Older blog entries for Bram (starting at number 19)

Axiomatic Bases

Raph quoted me as saying that ZF is a hack. I probably should explain.

PA seems logically compelling to me, even preexisting. I know what the number one is, what a successor is, and I absolutely believe in the principle of induction. ZF, on the other hand, has no obvious intuitive basis. What is a set? Is it a bag? A list? A data structure? A function? The inability of sets to contain themselves would seem to imply bag, but the ability to keep the same set in multiple other ones at once would seem to imply list. All around, ZF feels like something which was logically compelling but then had awkward restrictions placed on it to get rid of some paradoxes.

Perhaps if it were presented in some other way, using different names and metaphors, I wouldn't find ZF so awkward. I'm convinced of its practical utility for doing mathematics from the sheer amount of fiddling with has been done with it, but I'd still like for my intuition to naturally accept it as well.


Thanks, dmerrill! I think my work on BitTorrent is a reasonable qualification for master certification. I've spent over a year working on it, and it's now getting over a hundred downloads a day, as you can see on the statistics page.

26 Aug 2002 (updated 26 Aug 2002 at 20:13 UTC) »
Plugging a Hole

The proof of the non-computability of the halting of turing machines presented in most undergraduate programs contains a huge gaping hole. Fortunately it's not hard to fix.

The Broken Proof

Let us say that we have a turing machine which accepts an encoding of another turing machine as input and always prints out a 0 before its last move if the input halts or a 1 before its last move if the input doesn't halt, and never spends an infinite amount of time thinking about it.

We can then construct a turinng machine which calculates what whether its input would halt, and does the opposite. If we give that machine as input to itself, then we have a contradiction.

The problem here is that there's a distinction between machines which take input and machines which don't take input. The machine which does the opposite takes input, when we fed it to itself the emulated one is given no input, hence it's operating on something else entirely, and there is no contradiction.

The Fixed Proof

Let us say that we have a turing machine which accepts an encoding of another turing machine as input and always prints out a 0 before its last move if the input halts or a 1 before its last move if the input doesn't halt, and never spends an infinite amount of time thinking about it.

We can then construct a turing machine which spits out an encoding of itself, calculates whether that machine will halt, and then does the opposite. The spitting out is based on the standard quine trick - the program contains a blueprint of itself, with a special notation where the blueprint goes saying 'blueprint goes here', it then follows the blueprint exactly and spits out a copy of the blueprint where the notation goes.

Clearly, this machine leads to a contradiction.

Why there is such an obvious and easy to fix error as part of standard basic CS curriculum is beyond me.

No Mix and Matching

While I'm complaining about common errors, I'd like to point out something about log(n) multipliers in runtimes. There are basically two ways of calculating the runtime of an algorithm - the number gates required to construct it using a directed circuit, and the number of operations it requires on a realistic machine with parallel memory.

If you calculate runtime in the first way, you're perfectly justified in saying that adding two numbers on the scale of n takes log(n) time, but since many algorithms undergo a quadratic blow-up in size in this model due to the lack of parallel memory, it isn't applicable to many problems.

If you calculate runtime in the second way, you have no justification for saying that adding two numbers on the scale of n takes log(n) time unless you also take into account that simple memory retrieval technically takes log(n) time, which nobody does.

When calculating runtime, don't mix and match models - if you allow for constant-time memory retrieval, allow for constant-time addition of small numbers.

24 Aug 2002 (updated 26 Aug 2002 at 17:57 UTC) »
wardv: I used to play corewars all the time, in an integrated development environment on the Commodore 64 which my dad wrote in his spare time. Notably, he was a journalist and not a programmer back then.

There was one tournament in which I entered a top contender, Powerbomb, which won almost all of its battles before the finals. Unfortunately, Only five programs were put into the final round, including all the ones which Powerbomb didn't do all that well against. Ah well, I should probably be happy just to have gotten two programs into the finals. I was only twelve at the time.

Mathematical Foundations

I think this post claims an axiomatic system has been discovered which essentially declares its own consistency but manages to avoid the cheap trick of diagonalization.

Update: I'm told I read that all wrong. Ah well.

Intermediate Automata

I believe some of these papers demonstrate that there are (very artificial) cellular automata which exhibit nontrivial behavior but aren't universal, contrary to Wolfram's thesis. I'm increasingly coming to believe that some of the simple rules, especially 18 and 22, are of intermediate degree.

Second Best

I figured out a coherent strategy for Second Best - all players agree on what the two most common words will be, and each player picks one of them at random.

Tweaking Spam

I've been thinking about the spam filtering code I gave earlier. It can be improved a lot.

For starters, multiplying the number of nonspam appearances by two is kind of a hack. It's almost exactly equivalent to subtracting .7 from each token's value. A much more robust approach is to increase the spam threshold from 2.6 to 5.

