Here is very short introduction to functional programming language Haskel and functional reasoning. For this we will use CλaSH compiler, that is extension of Haskell language being able to convert large set of Haskell code in either VHLD, Verilog or SystemVerilog for now. Installation instructions are available in hackage doc.

If you need just quick peek in examples, there is source code available of all examples presented here and is compiled in VHLD, Verilog and SystemVerilog. Code is automatically generated, but can be used as quick check of what kind of circuit code describes and what to expect from CλaSH compiler.

Premise is that circuits described functional way are considered simpler to reason about, compose and reuse. We are able to describe topology trough functional composition on abstract way without sacrificing about reasoning, how may registries are in use and where logic gates are. This approach allows describing similar if not same circuits as with Verilog and VHDL using new and fresh approach allowing composing more ambitious synchronous digital circuits.

Descriptions is mathematical in way that is trying to use tools from mathematical theory about composition. It is common to see relations described with mathematical laws.

# Function application

Before running coding we should explain how calling functions in Haskell works. In most languages if function takes 2 arguments `x` and `y` it is called like `f(x,y)` where equivalent for this in Haskell is `f\ x\ y` . This come with advantage because Haskell supports partially applied functions where only one out of two arguments is applied. Lets say that signature of `f` is `Int -> Int -> Bool`. We can interpret this as - it takes two integers `x` an `y` and it returns boolean. When only first argument `x` is applied it returns function whit hremaining signature. When applying on function `Int -> Int -> Bool` first `Int` remaining is `Int -> Bool`. That is function that takes `Int` and it returns `Bool`.

``````-- function of 2 Int, that telse if
-- 1st argument is greater then 2nd
greater :: Int -> Int -> Bool
greater x y = x > y

-- takes y and tels whatever it is greater than 2
isgt2 :: Int -> Bool
isgt2 y = greater 2 y
``````

or less verbose as partially applied

``````isgt2 :: Int -> Bool
isgt2 = greater 2
``````

One way to think about functions in Haskell is that each function takes only one argument and it returns one value that can be function. Signature of `greater` can be seen as `greater :: Int -> ( Int -> Bool )`. So it takes an `Int` and it returns function `Int -> Bool`. That is function that takes `Int` and returns `Bool`.

In Haskell every function returns something. Pure functions, does not execute any action, but just transform input data and return relevant result. Pure functions, has important property that they always returns same output for same input and so they don’t have access to some external time changing data or internal state. In Haskell there is strong force to separate effects from pure functions.

# Function composition

Functional programming is about composition. Here it is example of signatures of 2 functions.

``````funcA2B :: a -> b
funcB2C :: b -> c
``````

`funcA2B` is function that takes type `a` and returns `b`, one example is that it takes `Int` and returns `Bool`.`funcB2C` takes `b` and it returns `c`. For example it takes `Bool` and returns `Bool`. Having this we are able to create function

``````funcA2C :: a -> c
``````

by composing together `funcA2B` and `funcB2C` to get `funcA2C`.

For composing two functions we need function

``````composeFunc :: (b->c) -> (a->b) -> a -> c
composeFunc f g x = f (g x)
``````

First we apply `x` on `g` and then we apply result on `f`.
Moving back to `composeFunc` it takes 3 arguments. First two are functions, `f` with signature `b->c` and `g` with signature `a->b`, 3rd param `x` is of type `a`. Function returns type `c` and is same type that `f` returns. Looking differently it takes two functions that is `b->c` and `a->b` and it returns function `a -> c`.

For example

``````minus1 x = x - 1
largeNum x = x > 3

superLargeNum = composeFunc largeNum minus1
``````

`superLargeNum` is function composed by `largeNum` and `minus1`.Type of each function is well defined, but we can either define it manually or compiler can automatically deduce it.

``````-- simplified types
minus1 :: a -> a
largeNum :: a -> Bool
superLargeNum :: a -> Bool

-- actual types
minus1 :: Num a => a -> a
largeNum :: (Num a, Ord a) => a -> Bool
superLargeNum :: (Num a, Ord a) => a -> Bool
``````

We can read this, that function `superLargeNum` transforms type `a` into `Bool`. Type `a` has to be defined for class `Num` and `Ord`. `Num` is class that has defined function `(-)` as minus operator. (among others) and class `Ord` has defined `(>)`.

