# Theory of Computation (CS422) in Fall 2021 at KAIST

The Theory of Computation provides a sound logical foundation to computer science. By comparing various formal models of computation with respect to their capabilities, it identifies both fundamental features and ultimate limitations of contemporary digital computing machinery. Rigorous notions of efficiency are captured by famous complexity classes such as P and PSPACE; and concepts like oracles or polynomial-time reduction permit to compare computational problems with respect to their algorithmic cost: NP-hardness results thus serve as 'beacons' of intractability.

• Lecturer: Martin Ziegler
• Schedule: Tuesdays and Thursdays, 16:00 to 17:30
• Location: online, via Zoom
• Language: English only
• Grading: Homework 30%, Quiz 10%, Midterm exam 30%, Final exam 30%

We make a special pedagogical effort to avoid the arduous Turing machine formalism and instead employ a variant of WHILE programs.

0. Motivation

I. Basic Computability Theory (ppt, pdf):

• Un-/Semi-Decidability and Enumerability
• Reduction, degrees of undecidabiliy
• (Post's Correspondence Problem, truth of arithmetic formulae)
• Busy Beaver function
• Ackermann's Function
• LOOP programs
• and their capabilities

• WHILE programs
• UTM Theorem
• Normal Form Theorem
• SMN Theorem / Currying (Schönfinkeling)
• Recursion Theorem, Fixedpoint Theorem, QUINES
• Oracle WHILE Programs
• Arithmetic Hierarchy
• Post/Friedberg/Muchnik

III. Computational Complexity (ppt, pdf):

• Model of computation with (bit) cost: WHILE+
• Complexity classes P, NP, PSPACE, EXP
• and their inclusion relations
• Example problems: Euler/Hamiltonian Circuit, Edge/Vertex Cover
• Boolean formulas and Satisfiability
• nondeterministic WHILE+ programs
• Time Hierarchy Theorem: P≠EXP

IV. Structural Complexity / NPc (ppt, pdf):

• polynomial-time reduction
• equivalent problems: Clique = Independent Set ⇐ Boolean Satisfiability ⇐ 3-Satisfiability ⇐ Independent Set
• Cook-Levin Theorem, Master Reduction
• Bin Packing is NP-complete
• Scenarios for P/NP, Ladner's Theorem

V. PSPACE and Polynomial Hierarchy (ppt, pdf):

• PSPACE and QBF
• Master Reduction
• A PSPACE-complete Two-Player Game on Digraphs
• Savitch's Theorem: NSPACE(f) in SPACE(f²)
• Immerman-Szelepcsényi: NSPACE(f) = coNSPACE(f)
• Oracle Complexity and Polynomial Hierarchy, Semantically and Syntactically
• Limitations of Relativizing Proofs: Baker, Gill & Solovay; Bennett & Gill

VI. Perspectives (ppt, pdf):

• complexity of cryptography: UP and one-way functions
• counting problems: #P and Toda's Theorem
• Approximation algorithms and hardness
• randomized algorithms, probability amplification, BPP, Adleman and Sipser-Gacs-Lautemann Theorems
• fixed-parameter tractability and hardness

Regularly recalling, applying, and extending the definitions, theorems, and proofs from the lecture is essential for comprehension and successful study. Therefore consider it as a courtesy that we will create homework assignments and publish them on this web page.

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