Also, max value of 4.6 (equivalent to .99 bayesian) seems like a bit much, since just one or two spam words could easily label an otherwise neutral message as spam. Reducing it to 3 (bayesian about .95) seems much more reasonable. Someone whose entry has scrolled off advogato recentlog mentioned that a single token, '2002' threw off his filtering quite a bit.

Several other subtle improvements and code cleanups are possible. I've included them all in the following code -

from math import log
from re import findall

class spamtrap: def __init__(self): self.good = {} self.bad = {}

def add_spam(self, message): for t in _maketokens(message): self.bad[t] = self.bad.get(t, 0) + 1

def add_nonspam(self, message): for t in _maketokens(message): self.good[t] = self.good.get(t, 0) + 1 def is_spam(self, message): ss = [] for token in _maketokens(message): ss.append(min(3, max(-3, log(self.bad.get(token, 0) + 1) - log(self.good.get(token, 0) + 1)))) sum = 0 if len(ss) > 16: ss.sort() for v in ss[:8] + ss[-8:]: sum += v else: for v in ss: sum += v return sum > 5

def _maketokens(message): ts = {} for t in findall("[a-zA-Z0-9'$]+", message): ts[t.lower()] = 1 return ts.keys()

Sorry again about whitespace mangling which interferes with cut'n'paste - it's advogato's doing.

Hot Potato

In the game Hot Potato, each player starts with 100 points. One unlucky player, chosen at random, is initially given the hot potato. On each turn, the player with the hot potato decides which player gets it next, and the recepient's score is decremented by one. The hot potato may not be passed to a player with a score of zero. The winner is the last player with a positive score.

I cannot fathom any coherent strategy for Hot Potato. How to approach it formally is a complete mystery to me.

Second Best

In the game Second Best, each player writes down a word and then all words are revealed. All players who wrote down the second most commonly occuring word are winners, everybody else are losers. In the case of a tie for second, everybody loses.

I can't make heads or tails of the strategy of Second Best either.

20 Aug 2002 (updated 20 Aug 2002 at 22:30 UTC) »
fxn: Using a persistent connection won't help with throughput much unless you implement pipelining.
barryb: Those numbers were derived by plugging in Raph's formula -

  • log(.01) - log(1 - .01) = -4.6
  • log(.99) - log(1 - .99) = 4.6
  • log(.2) - log(1 - .2) = -1.4
  • log(.9) - log(1 - .9) = 2.2

The formulas for probabilities were likewise converted -

log(a / (a + b)) - log(1 - (a / (a + b))) = log(a) - log(b)

Note how much simpler the formula on the right is. I think it's best to think of this heuristic as based on scores, with bayesian analysis being an intuitive motivation, rather than a rigorous basis.

ifile is a noteworthy related project.


Since there's been discussion on advogato diaries lately about which projects are worth donatiing to, I'd like to point out that I'm accepting donations for my work on BitTorrent, a project I've been working on full-time for over a year, with a mature, deployed piece of code to show for it.

Retroactive Spam

I realized that tiebreak behavior was unspecified in the spam filtering code I gave yesterday. The way I implemented it happened to work out that if there were 15 max probability spam and 15 max probability not spam, it assumed spam. A much more robust approach is, rather than summing up the fifteen scores whose absolute values are the highest, sum up the eight largest positive and eight most negative ones. I've now retroactively changed by previous diary entry to work that way. You can do that in a weblog.

17 Aug 2002 (updated 19 Aug 2002 at 02:09 UTC) »
Translating into English

Paul Graham has some interesting ideas about how to filter spam. Unfortunately he gives code samples in some weird language, so I've translated into Python and fleshed them out a bit.

This code ignores duplicates of a single token in a message, since they resulted in some rather ugly hacks. It also implements Raph's suggestion for using scores. Also, rather than summing the fifteen scores whose absolute values are highest, it sums the eight largest positive and eight most negative, which is much more robust behavior.

This code has no support for serialization and doesn't strip binary attachments, but other than that should work well.

from math import log
from re import findall

class spamtrap: def __init__(self): self.good = {} self.bad = {} self.scores = {}

def _recompute(self, token): g = 2 * self.good.get(token, 0) b = self.bad.get(token, 0) if g + b >= 5: self.scores[token] = min(4.6, max(-4.6, log(b + .00001) - log(g + .00001)))

def add_spam(self, message): for t in _maketokens(message): self.bad.setdefault(t, 0) self.bad[t] += 1 self._recompute(t) def add_nonspam(self, message): for t in _maketokens(message): self.good.setdefault(t, 0) self.good[t] += 1 self._recompute(t) def is_spam(self, message): ss = [self.scores.get(t, -1.4) for t in _maketokens(message)] ssp = [i for i in ss if i > 0] ssn = [i for i in ss if i < 0] ssp.sort() ssn.sort() sum = 0 for v in ssp[-8:] + ssn[:8]: sum += v return sum > 2.2

def _maketokens(message): ts = {} for t in findall("[a-zA-Z0-9'$]+", message): ts[t.lower()] = 1 return ts.keys()

Sorry if there are indentation problems on cut and paste - advogato inserts <p> tags even between <pre> tags.

lukeg: Going random when you've got a statistically insurmountable lead might help if you're optimizing for match wins over total score, but in practice it's very rare for an opponent you were trouncing to suddenly snap out of it and start beating you.