`composeFunc` is commonly used and in Haskell it is defined as operator `(.)`.

``````(.) = composeFunc
``````

Using this, one can alternatively define `superLargeNum` as

``````superLargeNum = largeNum . minus1
-- or
superLargeNum = (> 3) . (- 1)
-- or
superLargeNum x = (x - 1) > 3
``````

Operator `.` comes from function composition as defined in mathematics f ∘ g . Here we have showed implemention of `.` as in `composeFunc`, but sometimes instead of implementation we get only laws. For function `(.)` holds.

If `id` is identity function that returns argument

``````id :: a -> a
id x = x
``````

then function composition laws for ∘ are

``````id . f == f . id == f
f . g . h == (f . g) . h == f . (g . h)
``````

Instead of having function implementation available we sometimes have available laws that function obeys. It helps reasoning about how function behaves without knowing what it does. Laws also goes hand to hand with implementation.

# Functor

Lets take for example type in CλaSH called `Signal a`. This represent type `a` changing with clock. `Signal` is concrete example of functor and `a` is type embedded in `Signal`.

Other than function composition of `Signal a` there is different composition available for functor.

``````funcA2B :: a -> b
signalA :: Signal a
``````

If we have function of type `a -> b` and `Signal a`, than functor enables us to get `Signal b`

``````fmap :: (a -> b) -> Signal a -> Signal b
``````

Functor obeys two laws

``````fmap id == id
fmap (p . q) == (fmap p) . (fmap q)
``````

Or equivalently when applying signal.

``````fmap id signal == signal
fmap (p . q) signal == fmap p (fmap q signal)
``````

Knowing this we can aswer next question. Does `fmap` over `Signal` uses any registries as for example D flip-flop?

Answer is, if that would be case, neither of laws would hold. Form first law it would imply that on left we have signal delayed by 1 clock, and on right non delayed signal and they are not same. Second law would imply delay by 1 clock on left where right side made 2 delays, each by one `fmap` calls.

To generate core we define function `topEntity`. Generic type `a` should be concrete and in this case we will use `Signed 7` what is 7 bit signed integer.

``````topEntity :: Signal (Signed 7) -> Signal Bool
topEntity signal = fmap superLargeNum signal
``````

Executing command fmap can be used as operator. That is `<\$>`. Alternatively we can implement topEntity also as

``````topEntity signal = superLargeNum <\$> signal
-- or point free version
topEntity = fmap superLargeNum
-- or using fmap as operator
topEntity = superLargeNum `fmap` signal
``````

It is hard to tell witch version is most common among haskellers, but we can be sure, that due to laws all of them behaves same.

Compiling this exact into Verilog generates code as expected.

# Mealy machinery

For generating output based on previous stored value we can use `mealy` function. Lets implement integrator using mealy machine. With integrator we have in mind function that takes `Signal a` and produces `Signal a` and is representing sum of all previous values including current.

``````integrator signal = mealy mf 0 signal
where
mf state input = (nextstate,output)
where
nextstate = state + input
output = nextstate
``````

`mealy` is function that takes 3 arguments. First parameter is a function on how to generate next state from state and signal value. Signature of this function is `s -> i -> (s,o)`. In our code that is function called `mf`. `mf` is defined in `where` block for local visibility so can not be used out of
`integrator` . What this `mf` function does it takes 2 arguments `state` and `input` and it returns tuple `(nextstate,output)`. Next clock cycle `nextstate` will be passed again in `mf` as argument `state`. 2nd argument of integrator `0` is initial state value. Generated core will provide reset signal that sets state to initial value.

