This is the second post on the Gagallium participation to the ICFP programming context last year -- 2014. See the first post for the introduction and a link to the source code; this post details the design of the ghost code.

The ghost we submitted is the last of a series of increasingly sophisticated designs, each of which is an incremental enhancement of the previous one. Given the short amount of time available, we did not do a rewrite from scratch, no matter how much we wanted to. This method let us make a pretty good ghost with a small amount of work (about 3 man-days) but it's not without pitfalls: because serious testing was not an option, some huge bugs stayed undiscovered for quite some time.

Like almost everyone in the competition, we wrote the ghosts directly in assembly code, with only a small tool to make the GHC assembly language more palatable: a macro-processor that implements symbolic labels (for jumps) and names (for arguments). The first feature allowed us to retain our sanity when inserting and deleting instructions in the code, while the second one let us use names for our variables stored in memory.

Stupid

Our first version is called stupid. It starts with 6 steps up, 6 steps right, 6 steps down, 6 steps left (in an attempt to get out of the initial box) then it always tries to move toward lambda-man. It restarts this algorithm every time it gets eaten.

To implement it, we had to solve two interesting problems: 1. Detect when the ghost gets eaten or a new round has started. 2. Find out the direction closest to lambda-man's position.

Detecting restarts

Detecting when the ghost gets eaten or a new round is started is almost easy: just detect any sudden jump in position that is inconsistent with normal movement. This is done by getting the current position in (a,b) and having the previous position stored in ([1],[2]). We subtract [1] from a and [2] from b, which yields two numbers in {-1, 0, 1}. Adding them together yields either -1 or 1 because one of them must be zero and the other nonzero (the ghost has moved either vertically or horizontally). Adding one yields 0 or 2. Any other value indicates that the ghost's position has jumped suddenly. An unsigned comparison with 3 is enough to check this condition. This algorithm is not exact: it will fail whenever the ghost gets eaten at a position very close to its starting point. We decided we could live with this imprecision.

Computing the direction

Computing the direction of lambda-man is a bit harder as it involves trigonometry. Since GHC lacks instructions to compute sines, cosines, tangents, and the like, we do it with addition, subtraction, and exclusive-or instead.

We start by defining an angle alpha, encoded in units of 45 degrees. Alpha will be the middle of the quadrant where lambda-man is found (1, 3, 5, or 7).

The origin and orientation for alpha are the ones used by the INT 0 instruction: 0 is up, 1 is upper-right, 2 is right, etc.

+-----------> X
|   7  0  1
|    \ | /
|  6 - . - 2
|    / | \
|   5  4  3
Y

Consider dx = lambdaman.x - ghost.x and dy = lambdaman.y - ghost.y

The correct answer depends on the signs of dx and dy (to determine in which quadrant lambda-man is) and the comparison of |dx| and |dy| (to determine which octant in this quadrant). The following table enumerates all the cases:

dx sign | dy sign | |dx|>|dy| | direction
   +    |    +    |     T     |  2 = 3-1
   +    |    +    |     F     |  4 = 3+1
   +    |    -    |     T     |  2 = 1+1
   +    |    -    |     F     |  0 = 1-1
   -    |    +    |     T     |  6 = 5+1
   -    |    +    |     F     |  4 = 5-1
   -    |    -    |     T     |  6 = 7-1
   -    |    -    |     F     |  0 = 7+1 (modulo 8)

The result can be decomposed in two values: alpha (1, 3, 5 or 7) and phi (+1 or -1). alpha can be computed by:

  alpha = 1 + (6 if dx is negative) XOR (2 if dy is positive)

and phi is simply a XOR of all three inputs: it is the product of the three signs of the input, taking - for T and + for F. The code simply computes alpha, phi, and the absolute values simultaneously, by testing the signs of dx and dy only once. Then just dividing the result by two yields the direction of lambda-man.

We checked by hand that (as expected because the functions are continuous) we get correct results when lambda-man lies exactly in one of our 8 directions (i.e. when a sign is 0 or |dx| is equal to |dy|).

