Domain Models - Late Evaluation buys you better Composition

In the last post we talked about early abstractions that allow you to design generic interfaces which can be polymorphic in the type parameter. Unless you abuse the type system of a permissive language like Scala, if you adhere to the principles of parametricity, this approach helps you implement abstractions that are reusable under various contexts. We saw this when we implemented the generic contract of the mapReduce function and its various specializations by supplying different concrete instances of the Monoid algebra.

In this post we will take a look at the other end of the spectrum in designing functional domain models. We will discuss evaluation semantics of model behaviors - the precise problem of when to commit to specific concrete evaluation semantics. Consider the following definition of a domain service module ..

type ErrorOr[A] = Either[String, A]

trait PaymentService {
  def paymentCycle: ErrorOr[PaymentCycle]
  def qualifyingAccounts(paymentCycle: PaymentCycle): ErrorOr[Vector[Account]]
  def payments(accounts: Vector[Account]): ErrorOr[Vector[Payment]]
  def adjustTax(payments: Vector[Payment]): ErrorOr[Vector[Payment]]
  def postToLedger(payments: Vector[Payment]): ErrorOr[Unit]
} 

Such definitions are quite common these days. We have a nice monadic definition going on which can be composed as well to implement larger behaviors out of smaller ones ..

def processPayments() = for {
  p <- paymentCycle
  a <- qualifyingAccounts(p)
  m <- payments(a)
  a <- adjustTax(m)
  _ <- postToLedger(a)
} yield ()

Can we improve upon this design ?

Committing to the concrete early - the pitfalls ..

One of the defining aspects of reusable abstractions is the ability to run it under different context. This is one lesson that we learnt in the last post as well. Make the abstractions depend on the least powerful algebra. In this example our service functions return Either, which is a monad. But it's not necessarily the least powerful algebra in the context. Users may choose to use some other monad or may be even applicative to thread through the context of building larger behaviors. Why not keep the algebra unspecified at the service definition level and hope to have specializations in implementations or even in usages at the end of the world ? Here's what we can do ..

// algebra
trait PaymentService[M[_]] {
  def paymentCycle: M[PaymentCycle]
  def qualifyingAccounts(paymentCycle: PaymentCycle): M[Vector[Account]]
  def payments(accounts: Vector[Account]): M[Vector[Payment]]
  def adjustTax(payments: Vector[Payment]): M[Vector[Payment]]
  def postToLedger(payments: Vector[Payment]): M[Unit]
} 

A top level service definition that keeps the algebra unspecified. Now if we want to implement a larger behavior with monadic composition, we can do this ..

// weaving the monad
def processPayments()(implicit me: Monad[M]) = for {
  p <- paymentCycle
  a <- qualifyingAccounts(p)
  m <- payments(a)
  a <- adjustTax(m)
  _ <- postToLedger(a)
} yield p

Note that we are using only the monadic bind in composing the larger behavior - hence the least powerful algebra that we can use is that of a Monad. And we express this exact constraint by publishing the requirements of the existence of an instance of a Monad for the type constructor M.

What about Implementation ?

Well, we could avoid the committment to a concrete algebra in the definition of the service. What about the implementation ? One of the core issues with the implementation is how you need to handle errors. This is an issue which often makes you commit to an implementation when you write the interpreter / implementation of a service contract. You may use Failure for a Try based implementation, or Left for an Either based implementation etc. Can we abstract over this behavior through a generic error handling strategy ? Some libraries like cats offers you abstractions like MonadError that helps you implement error reporting functionality using generic monadic APIs. Here's how we can do this ..

class PaymentServiceInterpreter[M[_]](implicit me: MonadError[M, Throwable])
  extends PaymentService[M] {

  //..

  def payments(accounts: Vector[Account]): M[Vector[Payment]] = 
    if (accounts.isEmpty) me.raiseError(
      new IllegalArgumentException("Empty account list"))
    else //..
    //..
  }
  //..
}

Note we needed a monad with error handling capabilities and we used MonadError for that. Note that we have kept the error type in MonadError as Throwable, which may seem a bit unethical in the context of pure functional programming. But it's also true that many libraries (especially Java ones) or underlying abstractions like Future or Try play well with exceptions. Anyway this is just a red herring though it has nothing to do with the current topic of discussion. The moot point is that you need to supply a MonadError that you have instances of.

Here's how cats defines the trait MonadError ..

trait MonadError[F[_], E] extends ApplicativeError[F, E] with Monad[F] { //..

.. and that's exactly what we will commit to. We are still dealing with a generic Monad even in the implementation without committing to any concreate instance.

End of the World!

The basic purpose why we wanted to delay committing to the concrete instance was to allow the users the flexibility to choose their own implementations. This is what we call the principle of delayed evaluation. Abstract early, evaluate late and decouple the concerns of building and the evaluation of the abstractions. We have already seen the 2 of these principles - we will see that our design so far will accommodate the third one as well, at least for some instances of M.

The user of our API has the flexibility to choose the monad as long as she supplies the MonadError[M, Throwable] instance. And we have many to choose from. Here's an example of the above service implementation in use that composes with another service in a monadic way and choosing the exact concrete instance of the Monad at the end of the world ..

import cats._
import cats.data._
import cats.implicits._

// monix task based computation
object MonixTaskModule {
  import monix.eval.Task
  import monix.cats._

  val paymentInterpreter = new PaymentServiceInterpreter[Task]
  val emailInterpreter = new EmailServiceInterpreter[Task]

  for {
    p <- paymentInterpreter.processPayments
    e <- emailInterpreter.sendEmail(p)
  } yield e
}

// future based computation
object FutureModule {
  import scala.concurrent.Future
  import scala.concurrent.ExecutionContext.Implicits.global
  
  val paymentInterpreter = new PaymentServiceInterpreter[Future]
  val emailInterpreter = new EmailServiceInterpreter[Future]

  for {
    p <- paymentInterpreter.processPayments
    e <- emailInterpreter.sendEmail(p)
  } yield e
}

// try task based computation
object TryModule {
  import scala.util.Try

  val paymentInterpreter = new PaymentServiceInterpreter[Try]
  val emailInterpreter = new EmailServiceInterpreter[Try]

  for {
    p <- paymentInterpreter.processPayments
    e <- emailInterpreter.sendEmail(p)
  } yield e
}

Monix Task is an abstraction that decouples the building of the abstraction from execution. So the Task that we get from building the composed behavior as in the above example can be executed in a deferred way depending of the requirements of the application. It can also be composed with other Tasks to build larger ones as well.

Vertical Composition - stacking abstractions

When you have not committed to an implementation early enough, all you have is an unspecified algebra. You can do fun stuff like stacking abstractions vertically. Suppose we want to implement auditability in some of our service methods. Here we consider a simple strategy of logging as a means to audit the behaviors. How can we take an existing implementation and plugin the audit function selectively ? The answer is we compose algebras .. here's an example that stacks the Writer monad with an already existing algebra to make the payments function auditable ..

final class AuditablePaymentService[M[_]: Applicative](paymentService: PaymentService[M]) 
  extends PaymentService[WriterT[M, Vector[String], ?]] {

  def paymentCycle: WriterT[M, Vector[String], PaymentCycle] =
    WriterT.lift(paymentService.paymentCycle)

  def qualifyingAccounts(paymentCycle: PaymentCycle): WriterT[M, Vector[String], Vector[Account]] =
    WriterT.lift(paymentService.qualifyingAccounts(paymentCycle))

  def payments(accounts: Vector[Account]): WriterT[M, Vector[String], Vector[Payment]] =
    WriterT.putT(paymentService.payments(accounts))(accounts.map(_.no))

  //..
}

val auditablePaymentInterpreter = new AuditablePaymentService[Future](
  new PaymentServiceInterpreter[Future]
)

We took the decision to abstract the return type in the form of a type constructor early on. But committed to the specific type only during the actual usage of the service implementation. Early abstraction and late committment to implementation make great composition possible and often in ways that may pleasantly surprise you later ..

Domain Models - Early Abstractions and Polymorphic Domain Behaviors

Let's talk genericity or generic abstractions. In the last post we talked about an abstraction Money, which, BTW was not generic. But we expressed some of the operations on Money in terms of a Money[Monoid], where Monoid is a generic algebraic structure. By algebraic we mean that a Monoid

1. is generic in types
2. offers operations that are completely generic on the types
3. all operations honor the algebraic laws of left and right identities and associativity

But when we design a domain model, what does this really buy us ? We already saw in the earlier post how law abiding abstractions save you from writing some unit tests just through generic verification of the laws using property based testing. That's just a couple of lines in any of the available libraries out there.

Besides reducing the burden of your unit tests, what does Money[Monoid] buy us in the bigger context of things ? Let's look at a simple operation that we defined in Money ..

