Exploring infinite sequences

I was recently watching John Hughes's talk Why Functional Programming Matters. One thing that jumped out at me was his definition of the Newton-Raphson square root approximation algorithm using a lazy sequence of estimates. This is an example from his influential paper of the same name.

The iterative algorithm looks like this:

def sqrt(n)
  # terminate when two successive guesses differ by less than epsilon
  # Initial guess can be anything as long as abs(x - y) > epsilon
  epsilon = 0.0001
  x = 1.0
  y = x + 2.0 * epsilon

  while (x - y).abs > epsilon
    y = x
    x = (x + n/x)/2.0


Hughes describes this program as "indivisible" -- the estimation is coupled with the iteration and the selection of the final estimate.

What impressed me about his example of doing this lazily in Haskell was how simple it was:

sqrt n = limit eps (iterate next 1.0)
  where next x = (x + n/x) / 2

I wanted to learn more, so I found his paper which has a more verbose version.

First, he defines a next function that produces an estimate:

next n x = (x + n/x)/2

Then, he defines an infinite sequence of better and better estimates. If the first estimate is 1.0 and the estimation function is called f, the sequence is:

[1.0, f 1.0, f (f 1.0), f (f (f 1.0)), ...]

This sequence can be generated using the Haskell function iterate applied to the curried function next n:

iterate (next n) 1.0

This is fun to play with. Here's the first 5 approximations of the square root of 1000 starting with an estimate of 1.0:

take 5 $ iterate (next 1000) 1.0

Next, Hughes defines a limit function (called within in the original paper) to compare the difference of the first two estimates in the sequence to the tolerance, and return one if it's close enough, otherwise, call itself recursively with the rest of the sequence (which, again, is infinite). As long as the function converges, this will eventually return a result.

-- (x:y:ys) is destructuring. 
-- `x` is the first element, `y` is the second element, and `ys` is the rest
limit epsilon (x:y:ys) = 
  if abs (x - y) <= epsilon then 
    limit (y:ys)

Putting it together, square root can be defined as:

epsilon = 0.0001
sqrt n = limit epsilon (iterate (next n) 1.0)

The really cool thing about this is that it separates the generation of the sequence from the selection of the value. The limit function can be used for other estimation problems with the same algorithm, such as calculating the derivative:

deriv f x = 
  limit epsilon (map slope (iterate (/2) 1.0)) 
  where slope h = (f (x+h) - f x) / h

-- (^2) is partial application: https://wiki.haskell.org/Section_of_an_infix_operator
-- So this says "compute the derivative of x^2 at 5"
deriv (^2) 5

-- f(x) = x^2
-- f'(x) = 2x
-- f'(5) => 10
-- It works!

I have found lazy evaluation confusing, but this is a powerful example of how it can simplify code. I wanted to see how it could be applied in a language with eager evaluation, so I translated the code to Ruby.

def iterate(a, &f)
  Enumerator.new do |yielder|
    loop do
      yielder << a
      a = f.call(a)

def limit(epsilon, seq)
  # NOTE: To maintain the same behavior as the Haskell code,
  #       peek at the second estimate so it's not consumed.
  a = seq.next
  b = seq.peek
  if (a - b).abs <= epsilon
    limit(epsilon, seq)

def sqrt(n)
  limit(0.0001, iterate(1.0) { |a| (a + n/a.to_f)/2.0 })

This is not as simple as the Haskell code!

Most of the complexity is due to the difficulty of generating an infinite sequence. Ruby is eagerly evaluated, so the iterate method uses Enumerator with a loop. But as seen in the limit method, the Enumerator takes special care in order to produce the same behavior as the Haskell code. A method to split an infinite sequence into first and rest would make things a lot easier.

This was an enlightening exercise. I learned more about laziness -- and how hard it can be to shoehorn it into an eagerly evaluated language. Ruby doesn't lend itself towards lazy sequences, so its support for programming with lazy sequences is weak.

Further reading

I posted the source code from this blog post if you'd like to play around with it.

I adapted the Haskell code for this post from John Hughes's slides, original paper, and Heinrich Apfelmus's post about lazy evaluation.

For the Ruby code, I learned a lot by reading Joël Quenneville's Functional Ciphers in Ruby and Ross Kaffenberger's Infinite Sequences in Ruby (he also has several more articles about Enumerator).

Thanks to Paul Cantrell and Charles Capps for reviewing drafts of this post. Any errors are my own.

— May 24, 2017

Goodnight Dune

Goodnight Dune

— May 18, 2017

You can't average percentiles. Period.

Gil Tene on the common practice of averaging percentiles for monitoring:

At first blush, you may think that the hourly average of the per-5-second-interval 99%ile value has some meaning. you know averages can be inaccurate and misrepresenting at times, but this value could still give you some "feel" for the 99%'lie system, right?


