In this blog post we will explore the consequences of postulating 0 = 1 in an algebraic structure with two binary operations (S, +, 0) and (S, ⋅, 1). Such *united monoids* have a few interesting properties, which are not immediately obvious. For example, we will see that the axiom 0 = 1 is equivalent to a seemingly less extravagant axiom ab = ab + a, which will send us tumbling down the rabbit hole of algebraic graphs and topology.

# Category Archives: thinking

# Selective applicative functors

**Update:** I have written a paper about selective applicative functors, and it completely supersedes this blog post (it also uses a slightly different notation). You should read the paper instead.

I often need a Haskell abstraction that supports conditions (like Monad) yet can still be statically analysed (like Applicative). In such cases people typically point to the Arrow class, more specifically ArrowChoice, but when I look it up, I find several type classes and a dozen of methods. Impressive, categorical but also quite heavy. Is there a more lightweight approach? In this blog post I’ll explore what I call *selective applicative functors*, which extend the Applicative type class with a single method that makes it possible to be selective about effects.

Please meet Selective:

class Applicative f => Selective f where handle :: f (Either a b) -> f (a -> b) -> f b

Think of handle as a *selective function application*: you apply a handler function of type a → b when given a value of type Left a, but can skip the handler (along with its effects) in the case of Right b. Intuitively, handle allows you to *efficiently* handle errors, i.e. perform the error-handling effects only when needed.

# Fun with Ngrams

After coming across a cool post on Google Ngrams by Randall Munroe, I decided to try it on something relevant to my research.

# Inverting the universe with two inverters

Recently I came across an interesting puzzle from MIT Tech Review March/April 2013:

*Howard Cohen has plenty of AND and OR gates but just two inverters. How can he invert three signals a, b, and c?*

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*More generally, he wonders if the ratio I/S can ever be less than 2/3, where I is the number of inverters and S is the number of signals to invert (once again, unlimited AND and OR gates are available).*

I couldn’t quickly solve the first part of the puzzle with a pen and paper, so I decided to write a brute force solver. And to make it a bit more fun I did it in Haskell (any suggestions for improving the code are welcome, by the way!).