Knights
Write a program to solve logic puzzles.
Background
In 1978, logician Raymond Smullyan published “What is the name of this book?”, a book of logical puzzles. Among the puzzles in the book were a class of puzzles that Smullyan called “Knights and Knaves” puzzles.
In a Knights and Knaves puzzle, the following information is given: Each character is either a knight or a knave. A knight will always tell the truth: if knight states a sentence, then that sentence is true. Conversely, a knave will always lie: if a knave states a sentence, then that sentence is false.
The objective of the puzzle is, given a set of sentences spoken by each of the characters, determine, for each character, whether that character is a knight or a knave.
For example, consider a simple puzzle with just a single character named A. A says “I am both a knight and a knave.”
Logically, we might reason that if A were a knight, then that sentence would have to be true. But we know that the sentence cannot possibly be true, because A cannot be both a knight and a knave – we know that each character is either a knight or a knave, but not both. So, we could conclude, A must be a knave.
That puzzle was on the simpler side. With more characters and more sentences, the puzzles can get trickier! Your task in this problem is to determine how to represent these puzzles using propositional logic, such that an AI running a model-checking algorithm could solve these puzzles for us.
Getting Started
- Download the distribution code from https://cdn.cs50.net/ai/2020/spring/projects/1/knights.zip and unzip it.
Understanding
Take a look at logic.py
, which you may recall from Lecture 1. No need to understand everything in this file, but notice that this file defines several classes for different types of logical connectives. These classes can be composed within each other, so an expression like And(Not(A), Or(B, C))
represents the logical sentence stating that symbol A
is not true, and that symbol B
or symbol C
is true (where “or” here refers to inclusive, not exclusive, or).
Recall that logic.py
also contains a function model_check
. model_check
takes a knowledge base and a query. The knowledge base is a single logical sentence: if multiple logical sentences are known, they can be joined together in an And
expression. model_check
recursively considers all possible models, and returns True
if the knowledge base entails the query, and returns False
otherwise.
Now, take a look at puzzle.py
. At the top, we’ve defined six propositional symbols. AKnight
, for example, represents the sentence that “A is a knight,” while AKnave
represents the sentence that “A is a knave.” We’ve similarly defined propositional symbols for characters B and C as well.
What follows are four different knowledge bases, knowledge0
, knowledge1
, knowledge2
, and knowledge3
, which will contain the knowledge needed to deduce the solutions to the upcoming Puzzles 0, 1, 2, and 3, respectively. Notice that, for now, each of these knowledge bases is empty. That’s where you come in!
The main
function of this puzzle.py
loops over all puzzles, and uses model checking to compute, given the knowledge for that puzzle, whether each character is a knight or a knave, printing out any conclusions that the model checking algorithm is able to make.
Specification
Add knowledge to knowledge bases knowledge0
, knowledge1
, knowledge2
, and knowledge3
to solve the following puzzles.
- Puzzle 0 is the puzzle from the Background. It contains a single character, A.
- A says “I am both a knight and a knave.”
- Puzzle 1 has two characters: A and B.
- A says “We are both knaves.”
- B says nothing.
- Puzzle 2 has two characters: A and B.
- A says “We are the same kind.”
- B says “We are of different kinds.”
- Puzzle 3 has three characters: A, B, and C.
- A says either “I am a knight.” or “I am a knave.”, but you don’t know which.
- B says “A said ‘I am a knave.’”
- B then says “C is a knave.”
- C says “A is a knight.”
In each of the above puzzles, each character is either a knight or a knave. Every sentence spoken by a knight is true, and every sentence spoken by a knave is false.
Once you’ve completed the knowledge base for a problem, you should be able to run python puzzle.py
to see the solution to the puzzle.
Hints
- For each knowledge base, you’ll likely want to encode two different types of information: (1) information about the structure of the problem itself (i.e., information given in the definition of a Knight and Knave puzzle), and (2) information about what the characters actually said.
- Consider what it means if a sentence is spoken by a character. Under what conditions is that sentence true? Under what conditions is that sentence false? How can you express that as a logical sentence?
- There are multiple possible knowledge bases for each puzzle that will compute the correct result. You should attempt to choose a knowledge base that offers the most direct translation of the information in the puzzle, rather than performing logical reasoning on your own. You should also consider what the most concise representation of the information in the puzzle would be.
- For instance, for Puzzle 0, setting
knowledge0 = AKnave
would result in correct output, since through our own reasoning we know A must be a knave. But doing so would be against the spirit of this problem: the goal is to have your AI do the reasoning for you.
- For instance, for Puzzle 0, setting
- You should not need to (nor should you) modify
logic.py
at all to complete this problem.
How to Submit
- Download your
puzzle.py
file (which contains your implementation of the elements of the specification of this project). Be sure it has that exact filename! If you upload a file with a different name, the autograder likely will fail when trying to run it, and ensuring you have uploaded files with the correct filename is your responsibility! - Go to CSCI E-80’s Gradescope page.
- Click Project 1: Knights.
- Drag and drop your downloaded file to the area that says “Drag & Drop”.
- Click Upload.
You should see a message that says “Project 1: Knights submitted successfully!” Contact your teaching fellow if not! Note that the autograder will not run on your
Per Step 1 above, after you submit, be sure to go back and double-check your filename is correct. puzzle.py
(not logic.py
or PUZZLE.py
or puzzle (1).py
or, by accident, minesweeper.py
) is the only acceptable filename for this project. Correctness in submissions entails everything from reading the specification, writing code that is compliant with it, and submitting files with the correct name. If you see this error, you should resubmit right away, making sure your submission is fully compliant with the specification. The staff will not adjust your filenames for you after the fact, and if the autograder cannot run on your code because of a misnamed file, it will result in a score of zero!
Note that when you submit your code, the autograder will award a score of 0 points. This is normal behavior; we do not run the autograder until after the deadline for normal and late submissions has passed. You should make sure to test your code thoroughly, of course, as you go, but in this course you do not receive immediate score feedback upon submitting.