I've been programming in more functional-style languages and have gotten to appreciate things like tuples, and higher-order functions such as maps, and folds/aggregates. Do either VHDL or Verilog have any of these kinds of constructs?
For example, is there any way to do even simple things like
divByThreeCount = count (\x -> x `mod` 3 == 0) myArray
or
myArray2 = map (\x -> x `mod` 3) myArray
or even better yet let me define my own higher-level constructs recursively, in either of these languages?
I think you're right that there is a clash between the imperative style of HDLs and the more functional aspects of combinatorial circuits. Describing parallel circuits with languages which are linear in nature is odd, but I think a full blown functional HDL would be a disaster.
The thing with functional languages (I find) is that it's very easy to write code which takes huge resources, either in time, memory, or processing power. That's their power; they can express complexity quite succinctly, but they do so at the expense of resource management. Put that in an HDL and I think you'll have a lot of large, power hungry designs when synthesised.
Take your map example:
myArray2 = map (\x -> x `mod` 3) myArray
How would you like that to synthesize? To me you've described a modulo operator per element of the array. Ignoring the fact that modulo isn't cheap, was that what you intended, and how would you change it if it wasn't? If I start breaking that function up in some way so that I can say "instantiate a single operator and use it multiple times" I lost a lot of the power of functional languages.
...and then we've got retained state. Retained state is everywhere in hardware. You need it. You certainly wouldn't use a purely functional language.
That said, don't throw away your functional design patterns. Combinatorial processes (VHDL) and "always blocks" (Verilog) can be viewed as functions that apply themselves to the data presented at their input. Pipelines can be viewed as chains of functions. Often the way you structure a design looks functional, and can share a lot with the "Actor" design pattern that's popular in Erlang.
So is there stuff to learn from functional programming? Certainly. Do I wish VHDL and Verilog took more from functional languages? Sometimes. The trouble is functional languages get too high level too quickly. If I can't distinguish between "use one instance of f() many times" and "use many instances of f()" then it doesn't do what a Hardware Description Language must do... describe hardware.
Have a look at http://clash-lang.org for an example of a higher-order language that is transpiled into VHDL/Verilog. It allows functions to be passed as arguments, currying etc., even a limited set of recursive data structures (Vec). As you would expect the pure moore function instantiates a stateful Moore machine given step/output functions and a start state. It has this signature:
moore :: (s -> i -> s) -> (s -> o) -> s -> Signal i -> Signal o
Signals model the time-evolved values in sequential logic.
BlueSpec is used by a small com. known by few people only - INTEL
It’s an extension of SystemVerilog (SV) and called BSV
But it’s NOT open - I don’t think you can try use or even learn it without paying BlueSpec.com BIG money
There’s also Lava that’s used by Xilinx but I don’t know if you can use it directly.
Note:
In Vhdl functions (and maybe Verilog too)
CAN’T have time delay (you can’t ask function to wait for clk event)
Vhdl has a std. lib. for complex num. cal.
But in Vhdl you can change or overload a system function like add (“+”) and define your implementation.
You can declare vector or matrix add sub mul or div (+ - * /) function,
use generic size (any) and recursive declarations - most synthesizers will "understand you" and do what you’ve asked.
I read the mapreduce at http://en.wikipedia.org/wiki/MapReduce ,understood the example of how to get the count of a "word" in many "documents". However I did not understand the following line:
Thus the MapReduce framework transforms a list of (key, value) pairs into a list of values. This behavior is different from the functional programming map and reduce combination, which accepts a list of arbitrary values and returns one single value that combines all the values returned by map.
Can someone elaborate on the difference again(MapReduce framework VS map and reduce combination)? Especially, what does the reduce functional programming do?
Thanks a great deal.
The main difference would be that MapReduce is apparently patentable. (Couldn't help myself, sorry...)
