}