Thursday, September 17, 2020

Game of Risk

The last assignment in the CIS194 Haskell class is to simulate dice-throwing battles in the game of Risk. The combat rules, taken from assignment specs, are:

  • There is an attacking army (containing some number of units) and a defending army (containing some number of units). 
  • The attacking player may attack with up to three units at a time. However, they must always leave at least one unit behind. That is, if they only have three total units in their army they may only attack with two, and so on. 
  • The defending player may defend with up to two units (or only one if that is all they have). 
  • To determine the outcome of a single battle, the attacking and defending players each roll one six-sided die for every unit they have attacking or defending. So the attacking player rolls one, two, or three dice, and the defending player rolls one or two dice. 
  • The attacking player sorts their dice rolls in descending order. The defending player does the same.
  • The dice are then matched up in pairs, starting with the highest roll of each player, then the second-highest.
  • For each pair, if the attacking player’s roll is higher, then one of the defending player’s units die. If there is a tie, or the defending player’s roll is higher, then one of the attacking player’s units die.
We assume it is optimal to always attack or defend with as many units as possible.


Given an initial number of attacking and defending units, what is the expected outcome (say, the probability that the attacker wins)? 

It's not obvious that there is a closed-form expression, but we can find programmatically the exact probabilities. 

First, from the game specs, we know that there are only a few primitive battle patterns (A, D), where A is attacked and D is defender: (1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2). For each pattern, we can generate all possible permutations of die values. Each permutation represent a battle, with an outcome ("loss scenario") in which A and D both lose some number of units (which can be zero units, if they win all dice rolls).  

For example, for (A,D) of (1,1), both the attacker and defender role one die each, yielding 36 pairs of possible die rolls: (1,1), (1,2), (1,3) ... (6,5), (6,6). Tallying up the proportion of outcomes: A loses 1 unit 21 / 36; D loses 1 unit 15 / 36.

(Incidentally, the closed-form expression for 1 vs 1 is relatively simple: for each N that the defender rolls, with 1/6 probability each, the attacker loses in N/6 scenarios. For example, if the defender rolls 4, the attacker loses for all die values <= 4. So we can evaluate the expression (1/6) * (sum [1..6]) / 6, which equals 21/36. See these notes for a more detailed explanation.)

Once we have the loss scenarios with proportions for each primitive battle pattern, finding the exact win probability for the attacker is recursive: for any given starting pattern (A, D), lookup the relevant primitive battle pattern. For each loss scenario in which the attacker does not lose the game, update the battlefield with the losses and recursively lookup the next loss scenario, chaining the proportions (probabilities) until the game ends for each branch. 

The win probabilities for the attacker for patterns up to (10, 10) are:



Using Rational numbers in Haskell allows the answers to be expressed exactly. For example for (4,2), the win probability for A is 6610505 % 10077696 (see the rightmost column here). 



Friday, September 11, 2020

Trying TCR

A fellow at the Recurse Center hosted a workshop on TCR (see Kent Beck's post for an introduction to the idea), using these templates

The short of it is: a watcher will automatically run tests upon file save; if any tests fail, it will revert to the prior commit; else, it will make a new commit. 

Despite some initial mental (and practical workflow) resistance, I've warmed very rapidly to the idea. 

By far the biggest change is the revert part of test && commit || revert. Some initial complaints:

  • I made a typo, and it revert to the last commit
  • I just lost a whole block of code
  • I can't just try this simple thing because it keeps failing a test and getting deleted 
The first one maybe retains some legitimacy, but in general, the idea is: 
  • slow down
  • write fragments that you understand
  • write smaller fragments if necessary 
The process also highlights the fact that tests may evolve with the code; as an interface changes, so too do the tests need to change (or TCR won't let you make the commits!).


A few implementation notes for macOS users, building on the Python example from these templates

This gist contains the edits discussed below.

watch.sh
  • Instead of inotifywait, on macOS fswatch can be installed from Homebrew (brew install fswatch)
  • fswatch should be called with the --one-event flag, since it's already called in an infinite loop (otherwise it blocks the next line)
commit.sh
  • By default, TCR will auto-commit with the same message; as one possible alternative, call AppleScript and ask for a commit message via a prompt
    

test.sh
  • The default example has some assertions in the main Python file; modify this to call a testing framework instead (in my case, py.test)


There's a good video here of doing TCR-style testing in Elm. An interesting aspect they mention is squashing a bunch of TCR-generated commits ("what changed") at a later point into more of an explanatory commit ("why the changes").


