# Course Page for CS5616 Computational Complexity
---

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

$
\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\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\AC{\mathsf{AC}}
\def\NC{\mathsf{NC}}
\def\SC{\mathsf{SC}}
\def\PAL{\rm PAL}
$

## Lectures

**Jan 18 (Lec 01)** Administrative details, Overview, objectives and learning outcomes of the course

**Jan 20 (Lec 02)** What is computation ? Formal definition - Turing machines, Church-Turing hypothesis

Reading: [Kozen] Lec. 1, Addl. reading: Turing's work on [Entsheidungsproblem](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf)

**Jan 21 (Lec 03)** One tape and two tape TMs - formal vs informal definition, configurations, examples

Reading: [Kozen] Lec. 1, [Arora-Barak] Chap. 1

**Jan 25 (Lec 04)** More machines and variants: two way infinite tape, two stack, Random access, Non-deterministic and their equivalence. Universal Turing machines.

Reading: [Kozen] Lec. 1, [Arora-Barak] Chap. 1

**Jan 27 (Lec 05)** Recursive and recursively enumerable languages and two simple properties. Enumeration machines and recursively enumerable languages.

Reading: [Kozen] Lec. 1, [Arora-Barak] Chap. 1

**Jan 28 (Lec 06)** Enumeration machines for RE languages. Undecidability. Existence by cardinality based argument. Halting problem. Proof sketch of undecidability using diagonalization.

Reading: [Arora-Barak] Chap. 1

**Jan 29 (Lec 07)** Proof of undecidability of Halting problem and membership problem in detail using diagonalization.

Reading: [Arora-Barak] Chap. 1

**Feb 01 (Lec 08)** Reduction between problems. Proof of $\mathsf{MP}$ reduces to $\mathsf{HP}$.

Reading: [Arora-Barak] Chap. 1.

**Feb 02 (Lec 09)** More examples of reductions. Reductions among languages in R, RE. $\overline{\mathsf{HP}}$ is not in RE

Reading: [Kozen] Lec. 1, [Arora-Barak] Chap. 1.

**Feb 04 (Lec 10)** Resource bounded computation. Notions of resources - Time and Space. Input representations. Broad goals and issues to be addressed.

Reading: [Arora-Barak] Chap. 1, Lecture 1,2 of [Deiter's course](https://pages.cs.wisc.edu/~dieter/Courses/2011f-CS710/Scribes/lectures.html)

**Feb 08 (Lec 11)** Crossing sequence arguments: $\PAL_n$ requires $\Omega(n^2)$ time for a one-tap Turing machine. $o(\log \log n)$ space Turing machines can only accept regular languages - overview.

Reading: [Kozen] Lec. 1, Addl. reading: On equivalence between two way DFA and NFA, see Theorem 1.1 of [this](https://www.classes.cs.uchicago.edu/archive/2015/winter/28000-1/Lec3.pdf) notes.

**Feb 10 (Lec 12)** Crossing sequence arguments: $o(\log \log n)$ space Turing machines can only accept regular languages - completed the proof.

Reading: [Kozen] Lec. 1

**Feb 11 (Lec 13)** Resource - axiomatic definition due to Manuel Blum. Time and space as resources. Examples of non-resources. Definitions of classes $\DTIME,\DSPACE$.

Reading: [Kozen] Lec. 1

**Feb 15 (Lec 14)** $g$-time decidability. Existence of a decidable language $L_f$ that are is $g$-time decidable via diagonalization argument. Time and space constructibility. Time complexity of deciding $L_f$. Tape reduction - overview of statement and proof idea.

Reading: [Arora-Barak] Chap. 1

**Feb 17 (Lec 15)** Tape reduction - completing the details. Hennie and Stearns : statement of simulating $t(n)$ time Turing machine in $O(t(n)\log t(n))$ time. Alphabet reduction - details.

Reading: [Arora-Barak] Chap. 1. Section 1.4.1 + Section 1.7

**Feb 18 (Lec 16)** Quick recap - Tape reduction, Alphabet reduction. Used them to show $L_f$ is $g^2(\log g)^2$-time decidable. Tape compression - statement and proof. Linear speed up - statement and proof idea.

Reading: [Arora-Barak] Chap. 1.

**Feb 22 (Lec 17)** Completed proof of linear speed up theorem. Statement of deterministic space hierarchy theorem. Proof overview.

Reading: [Arora-Barak] Chap. 1., [Kozen] Lect. 2,3

**Feb 24 (Lec 18)** Completed the full proof of space hierarchy theorem.

Reading: [Arora-Barak] Chap. 1., [Kozen] Lect. 3

**Feb 25 (Lec 19)** Time hierarchy theorem - statement and proof overview. Necessity of the constructibility assumption - Statement of Blum's speedup theorem and Borodin's gap theorem. Non-deterministic time and space - definitions. Basic containments involving $\DTIME, \DSPACE, \NTIME, \NSPACE$.

Reading: [Arora-Barak] Chap. 3, Section 3.1, Chap 4, Section 4.1, [Kozen] Lect. 2

**Mar 01 (Lec 20)** Configuration graphs of space bounded machines. Complexity classes $\P,\NP,\EXP,\NEXP,\L,\NL,\PSPACE,\NPSPACE$ - containments and separations. A quick summary of the lectures so far.

Reading: [Arora-Barak] Chap. 3, Section 3.1, Chap 4, Section 4.1, [Kozen] Lect. 2

**Mar 03 (Lec 21)** Review of complexity classes: containments+separations - what is known and what is not. The $\P$ vs $\NP$ question and $\L$ vs $\NL$ question.