Just to recapitulate, here's the definition of Money

class Money (val items: Map[Currency, BigDecimal]) { //..
}

object Money {
  final val zeroMoney = new Money(Map.empty[Currency, BigDecimal])

  def apply(amount: BigDecimal, ccy: Currency) = new Money(Map(ccy -> amount))

  // concrete naive implementation: don't
  def add(m: Money, n: Money) = new Money(
    (m.items.toList ++ n.items.toList)
      .groupBy(_._1)
      .map { case (k, v) => 
        (k, v.map(_._2).sum) 
      }
    )

  //..
}

add is a naive implementation though it's possibly the most frequent one that you will ever encounter in domain models around you. It picks up the Map elements and then adds the ones with the same key to come up with the new Money.

Why is this a naive implementation ?

First of all it deconstructs the implementation of Money, instead of using the algebraic properties that the implementation may have. Here we implement Money in terms of a Map, which itself forms a Monoid under the operations defined by Monoid[Map[K, V]]. Hence why don't we use the monoidal algebra of a Map to implement the operations of Money ?

object Money {

  //..

  def add(m: Money, n: Money) = new Money(m.items |+| n.items)

  //..
}

|+| is a helper function that combines the 2 Maps in a monoidal manner. The concrete piece of code that you wrote in the naive implementation is now delegated to the implementation of the algebra of monoids for a Map in a completely generic way. The advantage is that you need (or possibly someone else has already done that for you) to write this implementation only once and use it in every place you use a Map. Reusability of polymorphic code is not via documentation but by actual code reuse.

On to some more reusability of generic patterns ..

Consider the following abstraction that builds on top of Money ..

import java.time.OffsetDateTime
import Money._

import cats._
import cats.data._
import cats.implicits._

object Payments {
  case class Account(no: String, name: String, openDate: OffsetDateTime, 
    closeDate: Option[OffsetDateTime] = None)
  case class Payment(account: Account, amount: Money, dateOfPayment: OffsetDateTime)

  // returns the Money for credit payment, zeroMoney otherwise
  def creditsOnly(p: Payment): Money = if (p.amount.isDebit) zeroMoney else p.amount

  // compute valuation of all credit payments
  def valuation(payments: List[Payment]) = payments.foldLeft(zeroMoney) { (a, e) =>
    add(a, creditsOnly(e))
  }
  //..
}

valuation gives a standard implementation folding over the List that it gets. Now let's try to critique the implementation ..

1. The function does a foldLeft on the passed in collection payments. The collection only needs to have the ability to be folded over and List can do much more than that. We violate the principle of using the least powerful abstraction as part of the implementation. The function that implements the fold over the collection only needs to take a Foldable - that prevents misuse on part of a user feeling like a child in a toy store with something more grandiose than what she needs.

2. The implementation uses the add function of Money, which is nothing but a concrete wrapper over a monoidal operation. If we can replace this with something more generic then it will be a step forward towards a generic implementation of the whole function.

3. If we squint a bit, we can get some more light into the generic nature of all the components of this 2 line small implementation. zeroMoney is a zero of a Monoid, fold is a generic operation of a Foldable, add is a wrapper over a monoidal operation and creditsOnly is a mapping operation over every payment that the collection hands you over. In summary the implementation folds over a Foldable mapping each element using a function and uses the monoidal operation to collapse the fold.

Well, it's actually a concrete implementation of a generic map-reduce function ..

def mapReduce[F[_], A, B](as: F[A])(f: A => B)
  (implicit fd: Foldable[F], m: Monoid[B]): B = 
    fd.foldLeft(as, m.empty)((b, a) => m.combine(b, f(a)))

In fact the Foldable trait contains this implementation in the name of foldMap, which makes our implementation of mapReduce even simpler ..

def mapReduce1[F[_], A, B](as: F[A])(f: A => B)
  (implicit fd: Foldable[F], m: Monoid[B]): B = fd.foldMap(as)(f)

And List is a Foldable and our implementation of valuation becomes as generic as ..

object Payments {
  //..

  // generic implementation
  def valuation(payments: List[Payment]): Money = {
    implicit val m: Monoid[Money] = Money.MoneyAddMonoid
    mapReduce(payments)(creditsOnly)
  }
}

The implementation is generic and the typesystem will ensure that the Money that we produce can only come from the list of payments that we pass. In the naive implementation there's always a chance that the user subverts the typesystem and can play malice by plugging in some additional Money as the output. If you look at the type signature of mapReduce, you will see that the only way we can get a B is by invoking the function f on an element of F[A]. Since the function is generic on types we cannot ever produce a B otherwise. Parametricity FTW.

mapReduce is completely generic on types - there's no specific implementation that asks it to add the payments passed to it. This abstraction over operations is provided by the Monoid[B]. And the abstraction over the form of collection is provided by Foldable[F]. It's now no surprise that we can pass in any concrete operation or structure that honors the contracts of mapReduce. Here's another example from the same model ..

object Payments {
  //..

  // generic implementation
  def maxPayment(payments: List[Payment]): Money = {
    implicit val m: Monoid[Money] = Money.MoneyOrderMonoid
    mapReduce(payments)(creditsOnly)
  }
}

We want to compute the maximum credit payment amount from a collection of payments. A different domain behavior needs to be modeled but we can think of it as belonging to the same form as valuation and implemented using the same structure as mapReduce, only passing a different instance of Monoid[Money]. No additional client code, no fiddling around with concrete data types, just matching the type contracts of a polymorphic function.

Looks like our investment on an early abstraction of mapReduce has started to pay off. The domain model remains clean with much of the domain logic being implemented in terms of the algebra that the likes of Foldables and Monoids offer. I discussed some of these topics at length in my book Functional and Reactive Domain Modeling. In the next instalment we will explore some more complex algebra as part of domain modeling ..

Domain models, Algebraic laws and Unit tests

In a domain model, when you have a domain element that forms an algebraic abstraction honoring certain laws, you can get rid of many of your explicitly written unit tests just by checking the laws. Of course you have to squint hard and discover the lawful abstraction that hides behind your concrete domain element.

Consider this simple abstraction for Money that keeps track of amounts in various currencies.

scala> import Money._
import Money._

// 1000 USD
scala> val m = Money(1000, USD)
m: laws.Money = (USD,1000)

// add 248 AUD
scala> val n = add(m, Money(248, AUD))
n: laws.Money = (AUD,248),(USD,1000)

// add 230 USD more
scala> val p = add(n, Money(230, USD))
p: laws.Money = (AUD,248),(USD,1230)

// value of the money in base currency (USD)
scala> p.toBaseCurrency
res1: BigDecimal = 1418.48

// debit amount
scala> val q = Money(-250, USD)
q: laws.Money = (USD,-250)

scala> val r = add(p, q)
r: laws.Money = (AUD,248),(USD,980)

The valuation of Money is done in terms of its base currency which is usually USD. One of the possible implementations of Money is the following (some parts elided for future explanations) ..

sealed trait Currency
case object USD extends Currency
case object AUD extends Currency
case object JPY extends Currency
case object INR extends Currency

class Money private[laws] (val items: Map[Currency, BigDecimal]) {
  def toBaseCurrency: BigDecimal = 
    items.foldLeft(BigDecimal(0)) { case (a, (ccy, amount)) =>
      a + Money.exchangeRateWithUSD.get(ccy).getOrElse(BigDecimal(1)) * amount
    }

  override def toString = items.toList.mkString(",")
}

object Money {
  final val zeroMoney = new Money(Map.empty[Currency, BigDecimal])

  def apply(amount: BigDecimal, ccy: Currency) = new Money(Map(ccy -> amount))
  def add(m: Money, amount: BigDecimal, ccy: Currency) = ???

  final val exchangeRateWithUSD: Map[Currency, BigDecimal] = 
    Map(AUD -> 0.76, JPY -> 0.009, INR -> 0.016, USD -> 1.0)
}

Needless to say we will have quite a number of unit tests that check for addition of Money, including the boundary cases of adding to zeroMoney.

It's not very hard to see that the type Money forms a Monoid under the add operation. Or to speak a bit loosely we can say that Money is a Monoid under the add operation.

A Monoid has laws that every instance needs to honor - associativity, left identity and right identity. And when your model element needs to honor the laws of algebra, it's always recommended to include the verification of the laws as part of your test suite. Besides validating the sanity of your abstractions, one side-effect of verifying laws is that you can get rid of many of your explicitly written unit tests for the operation that forms the Monoid. They will be automatically verified when verifying the laws of Monoid[Money].

Here's how we define Monoid[Money] using Cats ..

val MoneyAddMonoid: Monoid[Money] = new Monoid[Money] {
  def combine(m: Money, n: Money): Money = add(m, n)
  def empty: Money = zeroMoney
}

and the implementation of the previously elided add operation on Money using Monoid on Map ..

object Money {
  //..

  def add(m: Money, amount: BigDecimal, ccy: Currency) = 
    new Money(m.items |+| Map(ccy -> amount))

  //..