If we now need to create FPGA core, we need to provide `topEntity`. Here it is full source instantiated with Fixed point arithmetics. We arbitrarily choose `SFixed 2 8`, This is 10 bit signed fixed point number. It is representing rational numbers from -2 <= x < 2.

``````module Integrator where
import CLaSH.Prelude

integrator signal = mealy mf 0 signal
where
mf state input = (nextstate,output)
where
nextstate = state + input
output = nextstate

topEntity :: Signal (SFixed 2 8) -> Signal (SFixed 2 8)
topEntity = integrator
``````

`integrator` can be used in other modules, to build larger and more ambitious circuits. For example to make integrator of integrated signal so that output of first integrator is feed in second one.

``````intInt = integrator . integrator
``````

# Vector

`Vec` is sized vector with compile type defined size holding elements of same type. It is similar to tuple, except that all elements are of same type.

`Vec 5 (Signed 7)` is type that holds 5 elements of type `Signed 7` what is 7 bit signed number. `Vec` is also an functor.

For example

``````vs :: Vec 6 (Signed 9)
vs = 1 :> 2 :> 3 :> 4 :> 5 :> 6 :> Nil

vb :: Vec 6 Bool
vb = fmap (> 3) vs  -- that is <False,False,False,True,True,True>
``````

Operator `:>` is right-associativity operator that prepends vector on right with value on left. `Nil` is empty vector and is used, because `:>` expects vector on right side.

Size of `Vec` is defined at compile time and can be generic. Lets take look on one of many functions from `Vec` libraray that is `replicate` and has prototype

``````replicate :: SNat n -> a -> Vec n a
-- "replicate n a" returns a vector that has `n` copies of `a`.
``````

`Vec 5 a` is not same as `Vec 6 a` and it means that `replicate` returns different type based on argument `n`. For this to work, `n` is of type `SNat x` and is different type for each x. Value that has type `Signed 5` as 5 bit signed value can be -2 or 7, where value `d3` that has type `SNat 3` represents number 3 as unique type.

# Moving average

Moving average is simple low pass FIR filter expressed as sum of last `n` values. We will deliberately skip division in our example, so we will in reality talk about moving sum.

Here is straight forward and naive option to describe moving average using `mealy`.

``````movingAvarage n signal = mealy mf (replicate n 0) signal
where
mf state input = (nexts,output)
where
nexts = input :> init state
output = fold (+) nexts
``````

First argument `n` is number of accumulated value is also size of Vec used as internal state storage. Type of internal state is deduced as `Vec n a` and is deduced from type of initial state value `replicate n 0` that returns `Vec` that returns `n` copies of 0.

`init :: Vec (n + 1) a -> Vec n a` is function that takes `Vec` with size `n+1` and returns all except last element that is `Vec` of size `n`.

`nexts` expressed as next state is vector right shifted by previous value and perpended `:>` with current clock `input` value. Operation works as fifo shift register.

`fold` is function best expressed with same picture as in `manuals`

From signature we can conclude that `fold :: (a -> a -> a) -> Vec (n + 1) a -> a` takes 2 arguments. First is function of type `(a -> a -> a)` an second argument is of type `Vec (n + 1) a`. Function is than applied over vector in tree like structure, enabling minimum latency. Function f from out case is function `(+)` (in haskell operator in brakets is used as function). and has prototype `a -> a -> a` and is applied over `state`.

# There is better

Lets try some optimization of moving average from previous section. Observing totals we can conclude we don’t need that many adders. Total output is changed at each clock only by value that enters fifo from one side and value that pops out of fifo on other. At each clock we need to subtract value that pops out and add value that enters from total. This comes at expanse of 2 extra sized registers (previous solution can be reduced by 1 sized register) but requires only 2 sized adders.

``````movingAvarage2 n signal = mealy mf ((replicate n 0),0) signal
where
mf (vec,total) input = ((nextVec,nextTot),output)
where
output = total + input - (last vec)
nextVec = input :> init vec
nextTot = output
``````