Your champion would be soundly trounced by the more sophisticated strategies I described. Since even sword and shield is fairly simple to implement and plenty effecient enough to run in idel, just a bit of coding could bring your Roshambo competition from amateur to world-class.

thraxil: Roshambo and Prisoner's Dilemma have distinctly different flavors because Roshambo is a zero-sum game, in which no cooperative strategies are possible. Tit-for-tat is now getting practical application in the choking algorithms for my project BitTorrent, which are very analagous.

isenguard: That paper is interesting, but I'm very leery of empirical measurements of constraint satisfaction heuristics which don't set the actual record. Experimenting with 10 when the record is 21 is pretty bad (it's even more now). There is only one fair measure of time, and that's minutes and seconds.

Knight Moves

If we mark the minimum number of moves it takes for a knight to get from the origin to each square on an infinite chessboard, we find there are four local maxima - exactly two squares away from the origin along each of the four diagonals. Generalizing to knights which move x in one direction then y at a right angle, with x and y relatively prime (the standard knight has x = 2 and y = 1) it appears that there are always four local maxima, and they're always on the diagonals. I don't know why this is.


Today I'll be writing about computer strategies for the classic game Roshambo, also known as paper scissors stone.

But ... but ... There's no strategy in Roshambo!

While it is true that playing random can reliably do dead average, in any tournament everyone will be trying to win, hence everyone will have bias, and strategies which do better than average against a wide range of opponents will rise to the top.

I've spent far more time than is reasonable reading about the computer Roshambo tournaments, and will present a distilled explanation of the strategies here.


Henny exemplifies the simplest strategy which works well in practice. It picks a random move which the opponent played previously, and plays the response which beats it. If the opponent has any bias toward playing the same move over time, Henny will win. Henny is also very defensive - optimal play against it will only get an edge very slowly over time, and even that might get swallowed by noise.

Markov Chains

Another winning in practice strategy is to use markov chains - look at the last n moves your opponent made, and predict that they'll continue playing as they have most frequently in the past. Considerable variation is possible in length of chains, which brings us to...

Iocaine Powder

The competitor Iocaine Powder magically combines the best qualities of Henny and markov chaining by looking for the longest match between the moves your opponent just played and any sequence they've played in the past, and assuming they'll play the same next move again. If the opponent has a bias based on the last n moves for an n, this strategy will essentially pick a random time they played the same last n moves they just did, which is the defensive strategy used by Henny.

Ironically, this algorithm is completely deterministic.

Note that it's slightly better when there are equal length matches to go with the first matching sequence rather than the last matching sequence, to perform well when an opponent plays a string of paper, then a scissors, then a string of paper again.

It's possible to implement this technique very efficiently using trees, which strangely Iocaine Powder doesn't do (MemBot does, though).

But this strategy still has a major weakness, and Iocaine Powder has another trick up its sleeve...

Sicilian Reasoning

A braindead, fixed sequence entry named DeBruijnize nearly cleaned up in the first tournament. DeBruijnize sequences are biased against sequences they've done in the past, rather than in favor of them. A meta-strategy called sicilian reasoning beats it nicely. Compute how well you would have done to date predicting the opponents move straightforwardly, then always shifting it up one (paper->scissors, scissors->rock, rock->paper), then down one, then the same three predicting instead your own side. Play whichever one would have the highest score so far.

Sicilian reasoning cleanly defeats not only DeBruijnize, but all manner of cheap variants on it.

Sword and Shield

This is my own idea, which is untested, but seems reasonable.

Generate predictions via whatever method (pre-sicilian reasoning) for both your move and the opponent's move, then play a move based on the following table. The top row is your predicted move, the left row is the opponent's (t = stone) -

        p  s  t 
   p |  s  p  s
   s |  t  t  s
   t |  t  p  p

This algorithm plays defensively by predicting both sides simultaneously, and hence may beat any opponent which bases each specific move on a prediction for only one side. It has nine sicilian reasoning variants, found by offseting the prediction for either side one of the three possible ways.

Wimping Out

A tempting strategy is to make your bot 'wimp out' and start playing randomly if it isn't doing well. Tournaments play two programs against each other many times with no persistent information between runs to keep this strategy from being effective.

Optimizing for Score

The winning entry, Greenberg, looks at historical performance if its history buffer was various sizes, and has exponential falloff of their performance. This makes it score very well against weaker opponents, but makes it more vulnerable to better opponents. Score, match wins, and worst-case lossage are all subtly different things to optimize for.

The Future

Sadly, there appear to be no plans for future computer Roshambo tournaments. It would be interesting to see how strategies continue to evolve.

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