Mathias Brandewinder on .NET, F#, VSTO and Excel development, and quantitative analysis / machine learning.
12. May 2015 13:50

As much as we people who write code like to talk about code, the biggest challenge in a software project is not code. A project rarely fails because of technology – it usually fails because of miscommunications: the code that is delivered solves a problem (sometimes), but not the right one. One of the reasons we often deliver the wrong solution is that coding involves translating the world of the original problem into a different language. Translating one way is hard enough as it is, but then, rarely are users comfortable with reading and interpreting code – and as a result, confirming whether the code “does the right thing” is hard, and errors go un-noticed.

This is why the idea of Ubiquitous Language, coined by Eric Evans in his Domain Driven Design book, always appealed to me. The further apart the languages of the domain expert and the code are, the more likely it is that something will be lost in translation.

However, achieving this perfect situation, with “a language structured around the domain model and used by all team members to connect all the activities of the team with the software” [source], is hard. I have tried this in the past, mainly through tests. My idea at the time was that tests, especially BDD style, could perhaps provide domain experts with scenarios similar enough to their worldview that they could serve as a basis for an active dialogue. The experience wasn’t particularly successful: it helped some, but in the end, I never got to the point where tests would become a shared, common ground (which doesn’t mean it’s not possible – I just didn’t manage to do it).

Fast forward a bit to today – I just completed a project, and it’s the closest I have ever been to seeing Ubiquitous Language in action. It was one of the most satisfying experiences I had, and F# had a lot to do with why it worked.

The project involved some pretty complex modeling, and only two people – me and the client. The client is definitely a domain expert, and on the very high end of the “computer power user” spectrum: he is very comfortable with SQL, doesn’t write software applications, but has a license to Visual Studio and is not afraid of code.

The fact that F# worked well for me isn’t a surprise – I am the developer in that equation, and I love it, for all the usual technical reasons. It just makes my life writing code easier. The part that was interesting here is that F# worked well for the client, too, and became our basis for communication.

What ended up happening was the following: I created a GitHub private repository, and started coding in a script file, fleshing out a domain model with small, runnable pieces of code illustrating what it was doing. We would have regular Skype meetings, with a screen share so that I could walk him through the code in Visual Studio, and explain the changes I made - and we would discuss. Soon after, he started to run the code himself, and even making small changes here and there, not necessarily the most complicated bits, but more domain-specific parts, such as adjusting parameters and seeing how the results would differ. And soon, I began receiving emails containing specific scenarios he had experimented with, using actual production data, and pointing at possible flaws in my approach, or questions that required clarifications.

So how did F# make a difference? I think it’s a combination of at least 2 things: succinctness, and static typing + scripts. Succinctness, because you can define a domain with very little code, without loosing expressiveness. As a result, the core entities of the domain end up taking a couple of lines at the top of a single file, and it’s easy to get a full picture, without having to navigate around between files and folders, and keep information in your head. As an illustration, here is a snippet of code from the project:

type Window = { Early:DateTime; Target:DateTime; Late:DateTime }

type Trip = {
ID:TripID
Origin:Location
Destination:Location
Pickup:Window
Dropoff:Window
Dwell:TimeSpan }

type Action =
| Pickup of Trip
| Dropoff of Trip
| CompleteRoute of Location

This is concise, and pretty straightforward – no functional programming guru credentials needed. This is readable code, which we can talk about without getting bogged down in extraneous details.

The second ingredient is static typing + scripts. What this creates is a safe environment for experimentation.  You can just change a couple of lines here and there, run the code, and see what happens. And when you break something, the compiler immediately barks at you – just undo or fix it. Give someone a running script, and they can start playing with it, and exploring ideas.

In over 10 years writing code professionally, I never had such a collaborative, fruitful, and productive interaction cycle with a client. Never. This was the best of both worlds – I could focus on the code and the algorithms, and he could immediately use it, try it out, and send me invaluable feedback, based on his domain knowledge. No noise, no UML diagrams, no slides, no ceremony – just write code, and directly communicate around it, making sure nothing was amiss. Which triggered this happy tweet a few weeks back:

We were looking at the code together, and my client spotted a domain modeling mistake, right there. This is priceless.

