monads

Yet another clojure library for monads, focussing on expressivity and correctness.

For Leiningen:

[bwo/monads "0.2.2"]

The idioms and terminology for this library are unabashedly Haskell-derived: there is a special syntax for monad computations, mdo, which is similar to Haskell's do-notation, and the names (and selection) of monads which have implementations provided out of the box are influenced by the mtl.

Improvements from 0.1.0

• All monad implementations interoperate with clojure.algo.generic.functor.

• Internals rewritten to be faster and more flexible.

• Automatic lifting in monad transformers

• Applicative functors introduced; all monads support its interface.

• Combined reader/writer/state monad implementation

Usage

There are some code examples, and some benchmarks, on the wiki; the examples show building up a simple expression evaluator.

Monadic computations are built up using return and >>=. For instance, one could define lift-m-2 (which enables the application of a function to monadic values) as follows:

(defn lift-m-2
  "Take a function a -> b -> c and two values m a and m b, and return
  m c."
  [f m1 m2]
  (>>= m1 (fn [v1] (>>= m2 (fn [v2] (return (f v1 v2)))))))

Since writing functions this way is cumbersome, a macro is provided that mimics Haskell's do-notation:

(defn lift-m-2
  "Take a function a -> b -> c and two values m a and m b, and return
  m c."
  [f m1 m2]
  (mdo v1 <- m1
       v2 <- m2
       (return (f v1 v2))))

(The actual implementation of lift-m-2 in monads.util is slightly different again, due to being curried.)

However, with only return and >>=, we can't do anything that we couldn't do with ordinary functions. There are also several protocols that specific monads can implement, which bring with them specific operations allowing more interesting things. Monad transformers can be used to conveniently add capabilities together.

Implementations are provided for several monads:

Monad	Transfomer provided?	Example use case	Protocols supported	Specific operations
reader	yes	read-only access to global environment	monadreader	ask, local
state	yes	simulate mutable state	monadstate	get-state , put-state, modify
writer	yes	log messages during a computation	monadwriter	tell, pass, listen, listens, censor
maybe	yes	computations may fail	monadfail, monadplus	fail, mzero, mplus
error	yes	computations that may fail, error handling and recovery	monadfail, monadplus, monaderror	fail, mzero, mplus, throw-error, catch-error
list	no	computations that may produce multiple results	monadfail, monadplus	fail, mzero, mplus
cont	yes	arbitrary manipulation of control, emulate CPS transform	(none---not yet abstracted out)	callcc, shift, reset
rws	yes	inline combination of reader, writer, and state	monadstate, monadwriter, monadreader	union of reader, state, writer
identity	yes	trivial monad	(none)	None

The specific operations are documented in monads.core, or for the continuation operations, in monads.cont.

The protocols are defined in monads.types and the functions to take advantage of them are defined in monads.core (with the exception of shift and reset, which are defined in monads.cont).

The "base" monads are named by suffixing -m to the names in the table above (e.g. state-m, cont-m). If there is a transformer version of a monad, it is a function named by suffixing -t instead of -m. The monad and transformer implementations are found in namespaces given by the names in the table, so, e.g., state-m and state-t are defined in monads.state. Each such namespace also defines vars named m and t as shortcuts, so you can refer to state/m instead of stuttering out state/state-m.

Giving the transformer function a monad as an argument returns a new monad. The resulting "monad transformer stack" implements the MonadTrans protocol and supports two additional operation, lift and inner. inner returns the monad that was originally passed in as an argument; lift can be used to run operations specific to a base monad in the stack. In general, explicit lifting is not necessary with the monads and transformers defined in this library, as the transformers will automatically support the operations their arguments do. Explicit lifting is only necessary for disambiguation if more than one monad supports the same operation:

monads.core> (require '[monads.state :as st] '[monads.error :as e] '[monads.maybe :as m])
nil
;; the next two lines are equivalent, because `state-t` will auto-lift `fail`
monads.core> (st/run-state-t (st/t e/m) (lift (fail "oops")) :initial-state)
#<Either [:left oops]>
monads.core> (st/run-state-t (st/t e/m) (fail "oops") :initial-state)
#<Either [:left oops]>
;; and the next two lines are also equivalent, because the auto-lifting goes down one level in the stack
monads.core> (st/run-state-t (st/t (e/t m/m)) (fail "oops") :initial-state)
#<Just #<Either [:left oops]>>
monads.core> (st/run-state-t (st/t (e/t m/m)) (lift (fail "oops")) :initial-state)
#<Just #<Either [:left oops]>>
;; so if we explicitly want to use `maybe-m`'s fail, we need to lift down from the bottom.
monads.core> (st/run-state-t (st/t (e/t m/m)) (lift (lift (fail "oops"))) :initial-state)
nil

