Ma/CS117a : Computability Theory

Su Gao
Sloan 159, MWTh 1-2
Office: Sloan 384
Office Hours: MTh 2-3
Extension: 4367
E-mail: sugao@cco.caltech.edu

Announcement

The last homework (assignment 7) will be due Wednesday, December 2 by 2pm.

Since I intend to handout solutions and return the graded papers the following Thursday, no extensions will be made to this homework set.

The final will be administered Thursday, December 3.

For now, Happy Thansgiving!

Lectures

Date

  

Section#

Title

9/28

 

0

Introduction

9/30

 

1

Primitive Recursive Functions on Words

10/1

 

 

Closure Properties

10/5

 

2

Primitive Recursive Functions on Numbers

10/7

 

3

Partial Recursive Functions

10/8

 

4

While Programs

10/12

 

 

For Programs

10/14

 

5

The Grzegorczyk Hierarchy

10/15

 

6

Turing Machines

10/19

 

 

Computation by Turing Machines

10/21

 

7

Coding of Turing Machines

10/22

 

 

Basic Results in Recursion Theory

10/26

 

8

Church-Turing Thesis

10/28

 

9

Indexing of Partial Recursive Functions

10/29

 

10

The Halting Problem

11/2

 

11

Recursively Enumerable Sets

11/4

 

 

Examples of R.E. and Non-R.E. Sets

11/5

 

12

Recursively Enumerable Relations

11/9

 

13

More Undecidability: Rice's Theorem

11/11

 

14

Recursion Theorems

11/12

 

15

Acceptable Coding Systems

11/16

 

 

Rogers and Blum Isomorphism Theorems

11/18

 

16

Partial Recursive Functionals

11/19

 

 

First Recursion Theorem

11/23

 

17

Relative Recursion

11/25

 

18

Reducibility Notions

11/30

 

19

The Arithmetical Hierarchy

Other

People whose work had great impact on the field:

A list of books available in the Caltech libraries that is one way or the other related to the course (neither of them is recommended as a textbook or standard reference, or guanranteed to be helpful for the homework problems):

