# Evolvability and Learning

This post investigates the paper Evolvability by computer scientist Leslie Valiant analyzing evolution and its limits through the lens of machine learning theory.

The first section gives an overview of Valiant’s now famous Probably Approximately Correct learning model, and the material is derived from Kearns and Vazirani’s short textbook Introduction to Computational Learning Theory. Valiant’s more recent notion of evolvability (explored in the second section) implies PAC learnability, but not conversely. In particular, the parity functions (i.e. the mod 2 linear functions ${\{0,1\}^n\rightarrow\{0,1\}}$) are learnable, but provably not evolvable (under Valiant’s or any similar notion of evolvability).

Throughout, I’ve included some fun pictures, produced by simple simulations in Python with matplotlib.

# Making change

Consider the classic problem: How many ways can one make change for one dollar using half-dollars, quarters, dimes, nickels, and pennies? Or more generally, how many ways can one make change for a given amount using arbitrary (positive integer) denominations?

This post chronicles a series of incremental improvements to solutions to this problem. In the first section, we attack the problem with dynamic programming in Python, and we’re able to count the ways to change quite large amounts of money (eventually up to about \$100M for the given set of five coins). The second section explores a nice method for deriving closed form solutions, due to Lee Newberg, and the third is a synthesis of the previous two, automating the closed form derivation in the general setting.

# Sam and Polly and Python

The Sam and Polly Puzzle is a beautiful riddle in which two logicians Sam and Polly must simulate the other’s thought processes several layers deep, deriving information from seemingly unhelpful statements. While many variations have been proposed, I first heard of the puzzle from the excellent list at the XKCD wiki, where it takes the following form:

Sam and Polly are perfectly logical mathematicians. A student walks in and says: “I’m thinking of two numbers x and y, with 3 <= x <= y <= 97. I’ll tell their sum to Sam, and their product to Polly.” She does so, then leaves, and the following conversation takes place:

Sam (to Polly): You can’t know what x and y are.

Polly (to Sam): That was true, but now I know.

Sam (to Polly): Now I know too.

Find x and y.

If you haven’t solved this riddle or one of its variations before, I absolutely recommend giving it a try before reading on. It took me about three hours to solve, with a whiteboard and the Python programming language.

This post consists of:

• Section 1: A clean Python solution
• Section 2: An investigation of the (somewhat weak) dependence on the bounds 3 and 97
• Section 3: A discussion of unbounded variations of the riddle.

The most common variant on the puzzle (and the original version published by Freudenthal in 1969, available here) uses the following bounds instead: 2 <= x < y and x + y <= 100, leading to a different solution. I’m sticking with the above formulation just because that’s the version that I saw and started thinking about before looking around to see what was known. The solution in the following section can be easily modified to solve Freudenthal’s version.

This riddle also known as the Sum and Product Puzzle, or as the Impossible Puzzle (not to be confused with the Hardest Logic Puzzle Ever, another excellent riddle).