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+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Monad application examples
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+(ns
+ #^{:author "Konrad Hinsen"
+ :skip-wiki true
+ :doc "Examples for using monads"}
+ examples.monads
+ (:use [clojure.contrib.monads
+ :only (domonad with-monad m-lift m-seq m-reduce m-when
+ sequence-m
+ maybe-m
+ state-m fetch-state set-state
+ writer-m write
+ cont-m run-cont call-cc
+ maybe-t)])
+ (:require (clojure.contrib [accumulators :as accu])))
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Sequence manipulations with the sequence monad
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+; Note: in the Haskell world, this monad is called the list monad.
+; The Clojure equivalent to Haskell's lists are (possibly lazy)
+; sequences. This is why I call this monad "sequence". All sequences
+; created by sequence monad operations are lazy.
+
+; Monad comprehensions in the sequence monad work exactly the same
+; as Clojure's 'for' construct, except that :while clauses are not
+; available.
+(domonad sequence-m
+ [x (range 5)
+ y (range 3)]
+ (+ x y))
+
+; Inside a with-monad block, domonad is used without the monad name.
+(with-monad sequence-m
+ (domonad
+ [x (range 5)
+ y (range 3)]
+ (+ x y)))
+
+; Conditions are written with :when, as in Clojure's for form:
+(domonad sequence-m
+ [x (range 5)
+ y (range (+ 1 x))
+ :when (= (+ x y) 2)]
+ (list x y))
+
+; :let is also supported like in for:
+(domonad sequence-m
+ [x (range 5)
+ y (range (+ 1 x))
+ :let [sum (+ x y)
+ diff (- x y)]
+ :when (= sum 2)]
+ (list diff))
+
+; An example of a sequence function defined in terms of a lift operation.
+(with-monad sequence-m
+ (defn pairs [xs]
+ ((m-lift 2 #(list %1 %2)) xs xs)))
+
+(pairs (range 5))
+
+; Another way to define pairs is through the m-seq operation. It takes
+; a sequence of monadic values and returns a monadic value containing
+; the sequence of the underlying values, obtained from chaining together
+; from left to right the monadic values in the sequence.
+(with-monad sequence-m
+ (defn pairs [xs]
+ (m-seq (list xs xs))))
+
+(pairs (range 5))
+
+; This definition suggests a generalization:
+(with-monad sequence-m
+ (defn ntuples [n xs]
+ (m-seq (replicate n xs))))
+
+(ntuples 2 (range 5))
+(ntuples 3 (range 5))
+
+; Lift operations can also be used inside a monad comprehension:
+(domonad sequence-m
+ [x ((m-lift 1 (partial * 2)) (range 5))
+ y (range 2)]
+ [x y])
+
+; The m-plus operation does concatenation in the sequence monad.
+(domonad sequence-m
+ [x ((m-lift 2 +) (range 5) (range 3))
+ y (m-plus (range 2) '(10 11))]
+ [x y])
+
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Handling failures with the maybe monad
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+; Maybe monad versions of basic arithmetic
+(with-monad maybe-m
+ (def m+ (m-lift 2 +))
+ (def m- (m-lift 2 -))
+ (def m* (m-lift 2 *)))
+
+; Division is special for two reasons: we can't call it m/ because that's
+; not a legal Clojure symbol, and we want it to fail if a division by zero
+; is attempted. It is best defined by a monad comprehension with a
+; :when clause:
+(defn safe-div [x y]
+ (domonad maybe-m
+ [a x
+ b y
+ :when (not (zero? b))]
+ (/ a b)))
+
+; Now do some non-trivial computation with division
+; It fails for (1) x = 0, (2) y = 0 or (3) y = -x.
+(with-monad maybe-m
+ (defn some-function [x y]
+ (let [one (m-result 1)]
+ (safe-div one (m+ (safe-div one (m-result x))
+ (safe-div one (m-result y)))))))
+
+; An example that doesn't fail:
+(some-function 2 3)
+; And two that do fail, at different places:
+(some-function 2 0)
+(some-function 2 -2)
+
+; In the maybe monad, m-plus selects the first monadic value that
+; holds a valid value.
+(with-monad maybe-m
+ (m-plus (some-function 2 0) (some-function 2 -2) (some-function 2 3)))
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Random numbers with the state monad
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+; A state monad item represents a computation that changes a state and
+; returns a value. Its structure is a function that takes a state argument
+; and returns a two-item list containing the value and the updated state.
+; It is important to realize that everything you put into a state monad
+; expression is a state monad item (thus a function), and everything you
+; get out as well. A state monad does not perform a calculation, it
+; constructs a function that does the computation when called.