}

Now we can verify the laws of Monoid[Money] using specs2 and ScalaCheck and the helper classes that Cats offers ..

import cats._
import kernel.laws.GroupLaws
import org.scalacheck.{ Arbitrary, Gen }
import Arbitrary.arbitrary

class MoneySpec extends CatsSpec { def is = s2"""

  This is a specification for validating laws of Money

  (Money) should
     form a monoid under addition    $e1 
  """

  implicit lazy val arbCurrency: Arbitrary[Currency] = Arbitrary { 
    Gen.oneOf(AUD, USD, INR, JPY) 
  }

  implicit def moneyArbitrary: Arbitrary[Money] = 
    Arbitrary {
      for {
        i <- Arbitrary.arbitrary[Map[Currency, BigDecimal]]
      } yield new Money(i)
    }

  def e1 = checkAll("Money", GroupLaws[Money].monoid(Money.MoneyAddMonoid))
}

and running the test suite will verify the Monoid laws for Monoid[Money] ..

[info] This is a specification for validating laws of Money
[info]
[info] (Money) should
[info] form a monoid under addition monoid laws must hold for Money
[info] + monoid.associativity
[info] + monoid.combineAll
[info] + monoid.combineAll(Nil) == id
[info] + monoid.combineAllOption
[info] + monoid.combineN(a, 0) == id
[info] + monoid.combineN(a, 1) == a
[info] + monoid.combineN(a, 2) == a |+| a
[info] + monoid.isEmpty
[info] + monoid.leftIdentity
[info] + monoid.rightIdentity
[info] + monoid.serializable

In summary ..

• strive to find abstractions in your domain model that are constrained by algebraic laws
• check all laws as part of your test suite
• you will find that you can get rid of quite a few explicitly written unit tests just by checking the laws of your abstraction
• and of course use property based testing for unit tests

In case you want to take a look at the full code base, it's there on my Github repo. In the next post we will take the next step towards modeling with generic algebraic code using the Monoid pattern from this example. Code written in parametric form without depending on specialized concrete types can be more robust, easier to test and easier to reason about. I have also discussed this at length in my book Functional and Reactive Domain Modeling. I plan to supplement the materials covered there with more examples and code patterns ..

Baking a π can teach you a bit of Parametricity

Even though I got my copy of Prof. Eugenia Cheng's awesome How to Bake π a couple of weeks back, I started reading it only over this weekend. I am only on page 19 enjoying all the stuff regarding cookies that Prof. Cheng is using to explain abstraction. This is a beautiful piece of explanation and if you are a programmer you may get an extra mile out of the concepts that she explains here. Let's see if we can unravel a few of them ..

She starts with a real life situation such as:

If Grandma gives you five cookies and Grandpa gives you five cookies, how many cookies will you have ?

Let's model this as box of cookies that you get from your Grandma and Grandpa and you need to count them and find the total. Let's model this in Scala and we may have something like the following ..

case class CookieBox(count: Int)

and we can define a function that gives you a CookieBox containing the total number of cookies from the 2 boxes that we pass to the function ..

def howManyCookies(gm: CookieBox, gp: CookieBox) = {
  CookieBox(gm.count + gp.count)
}

and we use howManyCookies to find the count ..

scala> val gm = CookieBox(5)
gm: CookieBox = CookieBox(5)

scala> val gp = CookieBox(5)
gp: CookieBox = CookieBox(5)

scala> howManyCookies(gm, gp)
res5: CookieBox = CookieBox(10)

.. so we have 10 cookies from our Grandma & Grandpa .. Perfect!

The problem is .. the child answers: "None, because I'll eat them all". To model this let's add a function eat to our CookieBox abstraction ..

case class CookieBox(count: Int) {
  // let's assume n < count for simplicity
  def eat(n: Int): CookieBox = CookieBox(count - n)
}

So instead of the correct way to answer the question, the child cheats and implements howManyCookies as ..

def howManyCookies(gm: CookieBox, gp: CookieBox) = {
  CookieBox(gm.eat(gm.count).count + gp.eat(gp.count).count)
}

and we get the following ..

scala> howManyCookies(gm, gf)
res6: CookieBox = CookieBox(0)

Prof. Cheng continues ..

The trouble here is that cookies do not obey the rules of logic, so using math to study them doesn't quite work. .. We could impose an extra rule on the situation by adding "... and you're not allowed to eat the cookies". If you're not allowed to eat them, what's the point of them being cookies ?

This is profound indeed. When we are asked to count some stuff, it really doesn't matter if they are cookies or stones or pastries. The only property we need here is to be able to add together the 2 stuff that we are handed over. The fact that we have implemented howManyCookies in terms of CookieBox gives the little child the opportunity to cheat by using the eat function. More information is actually hurting us here, being concrete with data types is actually creating more avenues for incorrect implementation.

Prof. Cheng is succinct here when she explains ..

We could treat the cookies as just things rather than cookies. We lose some resemblance to reality, but we gain scope and with it efficiency. The point of numbers is that we can reason about "things" without having to change the reasoning depending on what "thing" we are thinking about.

Yes, she is talking about generalization, being polymorphic over what we count. We just need the ability to add 2 "things", be it cookies, monkeys or anchovies. In programming we model this with parametric polymorphism, and use a universal quantification over the set of types for which we implement the behavior.

def howMany[A](gm: A, gp: A) = //..

We have made the implementation parametric and got rid of the concrete data type CookieBox. But how do we add the capability to sum the 2 objects and get the result ? You got it right - we already have an abstraction that makes this algebra available to a generic data type. Monoids FTW .. and it doesn't get simpler than this ..

trait Monoid[T] {
  def zero: T
  def append(t1: T, t2: T): T
}

zero is the identity function and append is a binary associative function over 2 objects of the type. So given a monoid instance for our data type, we can model howMany in a completely generic way irrespective of whether A is a CookieBox or Monkey.

def howMany[A : Monoid](gm: A, gp: A): A = gm append gp

Implementing a monoid for CookieBox is also simple ..

object CookieBox {
  implicit val CookieBoxMonoid = new Monoid[CookieBox] {
    val zero = CookieBox(0)
    def append(i: CookieBox, j: CookieBox) = CookieBox(i.count + j.count)
  }
}
 

With the above implementation of howMany, the little child will not be able to cheat. By providing a simpler data type we have made the implementation more robust and reusable across multiple data types.

Next time someone wants me to explain parametricity, I will point them to Page 19 of How to Bake π.

Randomization and Probabilistic Techniques to scale up Machine Learning

Some time back I blogged about the possibilities that probabilistic techniques and randomization bring on to the paradigm of stream computing. Architectures based on big data not only relate to high volume storage, but also on low latency velocities, and this is exactly where stream computing has a role to play. I discussed a few data structures like bloom filters, count min sketch and hyperloglog and algorithms like Locality Sensitive Hashing that use probabilistic techniques to reduce the search and storage space while processing huge volumes of data.

Of late, I have been studying some of the theories behind machine learning algorithms and how they can be used in conjunction with the petabytes of data that we generate everyday. And the same thing strikes here - there are algorithms that can model the most perfect classifier. But you need randomization and probabilistic techniques to make them scale, even at the expense of a small amount of inaccuracy creeping within your model. In most cases we will see that the small inaccuracy that comes within your algorithm because of probabilistic bounds can be compensated by the ability to process more data within the specified computational timeline. This is true even for some of the basic algorithms like matrix multiplication that form the core of machine learning models.

The contents of this post is nothing original or new. It's just to share some of my thoughts in learning the usage of approximation techniques in building machine learning classifiers.

Matrix Multiplication

Not only these specialized data structures or algorithms, randomization has been found to be quite effective for processing large data sets even for standard algorithms like matrix multiplication, polynomial identity verification or min cut identification from large graphs. In all such cases the best available algorithms have computational complexity which works well for a small data set but doesn't scale well enough with the volumes of data.

Consider a case where we are given 3 matrices, $A$, $B$ and $C$ and we need to verify if $AB = C$. The standard algorithm for matrix multiplication takes $\Theta(n^3)$ operations and there's also a sophisticated algorithm that works in $\Theta(n^{2.37})$ operations. Instead let's consider some randomization and choose a random vector $\bar{r} = (r_1, r_2, .. r_n) \in \{0, 1\}^n$. Now we can compute $AB\bar{r}$ by first computing $B\bar{r}$ and then $A(B\bar{r})$. And then we compute $C\bar{r}$. If we find $A(B\bar{r}) \neq C\bar{r}$, then $AB \neq C$. Otherwise we return $AB = C$. Instead of matrix-matrix multiplication our randomized algorithm uses matrix-vector multiplication, which can be done in $\Theta(n^2)$ operations the standard way.

Obviously a $\Theta(n^2)$ algorithm has a lower computational complexity than $\Theta(n^3)$ and scales better with larger data sets. Now the question is how accurate is this algorithm ? Is it guaranteed to give the correct answer every time we run it ? As with other probabilistic algorithms, there's a chance that our algorithm will return a wrong result. But as long as we can show that the chance is minimal and can be reduced by tuning some parameters, we should be fine.

It can be shown that if $AB \neq C$ and if $\bar{r}$ is chosen uniformly at random from $\{0, 1\}^n$ then $Pr(AB\bar{r} = C\bar{r}) <= 1/2$. But the trick is that we can run our randomized algorithm many times choosing $\bar{r}$ with replacement from $\{0, 1\}^n$. If for any of these trials we get $AB\bar{r} \neq C\bar{r}$, then we can conclude $AB \neq C$. And the probability that we get $AB\bar{r} = C\bar{r}$ for all $k$ trials despite $AB \neq C$ is $2^{-k}$. So for $100$ trials, the chance of error is $2^{-100}$, which we can see is really small. The detailed proof of this analysis can be found in the excellent book Probability and Computing by Michael Mitzenmacher & Eli Upfal.