`mf` has at this point signature `(Vec n a,a) -> a -> ((Vec n a,a),a)`. Internal state type is tuple where first value is `vec` used as fifo and `total accumulated in fifp.`vec`and`total`are unpacked from tuple in input argument, so we are able to access this value, without unpacking them in body. Equivalent, but more verbose`mf` implemetaion is sketched as

``````mf state input = ...
where
vec = fst state -- first element of tuple
total = snd state -- second element of tuple
output = ...
``````

# Circuit composition

Great power of composition comes from separating logic into reusable parts and then composing them together in larger blocks. Function `(.)` and `fold` are good examples we have used so far. Now lets try something similar, by designing feedback loop.

``````          +---+    +---+
input -->-| f |-->-| g |-->-+-->-- output
+---+    +---+    |
|               ∨
∧      +---+    |
+----<-| h |-<--+
+---+
``````

First lets figure out prototypes of each function `f`, `g` and `h`.

``````f :: Signal a -> Signal b -> Signal c
g :: Signal c -> Signal d
h :: Signal d -> Signal b
``````

and feedback loop can be written directly from circuit diagram as

``````feedbackLoop :: (Signal a -> Signal b -> Signal c)
-> (Signal c -> Signal d)
-> (Signal d -> Signal b)
-> Signal a
-> Signal d
feedbackLoop f g h input = out where
out = g fout
fout = f input (h out)
``````

If user has available functions that works over primitive values like `Signed` instead `Signal (Signed 16)` for example, he can use `fold`. For example if

``````g :: Signed 16 -> Signed 16
``````

Then `gs` working over signal would be constructed from `g`

``````gs :: Signal (Signed 16) -> Signal (Signed 16)
gs = fmap g
-- or more verbose
gs sig = fmap g sig
``````

For function over 2 input arguments we can use library function liftA2

``````liftA2 :: (a -> b -> c) -> functor a -> functor b -> functor c
``````

Where `functor` is in our example `Signal` For example to get function minus `(-)` to operate over `Signal` we need to apply `liftA2`.

``````signalMinus :: Signal a -> Signal a -> Signal a
signalMinus = liftA2 (-)
``````

To construct negative feedback looking like

``````                   +---+
input -->-( - )-->-| g |-->-+-->-- output
|      +---+    |
∧               ∨
+--------<------+
``````

Using `feedbackLoop`,

``````negativeFeedback :: Num a =>
(Signal a -> Signal a) ->
Signal a ->
Signal a
negativeFeedback g = feedbackLoop (liftA2 (-)) g id
``````

We have replaced `h` by `id` and `f` argument is `liftA2 (-)`

There is subtle issue with this and in some cases this circuit can not be synthesized. That is because we have feedback loop, so we have made sure that there is delay in g function. If `g` function is for example `id` or `fmap id` so it works like wire this circuit is not stable any more. To overcome this, we have to introduce delay in feedback. Function `h` from previous diagram, becomes `register 0` instead of `id`. function

``````negativeFeedback :: Num a => (Signal a -> Signal a) -> Signal a -> Signal a
negativeFeedback g = feedbackLoop (liftA2 (-)) g (register 0)
``````

`register` :: a -> Signal a -> Signal a is a function, that takes inital value, that outputs this value before remaining of stream. `register` is available in CλaSH libraray, but can be implemented using `mealy` as

``````register initialValue signal = mealy mf initialValue
where
mf prevS input = (input,prevS)
``````

What is even more surprising is that it can be done even vice versa. That is `mealey` can be expressed in terms of `register`. For details check examples on github

# Conclusion

There are lot of features that were no covered. But are invaluable during development

• repl, (read eval print loop) that allows interactive development, type deduction
• ability to simulate in repl or compile in CPU native executable
• testebenches that generates VHDL / Verilog / Systemverilog and can be used directly in IDE
• full featured library, (ram and rom access and initalisation, multiple clock domains, type safe delays … )

From my and experience of many it takes lot of time and practice to grasp new programming language, what Haskell for those not familiar with functional programming certainly is. Reader with interest will be able to find lot of free resources available online for all levels of experience with open and active community.