As a side-note, another thing that is priceless is “F# for Fun and Profit”. Scott Wlaschin has been doing an incredible work with this website. It’s literally a gold mine, and I picked up a lot of ideas there. If you haven’t visited it yet, you probably should.

3. May 2015 16:34

I got curious the other day about how to measure the F# community growth, and thought it could be interesting to take a look at this through StackOverflow. As it turns out, it’s not too hard to get some data, because StackExchange exposes a nice API, which allows you to make all sorts of queries and get a JSON response back.

As a starting point, I figured I would just try to get the number of questions asked per month. The API allows you to retrieve questions on any site, by tag, between arbitrary dates. Responses are paged: you can get up to 100 items per page, and keep asking for next pages until there is nothing left to receive. That sounds like a perfect job for the FSharp.Data JSON Type Provider.

First things first, we create a type, Questions, by pointing the JSON Type Provider to a url that returns questions; based on the structure of the JSON document it receives, the Type Provider creates a type, which we will then be able to use to make queries:

#I"../packages"
#r @"FSharp.Data.2.2.0\lib\net40\FSharp.Data.dll"
open FSharp.Data
open System

[<Literal>]
let sampleUrl = "https://api.stackexchange.com/2.2/questions?site=stackoverflow"
type Questions = JsonProvider<sampleUrl>

Next, we’ll need to grab all the questions tagged F# between 2 given dates. As an example, the following would return the second page (questions 101 to 200) from all F# questions asked between January 1, 2014 and January 31, 2015:

https://api.stackexchange.com/2.2/questions?page=2&pagesize=100&fromdate=1420070400&todate=1422662400&tagged=F%23&site=stackoverflow

There are a couple of quirks here. First, the dates are in UNIX standard, that is, the number of seconds elapsed from January 1, 1970. Then, we need to keep pulling pages, until the response indicates that there are no more questions to receive, which is indicated by the HasMore property. That’s not too hard: let’s create a couple of functions, first to convert a .NET date to a UNIX date, and then to build up a proper query, appending the page and dates we are interested in to our base query – and finally, let’s build a request that recursively calls the API and appends results, until there is nothing left:

let fsharpQuery = "https://api.stackexchange.com/2.2/questions?site=stackoverflow&tagged=F%23&pagesize=100"

let unixEpoch = DateTime(1970,1,1)
let unixTime (date:DateTime) =
(date - unixEpoch).TotalSeconds |> int64

let page (page:int) (query:string) =
sprintf "%s&page=%i" query page
let between (from:DateTime) (to:DateTime) (query:string) =
sprintf "%s&&fromdate=%i&todate=%i" query (unixTime from) (unixTime to)

let questionsBetween (from:DateTime) (to:DateTime) =
let baseQuery = fsharpQuery |> between from to
let rec pull results p =
let nextPage = Questions.Load (baseQuery |> page p)
let results = results |> Array.append nextPage.Items
if (nextPage.HasMore)
then pull results (p+1)
else results
pull Array.empty 1

And we are pretty much done. At that point, we can for instance ask for all the questions asked in January 2015, and check what percentage were answered:

let january2015 = questionsBetween (DateTime(2015,1,1)) (DateTime(2015,1,31))

january2015
|> Seq.averageBy (fun x -> if x.IsAnswered then 1. else 0.)
|> printfn "Average answer rate: %.3f"

… which produces a fairly solid 78%.

If you play a bit more with this, and perhaps try to pull down more data, you might experience (as I did) the Big StackExchange BanHammer. As it turns out, the API has usage limits (which is totally fair). In particular, if you ask for too much data, too fast, you will get banned from making requests, for a dozen hours or so.

This is not pleasant. However, in their great kindness, the API designers have provided a way to avoid it. When you are making too many requests, the response you receive will include a field named “backoff”, which indicates for how many seconds you should back off until you make your next call.