On a more serious note, the MapReduce paper, as I remember it, describes a methodology of performing calculations in a massively parallelised fashion. This methodology builds upon the map / reduce construct which was well known for years before, but goes beyond into such matters as distributing the data etc. Also, some constraints are imposed on the structure of data being operated upon and returned by the functions used in the map-like and reduce-like parts of the computation (the thing about data coming in lists of key/value pairs), so you could say that MapReduce is a massive-parallelism-friendly specialisation of the map & reduce combination.
As for the Wikipedia comment on the function being mapped in the functional programming's map / reduce construct producing one value per input... Well, sure it does, but here there are no constraints at all on the type of said value. In particular, it could be a complex data structure like perhaps a list of things to which you would again apply a map / reduce transformation. Going back to the "counting words" example, you could very well have a function which, for a given portion of text, produces a data structure mapping words to occurrence counts, map that over your documents (or chunks of documents, as the case may be) and reduce the results.
In fact, that's exactly what happens in this article by Phil Hagelberg. It's a fun and supremely short example of a MapReduce-word-counting-like computation implemented in Clojure with map and something equivalent to reduce (the (apply + (merge-with ...)) bit -- merge-with is implemented in terms of reduce in clojure.core). The only difference between this and the Wikipedia example is that the objects being counted are URLs instead of arbitrary words -- other than that, you've got a counting words algorithm implemented with map and reduce, MapReduce-style, right there. The reason why it might not fully qualify as being an instance of MapReduce is that there's no complex distribution of workloads involved. It's all happening on a single box... albeit on all the CPUs the box provides.
For in-depth treatment of the reduce function -- also known as fold -- see Graham Hutton's A tutorial on the universality and expressiveness of fold. It's Haskell based, but should be readable even if you don't know the language, as long as you're willing to look up a Haskell thing or two as you go... Things like ++ = list concatenation, no deep Haskell magic.
Using the word count example, the original functional map() would take a set of documents, optionally distribute subsets of that set, and for each document emit a single value representing the number of words (or a particular word's occurrences) in the document. A functional reduce() would then add up the global counts for all documents, one for each document. So you get a total count (either of all words or a particular word).
In MapReduce, the map would emit a (word, count) pair for each word in each document. A MapReduce reduce() would then add up the count of each word in each document without mixing them into a single pile. So you get a list of words paired with their counts.
MapReduce is a framework built around splitting a computation into parallelizable mappers and reducers. It builds on the familiar idiom of map and reduce - if you can structure your tasks such that they can be performed by independent mappers and reducers, then you can write it in a way which takes advantage of a MapReduce framework.
Imagine a Python interpreter which recognized tasks which could be computed independently, and farmed them out to mapper or reducer nodes. If you wrote
reduce(lambda x, y: x+y, map(int, ['1', '2', '3']))
or
sum([int(x) for x in ['1', '2', '3']])
you would be using functional map and reduce methods in a MapReduce framework. With current MapReduce frameworks, there's a lot more plumbing involved, but it's the same concept.
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I've read the Wikipedia article on reactive programming. I've also read the small article on functional reactive programming. The descriptions are quite abstract.
What does functional reactive programming (FRP) mean in practice?
What does reactive programming (as opposed to non-reactive programming?) consist of?
My background is in imperative/OO languages, so an explanation that relates to this paradigm would be appreciated.
If you want to get a feel for FRP, you could start with the old Fran tutorial from 1998, which has animated illustrations. For papers, start with Functional Reactive Animation and then follow up on links on the publications link on my home page and the FRP link on the Haskell wiki.
Personally, I like to think about what FRP means before addressing how it might be implemented.
(Code without a specification is an answer without a question and thus "not even wrong".)
So I don't describe FRP in representation/implementation terms as Thomas K does in another answer (graphs, nodes, edges, firing, execution, etc).
There are many possible implementation styles, but no implementation says what FRP is.
I do resonate with Laurence G's simple description that FRP is about "datatypes that represent a value 'over time' ".
Conventional imperative programming captures these dynamic values only indirectly, through state and mutations.
