Last 10 minutes of class on Wednesday, March 11, 2026.
Everyone takes the exam at the same time.
Computability Theory.
You will take the exam on Brightspace. There is 10 minute time limit, or 15 minutes for those with time-and-a-half accommodations.
Do each of the following to maximize your preparation:
Were you able to check off every box?
The fact that active recall is better for acquiring long-term knowledge does not mean that outlines and concept maps are not useful. Learners should use multiple techniques—think “both and” rather than “either or.”
Computability Theory
Concerned with what can and cannot be computed
How Hilbert kind of started this all off
Wadler’s video on the history of computability
Bernhardt's Book
Equivalent models of computation
Turing Machines
Register Machines
Lambda Calculus
Recursive Functions
Semi-Thue Systems
Post’s Production Systems
Church-Turing Thesis
Why is there a limit to what can be computed?
Counting argument
# of possible programs is countable; # of possible functions is uncountable
Diagonalization argument
Works for both functions in N->N and N->B
The first non-decidable function
Halting Problem
Demonstration in Python
Setting up the contradiction
Decidability
Definition of decidable
Decidability is a property of languages
Recognizability
Recognizable = algorithm must always halt and answer yes for utterances
Decidable = algorithm must always halt and answer yes or no for utterances
Recognizers may loop forever on rejections
Deciders must always halt on all inputs
Halting Language H is recognizable but not decidable
Recognizable language == Recursively enumerable language
Decidable language == Recursive language
Nonrecognizability
First nonrecognizable language is language of machines that don’t recognize themselves
What does this really mean
Diagonalization proof
Reductions
Reducing A to B
means solving B to solve A
If A is undecidable and A reduces to B, then B is undecidable
Example with the empty language
Rice’s Theorem
Any nontrivial property of the language recognized by a Turing machine is undecidable
Nontrivial means some machines have the property and some don’t
Examples of properties that are undecidable
Is the language empty?
Is the language finite?
Is the language regular?
Is the language context-free?
Does the language contain a particular string?
Does the language contain all strings?
Applications in real life (esp correctness and security)
Busy Beavers
Why Study
Different definitions
but we focus on the “number of steps” variety, originally S, now called BB
Number of n-state Turing Machines
A list of the first few champions
BB is not computable (can show via reduction from the Halting Problem)
Fun fact: BB grows faster than any computable function
Computable numbers
Oracles
Connections to Complexity Theory
This is a mini-quiz which tests for immediate understanding of topics and not your ability to work out problems over an extended duration of time. There is a strict time limit so that your immediate fluency is tested rather than your ability to search the web (or worse, ask a chatbot), since these things take time. There will be 5–10 questions, some will be multiple choice, some multi-select, and some matching. There are no free-form answers, so the exam will be autograded and you will see your score immediately after it is submitted.
All content on the assigned readings is fair game for questions, so do not neglect the readings, and by all means do the recall questions!