This got me stumped for a bit, because that field doesn’t show up by default on the response – only when you are hitting the limit. As a result, I wasn’t sure how to pass that information to the JSON Type Provider, until Max Malook helped me out (thanks so much, Max!). The trick here is to supply not one sample response to the type provider, but a list of samples, in that case, one without the backoff field, and one with it.

I carved out an artisanal, hand-crafted sample for the occasion, along these lines:

[<Literal>]
let sample = """
[{"items":[
{"tags":["f#","units-of-measurement"],//SNIPPED FOR BREVITY}],
"has_more":false,
"quota_max":300,
"quota_remaining":294},
{"items":[
{"tags":["f#","units-of-measurement"],//SNIPPED FOR BREVITY}],
"has_more":false,
"quota_max":300,
"quota_remaining":294,
"backoff":10}]"""

type Questions = JsonProvider<sample,SampleIsList=true>

… and everything is back in order – we can now modify the recursive request, causing it to sleep for a bit when it encounters a backoff. Not the cleanest solution ever, but hey, I just want to get data here:

let questionsBetween (from:DateTime) (to:DateTime) =
let baseQuery = fsharpQuery |> between from to
let rec pull results p =
let nextPage = Questions.Load (baseQuery |> page p)
let results = results |> Array.append nextPage.Items
if (nextPage.HasMore)
then
match nextPage.Backoff with
| Some(seconds) -> System.Threading.Thread.Sleep (1000*seconds + 1000)
| None -> ignore ()
pull results (p+1)
else results
pull Array.empty 1

So what were the results? I decided, quite arbitrarily, to count questions month by month since January 2010. Here is how the results looks like:

Clearly, the trend is up – it doesn’t take an advanced degree in statistics to see that. It’s interesting also to see the slump around 2012-2013; I can see a similar pattern in the Meetup registration numbers in San Francisco. My sense is that after a spike in interest in 2010, when F# launched with Visual Studio, there hasn’t been much marketing push for the language, and interest eroded a bit, until serious community-driven efforts took place. However, I don’t really have data to back that up – this is speculation.

How this correlates to overall F# adoption is another question: while I think this curves indicates growth, the number of questions on StackOverflow is clearly a very indirect measurement of how many people actually use it, and StackOverflow itself is a distorted sample of the overall population. Would be interesting to take a similar look at GitHub, perhaps…

22. March 2015 17:27

I will admit it, I got a bit upset by James McCaffrey’s column in MSDN magazine this month, “Gradient Descent Training Using C#”. While the algorithm explanations are quite good, I was disappointed by the C# sample code, and kept thinking to myself “why oh why isn’t this written in F#”. This is by no means intended as a criticism of C#; it’s a great language, but some problems are just better suited for different languages, and in this case, I couldn’t fathom why F# wasn’t used.

Long story short, I just couldn’t let it go, and thought it would be interesting to take that C# code, and do a commented rewrite in F#. I won’t even go into why the code does what it does – the article explains it quite well – but will instead purely focus on the implementation, and will try to keep it reasonably close to the original, at the expense of some additional nifty things that could be done.

The general outline of the code follows two parts:

• Create a synthetic dataset, creating random input examples, and computing the expected result using a known function,
• Use gradient descent to learn the model parameters, and compare them to the true value to check whether the method is working.

You can download the original C# code here. Today we’ll focus only on the first part, which is mainly contained in two methods, MakeAllData and MakeTrainTest:

static double[][] MakeAllData(int numFeatures, int numRows, int seed)
{
Random rnd = new Random(seed);
double[] weights = new double[numFeatures + 1]; // inc. b0
for (int i = 0; i < weights.Length; ++i)
weights[i] = 20.0 * rnd.NextDouble() - 10.0; // [-10.0 to +10.0]

double[][] result = new double[numRows][]; // allocate matrix
for (int i = 0; i < numRows; ++i)
result[i] = new double[numFeatures + 1]; // Y in last column