The complete history (past, present, future) has no first class representation.
Moreover, only discretely evolving values can be (indirectly) captured, since the imperative paradigm is temporally discrete.
In contrast, FRP captures these evolving values directly and has no difficulty with continuously evolving values.
FRP is also unusual in that it is concurrent without running afoul of the theoretical & pragmatic rats' nest that plagues imperative concurrency.
Semantically, FRP's concurrency is fine-grained, determinate, and continuous.
(I'm talking about meaning, not implementation. An implementation may or may not involve concurrency or parallelism.)
Semantic determinacy is very important for reasoning, both rigorous and informal.
While concurrency adds enormous complexity to imperative programming (due to nondeterministic interleaving), it is effortless in FRP.
So, what is FRP?
You could have invented it yourself.
Start with these ideas:
Dynamic/evolving values (i.e., values "over time") are first class values in themselves. You can define them and combine them, pass them into & out of functions. I called these things "behaviors".
Behaviors are built up out of a few primitives, like constant (static) behaviors and time (like a clock), and then with sequential and parallel combination. n behaviors are combined by applying an n-ary function (on static values), "point-wise", i.e., continuously over time.
To account for discrete phenomena, have another type (family) of "events", each of which has a stream (finite or infinite) of occurrences. Each occurrence has an associated time and value.
To come up with the compositional vocabulary out of which all behaviors and events can be built, play with some examples. Keep deconstructing into pieces that are more general/simple.
So that you know you're on solid ground, give the whole model a compositional foundation, using the technique of denotational semantics, which just means that (a) each type has a corresponding simple & precise mathematical type of "meanings", and (b) each primitive and operator has a simple & precise meaning as a function of the meanings of the constituents.
Never, ever mix implementation considerations into your exploration process. If this description is gibberish to you, consult (a) Denotational design with type class morphisms, (b) Push-pull functional reactive programming (ignoring the implementation bits), and (c) the Denotational Semantics Haskell wikibooks page. Beware that denotational semantics has two parts, from its two founders Christopher Strachey and Dana Scott: the easier & more useful Strachey part and the harder and less useful (for software design) Scott part.
If you stick with these principles, I expect you'll get something more-or-less in the spirit of FRP.
Where did I get these principles? In software design, I always ask the same question: "what does it mean?".
Denotational semantics gave me a precise framework for this question, and one that fits my aesthetics (unlike operational or axiomatic semantics, both of which leave me unsatisfied).
So I asked myself what is behavior?
I soon realized that the temporally discrete nature of imperative computation is an accommodation to a particular style of machine, rather than a natural description of behavior itself.
The simplest precise description of behavior I can think of is simply "function of (continuous) time", so that's my model.
Delightfully, this model handles continuous, deterministic concurrency with ease and grace.
It's been quite a challenge to implement this model correctly and efficiently, but that's another story.
In pure functional programming, there are no side-effects. For many types of software (for example, anything with user interaction) side-effects are necessary at some level.
One way to get side-effect like behavior while still retaining a functional style is to use functional reactive programming. This is the combination of functional programming, and reactive programming. (The Wikipedia article you linked to is about the latter.)
The basic idea behind reactive programming is that there are certain datatypes that represent a value "over time". Computations that involve these changing-over-time values will themselves have values that change over time.
For example, you could represent the mouse coordinates as a pair of integer-over-time values. Let's say we had something like (this is pseudo-code):
x = <mouse-x>;
y = <mouse-y>;
At any moment in time, x and y would have the coordinates of the mouse. Unlike non-reactive programming, we only need to make this assignment once, and the x and y variables will stay "up to date" automatically. This is why reactive programming and functional programming work so well together: reactive programming removes the need to mutate variables while still letting you do a lot of what you could accomplish with variable mutations.