+
+; First, we define a simple random number generator with explicit state.
+; rng is a function of its state (an integer) that returns the
+; pseudo-random value derived from this state and the updated state
+; for the next iteration. This is exactly the structure of a state
+; monad item.
+(defn rng [seed]
+ (let [m 259200
+ value (/ (float seed) (float m))
+ next (rem (+ 54773 (* 7141 seed)) m)]
+ [value next]))
+
+; We define a convenience function that creates an infinite lazy seq
+; of values obtained from iteratively applying a state monad value.
+(defn value-seq [f seed]
+ (lazy-seq
+ (let [[value next] (f seed)]
+ (cons value (value-seq f next)))))
+
+; Next, we define basic statistics functions to check our random numbers
+(defn sum [xs] (apply + xs))
+(defn mean [xs] (/ (sum xs) (count xs)))
+(defn variance [xs]
+ (let [m (mean xs)
+ sq #(* % %)]
+ (mean (for [x xs] (sq (- x m))))))
+
+; rng implements a uniform distribution in the interval [0., 1.), so
+; ideally, the mean would be 1/2 (0.5) and the variance 1/12 (0.8333).
+(mean (take 1000 (value-seq rng 1)))
+(variance (take 1000 (value-seq rng 1)))
+
+; We make use of the state monad to implement a simple (but often sufficient)
+; approximation to a Gaussian distribution: the sum of 12 random numbers
+; from rng's distribution, shifted by -6, has a distribution that is
+; approximately Gaussian with 0 mean and variance 1, by virtue of the central
+; limit theorem.
+; In the first version, we call rng 12 times explicitly and calculate the
+; shifted sum in a monad comprehension:
+(def gaussian1
+ (domonad state-m
+ [x1 rng
+ x2 rng
+ x3 rng
+ x4 rng
+ x5 rng
+ x6 rng
+ x7 rng
+ x8 rng
+ x9 rng
+ x10 rng
+ x11 rng
+ x12 rng]
+ (- (+ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) 6.)))
+
+; Let's test it:
+(mean (take 1000 (value-seq gaussian1 1)))
+(variance (take 1000 (value-seq gaussian1 1)))
+
+; Of course, we'd rather have a loop construct for creating the 12
+; random numbers. This would be easy if we could define a summation
+; operation on random-number generators, which would then be used in
+; combination with reduce. The lift operation gives us exactly that.
+; More precisely, we need (m-lift 2 +), because we want both arguments
+; of + to be lifted to the state monad:
+(def gaussian2
+ (domonad state-m
+ [sum12 (reduce (m-lift 2 +) (replicate 12 rng))]
+ (- sum12 6.)))
+
+; Such a reduction is often quite useful, so there's m-reduce predefined
+; to simplify it:
+(def gaussian2
+ (domonad state-m
+ [sum12 (m-reduce + (replicate 12 rng))]
+ (- sum12 6.)))
+
+; The statistics should be strictly the same as above, as long as
+; we use the same seed:
+(mean (take 1000 (value-seq gaussian2 1)))
+(variance (take 1000 (value-seq gaussian2 1)))
+
+; We can also do the subtraction of 6 in a lifted function, and get rid
+; of the monad comprehension altogether:
+(with-monad state-m
+ (def gaussian3
+ ((m-lift 1 #(- % 6.))
+ (m-reduce + (replicate 12 rng)))))
+
+; Again, the statistics are the same:
+(mean (take 1000 (value-seq gaussian3 1)))
+(variance (take 1000 (value-seq gaussian3 1)))
+
+; For a random point in two dimensions, we'd like a random number generator
+; that yields a list of two random numbers. The m-seq operation can easily
+; provide it:
+(with-monad state-m
+ (def rng2 (m-seq (list rng rng))))
+
+; Let's test it:
+(rng2 1)
+
+; fetch-state and get-state can be used to save the seed of the random
+; number generator and go back to that saved seed later on:
+(def identical-random-seqs
+ (domonad state-m
+ [seed (fetch-state)
+ x1 rng
+ x2 rng
+ _ (set-state seed)
+ y1 rng
+ y2 rng]
+ (list [x1 x2] [y1 y2])))
+
+(identical-random-seqs 1)
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Logging with the writer monad
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+; A basic logging example
+(domonad (writer-m accu/empty-string)
+ [x (m-result 1)
+ _ (write "first step\n")
+ y (m-result 2)
+ _ (write "second step\n")]
+ (+ x y))
+
+; For a more elaborate application, let's trace the recursive calls of
+; a naive implementation of a Fibonacci function. The starting point is:
+(defn fib [n]
+ (if (< n 2)
+ n
+ (let [n1 (dec n)
+ n2 (dec n1)]
+ (+ (fib n1) (fib n2)))))
+
+; First we rewrite it to make every computational step explicit
+; in a let expression:
+(defn fib [n]
+ (if (< n 2)
+ n
+ (let [n1 (dec n)
+ n2 (dec n1)
+ f1 (fib n1)
+ f2 (fib n2)]
+ (+ f1 f2))))
+
+; Next, we replace the let by a domonad in a writer monad that uses a
+; vector accumulator. We can then place calls to write in between the
+; steps, and obtain as a result both the return value of the function
+; and the accumulated trace values.