for (int i = 0; i < numRows; ++i) // for each row
{
double z = weights[0]; // the b0
for (int j = 0; j < numFeatures; ++j) // each feature / column except last
{
double x = 20.0 * rnd.NextDouble() - 10.0; // random X in [10.0, +10.0]
result[i][j] = x; // store x
double wx = x * weights[j + 1]; // weight * x
z += wx; // accumulate to get Y
}
double y = 1.0 / (1.0 + Math.Exp(-z));
if (y > 0.55)  // slight bias towards 0
result[i][numFeatures] = 1.0; // store y in last column
else
result[i][numFeatures] = 0.0;
}
Console.WriteLine("Data generation weights:");
ShowVector(weights, 4, true);

return result;
}

MakeAllData takes a number of features and rows, and a seed for the random number generator so that we can replicate the same dataset repeatedly. The dataset is represented as an array of array of doubles. The first columns, from 0 to numFeatures – 1, contain random numbers between –10 and 10. The last column contains a 0 or a 1. What we are after here is a classification model: each row can take two states (1 or 0), and we are trying to predict them from observing the features. In our case, that value is computed using a logistic model: we have a set of weights (which we also generate randomly), corresponding to each feature, and the output is

logistic [ x1; x2; … xn ] = 1.0 / (1.0 + exp ( - (w0 * 1.0 + w1 * x1 + w2 * x2 + … + wn * xn))

Note that w0 plays the role of a constant term in the equation, and is multiplied by 1.0 all the time. This is adding some complications to a code where indices are already flying left and right, because now elements in the weights array are mis-aligned by one element with the elements in the features array. Personally, I also don’t like adding another column to contain the predicted value, because that’s another implicit piece of information we have to remember.

In that frame, I will make two minor changes here, just to keep my sanity. First, as is often done, we will insert a column containing just 1.0 in each observation, so that the weights and features are now aligned. Then, we will move the 0s and 1s outside of the features array, to avoid any ambiguity.

Good. Instead of creating a Console application, I’ll simply go for a script. That way, I can just edit my code and check live whether it does what I want, rather than recompile and run every time.

Let’s start with the weights. What we are doing here is simply creating an array of numFeatures + 1 elements, populated by random values between –10.0 and 10.0. We’ll go a bit fancy here: given that we are also generating random numbers the same way a bit further down, let’s extract a function that generates numbers uniformly between a low and high value:

let rnd = Random(seed)
let generate (low,high) = low + (high-low) * rnd.NextDouble()
let weights = Array.init (numFeatures + 1) (fun _ -> generate(-10.0,10.0))

The next section is where things get a bit thornier. The C# code creates an array, then populates it row by row, first filling in the columns with random numbers, and then applying the logistic function to compute the value that goes in the last column. We can make that much clearer, by extracting that function out. The logistic function is really doing 2 things:

• first, the sumproduct of 2 arrays,
• and then, 1.0/(1.0 + exp ( – z ).

That is easy enough to implement:

let sumprod (v1:float[]) (v2:float[]) =
Seq.zip v1 v2 |> Seq.sumBy (fun (x,y) -> x * y)

let sigmoid z = 1.0 / (1.0 + exp (- z))

let logistic (weights:float[]) (features:float[]) =
sumprod weights features |> sigmoid

We can now use all this, and generate a dataset by simply first creating rows of random values (with a 1.0 in the first column for the constant term), applying the logistic function to compute the value for that row, and return them as a tuple:

open System

let sumprod (v1:float[]) (v2:float[]) =
Seq.zip v1 v2 |> Seq.sumBy (fun (x,y) -> x * y)

let sigmoid z = 1.0 / (1.0 + exp (- z))

let logistic (weights:float[]) (features:float[]) =
sumprod weights features |> sigmoid

let makeAllData (numFeatures, numRows, seed) =

let rnd = Random(seed)
let generate (low,high) = low + (high-low) * rnd.NextDouble()
let weights = Array.init (numFeatures + 1) (fun _ -> generate(-10.0,10.0))

let dataset =
[| for row in 1 .. numRows ->
let features =
[|
yield 1.0
for feat in 1 .. numFeatures -> generate(-10.0,10.0)
|]
let value =
if logistic weights features > 0.55
then 1.0
else 0.0
(features, value)
|]

weights, dataset

Done. Let’s move to the second part of the data generation, with the MakeTrainTest method. Basically, what this does is take a dataset, shuffle it, and split it in two parts, 80% which we will use for training, and 20% we leave out for validation.