If we then do some computations based on this the resulting values will also be values that change over time. For example:
minX = x - 16;
minY = y - 16;
maxX = x + 16;
maxY = y + 16;
In this example, minX will always be 16 less than the x coordinate of the mouse pointer. With reactive-aware libraries you could then say something like:
rectangle(minX, minY, maxX, maxY)
And a 32x32 box will be drawn around the mouse pointer and will track it wherever it moves.
Here is a pretty good paper on functional reactive programming.
An easy way of reaching a first intuition about what it's like is to imagine your program is a spreadsheet and all of your variables are cells. If any of the cells in a spreadsheet change, any cells that refer to that cell change as well. It's just the same with FRP. Now imagine that some of the cells change on their own (or rather, are taken from the outside world): in a GUI situation, the position of the mouse would be a good example.
That necessarily misses out rather a lot. The metaphor breaks down pretty fast when you actually use a FRP system. For one, there are usually attempts to model discrete events as well (e.g. the mouse being clicked). I'm only putting this here to give you an idea what it's like.
To me it is about 2 different meanings of symbol =:
In math x = sin(t) means, that x is different name for sin(t). So writing x + y is the same thing as sin(t) + y. Functional reactive programming is like math in this respect: if you write x + y, it is computed with whatever the value of t is at the time it's used.
In C-like programming languages (imperative languages), x = sin(t) is an assignment: it means that x stores the value of sin(t) taken at the time of the assignment.
OK, from background knowledge and from reading the Wikipedia page to which you pointed, it appears that reactive programming is something like dataflow computing but with specific external "stimuli" triggering a set of nodes to fire and perform their computations.
This is pretty well suited to UI design, for example, in which touching a user interface control (say, the volume control on a music playing application) might need to update various display items and the actual volume of audio output. When you modify the volume (a slider, let's say) that would correspond to modifying the value associated with a node in a directed graph.
Various nodes having edges from that "volume value" node would automatically be triggered and any necessary computations and updates would naturally ripple through the application. The application "reacts" to the user stimulus. Functional reactive programming would just be the implementation of this idea in a functional language, or generally within a functional programming paradigm.
For more on "dataflow computing", search for those two words on Wikipedia or using your favorite search engine. The general idea is this: the program is a directed graph of nodes, each performing some simple computation. These nodes are connected to each other by graph links that provide the outputs of some nodes to the inputs of others.
When a node fires or performs its calculation, the nodes connected to its outputs have their corresponding inputs "triggered" or "marked". Any node having all inputs triggered/marked/available automatically fires. The graph might be implicit or explicit depending on exactly how reactive programming is implemented.
Nodes can be looked at as firing in parallel, but often they are executed serially or with limited parallelism (for example, there may be a few threads executing them). A famous example was the Manchester Dataflow Machine, which (IIRC) used a tagged data architecture to schedule execution of nodes in the graph through one or more execution units. Dataflow computing is fairly well suited to situations in which triggering computations asynchronously giving rise to cascades of computations works better than trying to have execution be governed by a clock (or clocks).
Reactive programming imports this "cascade of execution" idea and seems to think of the program in a dataflow-like fashion but with the proviso that some of the nodes are hooked to the "outside world" and the cascades of execution are triggered when these sensory-like nodes change. Program execution would then look like something analogous to a complex reflex arc. The program may or may not be basically sessile between stimuli or may settle into a basically sessile state between stimuli.
"non-reactive" programming would be programming with a very different view of the flow of execution and relationship to external inputs. It's likely to be somewhat subjective, since people will likely be tempted to say anything that responds to external inputs "reacts" to them. But looking at the spirit of the thing, a program that polls an event queue at a fixed interval and dispatches any events found to functions (or threads) is less reactive (because it only attends to user input at a fixed interval). Again, it's the spirit of the thing here: one can imagine putting a polling implementation with a fast polling interval into a system at a very low level and program in a reactive fashion on top of it.
After reading many pages about FRP I finally came across this enlightening writing about FRP, it finally made me understand what FRP really is all about.