In general, monadic computations are run using run-monad, which takes two arguments: a monad and a monadic computation. However, as the above example, using run-state-t, suggests, there are helper functions for some specific monads (any of those that require extra initial data):

Monad	Run function	Extra arguments
state-{m,t}	monads.state/run-state{,-t}	Initial state
reader-{m,t}	monads.reader/run-reader{,-t}	Starting environment
cont-{m,t}	monads.reader/run-cont{,-t}	None*
rws-{m,t}	monads.rws/run-rws{,-t}	Initial state and starting argument

(* In principle the extra argument should be the final continuation, but this is actually chosen by the implementation to be return for cont-t and identity for cont-m.)

run-state, run-reader, run-cont, and run-rws do not need the monad passed as their first argument, since it is assumed that the computation should be run in the state, reader, cont, or rws monads, respectively.

Utility functions

The function lift-m, which lifts a function defined over types a -> b to one defined over types m a -> m b for any monad m, is provided in monads.core; importing this file also makes all monads correctly treat the fmap defined in algo.generic correctly.

A (not very systematic) selection of monad functions is provided in monads.util:

• (sequence-m ms): transform a sequence of monadic actions into a monadic action yielding a sequence. (That is, go from [m a] to m [a].)

• (mwhen p m): execute monadic computation m if p is truthy.

• (guard p): exit from the computation if p is falsy (requires mzero).

• (lift-m-2 f [m [m2]]): as lift-m but for binary functions.

  There are also lift-m-3 through lift-m-8. All the lift-m-n functions are fully curried and can take at any stage anywhere from one to the remaining number of arguments, e.g. ((lift-m-3 + a b) c), (((lift-m-3 +) a) b c), etc. In the unlikely event that a lifting function of yet greater arity is needed, the deflift-m-n macro can be used to create one. deflift-m-ns can be used to create a range of such functions.

• (lift-m* f [& args]): as lift-m but for arbitrary arities. (N.B. this is implemented using sequence-m and each appears to behave unexpectedly in the context of the continuation monad's shift and reset, but those should probably be considered experimental for the time being).

• ap: lifts function application, but only for curried functions:

  (run-monad maybe-m (ap (ap (return (curryfn [a b] (+ a b))) (return 1)) (return 2)))
  #<Just 3>

  lift-m* is likelier to be useful, unless you happen to have a lot of curried functions lying around.

• (fold-m f init xs): apply a reduction within a monad. NB: the arguments here are as in Haskell's foldM, and not as in algo.monads' m-reduce! fold-m expects f to have type a -> b -> m a, init to have type a, and xs to have type [b], whereas m-reduce expects f to have type a -> b -> a, init to have type a, and xs to have type [m b].

• (msum [...]) "adds" the elements of its argument list with mplus.

Further such functions are easily defined. This, for instance, is the definition of guard:

(defn guard [p]
  (if p
    (return nil)
    mzero))

These are just ordinary Clojure functions that need not know anything about the context in which they will eventually be used.

Special syntax

While it is perfectly possible to write monadic computations as chains of >>= and anonymous functions, this quickly becomes tedious; a macro, mdo, is provided to make things simpler. As noted above, the syntax is very much derived from Haskell.

There are three types of elements of an mdo form:

• binding elements, which have the form destructure <- expression;

• plain elements, which are just expressions (except that no such expression can consist solely of the symbol <- or the symbol let);

• let elements, which have the form let destructure = expression (or let destructure1 = expression1, destructure2 = expression2, .... The commas here are just for presentation; since the reader gobbles them up, they aren't (and can't be) necessary to the syntax)

  let elements may also be written with a more conventional binding vector: let [destructure expression ...].

The final element of an mdo form must be a plain element.

In the above destructure can be any valid Clojure binding form. The expression on the left-hand side of a binding element, and the expression in a plain element, should have a monadic value; these are unwrapped and bound to the binding form on the right-hand side of the binding element, if there is one. Bindings established with let forms are, by contrast, pure (or at least treated as pure). Both forms of bindings are visible in all following statements (if not shadowed, of course).

