# Real World Applications of Automaton Theory

This term, I was teaching a course on intro to theory of computation, and one of the topics was finite automatons — DFAs, NFAs, and the like. I began by writing down the definition of a DFA:

A deterministic finite automaton (DFA) is a 5-tuple $(Q, \Sigma, \delta, q_0, F)$ where: $Q$ is a finite set of states…

I could practically feel my students falling asleep in their seats. Inevitably, a student asked the one question you should never ask a theorist:

“So… how is this useful in real life?”

## DFAs as a model of computation

I’ve done some theoretical research on formal language theory and DFAs, so my immediate response was why DFAs are important to theorists.

Above: A DFA requires O(1) memory, regardless of the length of the input.

You might have heard of Turing machines, which abstracts the idea of a “computer”. In a similar vein, regular languages describe what is possible to do with a computer with very little memory. No matter how long the input is, a DFA only keeps track of what state it’s currently in, so it only requires a constant amount of memory.

By studying properties of regular languages, we gain a better understanding of what is and what isn’t possible with computers with very little memory.

This explains why theorists care about regular languages — but what are some real world applications?

## DFAs and regular expressions

Regular expressions are a useful tool that every programmer should know. If you wanted to check if a string is a valid email address, you might write something like:

### Can you really develop without a device?

All across internet forums, people advocate that you should test your app thoroughly across many devices before submitting to the app store. It seems that trying to develop without a device is an edge case, often the instructions for a task assumes you have a device, and you have to find a workaround if you don’t.

In my case, it was successful, in the sense that I produced an app that didn’t crash and got past app review. But I don’t know if I got lucky, because things could have turned out badly.

Throughout the whole process, I was worried that running the app on a real device would exhibit bugs that aren’t producible in the simulator. In that case, I’d have no way to debug the problem, and the project would be done. My app uses nothing but the most basic functionality, so I had a good chance of dodging this bullet. But still, the possibility loomed over me, threatening to kill the project just as it crosses the finish line.

A second problem is by copying my original Android app feature by feature, the resulting iOS app looks and feels like an Android app. A friend pointed this out when I sent him the app. I hadn’t noticed it, but after looking at some other iOS apps, I have to agree with him. Actually in hindsight it shouldn’t surprise anyone that without seeing other iOS apps, I don’t really know how an iOS app should behave. But it just never occurred to me that the natural way to do things for me might be unnatural for iOS users.

Finally, this is subjective, but for me it wasn’t very fun to develop for a simulator. Without the tactile sensation of your creation running on an actual phone, the whole experience feels detached from reality. You feel like an unwelcome foreigner in a country where the customs are different, and you begin to question yourself, why am I doing this iOS port anyway?

Part of what kept me going was sunk cost fallacy. I paid \$119 to be an iOS developer, so I’d better get at least something on the app store, have something to show for it.

Now that the app is finished, I think I’m done with iOS development. Perhaps the app store is fertile ground for developers and startups looking to make a profit, but the cost of entry is unreasonable for someone making a few open source apps for fun.

# What’s the hardest bug you’ve ever debugged?

In a recent interview, I was asked this question: “what’s the most difficult bug you’ve encountered, and how did you fix it?” I thought this was an interesting question because there are so many answers you could give to this question, and the sort of answer you give demonstrated your level of experience with developing software.

I thought for a moment, recalling all the countless bugs I had seen and fixed. Which one was the most difficult and interesting? In this article I’m going to describe my most difficult bug to date.

It was an iOS app. I was working as a four-month intern at the time. “We’ve been seeing reports from our users that the app randomly display a black screen,” my boss explained one afternoon. “No error message, no crash log, nothing. The app is simply stuck at a black screen state until you kill it.”

“Fair enough. How do I reproduce it?”

He shrugged. “I don’t know. Users are reporting it happens randomly. Here’s what you gotta do: grab an iPad, download the game off the app store. Create an account and play the game until you hit the bug.”

So I did. I was reduced to one of these typewriter monkeys, banging away mindlessly at the keyboard until I stumble upon the sequence of button presses to trigger the undiscovered bug by sheer coincidence.

For an afternoon I monkeyed away, but no matter what buttons I pressed, the mythical black screen would not appear. I left the office, defeated and mentally exhausted.

The next morning I checked into the office, picked up the iPad, and resumed my monkeying. But this time my fortune was different: within 15 minutes, lo and behold, the screen flashed white, followed by an unrepentant screen of black.

