FORMAL
LANGUAGES, COMPUTABILITY, AND COMPLEXITY
Lecturer: Michel
Abdalla (michel.abdalla@ens.fr)
Teaching
Assistant: Nathan
Grosshans (nathan.grosshans@ens.fr)
Lectures: Thursdays
13:15-15:00 at Salle des Conférences, 46 rue d’Ulm
Classes/TD: After
each lecture from 15:15-17:00 (15 min break)
Other
Dates: Final exam on 23/01/2020. Projects subjects due on 15/11/2019. Project reports due
on 03/01/2020.
Lectures
will be in English. The classes by Nathan will be in French. An outline of the
course contents is given below.
By
November 15th, you will need to decide on a project topic. A preliminary list of
possible topics is given below and you are welcome to suggest topics of your
own choice. The project presentation and the project report can be in English
or French.
References
Exercise
Sheets: See here.
Lecture
Notes:
· Regular and context-free languages.
Contents
·
Lecture 01 (26/09): Course outline. Deterministic and
non-deterministic finite automaton. Equivalence of DFAs and NFAs.
·
Lecture 02 (03/10): Regular expressions and equivalence with
NFAs. Closure properties.
·
Lecture 03 (10/10): The pumping lemma. Minimization of DFAs.
·
Lecture 04 (17/10): Context-free grammars. Derivations and parse
trees. Ambiguity. Chomsky normal form.
·
Lecture 05 (24/10): Closure properties of CFLs. Pushdown
automata and equivalence with CFGs. Pumping Lemma for CFLs.
·
Vacances
Toussaint (31/10)
·
Lecture 06 (07/11): Turing machines. The Church-Turing thesis.
Universal TMs.
·
Lecture 07 (14/11): The halting problem and other
undecidable problems.
·
Lecture 08 (21/11): Rice’s theorem. Post correspondence problem. Decision
properties of regular and context-free languages.
·
Lecture 09 (28/11): Time complexity: P, NP, and coNP. NP-completeness.
·
Lecture 10 (05/12): NP-completeness of SAT. Other complete
problems.
·
Lecture 11 (12/12): Space complexity. Savitch’s
theorem. PSPACE-completeness. Classes L and NL, co-NL. NL=coNL.
·
Lecture 12 (19/12): Circuit complexity. Randomized
algorithms. Classes BPP, RP, coRP, and ZPP. BPP
in P/poly.
·
Lecture 13 (09/01): Cryptography. One-way functions.
Pseudorandom generators. The Goldreich-Levin theorem.
·
Presentations 13-17/01: Reports
are due on 03/01/2020. There will be a doodle page to decide on time slots for the
presentations.
·
Final exam (23/01): Salle de Conférences (3
Hours)
Presentation
Topics
Email
me if you’d like to propose your own topic(s).
1.
Repetition-Free Sequences
2.
Moore and Mealy Machines
3.
Other Algorithms for DFA Minimization
4.
Deterministic Context-Free Languages
5.
Context-Sensitive Languages
6.
Greibach
Normal Form
7.
Parikh’s Theorem
8.
The CYK Algorithm
9.
Tree Automata
10. WSkS and Monadic Logics
11. Infinite
Automata
12. Cellular
Automaton
13. Recursive
Functions
14. Random-Access
Machines
15. Lambda
Calculus
16. Fast-Growing
Functions
17. Kolmogorov
complexity
18. The
Arithmetic Hierarchy
19. Presburger Arithmetic
20. Undecidability
of the Generalized 3x + 1 Problem
21. Undecidability
of Tilings of the Plane
22. The
Unexpected Examination Paradox
23. Gödel’s
Incompleteness Theorems
24. Topics
in Kolmogorov Complexity
25. Turing
Degrees
26. Blum’s
Speedup Theorem
27. Ladner’s
Theorem
28. Complexity
of Decision versus Search
29. The
Polynomial Hierarchy
30. Barrington’s
Theorem
31. Class
#P
32. IP=PSPACE
33. Complexity
of Games
34. Parity
Is Not in AC0
35. Relativization and Natural Proofs
Barrier
36. Communication
Complexity
37. PAC
Learning
38. Learning
and Languages
39. Evolvability
40. Hardness
Amplification
41. Pseudorandom
Functions
42. Zero-Knowledge
Protocols
43. Quantum
Computation
44. Average-Case
Complexity
45. The
(Other) PCP Theorem