static void MakeTrainTest(double[][] allData, int seed,
out double[][] trainData, out double[][] testData)
{
Random rnd = new Random(seed);
int totRows = allData.Length;
int numTrainRows = (int)(totRows * 0.80); // 80% hard-coded
int numTestRows = totRows - numTrainRows;
trainData = new double[numTrainRows][];
testData = new double[numTestRows][];

double[][] copy = new double[allData.Length][]; // ref copy of all data
for (int i = 0; i < copy.Length; ++i)
copy[i] = allData[i];

for (int i = 0; i < copy.Length; ++i) // scramble order
{
int r = rnd.Next(i, copy.Length); // use Fisher-Yates
double[] tmp = copy[r];
copy[r] = copy[i];
copy[i] = tmp;
}
for (int i = 0; i < numTrainRows; ++i)
trainData[i] = copy[i];

for (int i = 0; i < numTestRows; ++i)
testData[i] = copy[i + numTrainRows];
}

Again, there is a ton of indexing going on, which in my old age I find very hard to follow. Upon closer inspection, really, the only thing complicated here is the Fischer-Yates shuffle, which takes an array and randomly shuffles the order. The rest is pretty simply – we just want to shuffle, and then split into two arrays. Let’s extract the shuffle code (which happens to also be used and re-implemented later on):

let shuffle (rng:Random) (data:_[]) =
let copy = Array.copy data
for i in 0 .. (copy.Length - 1) do
let r = rng.Next(i, copy.Length)
let tmp = copy.[r]
copy.[r] <- copy.[i]
copy.[i] <- tmp
copy

We went a tiny bit fancy again here, and made the shuffle work on generic arrays; we also pass in the Random instance we want to use, so that we can control / repeat shuffles if we want, by passing a seeded Random. Does this work? Let’s check in FSI:

> [| 1 .. 10 |] |> shuffle (Random ());;
val it : int [] = [|6; 7; 2; 10; 8; 5; 4; 9; 3; 1|]

Looks reasonable. Let’s move on – we can now implement the makeTrainTest function.

let makeTrainTest (allData:_[], seed) =

let rnd = Random(seed)
let totRows = allData.Length
let numTrainRows = int (float totRows * 0.80) // 80% hard-coded

let copy = shuffle rnd allData
copy.[.. numTrainRows-1], copy.[numTrainRows ..]

Done. A couple of remarks here. First, F# is a bit less lenient than C# around types, so we have to be explicit when converting the number of rows to 80%, first to float, then back to int. As an aside, this used to annoy me a bit in the beginning, but I have come to really like having F# as this slightly psycho-rigid friend who nags me when I am taking a dangerous path (for instance, dividing two integers and hoping for a percentage).

Besides that, I think the code is markedly clearer. The complexity of the shuffle has been nicely contained, and we just have to slice the array to get a training and test sets. As an added bonus, we got rid of the out parameters, and that always feels nice and fuzzy.

I’ll leave it at for today; next time we’ll look at the second part, the learning algorithm itself. Before closing shop, let me make a couple of comments. First, the code is a tad shorter, but not by much. I haven’t really tried, and deliberately made only the changes I thought were needed. What I like about it, though, is that all the indexes are gone, except for the shuffle. In my opinion, this is a good thing. I find it difficult to keep it all in my head when more than one index is involved; when I need to also remember what columns contain special values, I get worried – and just find it hard to figure out what is going on. By contrast, I think makeTrainTest, for instance, conveys pretty directly what it does. makeAllData, in spite of some complexity, also maps closely the way I think about my goal: “I want to generate rows of inputs” – this is precisely what the code does. There is probably an element of culture to it, though; looping over arrays has a long history, and is familiar to every developer, and what looks readable to me might look entirely weird to some.