What did I do to trigger this? I retraced my steps, trying to repeat the miracle. It happened again. Methodically I searched for a deterministic sequence of actions that brought our app to its knees. Go to the profile page. Hit button X. Go to page Y and back to the profile page. Hit button Z. The screen flickered for a millisecond, the black. Ten times out of ten.

With a sigh of relief, I jotted down this strange choreography and went for a walk. Returning with a fresh mind ready to tackle the next stage of the problem, I executed the sequence one more time, just to make sure. But the bug was nowhere to be seen.

I racked my brain for an explanation. The same sequence of actions now produced different results, I reasoned. Which meant something must have changed. But what?

It occurred to me that the page looked a little different now from when I was able to reproduce the bug. In the morning, when I came in, there was a little countdown timer in the corner of the screen that indicated the time until an upcoming event. The timer was not there anymore. Could it be the culprit…

To test this hypothesis, I produced a build that pointed the game to the dev server, and fired up a system event. The timer appeared. I executed my sequence — profile, tap, home screen, back to profile, tap, and sure enough, with a flicker the black screen appeared. I turned off the timer, repeated the sequence — profile, tap, home, profile, tap — no black screen. I had finally discovered the heart of the matter. There was some strange interaction going on between the timer and other things on the page.

At this point, with 100% reproducibility, the worst was over. It took a few more hours for me to investigate the issue and come up with a fix. My patch was quickly rolled out to production, and users stopped complaining about random black screens. Then my team went out for some celebratory beer.

I will now describe exactly what happened — and why did a timer cause such an insidious bug.

The timer widget was implemented using an NSTimer which made a callback every second. To do this, the timer holds a reference to the parent view which contains it. This is not too unusual, and is generally innocent and harmless — until you combine it with Objective C’s garbage collection system.

Objective C’s garbage collection system uses a reference counting algorithm. I’ll remind you what this means. The garbage collector maintains, for each object, a count of how many references lead to it. When this reference count reaches zero, it means your object is dead, since there is no way to reach it from anywhere in the system. Thus the garbage collector is free to delete it.

This doesn’t work for NSTimer, though. When two objects hold references to each other, their reference count remain at least 1, which means they can never get garbage collected. In our app, this meant that whenever the view with the timer goes out of view, it doesn’t get disposed, but remains in the background forever. A memory leak.

A memory leak, by itself, can go unnoticed for a long time with no impact. The last part of the puzzle that brought everything crashing down had to do with the way a certain button was implemented. This button, when pressed, broadcasted a message, which would then be received by the profile view.

When the timer is active, it is possible to get the system into a state with two profile views — a real one and a zombie one kept alive by a reference cycle with the timer.

Then when the message is broadcasted, both the real and zombie views receive the message in parallel. The button logic is executed twice in rapid succession, which understandably causes the whole system to give in.

With this mechanism in mind, the fix was easy. Just invalidate the timer when the view goes out of view. Without the reference cycle, the profile view is disposed of correctly and all is well again.

I think this story demonstrates a fundamental truth about debugging: in order to debug effectively, you need to have a deep understanding of your technology stack. This is not always true of programming in general — quite often you can write code that works yet not really understand what it’s doing. When developing a feature in an unfamiliar technology, the typical workflow is, if you don’t know how to do something, copy something similar from StackOverflow or a different part of your code base, make some changes until it works. And that’s a fine way to do things.

But debugging requires a more structured methodology. When many things are breaking in haphazard ways, you need to narrow down the problem to its very core, to identify precisely which component is broken. In this case it was a reference cycle that wouldn’t get released. The core of the problem may be buried within layers upon layers of an API, even an API you believe to be bulletproof. It might require digging into assembly code, even hardware.

To find that core requires an understanding of a mind-boggling stack of technologies that software today sits upon. That’s what it takes to become a master debugger.

So, what’s the hardest bug you’ve ever debugged?

# Algorithmic Trading Hackathon

The name of the hackathon was Code B: UW Algorithmic Trading Competition. It was hosted by Bloomberg and various UW student groups. It’s a 17 hour hackathon where you “create the best trading platform completely from scratch”. As far as I know, this is the first time the hackathon has been run, and in this article I’m going to write about my experience.

We were allowed teams of up to three, but my roommate Andrei and I signed up as a team of two. Like myself, Andrei is also a CS major. Neither of us had any experience with trading stocks, or anything finance related, for that matter. When asked to choose a team name, we named ourselves team /dev/rand (implying that we were so bad that we’d be no better than a random number generator)

The hackathon was scheduled to start Friday evening, running through the night until noon the next day. The goal was to write a program to autonomously trade stocks over a 20 minute period, battling other programs to earn as much money as possible. The programs communicated by connecting to a central server on Bloomberg’s side, so we could use any programming language we wanted. It was decided that Andrei would come up with strategies, and I would implement them in Python.