Matrix multiplication is something that's used heavily especially in implementing machine learning classifiers. And if we can tolerate that little chance of error we get an algorithm with lower computational complexity that scales much better.

Stochastic Gradient Descent

Consider another use case from core machine learning classifier design. Gradient descent is a standard way to minimize the empirical risk for measuring training set performance. The empirical risk is given by the following equation:
$$E_n(f) = (1/n)\sum_i l(f_w(x_i),y_i)$$
where $l$ is the loss function that measures the cost of predicting $f_w(x_i)$ from $n$ training examples where the actual answer is $y$ and $f_w(x)$ is the function parameterized by the weight vector $w$. Each iteration of gradient descent updates the weights $w$ on the basis of the gradient of $E_n(f_w)$ according to the following iterative step:

$$w_{t+1} = w_t - \gamma (1/n) \sum_i \nabla_w Q(z_i, w_t)$$
where $\gamma$ is an adequately chosen gain. Note that a single update step for the parameter runs through all the training examples and this gets repeated for every update step that you do before convergence. Compare this with Stochastic Gradient Descent (SGD) where the update step is given by the following:

$$w_{t+1} = w_t - \gamma \nabla_w Q(z_t, w_t)$$
Note instead of running through all examples and compute the exact gradient, SGD computes the gradient based on one randomly picked example $z_t$. So, SGD does a noisy approximation to the true gradient. But since it does not have to process all the examples in every iteration it scales better with a large data set. In this paper on Large Scale Machine Learning With Stochastic Gradient Descent, Leon Bottou classifies the error in building the classifier into 3 components:

Approximation Error, which comes from the fact that the function $f$ that we choose is different from the optimal function $f^*$ and we approximate using a few examples

Estimation Error, which comes from the fact that we have a finite number of training examples and would have gone away with infinite number of them

Optimization Error, which comes from the fact that we are using an inferior algorithm to estimate the gradient

With normal gradient descent we will have low optimization error since we run through all the training examples in every iteration to compute the gradient, which is clearly superior to the algorithm of SGD that does a noisy approximation. But SGD will report a lower approximation and estimation error since we will be able to process a larger dataset within the stipulated computation time. So it's a tradeoff of that we make using SGD, but clearly we scale better with larger data sets.

Singular Value Decomposition

Singular Value Decomposition is a dimensionality reduction technique to unearth a smaller number of intrinsic concepts from a high dimensional matrix by removing unnecessary information. It does so by projecting the original matrix on to lower dimensions such that the reconstruction error is minimized. What this means is that given a matrix $A$ we decompose it into lower dimensional matrices by removing the lesser important information. And we do this in such a way that we can reconstruct a fairly close approximation to $A$ from those lower dimensional matrices. In theory SVD gives the best possible projection in terms of reconstruction error (optimal low rank approximation). But in practice it suffers from scalability problems with large data sets. It generates dense singular vectors even if the original matrix is a sparse one and hence is computationally inefficient, taking cubic time in the size of the data.

This can be addressed by another algorithm, the CUR algorithm which allows larger reconstruction error but lesser computation time. CUR decomposes the original matrix into ones of lesser dimensions but uses a randomized algorithm in selection of columns and rows based on their probability distribution. Now it can be shown that CUR reconstruction is just an additive term away from SVD reconstruction and it's a probabilistic bound subject to the condition that we select a specific range of columns and rows from $A$. The computational bound of CUR is of the order of the data set, which is much less than that of SVD (which as I mentioned earlier is cubic). This is yet another example where we apply randomization and probabilistic techniques to scale our algorithm better for larger data sets in exchange for a little amount of inaccuracy.

These are only a few instances of probabilistic bounds being applied to solve real world machine learning problems. There are a lots more. In fact I find that scalability of machine learning has a vey direct correlation with application of probabilistic techniques to the model. As I mentioned earlier the point of this post is to share some of my thoughts as I continue to learn techniques to scale up machine learning models. Feel free to share your ideas, thoughts and discussions in comments.

Functional Patterns in Domain Modeling - Composing a domain workflow with statically checked invariants

I have been doing quite a bit of domain modeling using functional programming mostly in Scala. And as it happens when you work on something for a long period of time you tend to identify more and more patterns that come up repeatedly within your implementations. You may ignore these as patterns the first time, get a feeling of mere coincidence the next time, but third time really gives you that aha! moment and you feel like documenting it as a design pattern. In course of my learnings I have started blogging on some of these patterns - you can find the earlier ones in the series in:

Functional Patterns in Domain Modeling - The Specification Pattern

Functional Patterns in Domain Modeling - Immutable Aggregates and Functional Updates

Functional Patterns in Domain Modeling - Anemic Models and Compositional Domain Behaviors

In this continuing series of functional patterns in domain modeling, I will go through yet another idiom which has been a quite common occurrence in my explorations across various domain models. You will find many of these patterns explained in details in my upcoming book on Functional and Reactive Domain Modeling, the early access edition of which is already published by Manning.

One of the things that I strive to achieve in implementing domain models is to use the type system to encode as much domain logic as possible. If you can use the type system effectively then you get the benefits of parametricity, which not only makes your code generic, concise and polymorphic, but also makes it self-testing. But that's another story which we can discuss in another post. In this post I will talk about a pattern that helps you design domain workflows compositionally, and also enables implementing domain invariants within the workflow, all done statically with little help from the type system.

As an example let's consider a loan processing system (simplified for illustration purposes) typically followed by banks issuing loans to customers. A typical simplified workflow looks like the following :-

The Domain Model

The details of each process is not important - we will focus on how we compose the sequence and ensure that the API verifies statically that the correct sequence is followed. Let's start with a domain model for the loan application - we will keep on enriching it as we traverse the workflow.

case class LoanApplication private[Loans](
  // date of application
  date: Date,
  // name of applicant
  name: String,
  // purpose of loan
  purpose: String,
  // intended period of repayment in years
  repayIn: Int,
  // actually sanctioned repayment period in years
  actualRepaymentYears: Option[Int] = None,
  // actual start date of loan repayment
  startDate: Option[Date] = None,
  // loan application number
  loanNo: Option[String] = None,
  // emi
  emi: Option[BigDecimal] = None
)

Note we have a bunch of attributes that are defined as optional and will be filled out later as the loan application traverses through the sequence of workflow. Also we have declared the class private and we will have a smart constructor to create an instance of the class.

Wiring the workflow with Kleisli

Here are the various domain behaviors modeling the stages of the workflow .. I will be using the scalaz library for the Kleisli implementation.

def applyLoan(name: String, purpose: String, repayIn: Int, 
  date: Date = today) =
  LoanApplication(date, name, purpose, repayIn)

def approve = Kleisli[Option, LoanApplication, LoanApplication] { l => 
  // .. some logic to approve
  l.copy(
    loanNo = scala.util.Random.nextString(10).some,
    actualRepaymentYears = 15.some,
    startDate = today.some
  ).some
}

def enrich = Kleisli[Option, LoanApplication, LoanApplication] { l => 
  //.. may be some logic here
  val x = for {
    y <- l.actualRepaymentYears
    s <- l.startDate
  } yield (y, s)

  l.copy(emi = x.map { case (y, s) => calculateEMI(y, s) }).some
}

applyLoan is the smart constructor that creates the initial instance of LoanApplication. The other 2 functions approve and enrich perform the approval and enrichment steps of the workflow. Note both of them return an enriched version of the LoanApplication within a Kleisli, so that we can use the power of Kleisli composition and wire them together to model the workflow ..

val l = applyLoan("john", "house building", 10)
val op = approve andThen enrich
op run l

When you have a sequence to model that takes an initial object and then applies a chain of functions, you can use plain function composition like h(g(f(x))) or using the point free notation, (h compose g compose f) or using the more readable order (f andThen g andThen h). But in the above case we need to have effects along with the composition - we are returning Option from each stage of the workflow. So here instead of plain composition we need effectful composition of functions and that's exactly what Kleisli offers. The andThen combinator in the above code snippet is actually a Kleisli composition aka function composition with effects.

So we have everything the workflow needs and clients use our API to construct workflows for processing loan applications. But one of the qualities of good API design is to design it in such a way that it becomes difficult for the client to use it in the wrong way. Consider what happens with the above design of the workflow if we invoke the sequence as enrich andThen approve. This violates the domain invariant that states that enrichment is a process that happens after the approval. Approval of the application generates some information which the enrichment process needs to use. But because our types align, the compiler will be perfectly happy to accept this semantically invalid composition to pass through. And we will have the error reported during run time in this case.

Remembering that we have a static type system at our disposal, can we do better ?