  1. M. Garey & D. Johnson, Computers and Intractability: A guide to the theory of NP-completeness, W.H.Freeman, 1980. (Millikan QA76.6 G35 1980)
  2. J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, 1977. (Millikan QA9 H32 1977)
  3. H. Hermes, Enumerability, Decidability, Computability: An introduction to the theory of recursive functions, (Translated from German by G.T.Hermann and O. Plassmann), Springer, 1969. (Millikan QA248.5 H413 1969)
  4. Y. Manin, A Course in Mathematical Logic, (Translated from Russian by N. Koblitz), Springer-Verlag, 1977. (Millikan QA9 M29613)
  5. J. Hopcroft & J. Ullman, Formal Languages and Their Relation to Automata, Addison-Wesley, 1969. (SFL QA267.3 H6 1969)
  6. J. Hopcroft & J. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, 1979. (SFL QA267 H56 1979)
  7. M. Machtey & P. Young, An Introduction to the General Theory of Algorithms, North-Holland, 1978. (Millikan SFL QA76.7 M3)
  8. G. S. Boolos & R. Jeffery, Computability and Logic, Cambridge University Press, 1989. (SFL BC135 B665 1989)
  9. D. Kozen, Automata and Computability, Spriger, 1997. (SFL QA267 K69 1997)
  10. Z. Manna, Mathematical Theory of Computation, McGraw-Hill, 1974. (Millikan SFL QA76.6 M356 1974)
  11. N. Cutland, Computability: An introduction to recursive function theory, Cambridge University Press, 1980. (Millikan QA9.59 C87)
  12. F. Beckman, Mathematical Foundations of Programming, Addison-Wesley, 1980. (SFL QA9 B36 1980)
  13. C. H. Papadimitriou & K. Sleiglitz, Combinatorial Optimization: Algorithms and complexity, Prentice-Hall, 1982. (SFL QA402.5 P37 1982)
  14. C. H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994. (SFL QA267.7 P36 1994)
  15. H. Lewis & C. H. Papadimitriou, Elements of the Theory of Computation, Prentice-Hall, 1981. (SFL QA267 L49)
  16. M. Davis, Computability and Unsolvability, McGraw-Hill, 1958. (Millikan SFL QA248.5 D3 1958)
  17. E. Engeler, Introduction to the Theory of Computation, Academic Press, 1973. (SFL QA267.5 S4 E53 1973)
  18. M. A. Harrison, Introduction to Formal Language Theory, Addison-Wesley, 1978. (SFL QA267.3 H37)
  19. F. Hennie, Introduction to Computability, Addison-Wesley, 1977. (SFL QA267 H46 1977)
  20. M. L. Minsky, Computation: Finite and infinite machines, Prentice-Hall, 1972. (SFL QA267 M55 1972)
  21. R. N. Moll, M. A. Arbib & A. J. Kfoury, An Introduction to Formal Language Theory, Springer-Verlag, 1988. (SFL QA76.7 M65 1988)
  22. A. J. Kfoury, R. N. Moll & M. A. Arbib, A Programming Approach to Computability, Springer-Verlag, 1982. (SFL QA9.59 K46 1982)
  23. M. A. Arbib, Theories of Abstract Automata, Prentice-Hall, 1969. (SFL QA267 A72)
  24. M. A. Arbib, A. J. Kfoury & R. N. Moll, A Basis for Theoretical Computer Science, Spriger-Verlag, 1981. (SFL QA267 A715 1981)
  25. B. A. Trakhtenbrot, Algorithms and Automatic Computing Machines, (Translated from Russian by J. Kristan, J. D. McCawley & S. A. Schmitt), Heath, 1963. (SFL QA76 T713 1963)
  26. J. M. Brady, The Theory of Computer Science, Chapman & Hall, 1977. (SFL QA76 B697 1977)
  27. A. I. Malcev, Algorithms and Recursive Functions, (Translated from Russian by L. F. Boron), Noordhoff, 1970. (Millikan MathRes QA266 M2713)
  28. A. V. Aho & J. D. Ullman, Foundations of Computer Science, Computer Science Press, 1992. (SFL QA76 A334 1992)
  29. A. V. Aho, J. E. Hopcroft & J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974. (Millikan SFL QA76.6 A36)
  30. J. Ferrante & C. W. Rackoff, The Computational Complexity of Logical Theories, Springer-Verlag, 1979. (Millikan SFL QA9.35 F47 1979)
  31. B. W. Arden (ed.), What Can Be Automated?: The computer science and engineering research study (COSERS), MIT Press, 1980. (SFL QA76 W49)
  32. N. D. Jones, Computability and Complexity: from a programming perspective, MIT Press, 1997. (SFL QA76.6 J6658 1997)
  33. H. Rogers, Theory of Recursive Functions and Effective Computability, MIT Press, 1987. (SFL QA248.5 R6 1987)
  34. G. J. Tourlakis, Computability, Reston, 1984. (Millikan SFL QA9.59 T68 1984)
  35. A. Fisher, Formal Number Theory and Computability: a workbook, Oxford University Press, 1982. (Millikan MathRes QA241 F57 1982)
  36. R. McNaughton, Elementary Computability, Formal Languages and Automata, Prentice-Hall, 1982. (Millikan QA9.59 M35)
  37. W. J. Savitch, Abstract Machines and Grammars, Little Brown, 1982. (SFL QA267 S29 1982)
  38. A. Salomaa, Computation and Automata, Cambridge University Press, 1985. (Millikan QA9.59 S25 1985)
  39. K. Weihrauch, Computability, Springer-Verlag, 1987. (SFL QA9.59 W45 1987)
  40. D. S. Bridges, Computability: a mathematical sketchbook, Springer-Verlag, 1994. (Millikan QA9.59 B75 1994)
  41. M. D. Davis, R. Sigal & E. J. Weyuker, Computability, Complexity, and Languages: Fundamentals of theoretical computer science, Academic Press, 1994. (SFL QA267 D38 1994)
  42. E. Borger, Computability, Complexity, Logic, North-Holland, 1989. (Millikan QA9.59 B6713 1989)
  43. R. L. Epstein & W. A. Carnielli, Computability: Computable functions, logic, and foundations of mathematics, Wadsworth & Brooks, 1989. (SFL QA9.59 E67 1989)
  44. M. B. Pour-El & J. I. Richards, Computability in Analysis and Physics, Springer-Verlag, 1989. (Millikan QA9.59 P68 1989)
  45. M. C. Fitting, Computability Theory, Semantics, and Logic Programming, Oxford University Press, 1987. (Millikan QA9.59 F58 1987)
  46. Yu. I. Ershov, Definability and Computability, Consultants Bureau, 1996. (Millikan QA9.59 E78 1996)
  47. R. W. Floyd & R. Beigel, The Language of Machines: An introduction to computability and formal languages, Computer Science Press, 1994. (SFL QA267.3 F56 1994)
  48. L. E. Sanchis, Reflexive Structures: An introduction to computability theory, Springer-Verlag, 1988. (SFL QA9.59 S26 1988)
  49. R. Sommerhalder & S. C. van Westrhenen, The theory of computability : programs, machines, effectiveness, and feasibility, Addison-Wesley, 1988. (Millikan QA9.58 S64 1988)
  50. P. Odifreddi, Classical recursion theory : the theory of functions and sets of natural numbers, North-Holland, 1989. (Millikan QA9.6 O35 1989)
  51. R. M. Smullyan, Recursion Theory for Metamathematics, Oxford University Press, 1993. (Millikan QA9.6 S68 1993)
  52. M. Sipser, Introduction to the Theory of Computation, PWS, 1997. (SFL QA267 S56 1997)
  53. H. E. Rose, Subrecursion: functions and hierarchies, Oxford University Press, 1984. (Millikan QA9.615 R67 1984)