I quote below Heinrich Apfelmus (author of reactive banana).
What is the essence of functional reactive programming?
A common answer would be that “FRP is all about describing a system in
terms of time-varying functions instead of mutable state”, and that
would certainly not be wrong. This is the semantic viewpoint. But in
my opinion, the deeper, more satisfying answer is given by the
following purely syntactic criterion:
The essence of functional reactive programming is to specify the dynamic behavior of a value completely at the time of declaration.
For instance, take the example of a counter: you have two buttons
labelled “Up” and “Down” which can be used to increment or decrement
the counter. Imperatively, you would first specify an initial value
and then change it whenever a button is pressed; something like this:
counter := 0 -- initial value
on buttonUp = (counter := counter + 1) -- change it later
on buttonDown = (counter := counter - 1)
The point is that at the time of declaration, only the initial value
for the counter is specified; the dynamic behavior of counter is
implicit in the rest of the program text. In contrast, functional
reactive programming specifies the whole dynamic behavior at the time
of declaration, like this:
counter :: Behavior Int
counter = accumulate ($) 0
(fmap (+1) eventUp
`union` fmap (subtract 1) eventDown)
Whenever you want to understand the dynamics of counter, you only have
to look at its definition. Everything that can happen to it will
appear on the right-hand side. This is very much in contrast to the
imperative approach where subsequent declarations can change the
dynamic behavior of previously declared values.
So, in my understanding an FRP program is a set of equations:
j is discrete: 1,2,3,4...
f depends on t so this incorporates the possiblilty to model external stimuli
all state of the program is encapsulated in variables x_i
The FRP library takes care of progressing time, in other words, taking j to j+1.
I explain these equations in much more detail in this video.
EDIT:
About 2 years after the original answer, recently I came to the conclusion that FRP implementations have another important aspect. They need to (and usually do) solve an important practical problem: cache invalidation.
The equations for x_i-s describe a dependency graph. When some of the x_i changes at time j then not all the other x_i' values at j+1 need to be updated, so not all the dependencies need to be recalculated because some x_i' might be independent from x_i.
Furthermore, x_i-s that do change can be incrementally updated. For example let's consider a map operation f=g.map(_+1) in Scala, where f and g are List of Ints. Here f corresponds to x_i(t_j) and g is x_j(t_j). Now if I prepend an element to g then it would be wasteful to carry out the map operation for all the elements in g. Some FRP implementations (for example reflex-frp) aim to solve this problem. This problem is also known as incremental computing.
In other words, behaviours (the x_i-s ) in FRP can be thought as cache-ed computations. It is the task of the FRP engine to efficiently invalidate and recompute these cache-s (the x_i-s) if some of the f_i-s do change.
The paper Simply efficient functional reactivity by Conal Elliott (direct PDF, 233 KB) is a fairly good introduction. The corresponding library also works.
The paper is now superceded by another paper, Push-pull functional reactive programming (direct PDF, 286 KB).
Disclaimer: my answer is in the context of rx.js - a 'reactive programming' library for Javascript.
In functional programming, instead of iterating through each item of a collection, you apply higher order functions (HoFs) to the collection itself. So the idea behind FRP is that instead of processing each individual event, create a stream of events (implemented with an observable*) and apply HoFs to that instead. This way you can visualize the system as data pipelines connecting publishers to subscribers.
The major advantages of using an observable are:
i) it abstracts away state from your code, e.g., if you want the event handler to get fired only for every 'n'th event, or stop firing after the first 'n' events, or start firing only after the first 'n' events, you can just use the HoFs (filter, takeUntil, skip respectively) instead of setting, updating and checking counters.
ii) it improves code locality - if you have 5 different event handlers changing the state of a component, you can merge their observables and define a single event handler on the merged observable instead, effectively combining 5 event handlers into 1. This makes it very easy to reason about what events in your entire system can affect a component, since it's all present in a single handler.
An Observable is the dual of an Iterable.