### Rules of the Game

The specifics of the API and mechanics of the game were not revealed until the official start of the hackathon. The 50-60 teams packed into an auditorium as the organizers started to explain the technical details.

The rules turned out to be fairly simple. The only actions allowed were to bid (attempt to purchase) on a stock for some price, or ask (attempt to sell) a stock for some price. If at any point someone’s bid is higher than someone else’s ask, the deal goes through and the stock changes hands.

Now all of this was fairly standard, but after this part, the rules diverged from real life. In order to encourage people to buy stocks (and not just hoard the initial money), each share of a stock paid dividends to its owner every second. And to prevent simply buying one stock and holding it for the entire duration, the dividends given out quickly diminishes the longer you own the stock.

This quirky dividends system turned out to be central to our strategy. Additionally, the differences from real stock markets meant that any previous experience with finance and stock trading was less useful — definitely a good thing for us because many of our competitors were seriously studying finance and we had no experience anyway.

### And it begins!

After the rules presentation, the hackathon kicked off. It was slightly past 7pm, and very quickly you could see teams buying and selling stocks. We decided to take it slow, discussing strategies over dinner.

We started work around 8pm. I began writing code to parse the input, while Andrei worked on deciphering the rather cryptic specifications document. Although API specs were clear enough, they were (intentionally) vague about how the system behaved behind the scenes. There were many formulas with lots of variables, many of which we had no idea what they meant.

So we took an experimental approach. Tentatively we put in a bid for a few shares of Google stock — and our net worth immediately took a nosedive. But the stock rapidly generated dividends, and before long, our net worth recovered to what it was initially, and it kept on going up! The success was short lasted, however, as quickly the dampening effects of the dividends started to kick in, and our rate of return quickly diminished to near zero.

We tried again, buying a few shares of the Twitter stock. The same thing happened — our value went down, quickly recovered, then gradually leveled to 50 dollars more than we started with.

With this information, we formulated a rough strategy. We didn’t know how to predict which stocks will go up; neither did we have a plan for buying and selling stocks at a favorable rate. Instead, we would take advantage of a stock’s “golden period”, where the stock initially pays massive dividends. It was crucial to buy as quickly as possible, since the clock started ticking as soon as you own one share of the stock. So we use all our money to buy as many shares of the stock as possible, doesn’t matter what price. Now we wait as the golden period payout multiplied by our entire bankroll makes us rich. Then, a few minutes later, when the golden period runs out, we would slowly sell, iteratively lowering our asking price until we found a buyer.

Once we sold the last share of a stock, the dividend clock doesn’t immediately reset, it slowly regenerates. So if we wait a while, say 5 minutes, then buy back the stock, we get another brief golden period. Taking this one step further, we decided on a strategy that cycled through the 10 stocks: at any given point, we would hold at most 4 of them, while the other 6 were left to “recharge”.

I proceeded to code the algorithm, while Andrei analyzed the spec document and brainstormed ways to improve the strategy. From the equations in the spec, he came up with a formula to determine what stock generated the highest dividends. Every half hour, the scoreboard would reset, and by 3am, I was basically done, and our algorithm consistently came either first or second by the end of each round. Our algorithm worked beautifully, simultaneously juggling a bunch of different stocks, some buying, others slowly selling. We watched the scoreboard as we earned hundreds of dollars every minute, ending with a ridiculous amount of money by the time it reset.

It seemed at this point that a lot of the teams were having implementation issues, like connecting to the network and parsing input, and only a handful were making any money at all, so I was pretty happy with our results.

But at 4am, disaster struck. A new round started, and our algorithm instantly plummeted to the bottom of the leaderboard. Every time we bought or sold anything, we lost money, and none of it was coming back through dividends. What happened?? It turns out that the parameters were changed, so that a very low amount of dividends were paid for owning a stock, and the only profits were made by buying low and selling high. This meant that our whole strategy, which centered around maximizing dividends, was rendered useless.

What’s worse — I discovered a bug in my implementation where our stocks were not being cycled properly: it would sell a certain stock, then instantly buy back the same stock, which didn’t allow the dividends clock to reset, meaning no dividends. Also, by this point a lot of teams were flooding the network with requests, making any network call have a small chance of throwing an exception and crashing the whole thing entirely.