Phantom Types in the Mix

Let's throw in some more types and see if we can tag in some more information for the compiler to help us. Let's tag each state of the workflow with a separate type ..

trait Applied
trait Approved
trait Enriched

Finally make the main model LoanApplication parameterized on a type that indicates which state it is in. And we have some helpful type aliases ..

case class LoanApplication[Status] private[Loans]( //..

type LoanApplied  = LoanApplication[Applied]
type LoanApproved = LoanApplication[Approved]
type LoanEnriched = LoanApplication[Enriched]

These types will have no role in modeling domain behaviors - they will just be used to dispatch to the correct state of the sequence that the domain invariants mandate. The workflow functions need to be modified slightly to take care of this ..

def applyLoan(name: String, purpose: String, repayIn: Int, 
  date: Date = today) =
  LoanApplication[Applied](date, name, purpose, repayIn)

def approve = Kleisli[Option, LoanApplied, LoanApproved] { l => 
  l.copy(
    loanNo = scala.util.Random.nextString(10).some,
    actualRepaymentYears = 15.some,
    startDate = today.some
  ).some.map(identity[LoanApproved])
}

def enrich = Kleisli[Option, LoanApproved, LoanEnriched] { l => 
  val x = for {
    y <- l.actualRepaymentYears
    s <- l.startDate
  } yield (y, s)

  l.copy(emi = x.map { case (y, s) => calculateEMI(y, s) }).some.map(identity[LoanEnriched])
}

Note how we use the phantom types within the Kleisli and ensure statically that the sequence can flow only in one direction - that which is mandated by the domain invariant. So now an invocation of enrich andThen approve will result in a compilation error because the types don't match. So once again yay! for having the correct encoding of domain logic with proper types.

Probabilistic techniques, data streams and online learning - Looking forward to a bigger 2015

I look forward to 2015 as the year when randomized algorithms, probabilistic techniques and data structures become more pervasive and mainstream. The primary driving factors for this will be more and more prevalence of big data and the necessity to process them in near real time using minimal (or constant) memory bandwidth. You are given data streams where possibly you will see every data only once in your lifetime and you need to churn out analytics from them in real time. You cannot afford to store all of them in a database on disk since it will incur an unrealistic performance penalty to serve queries in real time. And you cannot afford to store all information in memory even if you add RAM at your own will. You need to find clever ways to optimize your storage, employ algorithms and data structures that use sublinear space and yet deliver information in real time.

Many such data structures are already being used quite heavily for specialized processing of data streams ..

• Bloom Filters for set membership using sublinear storage
• HyperLogLog for efficient cardinality estimation
• Count Min Sketch for estimating frequency related properties of a data stream

These data structures are becoming more and more useful as we prepare to embrace and process larger data sets with fairly strict online requirements. And it has started making a difference. Take for example Impala, the open source analytic database from Cloudera that works on top of Hadoop. Impala's NDV aggregate function (number of distinct values) uses the HyperLogLog algorithm to estimate this number, in parallel, in a fixed amount of space. This blog post has the details of the performance improvement that it offers in comparison to the standard distinct count. The immensely popular NoSQL store Redis also offers a HyperLogLog implementation that you can use to get an approximation on the cardinality of a set using randomization. Salvatore has the details here on the implementation of HyperLogLog algorithm in Redis.

The most important reason these algorithms and data structures are becoming popular is the increased focus on our "online" requirements. We are not only processing bigger and bigger data set, we need results faster too. We just cannot afford to push all analytics to the batch mode and expect results coming out after an overnight batch processing. Various architectural paradigms like the lambda architecture also target to address this niche area. But before investing on such complex architectures, often some neat data structures that use probabilistic techniques and randomization may offer a much lighter weight solution that you are looking for.

Consider processing the Twitter stream and generating analytics (of whatever form) online. This means that immediately after seeing one twitter feed you must be able to predict something and update your model at the same time. Which means you need to memorize the data that you see in the feed, apply it to update your model and yet cannot store the entire hose that you have seen so far. This is online learning and is the essence of techniques like stochastic gradient descent that help you do this - the model is capable of making up to date predictions after every data that you see. John Myles White has an excellent presentation on this topic.

Consider this other problem of detecting similarities between documents. When you are doing this on a Web scale you will have to deal with millions of documents to find the similar sets. There are techniques like minhash which enable you to compress documents into signature matrices. But even then the scale becomes too big to be processed and reported to the user in a meaningful amount of time. As an example (from Mining Massive Datasets), if you process 1 million document using signatures of length 250, you still have to use 1000 bytes per document - the total comes to 1 gigabyte which very well fits into the memory of a standard laptop. But when you check for similar pairs, you need to process (1,000,000 choose 2) or half a trillion pairs of documents which will take almost 6 days to compute all similarities on a laptop. Enter probabilistic techniques and locality sensitive hashing (LSH) algorithm fits this problem like a charm. Detecting similarity is a problem that arises in recommender systems with collaborative filtering and LSH can be used there as well. The basic idea of LSH as applied to similarity detection is to use hashing multiple number of times and identify candidate pairs that qualify for similarity checking. The idea is to reduce the search space using probabilistic techniques so that we can eliminate a class of candidates which have very low chance of being similar.

Here I have only scratched the surface of the areas where we apply randomization and probabilistic techniques to solve problems that are very real today. There are plentiful other areas in data mining, graph clustering, machine learning and big data processing where similar techniques are employed to reduce the curse of dimensionality and provide practical solution at scale. 2014 has already seen a big surge in terms of popularizing these techniques. I expect 2015 to be bigger and more mainstream in terms of their usage.

Personally I have been exploring data stream algorithms a lot and have prepared a collection of some useful references. Feel free to share in case you find it useful. I hope to do something more meaningful with stream processing data structures and online learning in 2015. Have a very happy and joyous new year ..

Functional and Reactive Domain Modeling

Manning has launched the MEAP of my upcoming book on Domain Modeling.

The first time I was formally introduced to the topic was way back when I played around with Erik Evans' awesome text on the subject of Domain Driven Design. In the book he discusses various object lifecycle patterns like the Factory, Aggregate or Repository that help separation of concerns when you are implementing the various interactions between the elements of the domain model. Entities are artifacts with identities, value objects are pure values while services model the coarse level use cases of the model components.

Traditionally we followed the recommendations of Erik in our real world implementations and used the object oriented paradigm for modeling all interactions. We started talking about rich domain models and anemic domain models. The rich model espoused a richer agglomeration of state and behavior within the model, while the anemic model preferred to keep them decoupled. Martin Fowler sums this up in his post on Anemic Domain Models ..

"The basic symptom of an Anemic Domain Model is that at first blush it looks like the real thing. There are objects, many named after the nouns in the domain space, and these objects are connected with the rich relationships and structure that true domain models have. The catch comes when you look at the behavior, and you realize that there is hardly any behavior on these objects, making them little more than bags of getters and setters. Indeed often these models come with design rules that say that you are not to put any domain logic in the the domain objects. Instead there are a set of service objects which capture all the domain logic. These services live on top of the domain model and use the domain model for data."

Go Functional

In Functional and Reactive Domain Modeling I look at the problem with a different lens. The primary focus of the book is to encourage building domain models using the principles of functional programming. It's a completely orthogonal approach than OO and focuses on verbs first (as opposed to nouns first in OO), algebra first (as opposed to objects in OO), function composition first (as opposed to object composition in OO), lightweight objects as ADTs (instead of rich class models).

The book starts with the basics of functional programming principles and discusses the virtues of purity and the advantages of keeping side-effects decoupled from the core business logic. The book uses Scala as the programming language and does an extensive discussion on why the OO and functional features of Scala are a perfect fit for modelling complex domains. Chapter 3 starts the core subject of functional domain modeling with real world examples illustrating how we can make good use of patterns like smart constructors, monads and monoids in implementing your domain model. The main virtue that these patterns bring to your model is genericity - they help you extract generic algebra from domain specific logic into parametric functions which are far more reusable and less error prone. Chapter 4 focuses on advanced usages like typeclass based design and patterns like monad transformers, kleislis and other forms of compositional idioms of functional programming. One of the primary focus of the book is an emphasis on algebraic API design and to develop an appreciation towards ability to reason about your model.

Go Reactive

The term "reactive" has recently become quite popular in describing systems that are responsive, scalable and adaptive. In implementing complex domain models, we find many areas which can be made more responsive by implementing them as non blocking operations instead of the standard blocking function calls. Using higher level concurrency primitives like actors, futures and data flow based computing we can compose asynchronous operations and increase the net throughput of your model. The second part of the book discusses how you can combine the principles of functional programming with the reactive way of implementing behaviors. The paradigm of application design using the principles of functional programming together with an asynchronous non-blocking mode of communication between the participating entities promises to be a potent combination towards developing performant systems that are relatively easy to manage, maintain and evolve. But designing and implementing such models need a different way of thinking. The behaviors that you implement have to be composable using pure functions and they can form building blocks of bigger abstractions that communicate between them using non-blocking asynchronous message passing.