An Iterable is a lazily consumed sequence - each item is pulled by the iterator whenever it wants to use it, and hence the enumeration is driven by the consumer.
An observable is a lazily produced sequence - each item is pushed to the observer whenever it is added to the sequence, and hence the enumeration is driven by the producer.
Dude, this is a freaking brilliant idea! Why didn't I find out about this back in 1998? Anyway, here's my interpretation of the Fran tutorial. Suggestions are most welcome, I am thinking about starting a game engine based on this.
import pygame
from pygame.surface import Surface
from pygame.sprite import Sprite, Group
from pygame.locals import *
from time import time as epoch_delta
from math import sin, pi
from copy import copy
pygame.init()
screen = pygame.display.set_mode((600,400))
pygame.display.set_caption('Functional Reactive System Demo')
class Time:
def __float__(self):
return epoch_delta()
time = Time()
class Function:
def __init__(self, var, func, phase = 0., scale = 1., offset = 0.):
self.var = var
self.func = func
self.phase = phase
self.scale = scale
self.offset = offset
def copy(self):
return copy(self)
def __float__(self):
return self.func(float(self.var) + float(self.phase)) * float(self.scale) + float(self.offset)
def __int__(self):
return int(float(self))
def __add__(self, n):
result = self.copy()
result.offset += n
return result
def __mul__(self, n):
result = self.copy()
result.scale += n
return result
def __inv__(self):
result = self.copy()
result.scale *= -1.
return result
def __abs__(self):
return Function(self, abs)
def FuncTime(func, phase = 0., scale = 1., offset = 0.):
global time
return Function(time, func, phase, scale, offset)
def SinTime(phase = 0., scale = 1., offset = 0.):
return FuncTime(sin, phase, scale, offset)
sin_time = SinTime()
def CosTime(phase = 0., scale = 1., offset = 0.):
phase += pi / 2.
return SinTime(phase, scale, offset)
cos_time = CosTime()
class Circle:
def __init__(self, x, y, radius):
self.x = x
self.y = y
self.radius = radius
#property
def size(self):
return [self.radius * 2] * 2
circle = Circle(
x = cos_time * 200 + 250,
y = abs(sin_time) * 200 + 50,
radius = 50)
class CircleView(Sprite):
def __init__(self, model, color = (255, 0, 0)):
Sprite.__init__(self)
self.color = color
self.model = model
self.image = Surface([model.radius * 2] * 2).convert_alpha()
self.rect = self.image.get_rect()
pygame.draw.ellipse(self.image, self.color, self.rect)
def update(self):
self.rect[:] = int(self.model.x), int(self.model.y), self.model.radius * 2, self.model.radius * 2
circle_view = CircleView(circle)
sprites = Group(circle_view)
running = True
while running:
for event in pygame.event.get():
if event.type == QUIT:
running = False
if event.type == KEYDOWN and event.key == K_ESCAPE:
running = False
screen.fill((0, 0, 0))
sprites.update()
sprites.draw(screen)
pygame.display.flip()
pygame.quit()
In short: If every component can be treated like a number, the whole system can be treated like a math equation, right?
Paul Hudak's book, The Haskell School of Expression, is not only a fine introduction to Haskell, but it also spends a fair amount of time on FRP. If you're a beginner with FRP, I highly recommend it to give you a sense of how FRP works.
There is also what looks like a new rewrite of this book (released 2011, updated 2014), The Haskell School of Music.
According to the previous answers, it seems that mathematically, we simply think in a higher order. Instead of thinking a value x having type X, we think of a function x: T → X, where T is the type of time, be it the natural numbers, the integers or the continuum. Now when we write y := x + 1 in the programming language, we actually mean the equation y(t) = x(t) + 1.
Acts like a spreadsheet as noted. Usually based on an event driven framework.
As with all "paradigms", it's newness is debatable.
From my experience of distributed flow networks of actors, it can easily fall prey to a general problem of state consistency across the network of nodes i.e. you end up with a lot of oscillation and trapping in strange loops.