The network problem was easy to fix, but at 5am, I was really tired and had difficulty tracking down the bug that was causing it to buy back the same stock. Andrei suggested a new set of strategies for the “low dividends” scenario, but by now, I was too tired to implement another set of strategies. Instead, we tweaked various constants in our program to make it play more patiently and more predictably, so even in the worst case it would make marginal gains instead of finishing dead last. After 2 hours of debugging, we managed to track down the cycling bug.

It was 7am and I could hardly keep my eyes open so we found a couch and napped for two hours, until the mock competition began.

### Mock Competition and final tweaking

At 9am, a few hours before the final competition, there was a mock competition which was meant to be identical to the final competition. There were three rounds: a high dividends scenario, a low dividends, and one in the middle.

We won the high dividends round hands down, unsurprisingly as our entire strategy was designed around this set of parameters in mind. In the low dividends round, we didn’t do as well, but thanks to careful tweaking, we still made a modest amount of money, coming in fifth. In the medium round, we got second place. This was enough to win the mock competition.

Now, let me give you a summary of our competition. Most of the teams increased gradually in net worth, with their score slowly increasing as they slowly accumulated dividends. We were confident that we could play the dividends game, so it didn’t trouble us too much. What was really troubling was a team called “vlad” (I don’t quite remember what their name was, but it ended with vlad). Instead of gradually gaining money a few dollars at a time, “vlad” remained at a constant net worth for a long time, then suddenly gain hundreds of dollars instantly. This meant that their algorithm operated completely differently from ours, and we had absolutely no explanation of what was going on.

It didn’t help that the formula for net worth was complicated and we didn’t fully understand it. Our net worth clearly increased when we did well, but it fluctuated wildly, sometimes dipping by hundreds of points when we made a large transaction, only to bounce back when dividends started rolling in.

The next few hours were fairly unproductive, since we had no more ideas on how to improve our algorithm. Although Andrei had some ideas on strategies for the low dividends game, after pulling an all-nighter I was in no shape to try implementing them.

### The Final Game

It was soon time for the final competition, the cumulation of all our efforts. Having carefully noted down the parameters for the mock competition, we were ready to use this information to get every edge we could for the finals.

Round 1 was high dividends. We played with a highly aggressive set of parameters, dumping our bad stocks for very cheap in pursuit of the dividends regeneration. The early game was contentious, but by the 10 minute mark, we gained a solid lead over the competition and maintained the lead until the end. We won round 1, with “vlad” coming in third place.

Round 2 was low dividends. We deployed the patient strategy, which was less eager to dump anything, holding onto bad stocks until we get a good price for them, since there were little dividends to fight over anyway. We came fifth place, with “vlad” coming in fourth.

Round 3 was medium dividends. We started off uncertain — at the halfway mark we were still in the middle of the pack — but slowly we gained ground, and five minutes before the end, we were in third position. “vlad” was in first place, with a big enough lead that neither we nor the second place team were going to overtake him. But at this point, we knew that from our points in the first round, we only needed second place to beat “vlad” and win the competition — and with 3 minutes left on the clock, we overtook the second place team. We were going to win it!

Then, the whole scoreboard goes black.

It didn’t crash, no, it was the contest organizers’ tactic to increase suspense so the final winners are not known until the winners are announced. We waited anxiously as the final seconds ticked down, the organizers announcing fourth place, third place, UI award. We just needed second place in this round to win, if we get third place in this round, “vlad” beats us by a hair.

And the second place goes to… team /dev/rand. What? We stared in disbelief as we realize we lost to “vlad”.

### Going home

Turns out that in the last 2 minutes of competition, we got overtaken by not one, but two teams. So we actually finished round 3 in fourth place.

Our prize for winning second place? A playstation 4 (worth ~450) and a parrot drone (worth ~100), and most importantly, the satisfaction of winning a finance competition without knowing the first thing about finance. Team “vlad” got two playstations and a drone (well, they could have taken all 3 playstations but they were nice enough to leave us one)

Big thanks to all the organizers and volunteers for keeping everything running smoothly!

If you’re interested, our source code is in a git repo here. It’s 400 lines of hackathon-level-bad python code.

### What about real algo trading?

A natural question to ask is, can we get rich IRL with this algorithm? Answer is clearly no — we essentially gamed the system by greedily grabbing the golden period of dividends, a mechanic designed to encourage people to buy and sell stocks. Of course, in the real world, dividends don’t work like that.

Then other than this mechanic, how else is this competition different from real world algo trading? Unfortunately, I don’t know enough about this topic to answer that question.

Philosophically, I still don’t understand how it’s possible that they basically pull money out of thin air. I mean, a stock trader doesn’t intrinsically create value for society, but they get rich doing it? I don’t know.