So the following, for instance, is a not very interesting computation in the state monad:

(mdo {:keys [x y]} <- get-state
     let [z (+ (* x x) (* y y))]
     (modify #(assoc % :z z))
     (return z)

It does what you would expect:

> (def m (mdo {:keys [x y]} <- get-state
              let z = (+ (* x x) (* y y))
              (modify #(assoc % :z z))
              (return z)))
> (run-state m {:x 1 :y 3})
#<Pair [10 {:z 10, :y 3, :x 1}]>
> (run-state-t (state-t monads.maybe/maybe-m) m {:x 1 :y 3})
#<Just #<Pair [10 {:z 10, :y 3, :x 1}]>>

And expands into uses of >>= and anonymous functions:

(>>=
 get-state
 (fn [{:keys (x y)}]
     (let [z (+ (* x x) (* y y))]
       (>>= (modify #(assoc % :z z)) (fn [G__6125] (return z))))))

In fact, the "let" form is not really necessary; we could have omitted it and simply written this:

(mdo {:keys [x y]} <- get-state
     (let [z (+ (* x x) (* y y))]
       (mdo (modify #(assoc % :z z))
            (return z))))

And only suffered a little indentation. Similarly, there is no need for special syntax for if or when (and none is provided); just as we can write this code:

monads.list> (def pythags (mdo a <- (range 1 200)
                               b <- (range (inc a) 200)
                               let a2+b2 = (+ (* a a) (* b b))
                               c <- (range 1 200)
                               (monads.util/guard (== (* c c) a2+b2))
                               (return (list a b c))))
#'monads.list/pythags
monads.list> (take 3 (run-monad list-m pythags))
((3 4 5) (5 12 13) (6 8 10))

We could have taken advantage of the fact that the return is the only statement following the guard:

monads.list> (def pythags (mdo a <- (range 1 200)
                               b <- (range (inc a) 200)
                               let a2+b2 = (+ (* a a) (* b b))
                               c <- (range 1 200)
                               (if (== (* c c) a2+b2)
                                   (return (list a b c))
                                   mzero)

Applicative functors

monads.applicative defines a simple applicative functor interface, and gives default implementations for it to all monads, as well as for sequences, nil, the Just and Either types defined in monads.types, and Const and Id functors also defined in monads.applicative.

The applicative interface consists of pure, which is analogous to return for monads, and effectful function application, <*>. Since we don't assume that all arguments will be supplied immediately, however, the function argument to <*> must be curried, so that arguments can be fed in one by one. A convenience function cpure is supplied that takes an arity and a function and returns a curried function with the given arity wrapped in the Pure constructor:

monads.applicative> (require '[monads.types :as t])
nil
monads.applicative> (<*> (cpure 3 +) (t/just 3) (t/just 1) (t/just 2))
#<Just 6>
monads.applicative> (<*> (cpure 3 +) (t/just 3) t/nothing (t/just 2))
nil
monads.applicative> (<*> (<*> (cpure 3 +) (t/just 3)) (t/just 1) (t/just 2))
#<Just 6>

General utilities for currying functions can be found in monads.util: the macros curryfn and defcurryfn define curried functions, and the macro curry and function ecurry both take an arity and a function and create a curried function with the given arity. curry falls back to ecurry if the arity is not statically known; if it is known, curry is significantly faster:

monads.util> (time (dotimes [_ 10000] ((((ecurry 3 +) 1) 2) 3)))
"Elapsed time: 30.518729 msecs"
nil
monads.util> (time (dotimes [_ 10000] ((((curry 3 +) 1) 2) 3)))
"Elapsed time: 7.261895 msecs"
nil

Despite the inconvenience of manual currying, applicative functors are still useful; as an example, monads.examples.applicative-fold contains an implementation of the core of a streaming fold abstraction (though for complex computations something like babbage might be better).

Implementation

Monads are implemented with a protocol defining a binary mreturn and trinary bind operations; the additional parameter over return and >>= is for the carrier of the protocol. There are monad and defmonad macros which delegate to reify; the followuing definitions of the identity monad are equivalent:

(defmonad identity-m
  (mreturn [me x] x)
  (bind [me m f] (run-monad me (f m))))

(require '[monads.types :as types])
(def identity-m 
    (reify types/Monad
       (mreturn [me x] x)
       (bind [me m f] (run-monad me (f m)))))

However, monad and defmonad allow one to conditionally support other protocols as well, which is useful for defining monad transformers that support a protocol if the transformed, inner monad does.