Easier, or more complicated than before? Anything you like or don’t like – or find unclear? Always interested to hear your opinion! Ping me on Twitter if you have comments.

11. January 2015 18:37

I had the great pleasure to speak at CodeMash this week, and, on my way back, ended up spending a couple of hours at the Atlanta airport waiting for my connecting flight back to the warmer climate of San Francisco – a perfect opportunity for some light-hearted coding fun. A couple of days earlier, I came across this really nice tweet, rendering the results of an L-system:

I ended up looking up L-systems on Wikipedia, and thought this would make for some fun coding exercise. In a nutshell, a L-system is a grammar. It starts with an alphabet of symbols, and a set of rules which govern how each symbol can be transformed into another chain of symbols. By applying these rules to a starting state (the initial axiom), one can evolve it into a succession of states, which can be seen as the growth of an organism. And by mapping each symbol to operations in a logo/turtle like language, each generation can then be rendered as a graphic.

So how could we go about coding this in F#? If you are impatient, you can find the final result as a gist here.

First, I started with representing the core elements of an L-System with a couple of types:

type Symbol = | Sym of char

type State = Symbol list

type Rules = Map<Symbol,State>

type LSystem =
{ Axiom:State
Rules:Rules }

A symbol is a char, wrapped in a single-case discriminated union, and a State is simply a list of Symbols. We define the Rules that govern the transformation of Symbols by a Map, which associates a particular Symbol with a State, and an L-System is then an Axiom (the initial State), with a collection of Rules.

Let’s illustrate this on the second example from the Wikipedia page, the Pythagoras tree. Our grammar contains 4 symbols, 0, 1, [ and ], we start with a 0, and we have 2 rules, (1 → 11), and (0 → 1[0]0). This can be encoded in a straightforward manner in our domain, like this:

let lSystem =
{ Axiom = [ Sym('0') ]
Rules = [ Sym('1'), [ Sym('1'); Sym('1') ]
Sym('0'), [ Sym('1'); Sym('['); Sym('0'); Sym(']'); Sym('0') ]]
|> Map.ofList }

Growing the organism by applying the rules is fairly straightforward: given a State, we traverse the list of Symbols, look up for each of them if there is a matching rule, and perform a substitution if it is found, leaving it unchanged otherwise:

(*
Growing from the original axiom
by applying the rules
*)

let applyRules (rs:Rules) (s:Symbol) =
match (rs.TryFind s) with
| None -> [s]
| Some(x) -> x

let evolve (rs:Rules) (s:State) =
[ for sym in s do yield! (applyRules rs sym) ]

let forward (g:LSystem) =
let init = g.Axiom
let gen = evolve g.Rules
init |> Seq.unfold (fun state -> Some(state, gen state))

// compute nth generation of lSystem
let generation gen lSystem =
lSystem
|> forward
|> Seq.nth gen
|> Seq.toList

What does this give us on the Pythagoras Tree?

> lSystem |> generation 1;;
val it : Symbol list = [Sym '1'; Sym '['; Sym '0'; Sym ']'; Sym '0']

Nice and crisp – that part is done. Next up, rendering. The idea here is that for each Symbol in a State, we will perform a substitution with a sequence of instructions, either a Move, drawing a line of a certain length, or a Turn of a certain Angle. We will also have a Stack, where we can Push or Pop the current position of the Turtle, so that we can for instance store the current position and direction on the stack, perform a couple of moves with a Push, and then return to the previous position by a Pop, which will reset the turtle to the previous position. Again, that lends itself to a very natural model:

(*
Modelling the Turtle/Logo instructions
*)

type Length = | Len of float
type Angle = | Deg of float

// override operator later
let add (a1:Angle) (a2:Angle) =
let d1 = match a1 with Deg(x) -> x
let d2 = match a2 with Deg(x) -> x
Deg(d1+d2)

type Inst =
| Move of Length
| Turn of Angle
| Push
| Pop

let Fwd x = Move(Len(x))
let Lft x = Turn(Deg(x))
let Rgt x = Turn(Deg(-x))

We can now transform our L-system state into a list of instructions, and convert them into a sequence of Operations, in that case Drawing lines between 2 points:

type Pos = { X:float; Y:float; }
type Dir = { L:Length; A:Angle }

type Turtle = { Pos:Pos; Dir:Dir }
type ProgState = { Curr:Turtle; Stack:Turtle list }

let turn angle turtle =
let a = turtle.Dir.A |> add angle
{ turtle with Dir = { turtle.Dir with A = a } }

type Translation = Map<Symbol,Inst list>

type Ops = | Draw of Pos * Pos

let pi = System.Math.PI

let line (pos:Pos) (len:Length) (ang:Angle) =
let l = match len with | Len(l) -> l
let a = match ang with | Deg(a) -> (a * pi / 180.)
{ X = pos.X + l * cos a ; Y = pos.Y + l * sin a }

let execute (inst:Inst) (state:ProgState) =
match inst with
| Push -> None, { state with Stack = state.Curr :: state.Stack }
| Pop ->
let head::tail = state.Stack // assumes more Push than Pop
None, { state with Curr = head; Stack = tail }
| Turn(angle) ->
None, { state with Curr =  state.Curr |> turn angle }
| Move(len) ->
let startPoint = state.Curr.Pos
let endPoint = line startPoint len state.Curr.Dir.A
Some(Draw(startPoint,endPoint)), { state with Curr = { state.Curr with Pos = endPoint } }

let toTurtle (T:Translation) (xs:Symbol list) =

let startPos = { X = 400.; Y = 400. }
let startDir = { L = Len(0.); A = Deg(0.) }
let init =
{ Curr = { Pos = startPos; Dir = startDir }
Stack = [] }
xs
|> List.map (fun sym -> T.[sym])
|> List.concat
|> Seq.scan (fun (op,state) inst -> execute inst state) (None,init)
|> Seq.map fst
|> Seq.choose id

We simply map each Symbol to a List of instructions, transform the list of symbols into a list of instructions, and maintain at each step the current position and direction, as well as a Stack (represented as a list) of positions and directions. How does it play out on our Pythagoras Tree? First, we define the mapping from Symbols to Instructions:

let l = 1.
let T =
[ Sym('0'), [ Fwd l; ]
Sym('1'), [ Fwd l; ]
Sym('['), [ Push; Lft 45.; ]
Sym(']'), [ Pop; Rgt 45.; ] ]
|> Map.ofList

… and we simply send that toTurtle, which produces a list of Draw instructions:

> lSystem |> generation 1 |> toTurtle T;;
val it : seq<Ops> =
seq
[Draw ({X = 400.0;
Y = 400.0;},{X = 401.0;
Y = 400.0;}); Draw ({X = 401.0;
Y = 400.0;},{X = 401.7071068;
Y = 400.7071068;});
Draw ({X = 401.0;
Y = 400.0;},{X = 401.7071068;
Y = 399.2928932;})]

Last step – some pretty pictures. We’ll simply generate a html document, rendering the image using SVG, by creating one SVG line per Draw instruction:

let header = """
<!DOCTYPE html>
<html>
<body>
<svg height="800" width="800">"""

let footer = """
</svg>
</body>
</html>
"""

let toSvg (ops:Ops seq) =
let asString (op:Ops) =
match op with
| Draw(p1,p2) -> sprintf """<line x1="%f" y1="%f" x2="%f" y2="%f" style="stroke:rgb(0,0,0);stroke-width:1" />""" p1.X p1.Y p2.X p2.Y

for op in ops -> asString op
yield footer ]
|> String.concat "\n"

open System.IO

let path = "C:/users/mathias/desktop/lsystem.html"
let save template = File.WriteAllText(path,template)