If you had the Maximum value recorded for each 5 second interval over a 5000 second run, and want to deduce the Maximum value for the whole run, would an average of the 1,000 max values produce anything useful? No. You'd need to use the Maximum of the Maximums. Which will actually work, but only for the unique edge case of 100%...

The only thing you can deduce about the 99%'lie of a 5000 second run for which you have 1,000 5 second summary 99%'lie values is that it falls somewhere between the Maximum and Minimum of those 1,000 99%'lie values...

Bottom line: You can't average percentiles. It doesn't work. Period.

— May 15, 2017

Skipping Travis CI builds for snapshot release commits with sbt-release

If you use sbt-release with the MAJOR.MINOR.PATCH and MAJOR.MINOR.PATCH+1-SNAPSHOT style of release versioning, this creates two commits when you run sbt release that differ only in the contents of the version.sbt file.

Unfortunately, if you're using a continuous integration system like Travis CI, those two commits both trigger full builds. There's really no point for that, since there's no difference in the code. Fortunately, Travis CI (and other systems) support marking a commit with [ci skip] to skip the build.

You can customize the commit message by changing the releaseCommitMessage setting. However, releaseCommitMessage is used for both the commitReleaseVersion and commitNextVersion actions, so some conditional logic is needed.

The default value looks like this:

releaseCommitMessage := s"Setting version to ${if (releaseUseGlobalVersion.value) (version in ThisBuild).value else version.value}"

Change releaseCommitMessage so that if the version ends with "-SNAPSHOT" then " [ci skip]" is appended to the message:

releaseCommitMessage := "Setting version to " +
  s"${if (releaseUseGlobalVersion.value) (version in ThisBuild).value else version.value}" +
  s"${if (if (releaseUseGlobalVersion.value) (version in ThisBuild) else version).value.endsWith("-SNAPSHOT")) " [ci skip]" else ""}"

Now you'll end up with release commits that look like "Setting version to 1.2.3" and snapshot commits that look like "Setting version to 1.2.4-SNAPSHOT [ci skip]".

— March 18, 2017

Acing the technical interview

No time for descriptive variables, examples, or docstrings here. In the whiteboard interview, time is of the essence. Pretend you are a Haskell programmer, as your grandmother was, before her continuation passed.

— March 17, 2017

My favorite books of 2016

Time for my annual round up of books read in the past year with a few favorites noted.

Cadillac Desert

Cadillac Desert: The American West and Its Disappearing Water, Marc Reiser

This is an eye-opening story of man's hubris in the American West. Attempting to make the desert bloom led to uncountable environmental destruction. As the climate changes and water patterns change, will we attempt to hold on to the "Cadillac desert" at huge expense, or let farmers and communities wither? The present Oroville Dam disaster would fit right in as a chapter in this book.


Sapiens: A Brief History of Humankind, Yuval Noah Harari

Breezy and fun, but with a lot to think about. This book stuck with me.

My Ántonia

My Ántonia, Willa Cather

Based on reading this back in 2007, I consider it one of my favorite novels. So much so that we named our daughter after the title character. I was worried that I wouldn't like it, but, if anything, the nostogia of the novel hit me harder now that I'm older.

From Bauhaus to Our House

From Bauhaus to Our House, Tom Wolfe

I am sure Tom Wolfe is an enormous dick in real life, but wow is his skewering of modernist architecture funny and satisfying.

San Francisco: La grille sur les collines<

San Francisco: La grille sur les collines / San Francisco: The Grid meets the Hills, Florence Lipsky

This is a fascinating and unique book about how San Francisco's street grid is adapted to the landscape. Through a combination of greed, incompetence, hubris, and Yankee ingenuity, San Francisco's city planners overlaid the standard American gridiron street pattern onto the penninsula's topography. The result is San Francisco's wildly undulating streets, which sometimes dead end spectacularly. Despite their best efforts, sometimes the grid could not be made to fit the landscape. These deformations are a major focus of the book and it features photos and illustrations of the effects in many neighborhoods across the city:

Telegraph Hill

For more pages from the book, check out this post on Map Library.

Below is the complete list of books I read in 2015:

Cadillac Desert: The American West and Its Disappearing Water, Marc Reisner

Grounded: The Case for Abolishing the United States Air Force, Robert M. Farley

Sapiens: A Brief History of Humankind, Yuval Noah Harari

A Shortage of Engineers, Robert Grossbach

The Wise Man's Fear, Patrick Rothfuss

Snowpiercer: The Escape, Jacques Lob, Jean-Marc Rochette (illustrator), and Virginie Selavy (translator)

Hyperbole and a Half: Unfortunate Situations, Flawed Coping Mechanisms, Mayhem, and Other Things That Happened, Allie Brosh

My Ántonia, Willa Cather

The Alchemist, Paolo Bacigalupi

Straight Man, Richard Russo

Decisive: How to Make Better Choices in Life and Work, Chip Heath and Dan Heath

Alice's Adventures in Wonderland Lewis Carroll

Through the Looking-Glass and What Alice Found There, Lewis Carroll

Liar's Poker: Rising Through the Wreckage on Wall Street, Michael Lewis

The Night Sessions, Ken MacLeod

How We Got to Now: Six Innovations That Made the Modern World, Steven Johnson

Terminal World, Alastair Reynolds

Success and Luck: Good Fortune and the Myth of Meritocracy, Robert H. Frank

Deadpool: The Complete Collection Volume 1, Daniel Way et al.