This is hard to avoid as some semantics imply referential loops or broadcasting, and can be quite chaotic as the network of actors converges (or not) on some unpredictable state.
Similarly, some states may not be reached, despite having well-defined edges, because the global state steers away from the solution. 2+2 may or may not get to be 4 depending on when the 2's became 2, and whether they stayed that way. Spreadsheets have synchronous clocks and loop detection. Distributed actors generally don't.
All good fun :).
I found this nice video on the Clojure subreddit about FRP. It is pretty easy to understand even if you don't know Clojure.
Here's the video: http://www.youtube.com/watch?v=nket0K1RXU4
Here's the source the video refers to in the 2nd half: https://github.com/Cicayda/yolk-examples/blob/master/src/yolk_examples/client/autocomplete.cljs
This article by Andre Staltz is the best and clearest explanation I've seen so far.
Some quotes from the article:
Reactive programming is programming with asynchronous data streams.
On top of that, you are given an amazing toolbox of functions to combine, create and filter any of those streams.
Here's an example of the fantastic diagrams that are a part of the article:
It is about mathematical data transformations over time (or ignoring time).
In code this means functional purity and declarative programming.
State bugs are a huge problem in the standard imperative paradigm. Various bits of code may change some shared state at different "times" in the programs execution. This is hard to deal with.
In FRP you describe (like in declarative programming) how data transforms from one state to another and what triggers it. This allows you to ignore time because your function is simply reacting to its inputs and using their current values to create a new one. This means that the state is contained in the graph (or tree) of transformation nodes and is functionally pure.
This massively reduces complexity and debugging time.
Think of the difference between A=B+C in math and A=B+C in a program.
In math you are describing a relationship that will never change. In a program, its says that "Right now" A is B+C. But the next command might be B++ in which case A is not equal to B+C. In math or declarative programming A will always be equal to B+C no matter what point in time you ask.
So by removing the complexities of shared state and changing values over time. You program is much easier to reason about.
An EventStream is an EventStream + some transformation function.
A Behaviour is an EventStream + Some value in memory.
When the event fires the value is updated by running the transformation function. The value that this produces is stored in the behaviours memory.
Behaviours can be composed to produce new behaviours that are a transformation on N other behaviours. This composed value will recalculate as the input events (behaviours) fire.
"Since observers are stateless, we often need several of them to simulate a state machine as in the drag example. We have to save the state where it is accessible to all involved observers such as in the variable path above."
Quote from - Deprecating The Observer Pattern
http://infoscience.epfl.ch/record/148043/files/DeprecatingObserversTR2010.pdf
The short and clear explanation about Reactive Programming appears on Cyclejs - Reactive Programming, it uses simple and visual samples.
A [module/Component/object] is reactive means it is fully responsible
for managing its own state by reacting to external events.
What is the benefit of this approach? It is Inversion of Control,
mainly because [module/Component/object] is responsible for itself, improving encapsulation using private methods against public ones.
It is a good startup point, not a complete source of knowlege. From there you could jump to more complex and deep papers.
Check out Rx, Reactive Extensions for .NET. They point out that with IEnumerable you are basically 'pulling' from a stream. Linq queries over IQueryable/IEnumerable are set operations that 'suck' the results out of a set. But with the same operators over IObservable you can write Linq queries that 'react'.
For example, you could write a Linq query like
(from m in MyObservableSetOfMouseMovements
where m.X<100 and m.Y<100
select new Point(m.X,m.Y)).
and with the Rx extensions, that's it: You have UI code that reacts to the incoming stream of mouse movements and draws whenever you're in the 100,100 box...
FRP is a combination of Functional programming(programming paradigm built upon the idea of everything is a function) and reactive programming paradigm (built upon the idea that everything is a stream(observer and observable philosophy)). It is supposed to be the best of the worlds.
Check out Andre Staltz post on reactive programming to start with.