Apart from tracking the effect of randomness, a computational intelligence algorithm requires some additional information:

  • The optimization scheme (Opt) required, either minimization or maximization
  • The fitness function evaluation instance, Eval, which calculates the quality of the candidate solution

The result is a function, which uses a predefined set of values to generate a RVar computation to which randomness still needs to be applied. The function is therefore of the shape

(Opt,Eval[A]) => RVar[B]

The predefined set of values required for input is also referred to as the “environment”. Given that the environment is decided once, at the beginning of the execution process, we can factor out this common parameter using a Kleisli arrow, on this case the ReaderT monad transformer is applicable, resulting in the final type of:

ReaderT[RVar, (Opt,Eval[A]), B]

Because monad transformers are monads themselves, we can freely compose different Step instances to create a larger computation. The larger computation, being termed the “algorithm”. Due to the sequencing action of monads, Step instances are analogous to steps in algorithm pseudo-code.

Now that the intention of Step is clearer, lets have a look at some usage. Keep in mind that even though the final “shape” of the Step is defined, this does not limit the manner in which we can create Step instances. Within a PSO, each particle has several actions applied to it, in order to create a new particle that occupies a different location within the problem search space.

The canonical PSO algorithm iteratively alters the position (and state data) of each particle to create a new particle for each within the collection. The new collection essentially replaces the original collection and this process repeats until some kind of stopping condition is met. Stopping condition logic is intentionally not included within the Step. The type for a PSO is then

List[Particle[S,A]] => Particle[S,A] => Step[RVar,(Opt,Eval[A]),Particle[A]]

and the algorithm definition is:

scala> def pso[S](w: Double, c1: Double, c2: Double,
     |   cognitive: Guide[S,Double], social: Guide[S,Double]
     | )(implicit M: HasMemory[S,Double], V: HasVelocity[S,Double]
     | ): NonEmptyList[Particle[S,Double]] => Particle[S,Double] => Step[Double,Particle[S,Double]]  =
     |   collection => x => for {
     |     cog     <- cognitive(collection, x)
     |     soc     <- social(collection, x)
     |     vel     <- stdVelocity(x, soc, cog, w, c1, c2)
     |     p       <- stdPosition(x, vel)
     |     p2      <- evalParticle(p)
     |     p3      <- updateVelocity(p2, vel)
     |     updated <- updatePBest(p3)
     |   } yield updated
pso: [S](w: Double, c1: Double, c2: Double, cognitive: cilib.pso.Guide[S,Double], social: cilib.pso.Guide[S,Double])(implicit M: cilib.HasMemory[S,Double], implicit V: cilib.HasVelocity[S,Double])scalaz.NonEmptyList[cilib.pso.Particle[S,Double]] => (cilib.pso.Particle[S,Double] => cilib.Step[Double,cilib.pso.Particle[S,Double]])

That sure looks complex! Sure, there are a few things going on there, but it certainly is not complex - the syntax just makes the intention a little hidden due to the verbosity. The pso function defines that there are some parameters that it requires for the stdVelocity function, namely w, c1, c2. Next it requires two individual Guide instances which are the particle attactors for the velocity update equation. In the next parameter group is an implicit parameter that is constraining the types of data elements pso can work with. In this case, the pso requires that all Particle instances (which is just an alias for the type Entity[S,Position[Double]]) must have a HasMemory and HasVelocity instance available for the compiler to provide. As mentioned previously, in the section about lenses and optics, these classy lens instances prevent Entity types that do not have a memory nor a velocity present in their state parameter type. Again, we don’t know what the value of S is at the definition level, it will be made concrete when we actually use the pso function.

Every function within the for-comprehension is a function that yields a Step instance of the type Step[Double,Particle[S,Double]] (in this usage example). The individual Steps are then composed into a larger composition that ultimately results in the creation of a new piece of data: the particle which replaces the particle identified by the parameter x.


Algorithms can be executed either synchronously or asynchronously. Given an algorithm definition like the definition of pso above, it may be passed into an iteration function which converts the signature of pso function into the shape:

List[Particle[S,A]] => RVar[A,Particle[S,A]]

It is important to note that the manner of execution constrains the amount of parallelism that may be applied to the iteration process. The synchronous scheme can be completely parallelized as there is no reliance on the currently building collection. The asynchronous process on the other hand builds the new collection from a combination of the current and new collections. As a result of the manner in which the asynchronous process executes, parallelism is not possible due to the shared state that is managed during the execution process.