Minority Report, Philip K. Dick

From Bauhaus to Our House, Tom Wolfe

Wolf: The Lives of Jack London, James L. Haley

San Francisco: La grille sur les collines / San Francisco: The Grid Meets the Hills, Florence Lipsky

Pump Six and Other Stories, Paolo Bacigalupi

Bébé Day by Day: 100 Keys to French Parenting, Pamela Bruckerman

Glup: Adventures in the Alimentary Canal, Mary Roach

Wine Wars: The Curse of the Blue Nun, the Miracle of Two Buck Chuck, and the Revenge of the Terroirists, Mike Veseth

Disrupted: My Misadventure in the Start-up Bubble, Dan Lyons

The Swerve: How the World Became Modern, Stephen Greenblatt

You can also check out lists from previous years: 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, and 2015.

— March 9, 2017

A Low-Cost Solution to Traffic

In Governing magazine, William Fulton raises a consequence of compact urban areas that I had not realized fully. Even if a place is car dependent -- public transit is inadequate for its residents needs -- it can still reduce traffic if the people who live there don't have to drive as far because everything is relatively close together. Compact urban areas reduce traffic simply by being compact.

Which brings us to proximity. One of the few ways around this problem is to build more housing close to the urban cores -- or, at least, close to the dense suburban job centers. Urban planners often argue for locating more housing along high-frequency transit lines, which makes sense because many people can commute by transit.

What’s not well understood, however, is that well-located housing can cut down on the amount of driving -- and hence the need for additional road space -- even if people are still tethered to their cars. One famous study in the San Francisco Bay Area found that people living in Berkeley and Oakland drive only half as far as people in the outer suburbs -- not because they take transit more, but because the places they have to go are closer together.

— February 12, 2017

Grid corrections

Sometimes when riding gravel races (which mostly follow roads laid out on the Jeffersonian grid), I would come to a T intersection. I never really thought about why this happens, but Gerco de Ruijter has turned it into an art project:

By superimposing a rectangular grid on the earth surface, a grid built from exact square miles, the spherical deviations have to be fixed. After all, the grid has only two dimensions.

The north-south boundaries in the grid are on the lines of longitude, which converge to the north. The roads that follow these boundaries must dogleg every twenty-four miles to counter the diminishing distances: Grid Corrections.

Grid Corrections (a one minute) from Gerco de Ruijter on Vimeo.

— January 1, 2017

How to accept gifts for a Vanguard 529

We recently set up a 529 account to save for our daughter's college education. You can accept gifts from grandparents and others with a 529, but it's surprisingly difficult to find out how this works. It took me a while to figure out; hopefully writing this down will save someone out there some time.

To accept gifts for a Vanguard 529, you need to link the account to Ugift.

To do this with Vanguard, log in and click "Go to my 529 plan account area". This launches a new window for 529 account management.

Vanguard go to 529 account management

Next, click on "Ugift" in the side bar and fill out the form to generate your Ugift code.

Vanguard UGift integration

Once you do, you'll see instructions about how to share the code and can also download a PDF for relatives who prefer to contribute offline.

Once you have the code, you can provide it to benefactors to input at Ugift

UGift input code

Upon entering the code, they'll see the beneficiary's name, so they'll know they're contributing to the right kid.

UGift gift form

Hope this helps you. And Nia's code is really V6G-V7J as shown above, in case you feel like helping out.

— December 21, 2016

Fixing "Permission denied (publickey)" when connecting to a Travis CI build container

I was attempting to debug a build with Travis CI's SSH connection option (sadly, I can't find any documentation about this feature -- it's the "Debug job" button).

After the Debug job starts up, it prints:

Use the following SSH command to access the interactive debugging environment: ssh <random-string>@to2.tmate.io

But when I tried this, I got:

Permission denied (publickey).

It took me a while to figure out why this was happening. What public key should a temporary SSH connection use? After talking to my co-workers, who reported no problems connecting, I remembered I had set up a separate public key for GitHub. By specifying that key with -i I was able to connect:

ssh -i ~/.ssh/id_github_rsa <random-string>@to2.tmate.io

— December 13, 2016