And we are pretty much done:

> lSystem |> generation 8 |> toTurtle T |> toSvg |> save;;
val it : unit = ()

… which produces the following graphic:

Pretty neat! Just for fun, I replicated the Sierpinski Triangle example as well:

let sierpinski () =

let lSystem =
{ Axiom = [ Sym('A') ]
Rules = [ Sym('A'), [ Sym('B'); Sym('>'); Sym('A'); Sym('>'); Sym('B') ]
Sym('B'), [ Sym('A'); Sym('<'); Sym('B'); Sym('<'); Sym('A') ]]
|> Map.ofList }

let l = 1.
let T =
[ Sym('A'), [ Fwd l; ]
Sym('B'), [ Fwd l; ]
Sym('>'), [ Lft 60.; ]
Sym('<'), [ Rgt 60.; ] ]
|> Map.ofList

lSystem
|> generation 9
|> toTurtle T
|> toSvg
|> save

… which results in the following picture:

That’s it for tonight! I had a lot of fun coding this (it certainly made the flight less boring), and found the idea of converting code to turtle instructions, with a stack, pretty interesting. Hope you enjoyed it, and if you end up playing with this, share your creations on Twitter and ping me at @brandewinder!

Gist for the whole code here

31. December 2014 10:59

Well, we are in the last hours of 2014, and I am nearly recovered from the craziness that was the F# Europa Tour 2014, so here we go – the Tour, in cold, hard facts (after all, I am a numbers’ guy):

• 40 days of travelling across Europe.
• 16 talks.
• 5 workshops (about 50 hours total).
• 9 countries.
• 6991 miles (11,250 kilometers) travelled, roughly (this is straight-line city to city, so the actual number is probably a good deal larger).
• 14 hours of bus.
• roughly 50 hours of train.
• roughly 28 hours of plane.
• 12 cities visited (and spoken at!).
• I lost track of how many gallons of beer were ingested. This is big data.
• 500 attendees? Maybe more? See previous data point.
• Delivered hundreds of shiny fsharp.org stickers to F# Communities across Europe. [btw, in case you didn't hear - the F# Software Foundation is now a full-fledged, legally established entity, and YOU can be a member. Check it out!]

Now for the important qualitative questions:

• Where did I eat the best bacon? This came as a surprise to me, but I have to say, the bacon I ate in Dublin, Ireland was amazing. Twice.
• Where does one find the best beer in Europe? This is a hard one – I had a chance to sample great beers from all over the place. I would say, Munich and its Biergarten rules, but the live beers at BuildStuff in Vilnius, Lithuania, were a very nice surprise.
• What’s the weirdest thing I ate? This one goes to Norway and its Lutefisk, a traditional Christmas fish dish. It’s definitely a regional specialty, as in, a specialty which didn’t expand beyond a limited regional area, for good reasons. For the record, I actually enjoyed it!
• What was the worst travelling mistake? Booking a last minute train ticket from Paris to Aarhus, Denmark, to realize in the train that instead of a nice sleeping car, I would be spending 22 hours sitting in a train with no food on board.
• Biggest scare: every person who has given a talk will tell you, relying on the internet and anything live in a presentation is a rookie mistake. This is great advice, which is why I completely ignored it. It all worked just fine, but learning that Azure had been down for a couple of hours, right before a talk at BuildStuff which 100% required a live deployment to Azure to work, did give me some cold sweat.

Would I do it again? In a heartbeat! It was a bit crazy, and definitely exhausting, but a ton of fun. All of you who helped out making this happen, from the bottom of my heart, thank you! The F# Community is absolutely fantastic, packed with energy and a good, friendly vibe, and everywhere I went felt like family. You all kept me going, so again, thank you (you know who you are)! In the meanwhile, I wish you all a happy year 2015 ahead, let’s make that one even better than 2014, and I hope to see many of you again this year! And, as always, feel free to ping me on Twitter as @brandewinder.

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