Since the macros know that they are defining a monad, nothing special needs to be done to ensure that mreturn and bind find their homes in the right protocol; other protocols need to be given explicitly using reify-like syntax. For example, the reader-t transformer function looks like this:

(defn reader-t [inner]
  (monad
   (mreturn [me v] (constantly (types/mreturn inner v)))
   (bind [me m f] (fn [e]
                    (run-mdo inner
                             a <- (m e)
                             (run-reader-t me (f a) e))))
   types/MonadTrans
   (inner [me] inner)
   (lift [me c] (fn [e] (run-monad inner c)))
   types/MonadReader
   (ask [me] (fn [e] (types/mreturn inner e)))
   (local [me f m] (fn [e] (run-reader-t me m (f e))))
   (when (types/monadfail? inner)
     types/MonadFail
     (fail [me msg] (fn [e] (types/fail inner msg))))
   (when (types/monadplus? inner)
     types/MonadPlus
     (mzero [me] (constantly (types/mzero inner)))
     (mplus [me lr] (fn [e]
                      (types/mplus inner
                                   (lazy-pair (run-reader-t me (first lr) e)
                                              (run-reader-t me (second lr) e))))))))

Note the conditional support for MonadFail and MonadPlus.

A caveat about the stack

The use of the "bare" monads (maybe-m, error-m, etc.) is vulnerable to stack-blowing on deeply nested computations, e.g. (msum (repeat 4000 mzero)). This danger can be mostly obviated by using the transformer version of the monad with cont-m as the base monad:

monads.maybe> (require '[monads.util :as u] '[monads.cont :as c])
nil
monads.maybe> (run-monad maybe-m (u/msum (repeat 4000 mzero)))
; Evaluation aborted.
monads.maybe> (c/run-cont (run-monad (maybe-t c/m) (u/msum (repeat 4000 mzero))))
nil

On a less trivial computation:

monads.examples.treenumber> (require '[monads.cont :as c])
nil
monads.examples.treenumber> (def x (num-tree (longtree 10000)))
StackOverflowError   monads.core/fn--1769 (core.clj:63)
monads.examples.treenumber> (def x (c/run-cont (s/run-state-t (s/t c/m) (number-tree (longtree 10000)) {})))
#'monads.examples.treenumber/x
monads.examples.treenumber> (count (second x)) ;; check that we've actually got the right # of entries
10000

However, this doesn't get around the entire problem: msum is written to associate to the right. A left-associative version would still blow the stack:

monads.maybe> (c/run-cont (run-monad (maybe-t c/m) (reduce mplus mzero (repeat 4000 mzero))))
; Evaluation aborted.

The same thing happens with nested binds on the left:

monads.maybe> (monads.cont/run-cont (run-monad (t monads.cont/m)
                (reduce (fn [acc _] (>>= acc (fn [x] (return (inc x))))) 
                        (return 0)
                        (range 10000))))
StackOverflowError   monads.types.Bind (types.clj:33)

However, since we have a programmatically manipulable representation of the computation, this difficulty can be worked around:

monads.maybe> (monads.cont/run-cont (run-monad (t monads.cont/m) 
                (monads.cont/reorganize (reduce (fn [acc _] (>>= acc (fn [x] (return (inc x))))) 
                                                (return 0) 
                                                (range 10000)))))
#<Just 10000>
monads.maybe> (monads.cont/run-cont (run-monad (t monads.cont/m)
                (monads.cont/reorganize (reduce #(mplus %1 %2)
                                                mzero
                                                (repeat 10000 mzero)))))
nil

Monadic computations are required to ensure the behavioral identity of (>>= (>>= m f) g) and (>>= m (fn [x] (>>= (f x) g))), so the reorganize function can convert left-biased computations with the former shape to right-biased computations with the latter. Since mplus is similarly required to be associative, it does the same for left-biased mplus applications, rewriting (mplus (mplus a b) c) to (mplus a (mplus b c)).

Note that this reorganization at present doesn't descend into the monadic arguments of e.g. local, and (obviously) the contents of closures in the second argument of >>= are opaque to it. If the rewriting were baked into mplus and >>=, this would not be an issue, but I'm hesitant to carry the rewriting out if it's not asked for.

License

Copyright © 2014 Ben Wolfson

Distributed under the Eclipse Public License, the same as Clojure.