P.S. for best results with editors, autosave features should be disabled, and autoreload features should be enabled, e.g. in emacs: 

(global-auto-revert-mode t)
(setq auto-save-default nil)



Friday, August 28, 2020

Sudoku Visualization

The Sudoku solver visualization in Elm is done! 

https://tarokuriyama.com/projects/sudoku.php

Writing Elm feels similar to writing Haskell, with a bit of syntax that's more like F# (e.g. the pipeline operators <| and |>). Between the familiar format of Learn You and Elm and the examples in Beginning Elm, it's relatively easy to pick up given a bit of functional programming background.

All of the code for the solver and visualization is on GitHub.








 

Wednesday, August 26, 2020

Elm JSON Decoder

In the vein of functional programming, I've been learning Elm to visualize the progress of a Sudoku solver.

The JSON decoding tutorials are clear, and the JSON.Decode library is well-documented once you know how to read it (in general, the library documentation in Elm seems to be on the terse side). 

Two things were not very clear from initial read of the docs: how to parse lists to tuples, and how to parse strings to custom types. 

Leaning on SO for the below example,  tuples apparently used to be supported as first-class decoders, but in the current version of Json.Decoder, index seems to be the official way to construct decoders for tuples. 

arrayAsTuple2 a b c =

    Decode.index 0 a

        |> Decode.andThen (\ aVal -> Decode.index 1 b

            |> Decode.andThen (\ bVal -> Decode.succeed (aVal, bVal)))            

boardDecoder : Decoder Board

boardDecoder = Decode.list <| arrayAsTuple2 Decode.int Decode.int 

Elm 0.19 only supports up to three-value tuples (somewhat surprising coming from Haskell, though with records and custom data types, the decision to limit unnamed tuples has a certain logic...). So the helper construction isn't too bad.


Parsing strings to custom data types turns out to be more straightforward, though a bit repetitive (especially if you need to later get the string representation back out of custom data type -- there is no deriving Show in Elm). First decode the string, then chain a pattern match using andThen:

data Transform = Rows | Cols | Boxes

transformDecoder : Decoder Transform
transformDecoder =
    Decode.string
    |> Decode.andThen
       (\ s ->
            case s of
                "Rows" -> Decode.succeed Rows
                "Cols" -> Decode.succeed Cols
                "Boxes" -> Decode.succeed Boxes
                _ -> Decode.fail <| "Unknown transformation.")

Saturday, August 22, 2020

Recurse Center

 I've wanted to attend the Recurse Center (formerly known as Hacker School) or a while, and have finally found the time to do so. It's very consistent with the free education path (it's free!), and so far has been a great source of exposure to new topics and ideas, as well as of inspiration to work on different classes of problems. They've put a lot of effort into fostering a healthy, supportive community, and the nature of the program is such that self-motivated, inquisitive learners more or less self-select themeselves into attendance. Some topics I was exposed to or participated in so far -- audio programming, gaming, lower-level / systems programming languages (eg Zig), functional programming (eg Haskell, Elm), procedural generetion, Web Assembly compilation... There are folks of all walks of life, skills, and interests in the community, making it something that I expect will continue to yield rich experiences for the remainder of the three-month batch. 

Wednesday, July 29, 2020

Stochastic Writing

Bang Bang Con 2020 features a number of pithy, unexpected talks.

Following Andrew Yoon's talk about his work on BML (a "stochastic markup language"), here's a short poem that changes under the reader's eyes... 





Sunday, June 21, 2020

What goes on in a computer in a millisecond?

In a 2001 Q&A session (around 1h30m) Knuth is asked about unsolved challenges in the field of compute science, à la Hilbert's problems. Knuth's responds by referring to a prompt he posed to the World Computer Congress in 1989, in a keynote titled "Theory and Practice, IV" (Arxiv has a version, though this version appears to have the slides as well):

"Make a thorough analysis of everything your computer does during one second of computation"

Does the prompt require adjustment for context, some thirty years later? Knuth noted the practical challenges associated with monitoring and interpreting the computer's activities, whose difficulty would appear to remain on the same order of magnitude today. On the other hand, Knuth associated 1 second with approximately 250,000 instructions. Given that CPU clock speeds have jumped from standard units in MHz to GHz (wikipedia notes clock speeds in 1989 were in the 16-35 MHz range, in line with Knuth's approximation), maybe today's equivalent of the question is:

"What goes on in a computer in a millisecond"?

In the same keynote, Knuth provides a set of guiding questions, including the correctness of the observed activities and their correspondence to theory -- that is, whether they reflect existing theory, and whether new theory would yield practical improvements.

Despite the daunting scope of the problem, the fundamental idea of mapping theory to some atomic observations of computing practice seems widely applicable and useful for students and practitioners alike.