Functional meets Reactive

Functional and Reactive Domain Modeling takes you through the steps teaching you how to think of the domain model in terms of pure functions and how to compose them to build larger abstractions. You will start learning with the basics of functional programming principles and gradually progress to the advanced concepts and patterns that you need to know to implement complex domain models. The book demonstrates how advanced FP patterns like algebraic data types, typeclass based design and isolation of side-effects can make your model compose for readability and verifiability. On the subject of reactive modeling, the book focuses on the higher order concurrency patterns like actors and futures. It uses the Akka framework as the reference implementation and demonstrates how advanced architectural patterns like event sourcing and command-query-responsibility-segregation can be put to great use while implementing scalable models. You will learn techniques that are radically different from the standard RDBMS based applications that are based on mutation of records. You’ll also pick up important patterns like using asynchronous messaging for interaction based on non blocking concurrency and model persistence, which delivers the speed of in-memory processing along with suitable guarantees of reliability.

Looking forward to an exciting journey with the book. I am sure you will also find interest in the topics that I discuss there. And feel free to jump on to AuthorOnline and fire questions that we all can discuss. I am sure this will also lead to an overall improvement in the quality of the book.

Functional Patterns in Domain Modeling - Anemic Models and Compositional Domain Behaviors

I was looking at the presentation that Dean Wampler made recently regarding domain driven design, anemic domain models and how using functional programming principles help ameliorate some of the problems there. There are some statements that he made which, I am sure made many OO practitioners chuckle. They contradict popular beliefs that encourage OOP as the primary way of modeling using DDD principles.

One statement that resonates a lot with my thought is "DDD encourages understanding of the domain, but don't implement the models". DDD does a great job in encouraging developers to understand the underlying domain model and ensuring a uniform vocabulary throughout the lifecycle of design and implementation. This is what design patterns also do by giving you a vocabulary that you can heartily exchange with your fellow developers without influencing any bit of implementation of the underlying pattern.

On the flip side of it, trying to implement DDD concepts using standard techniques of OO with joined state and behavior often gives you a muddled mutable model. The model may be rich from the point of view that you will find all concepts related to the particular domain abstraction baked in the class you are modeling. But it makes the class fragile as well since the abstraction becomes more locally focused losing the global perspective of reusability and composability. As a result when you try to compose multiple abstractions within the domain service layer, it becomes too much polluted with glue code that resolves the impedance mismatch between class boundaries.

So when Dean claims "Models should be anemic", I think he means to avoid this bundling of state and behavior within the domain object that gives you the false sense of security of richness of the model. He encourages the practice that builds domain objects to have the state only while you model behaviors using standalone functions.

Sometimes, the elegant implementation is just a function. Not a method. Not a class. Not a framework. Just a function.

— John Carmack (@ID_AA_Carmack) March 31, 2011

One other strawman argument that I come across very frequently is that bundling state and behavior by modeling the latter as methods of the class increases encapsulation. If you are still a believer of this school of thought, have a look at Scott Meyer's excellent article which he wrote as early as 2000. He eschews the view that a class is the right level of modularization and encourages more powerful module systems as better containers of your domain behaviors.

As continuation of my series on functional domain modeling, we continue with the example of the earlier posts and explore the theme that Dean discusses ..

Here's the anemic domain model of the Order abstraction ..

case class Order(orderNo: String, orderDate: Date, customer: Customer, 
  lineItems: Vector[LineItem], shipTo: ShipTo, 
  netOrderValue: Option[BigDecimal] = None, status: OrderStatus = Placed)

In the earlier posts we discussed how to implement the Specification and Aggregate Patterns of DDD using functional programming principles. We also discussed how to do functional updates of aggregates using data structures like Lens. In this post we will use these as the building blocks, use more functional patterns and build larger behaviors that model the ubiquitous language of the domain. After all, one of the basic principles behind DDD is to lift the domain model vocabulary into your implementation so that the functionality becomes apparent to the developer maintaining your model.

The core idea is to validate the assumption that building domain behaviors as standalone functions leads to an effective realization of the domain model according to the principles of DDD. The base classes of the model contain only the states that can be mutated functionally. All domain behaviors are modeled through functions that reside within the module that represents the aggregate.

Functions compose and that's precisely how we will chain sequence of domain behaviors to build bigger abstractions out of smaller ones. Here's a small function that values an Order. Note it returns a Kleisli, which essentially gives us a composition over monadic functions. So instead of composing a -> b and b -> c, which we do with normal function composition, we can do the same over a -> m b and b -> m c, where m is a monad. Composition with effects if you may say so.

def valueOrder = Kleisli[ProcessingStatus, Order, Order] {order =>
  val o = orderLineItems.set(
    order,
    setLineItemValues(order.lineItems)
  )
  o.lineItems.map(_.value).sequenceU match {
    case Some(_) => right(o)
    case _ => left("Missing value for items")
  }
}

But what does that buy us ? What exactly do we gain from these functional patterns ? It's the power to abstract over families of similar abstractions like applicatives and monads. Well, that may sound a bit rhetoric and it needs a separate post to justify their use. Stated simply, they encapsulate effects and side-effects of your computation so that you can focus on the domain behavior itself. Have a look at the process function below - it's actually a composition of monadic functions in action. But all the machinery that does the processing of effects and side-effects are abstracted within the Kleisli itself so that the user level implementation is simple and concise.

With Kleisli it's the power to compose over monadic functions. Every domain behavior has a chance of failure, which we model using the Either monad - here ProcessingStatus is just a type alias for this .. type ProcessingStatus[S] = \/[String, S]. Using the Kleisli, we don't have to write any code for handling failures. As you will see below, the composition is just like the normal functions - the design pattern takes care of alternate flows.

Once the Order is valued, we need to apply discounts to qualifying items. It's another behavior that follows the same pattern of implementation as valueOrder.

def applyDiscounts = Kleisli[ProcessingStatus, Order, Order] {order =>
  val o = orderLineItems.set(
    order,
    setLineItemValues(order.lineItems)
  )
  o.lineItems.map(_.discount).sequenceU match {
    case Some(_) => right(o)
    case _ => left("Missing discount for items")
  }
}

Finally we check out the Order ..

def checkOut = Kleisli[ProcessingStatus, Order, Order] {order =>
  val netOrderValue = order.lineItems.foldLeft(BigDecimal(0).some) {(s, i) => 
    s |+| (i.value |+| i.discount.map(d => Tags.Multiplication(BigDecimal(-1)) |+| Tags.Multiplication(d)))
  }
  right(orderNetValue.set(order, netOrderValue))
}

And here's the service method that composes all of the above domain behaviors into the big abstraction. We don't have any object to instantiate. Just plain function composition that results in an expression modeling the entire flow of events. And it's the cleanliness of abstraction that makes the code readable and succinct.

def process(order: Order) = {
  (valueOrder andThen 
     applyDiscounts andThen checkOut) =<< right(orderStatus.set(order, Validated))
}

In case you are interested in the full source code of this small example, feel free to take a peek at my github repo.

Functional Patterns in Domain Modeling - Immutable Aggregates and Functional Updates

In the last post I looked at a pattern that enforces constraints to ensure domain objects honor the domain rules. But what exactly is a domain object ? What should be the granularity of an object that my solution model should expose so that it makes sense to a domain user ? After all, the domain model should speak the language of the domain. We may have a cluster of entities modeling various concepts of the domain. But only some of them can be published as abstractions to the user of the model. The others can be treated as implementation artifacts and are best hidden under the covers of the published ones.

An aggregate in domain driven design is a published abstraction that provides a single point of interaction to a specific domain concept. Considering the classes I introduced in the last post, an Order is an aggregate. It encapsulates the details that an Order is composed of in the real world (well, only barely in this example, which is only for illustration purposes :-)).

Note an aggregate can consist of other aggregates - e.g. we have a Customer instance within an Order. Eric Evans in his book on Domain Driven Design provides an excellent discussion of what constitutes an Aggregate.

Functional Updates of Aggregates with Lens

This is not a post about Aggregates and how they fit in the realm of domain driven design. In this post I will talk about how to use some patterns to build immutable aggregates. Immutable data structures offer a lot of advantages, so build your aggregates ground up as immutable objects. Algebraic Data Type (ADT) is one of the patterns to build immutable aggregate objects. Primarily coming from the domain of functional programming, ADTs offer powerful techniques of pattern matching that help you match values against patterns and bind variables to successful matches. In Scala we use case classes as ADTs that give immutable objects out of the box ..

case class Order(orderNo: String, orderDate: Date, customer: Customer, 
  lineItems: Vector[LineItem], shipTo: ShipTo, netOrderValue: Option[BigDecimal] = None, 
  status: OrderStatus = Placed)

Like all good aggregates, we need to provide a single point of interaction to users. Of course we can access all properties using accessors of case classes. But what about updates ? We can update the orderNo of an order like this ..

val o = Order( .. )
o.copy(orderNo = newOrderNo)

which gives us a copy of the original order with the new order no. We don't mutate the original order. But anybody having some knowledge of Scala will realize that this becomes pretty clunky when we have to deal with nested object updation. e.g in the above case, ShipTo is defined as follows ..

case class Address(number: String, street: String, city: String, zip: String)
case class ShipTo(name: String, address: Address)

So, here you go in order to update the zip code of a ShipTo ..

val s = ShipTo("abc", Address("123", "Monroe Street", "Denver", "80233"))
s.copy(address = s.address.copy(zip = "80231"))

Not really pleasing and can go off bounds in comprehensibility pretty soon.