+(with-monad (writer-m accu/empty-vector)
+
+ (defn fib-trace [n]
+ (if (< n 2)
+ (m-result n)
+ (domonad
+ [n1 (m-result (dec n))
+ n2 (m-result (dec n1))
+ f1 (fib-trace n1)
+ _ (write [n1 f1])
+ f2 (fib-trace n2)
+ _ (write [n2 f2])
+ ]
+ (+ f1 f2))))
+
+)
+
+(fib-trace 5)
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Sequences with undefined value: the maybe-t monad transformer
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+; A monad transformer is a function that takes a monad argument and
+; returns a monad as its result. The resulting monad adds some
+; specific behaviour aspect to the input monad.
+
+; The simplest monad transformer is maybe-t. It adds the functionality
+; of the maybe monad (handling failures or undefined values) to any other
+; monad. We illustrate this by applying maybe-t to the sequence monad.
+; The result is an enhanced sequence monad in which undefined values
+; (represented by nil) are not subjected to any transformation, but
+; lead immediately to a nil result in the output.
+
+; First we define the combined monad:
+(def seq-maybe-m (maybe-t sequence-m))
+
+; As a first illustration, we create a range of integers and replace
+; all even values by nil, using a simple when expression. We use this
+; sequence in a monad comprehension that yields (inc x). The result
+; is a sequence in which inc has been applied to all non-nil values,
+; whereas the nil values appear unmodified in the output:
+(domonad seq-maybe-m
+ [x (for [n (range 10)] (when (odd? n) n))]
+ (inc x))
+
+; Next we repeat the definition of the function pairs (see above), but
+; using the seq-maybe monad:
+(with-monad seq-maybe-m
+ (defn pairs-maybe [xs]
+ (m-seq (list xs xs))))
+
+; Applying this to a sequence containing nils yields the pairs of all
+; non-nil values interspersed with nils that result from any combination
+; in which one or both of the values is nil:
+(pairs-maybe (for [n (range 5)] (when (odd? n) n)))
+
+; It is important to realize that undefined values (nil) are not eliminated
+; from the iterations. They are simply not passed on to any operations.
+; The outcome of any function applied to arguments of which at least one
+; is nil is supposed to be nil as well, and the function is never called.
+
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+;;
+;; Continuation-passing style in the cont monad
+;;
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
+
+; A simple computation performed in continuation-passing style.
+; (m-result 1) returns a function that, when called with a single
+; argument f, calls (f 1). The result of the domonad-computation is
+; a function that behaves in the same way, passing 3 to its function
+; argument. run-cont executes a continuation by calling it on identity.
+(run-cont
+ (domonad cont-m
+ [x (m-result 1)
+ y (m-result 2)]
+ (+ x y)))
+
+; Let's capture a continuation using call-cc. We store it in a global
+; variable so that we can do with it whatever we want. The computation
+; is the same one as in the first example, but it has the side effect
+; of storing the continuation at (m-result 2).
+(def continuation nil)
+
+(run-cont
+ (domonad cont-m
+ [x (m-result 1)
+ y (call-cc (fn [c] (def continuation c) (c 2)))]
+ (+ x y)))
+
+; Now we can call the continuation with whatever argument we want. The
+; supplied argument takes the place of 2 in the above computation:
+(run-cont (continuation 5))
+(run-cont (continuation 42))
+(run-cont (continuation -1))
+
+; Next, a function that illustrates how a captured continuation can be
+; used as an "emergency exit" out of a computation:
+(defn sqrt-as-str [x]
+ (call-cc
+ (fn [k]
+ (domonad cont-m
+ [_ (m-when (< x 0) (k (str "negative argument " x)))]
+ (str (. Math sqrt x))))))
+
+(run-cont (sqrt-as-str 2))
+(run-cont (sqrt-as-str -2))
+
+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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