In retrospect, we could probably get shorter code by simply implementing:

  if |dx| > |dy| then
    if dx positive then 2 else 6
  else
    if dy positive then 4 else 0

Panic mode

To complete the ghost's code, it tests the "fright mode" flag returned by interrupt 6, and reverses the direction if it's in fright mode, thus fleeing away from lambda-man instead of chasing him.

Idiot

Idiot is a very small variation on the Stupid ghost: it just tries to avoid banging its head on walls. If there is a wall in the direction chosen by Stupid, Idiot will try the other side of the quadrant where lambda-man lies.

This new direction is easy to compute from the outputs of the Stupid code: the primary direction (the direction chosen by Stupid) is alpha + phi and the secondary direction (the other side of the quadrant) is alpha - phi.

To have a look at the square that lies in a given direction, we used our first subroutine. Since the GHC processor doesn't have a stack, we had to manage the return address by hand, storing it in a global variable.

Smart

The Smart ghost uses a trick from the ghosts in the real PacMan game: alternate between scatter and chase. Each ghost maintains a "clock" by counting its execution cycles, and in 32 cycles out of 128, the ghost goes into "scatter" mode. In this mode, it sets a target point that depends on the ghost number (a different corner of the map for each ghost). Otherwise, it is in "chase" mode, where the target point is the coordinates of lambda-man.

The goal of this system is to avoid having all the ghosts chasing behind lambda-man without ever blocking his path. By scattering from time to time, the ghosts have a better chance of surrounding lambda-man at some point.

Note that the scatter modes of the ghosts start synchronized, but their clocks run at different speeds, so they will become unsynchronized as the game progresses. In addition, there is no way to know the size of the map, so we used an approximate notion of "corner of the map".

Finally, when Smart is in panic mode (i.e. when lambda-man has eaten a power pill and is chasing the ghost) the ghost will run the scatter algorithm rather than just fleeing in the direction opposite to lambda-man. This is to disperse the ghosts, thus maximising the probability that some of them will escape lambda-man.

Brilliant

Brilliant is the ghost that adds the most important feature: before going into a given direction, it follows the corresponding corridor until it finds one of three things:

  1. lambda-man
  2. an intersection or a fork
  3. a dead end

In cases 1 and 2, the ghost goes into that direction, while in case 3 it will chose another direction: there's no point in going into a dead-end unless lambda-man is there.

Unlike its predecessors, Brilliant will not only test the principal direction to avoid banging its head on walls, it also tests it to make sure it is not a U-turn (the rules of the game forbid U-turns for ghosts, except at the bottom of a dead end). This is done at lines 146 to 178: if there is a wall in the primary direction, or the primary direction is a U-turn, or it's a dead-end without lambda-man, then take the secondary direction.

At least that's the theory, because there is a nasty bug: after following the corridor, Brilliant will just ignore this information and take the principal direction anyway... not really brilliant after all!

To follow a corridor from a starting point and a direction, we first make a step in that direction (lines 193 to 196) then look in the three allowed direction (excluding a U-turn) and count the number of non-wall tiles in %good_dir_num, recording one of them in %good_dir_val.

Line 220 has a nasty bug: it should jump to lwalling1 rather than lwalling2. As a result, the second direction (90 degrees clockwise from the current direction) is never tested and the whole corridor-following code seldom returns a correct result.

Genius

Genius is a small but important variation on the Brilliant code: when following the corridor, check for the presence of other ghosts in front of the current ghost. If there is already a ghost going down that corridor, avoid entering it.

This means that the ghosts will naturally disperse and try alternate routes to lambda-man, thus cutting of more of his escape paths.

There is a difficulty here: the ghost doesn't know how many other ghosts there are, so it cannot query their position. We solved it by only testing for lower-numbered ghosts, that is ghost 0 ignores all other ghosts, ghost 1 only tests for presence of ghost 0, etc. It works pretty well because lower-numbered ghosts walk faster, so they will tend to be in front anyway.

Genius2

Since Genius tends to scatter naturally around lambda-man, it was natural to try a version that didn't alternate between scatter and chase phases, but did chase all the time (except when in panic mode, obviously). This is what Genius2 does; otherwise it is identical to Genius.