In our domain model we use an abstraction called a Lens for updating Aggregates. In very layman's terms, a lens is an encapsulated get and set combination. The get extracts a small part from a larger whole, while the set transforms the larger abstraction with a smaller part taken as a parameter.

case class Lens[A, B](get: A => B, set: (A, B) => A)

This is a naive definition of a Lens in Scala. Sophisticated lens designs go a long way to ensure proper abstraction and composition. scalaz provides one such implementation out of the box that exploits the similarity in structure between the get and the set to generalize the lens definition in terms of another abstraction named Store. As it happens so often in functional programming, Store happens to abstract yet another pattern called the Comonad. You can think of a Comonad as the inverse of a Monad. But in case you are more curious, and have wondered how lenses form "the Coalgebras for the Store Comonad", have a look at the 2 papers here and here.

Anyway for us mere domain modelers, we will use the Lens implementation as in scalaz .. here's a lens that helps us update the OrderStatus within an Order ..

val orderStatus = Lens.lensu[Order, OrderStatus] (
  (o, value) => o.copy(status = value),
  _.status
)

and use it as follows ..

val o = Order( .. )
orderStatus.set(o, Placed)

will change the status field of the Order to Placed. Let's have a look at some of the compositional properties of a lens which help us write readable code for functionally updating nested structures.

Composition of Lenses

First let's define some individual lenses ..

// lens for updating a ShipTo of an Order
val orderShipTo = Lens.lensu[Order, ShipTo] (
  (o, sh) => o.copy(shipTo = sh),
  _.shipTo
)

// lens for updating an address of a ShipTo
val shipToAddress = Lens.lensu[ShipTo, Address] (
  (sh, add) => sh.copy(address = add),
  _.address
)

// lens for updating a city of an address
val addressToCity = Lens.lensu[Address, String] (
  (add, c) => add.copy(city = c),
  _.city
)

And now we compose them to define a lens that directly updates the city of a ShipTo belonging to an Order ..

// compositionality FTW
def orderShipToCity = orderShipTo andThen shipToAddress andThen addressToCity

Now updating a city of a ShipTo in an Order is as simple and expressive as ..

val o = Order( .. )
orderShipToCity.set(o, "London")

The best part of using such compositional data structures is that it makes your domain model implementation readable and expressive to the users of your API. And yet your aggregate remains immutable.

Let's look at another use case when the nested object is a collection. scalaz offers partial lenses that you can use for such composition. Here's an example where we build a lens that updates the value member within a LineItem of an Order. A LineItem is defined as ..

case class LineItem(item: Item, quantity: BigDecimal, value: Option[BigDecimal] = None, 
  discount: Option[BigDecimal] = None)

and an Order has a collection of LineItems. Let's define a lens that updates the value within a LineItem ..

val lineItemValue = Lens.lensu[LineItem, Option[BigDecimal]] (
  (l, v) => l.copy(value = v),
  _.value
)

and then compose it with a partial lens that helps us update a specific item within a vector. Note how we convert our lineItemValue lens to a partial lens using the unary operator ~ ..

// a lens that updates the value in a specific LineItem within an Order 
def lineItemValues(i: Int) = ~lineItemValue compose vectorNthPLens(i)

Now we can use this composite lens to functionally update the value field of each of the items in a Vector of LineItems using some specific business rules ..

(0 to lis.length - 1).foldLeft(lis) {(s, i) => 
  val li = lis(i)
  lineItemValues(i).set(s, unitPrice(li.item).map(_ * li.quantity)).getOrElse(s)
}

In this post we saw how we can handle aggregates functionally and without any in-place mutation. This keeps the model pure and helps us implement domain models that has sane behavior even in concurrent settings without any explicit use of locks and semaphores. In the next post we will take a look at how we can use such compositional structures to make the domain model speak the ubiquitous language of the domain - another pattern recommended by Eric Evans in domain driven design.

Functional Patterns in Domain Modeling - The Specification Pattern

When you model a domain, you model its entities and behaviors. As Eric Evans mentions in his book Domain Driven Design, the focus is on the domain itself. The model that you design and implement must speak the ubiquitous language so that the essence of the domain is not lost in the myriads of incidental complexities that your implementation enforces. While being expressive the model needs to be extensible too. And when we talk about extensibility, one related attribute is compositionality.

Functions compose more naturally than objects and In this post I will use functional programming idioms to implement one of the patterns that form the core of domain driven design - the Specification pattern, whose most common use case is to implement domain validation. Eric's book on DDD says regarding the Specification pattern ..

It has multiple uses, but one that conveys the most basic concept is that a SPECIFICATION can test any object to see if it satisfies the specified criteria.

A specification is defined as a predicate, whereby business rules can be combined by chaining them together using boolean logic. So there's a concept of composition and we can talk about Composite Specification when we talk about this pattern. Various literature on DDD implement this using the Composite design pattern so commonly implemented using class hierarchies and composition. In this post we will use function composition instead.

Specification - Where ?

One of the very common confusions that we have when we design a model is where to keep the validation code of an aggregate root or any entity, for that matter.

• Should we have the validation as part of the entity ? No, it makes the entity bloated. Also validations may vary based on some context, while the core of the entity remains the same.
• Should we have validations as part of the interface ? May be we consume JSON and build entities out of it. Indeed some validations can belong to the interface and don't hesitate to put them there.
• But the most interesting validations are those that belong to the domain layer. They are business validations (or specifications), which Eric Evans defines as something that "states a constraint on the state of another object". They are business rules which the entity needs to honor in order to proceed to the next stage of processing.

We consider the following simple example. We take an Order entity and the model identifies the following domain "specifications" that a new Order must satisfy before being thrown into the processing pipeline:

1. it must be a valid order obeying the constraints that the domain requires e.g. valid date, valid no of line items etc.
2. it must be approved by a valid approving authority - only then it proceeds to the next stage of the pipeline
3. customer status check must be passed to ensure that the customer is not black-listed
4. the line items from the order must be checked against inventory to see if the order can be fulfilled

These are the separate steps that are to be done in sequence by the order processing pipeline as pre-order checks before the actual order is ready for fulfilment. A failure in any of them takes the order out of the pipeline and the process stops there. So the model that we will design needs to honor the sequence as well as check all constraints that need to be done as part of every step.

An important point to note here is that none of the above steps mutate the order - so every specification gets a copy of the original Order object as input, on which it checks some domain rules and determines if it's suitable to be passed to the next step of the pipeline.

Jumping on to the implementation ..

Let's take down some implementation notes from what we learnt above ..

• The Order can be an immutable entity at least for this sequence of operations
• Every specification needs an order, can we can pull some trick out of our hat which prevents this cluttering of API by passing an Order instance to every specification in the sequence ?
• Since we plan to use functional programming principles, how can we model the above sequence as an expression so that our final result still remains composable with the next process of order fulfilment (which we will discuss in a future post) ?
• All these functions look like having similar signatures - we need to make them compose with each other

Before I present more of any explanation or theory, here are the basic building blocks which will implement the notes that we took after talking to the domain experts ..

type ValidationStatus[S] = \/[String, S]

type ReaderTStatus[A, S] = ReaderT[ValidationStatus, A, S]

object ReaderTStatus extends KleisliInstances with KleisliFunctions {
  def apply[A, S](f: A => ValidationStatus[S]): ReaderTStatus[A, S] = kleisli(f)
}

ValidationStatus defines the type that we will return from each of the functions. It's either some status S or an error string that explains what went wrong. It's actually an Either type (right biased) as implemented in scalaz.

One of the things which we thought will be cool is to avoid repeating the Order parameter for every method when we invoke the sequence. And one of the idioamtic ways of doing it is to use the Reader monad. But here we already have a monad - \/ is a monad. So we need to stack them together using a monad transformer. ReaderT does this job and ReaderTStatus defines the type that somehow makes our life easier by combining the two of them.

The next step is an implementation of ReaderTStatus, which we do in terms of another abstraction called Kleisli. We will use the scalaz library for this, which implements ReaderT in terms of Kleisli. I will not go into the details of this implementation - in case you are curious, refer to this excellent piece by Eugene.

So, how does one sample specification look like ?

Before going into that, here are some basic abstractions, grossly simplified only for illustration purposes ..

// the base abstraction
sealed trait Item {
  def itemCode: String
}

// sample implementations
case class ItemA(itemCode: String, desc: Option[String], 
  minPurchaseUnit: Int) extends Item
case class ItemB(itemCode: String, desc: Option[String], 
  nutritionInfo: String) extends Item

case class LineItem(item: Item, quantity: Int)

case class Customer(custId: String, name: String, category: Int)

// a skeleton order
case class Order(orderNo: String, orderDate: Date, customer: Customer, 
  lineItems: List[LineItem])

And here's a specification that checks some of the constraints on the Order object ..

