18.511: Introduction to Computability and Undecidability
1
Administrivia
Instructor: Dr. M. Minnes
Office: 2-172
Office hours: W 1pm-3pm
Course website: http://math.mit.edu/˜minnes/18.511
Email: minnes@math.mit.edu
2
Course format
Textbook
NJ Cutland, Computability: An Introduction to Recursive Function Theory. Cambridge University
Press, 1980.
Problem Sets
Weekly psets will be assigned based on the lectures and readings. They are due in class at the
beginning of each Thursday meeting. You are allowed and encouraged to work on the pset problems
together. However, you must write up the solutions individually. Make sure you are familiar with
MIT’s policy on Academic Integrity (see http://web.mit.edu/academicintegrity).
The lowest pset grade will be dropped. No late homework will be accepted.
Grading
Final grades in the course will be assigned according to the following approximate weighting.
• Homework: 50 %
• Midterm exam: 20 %
• Final exam: 30%
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Important Dates
Thursday, 9/17
Tuesday, 10/13
Thursday, 10/15
Thursday, 11/26
TBA
First pset due.
No class (Monday schedule).
Midterm exam.
No class (Thanksgiving).
Final exam.
A full calendar is available online at http://math.mit.edu/˜minnes/18.511/resources.html.
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Approximate Syllabus
9/10
9/15
9/17
9/22
9/24
9/29
{9/17
{9/22
{9/24
{9/29
{10/1
{10/6
10/1
10/6
10/8
10/13
10/15
10/20
10/22
10/27
10/29
11/3
11/5
11/10
11/12
11/17
11/19
11/24
11/26
12/1
12/3
12/8
12/10
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Introduction, overview
URMs, closure under composition
Closure under recursion, Def of TMs
Equivalence between URM and TM, Lambda calculus
Lambda calculus, Proof TM subset Lambda, Post systems
Partial recursive functions, Ackerman function, Equivalence proofs
Partial recursive functions
Equivalence of R, C , primitive recursive functions
TMs, Post systems, Equivalence proofs (I)
TMs, Post systems, Equivalence proofs (II)
TMs, Post systems, Equivalence proofs (III)
Other domains, Church’s thesis
Other domains, Church’s Thesis
Enumeration of programs, s − m − n theorem
Recursion theorem, universal programs
(Monday schedule)
Exam 1
1.1
1.2-1.3, 2.1-2.4
3.4
3.4
3.5
3.1-3.3, 3.5
3.2}
2.1-2.5, 3.3 }
3.4-3.6}
3.4-3.6}
3.4-3.6}
1.5, 3.6-3.7 }
1.5, 3.6-3.7
4.1-4.4
11.1, 5.1-5.2
Undecidability
Undecidability elsewhere
Recursively enumerable sets
Productive and creative sets, simple sets
Gödel’s incompleteness theorem (I)
Gödel’s incompleteness theorem (II)
≤m
≡m , m-complete sets
(Thanksgiving)
≤T
≡T
Priority constructions
Priority constructions
1.4, 6.1
6.2 - 6.6
6.6, 7.1-7.2
7.3-7.4
8
8
9.1
9.2-9.3
1.1 - 5.2, 11.1
9.4-9.5
9.4-9.5
Wikipedia list of undecidable problems
Problems in logic
- Hilbert’s Entscheidungsproblem: decide truth/falsity of all mathematical statements once
formalized. (1928; Church-Turing 1937)
-Type inference and type checking for the second-order lambda calculus (or equivalent).
Problems about abstract machines
- The halting problem (determining whether a Turing machine halts on specific input).
- Determining whether a Turing machine is a busy beaver champion (i.e., is the longest running
among halting Turing machines with the same number of states).
- The mortality problem (determining whether a Turing machine halts on any input).
- Rice’s theorem states that for all nontrivial properties of partial functions, it is undecidable
whether a machine computes a partial function with that property.
Problems about matrices [Paterson (70) Studies in Appl Math 59, 105-107]
- The mortal matrix problem: determining, given a finite set of n × n matrices with integer
entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero
matrix. (This is undecidable already for a set of 15 or more 3 × 3 matrices, or a set of two 45 × 45
matrices.)
- Determining whether a finite set of upper triangular 3 × 3 matrices with nonnegative integer
entries generates a free semigroup.
- Determining whether two finitely generated subsemigroups of Mn (Z) have a common element.
Problems in combinatorial group theory
- The word problem for groups: given a presentation of a group and two words of the
presentation, do they correspond to same group element
- The conjugacy problem: given a presentation of a group and two words of the presentation x, y,
is there group element such that x = z −1 yz.
- The group isomorphism problem.
Problems in topology
- Determining whether two finite simplicial complexes are homeomorphic.
- Determining whether a finite simplicial complex is (homeomorphic to) a manifold.
- Determining whether the fundamental group of a finite simplicial complex is trivial.
Other problems
−
→
→
- The Post correspondence problem: given two sequences of strings −
a , b over a finite alphabet, is
→
there a sequence of indices such that the concatenation of the −
a strings of those indices is equal
−
→
to the concatenation of the b strings of those indices.
- The word problem for certain formal languages.
- The problem of determining if a given set of Wang tiles can tile the plane.
- The problem of determining the Kolmogorov complexity of a string.
- Hilbert’s tenth problem: the problem of deciding whether a Diophantine equation (multivariable
polynomial equation) has a solution in integers.
- Determining if a context-free grammar generates all possible strings, or if it is ambiguous.
- Given two context-free grammars, determining whether they generate the same set of strings, or
whether one generates a subset of the strings generated by the other, or whether there is any
string at all that both generate.
- Determining whether a given initial point with rational coordinates is periodic, or whether it lies
in the basin of attraction of a given open set, in a piecewise-linear iterated map in two
dimensions, or in a piecewise-linear flow in three dimensions.