// a basic validation
private def validate = ReaderTStatus[Order, Boolean] {order =>
  if (order.lineItems isEmpty) left(s"Validation failed for order $order") 
  else right(true)
}

It's just for illustration and does not contain much domain rules. The important part is how we use the above defined types to implement the function. Order is not an explicit argument to the function - it's curried. The function returns a ReaderTStatus, which itself is a monad and hence allows us to sequence in the pipeline with other specifications. So we get the requirement of sequencing without breaking out of the expression oriented programming style.

Here are a few other specifications based on the domain knowledge that we have gathered ..

private def approve = ReaderTStatus[Order, Boolean] {order =>
  right(true)
}

private def checkCustomerStatus(customer: Customer) = ReaderTStatus[Order, Boolean] {order =>
  right(true)
}

private def checkInventory = ReaderTStatus[Order, Boolean] {order =>
  right(true)
}

Wiring them together

But how do we wire these pieces together so that we have the sequence of operations that the domain mandates and yet all goodness of compositionality in our model ? It's actually quite easy since we have already done the hard work of defining the appropriate types that compose ..

Here's the isReadyForFulfilment method that defines the composite specification and invokes all the individual specifications in sequence using for-comprehension, which, as you all know does the monadic bind in Scala and gives us the final expression that needs to be evaluated for the Order supplied.

def isReadyForFulfilment(order: Order) = {
  val s = for {
    _ <- validate
    _ <- approve
    _ <- checkCustomerStatus(order.customer)
    c <- checkInventory
  } yield c
  s(order)
}

So we have the monadic bind implement the sequencing without breaking the compositionality of the abstractions. In the next instalment we will see how this can be composed with the downstream processing of the order that will not only read stuff from the entity but mutate it too, of course in a functional way.

A Sketch as the Query Model of an EventSourced System

In my last post I discussed the count-min sketch data structure that can be used to process data streams using sub-linear space. In this post I will continue with some of my thoughts on how count-min sketches can be used in a typical event sourced application architecture. An event sourcing system typically has a query model which provides a read only view of how all the events are folded to provide a coherent view of the system. I have seen applications where the query model is typically rendered from a relational database. And the queries can take a lot of time to be successfully processed and displayed to the user if the data volume is huge. And when we are talking about Big Data, this is not a very uncommon use case.

Instead of rendering the query from the RDBMS, quite a few types of them can be rendered from a count-min sketch using sub-linear space. Consider the use case where you need to report the highest occuring user-ids in a Twitter stream. The stream is continuous, huge and non ending and you get to see each item once. So you get each item from where you parse out the user-id occurring in it and update the sketch. So each entry of the sketch contains the frequency of the user-id that hashes to that slot. And we can take the minimum of all the slots to which a user-id hashes to, in order to get the frequency of that user-id. The details of how this works can be found in my last post.

Consider the case where we need to find the heavy-hitters - those user-ids whose frequency exceeds a pre-determined threshold. For that, in addition to the sketch we can also maintain a data structure like heap or tree where we update the top-k heavy hitters. When a user-id appears, we update the sketch, get its estimated frequency from the sketch and if it exceeds the threshold, also record it in the data structure. So at any point in time we can probe this accessary data structure to get the current heavy-hitters. Spark examples contain a sample implementation of this heavy hitters query from a Twitter stream using the CountMinSketchMonoid of Algebird.

Can this be a viable approach of implementing the query model in an event sourced system if the use case fits the approximation query approach ? It can be faster, relatively cheap in space and can prove to be responsive enough to be displayed in dashboards in the form of charts or graphs.

Count-Min Sketch - A Data Structure for Stream Mining Applications

In today's age of Big Data, streaming is one of the techniques for low latency computing. Besides the batch processing infrastructure of map/reduce paradigm, we are seeing a plethora of ways in which streaming data is processed at near real time to cater to some specific kinds of applications. Libraries like Storm, Samza and Spark belong to this genre and are starting to get their share of user base in the industry today.

This post is not about Spark, Storm or Samza. It's about a data structure which is one of the relatively new entrants in the domain of stream processing, which is simple to implement, but has already proved to be of immense use in serving a certain class of queries over huge streams of data. I have been doing some readings about the application of such structures and thought of sharing them with the readers of my blog.

Using Sublinear Space

Besides data processing, these tools also support data mining over streams that include serving specialized queries over data using limited space and time. Ok, so once we store all data as they come we can always serve queries with O(n) space. But since we are talking about huge data streams, it may not even be possible to run algorithms on the full set of data - it simply will be too expensive. Even if we have the entire set of data in a data warehouse, the processing of the entire data set may take time and consume resources that we cannot afford to have, considering the fee charged under the evolving models of using the platform-as-a-service within the cloud based infrastructure. Also the fact that these algorithms will be working on data streams, there's a high likelihood that they will get to see these data only in a single pass. The bottom line is that we need to have algorithms that work on sub-linear space.

Working on sublinear space implies that we don't get to store or see all data - hence an obvious conclusion from this will be the fact that we also don't get to deliver an accurate answer to some queries. We rely on some approximation techniques and deliver an accuracy with a reasonably high probability bound. We don't store all data, instead we store a lossy compressed representation of the data and deliver user queries from this subset instead of the entire set.

One widely used technique for storing a subset of data is through Random Sampling, where the data stored is selected through some stochastic mechanism. There are various ways to determine which data we select for storing and how we build the estimator for querying the data. There are pros and cons with this approach, but it's one of the simplest ways to do approximation based queries on streaming data.

There are a few other options like Histograms and Wavelet based synopses. But one of the most interesting data structures that have been developed in recent times is the Sketch, which uses summary based techniques for delivering approximation queries, gets around the typical problems that sampling techniques have and are highly parallelizable in practice.

An important class of sketch is one where the sketch vector (which is the summary information) is a linear transform of the input vector. So if we model the input as a vector we can multiply it by a sketch matrix to obtain the sketch vector that contains the synopses data that we can use for serving approximation queries. Here's a diagrammatic representation of the sketch as a linear transform of the input data.

Count-Min Sketch

One of the most popular forms of the sketch data structure is the Count-Min Sketch introduced by Muthukrishnan and Cormode in 2003. The idea is quite simple and the data structure is based on probabilistic algorithms to serve various types of queries on streaming data. The data structure is parameterized by two factors - ε and δ, where the error in answering the query is within a factor of ε with probability δ. So you can tune these parameters based on the space that you can afford and accordingly amortize the accuracy of results that the data structure can serve you.

Consider this situation where you have a stream of data (typically modeled as a vector) like updates to stock quotes in a financial processing system arriving continuously that you need to process and report statistical queries on a real time basis.

We model the data stream as a vector a[1 .. n] and the updates received at time t are of the form (i_t, c_t) which mean that the stock quote for a[i_t] has been incremented by c_t. There are various models in which this update can appear as discussed in Data Streams: Algorithms and Applications by Muthukrishnan which includes negative updates as well and the data structure can be tuned to handle each of these variants.

The core of the data structure is a 2 dimensional array count[w, d] that stores the synopses of the original vector and which is used to report results of queries using approximation techniques. Hence the total space requirement of the data structure is (w * d). We can bound each of w and d in terms of our parameters ε and δ and control the level of accuracy that we want our data structure to serve.

The data structure uses hashing techniques to process these updates and report queries using sublinear space. So assume we have d pairwise-independent hash functions {h₁ .. h_d} that hash each of our inputs to the range (1 .. w). Just for the more curious mind, pairwise independence is a method to construct a universal hash family, a technique that ensures lower number of collisions in the hash implementation.

When an update (i_t, c_t) comes for the stream, we hash a[i_t] through each of the hash functions h₁ .. h_d and increment each of the w entries in the array that they hash to.

for i = 1 to d
  v = h(i)(a[i_t]) // v is between 1 and w
  count[i, v] += c_t // increment the cell count by c_t
end

At any point in time if we want to know the approximate value of an element a[i] of the vector a, we can get it from computing the minimum of all values in each of the d cells of count where i hashes to. This can be proved formally. But the general intuition is that since we are using hash functions there's always a possibility of multiple i's colliding on to the same cell and contributing additively to the value of the cell. Hence the minimum among all hash values is the closest candidate to give us the correct result for the query.

The figure above shows the processing of the updates in a Count-Min sketch. This is typically called the Point Query that returns an approximation of a[i]. Similarly we can use a Count-Min sketch to get approximation queries for ranges which is typically a summation over multiple point queries. Another interesting application is to serve inner product queries where the data structure is used to query inner products of 2 vectors, a typical application of this being the estimation of join sizes in relational query processing. The paper Statistical Analysis of Sketch Estimators gives all details of how to use sketching as a technique for this.

Count-Min sketches have some great properties which make them a very useful data structure when processing distributed streams. They have associativity properties and can be modelled as monoids and hence terribly performant in a distributed environment where you can parallelize sketch operations. In a future post I will discuss some implementation techniques and how we can use count-min sketches to serve some useful applications over data streams. Meanwhile Twitter's algebird and ClearSpring's stream-lib offer implementations of Count-Min sketch and various other data structures applicable for stream mining applications.