forked from pennstdropout/05-quickcheck
-
Notifications
You must be signed in to change notification settings - Fork 0
/
QuickCheck.hs
1199 lines (907 loc) · 33.2 KB
/
QuickCheck.hs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
{-
---
fulltitle: "Type-directed Property Testing"
date: October 2, 2023
---
-}
module QuickCheck where
{-
In this lecture, we will look at [QuickCheck][1], a technique that
cleverly exploits typeclasses and monads to deliver a powerful
automatic testing methodology.
Quickcheck was developed by [Koen Claessen][0] and [John Hughes][11]
more than ten years ago, and has since been ported to other languages
and is currently used, among other things to find subtle [concurrency
bugs][3] in [telecommunications code][4]. In 2010, it received the
[most influential paper award](http://www.sigplan.org/award-icfp.htm)
for the ICFP 2000 conference.
The key idea on which QuickCheck is founded is *property-based
testing*. That is, instead of writing individual test cases (eg unit
tests corresponding to input-output pairs for particular functions)
one should write *properties* that are desired of the functions, and
then *automatically* generate *random* tests which can be run to
verify (or rather, falsify) the property.
By emphasizing the importance of specifications, QuickCheck yields
several benefits:
1. The developer is forced to think about what the code *should do*,
2. The tool finds corner-cases where the specification is violated,
which leads to either the code or the specification getting fixed,
3. The specifications live on as rich, machine-checkable documentation
about how the code should behave.
In this module, we'll import some of QuickCheck's types, type classes
and operators without qualification for convenience. But all of the functions
that we use from this module will be marked by `QC.`
-}
import Control.Monad (liftM2, liftM3)
import qualified Data.DList as DL
import qualified Data.Foldable as Foldable
import qualified Data.List as List
import Test.QuickCheck
( Arbitrary (..),
Gen,
Property,
Testable (..),
(==>),
)
import qualified Test.QuickCheck as QC
{-
While you will be able to run some of the examples in this module
directly in the IDE, you will need to have a terminal with this
module loaded into ghci in order to work with QuickCheck.
Properties
==========
A QuickCheck property is essentially a function whose output is a
boolean. A standard "hello-world" QC property might be something
about common functions on lists.
-}
prop_revapp :: [Int] -> [Int] -> Bool
prop_revapp xs ys = reverse (xs ++ ys) == reverse xs ++ reverse ys
{-
That is, a property looks a bit like a mathematical theorem that the
programmer believes is true. A QC convention is to use the prefix `"prop_"`
for QC properties. Note that the type signature for the property is not the
usual polymorphic signature; we have given the concrete type `Int` for the
elements of the list. This is because QC uses the types to generate random
inputs, and hence is restricted to monomorphic properties (those that don't
contain type variables.)
To *check* a property, we simply invoke the `quickCheck` action with the
property. Note that only certain types of properties can be tested, these
properties are all in the 'Testable' type class.
~~~~~{.haskell}
quickCheck :: (Testable prop) => prop -> IO ()
-- Defined in Test.QuickCheck.Test
~~~~~
`[Int] -> [Int] -> Bool` is a Testable property, so
let's try quickCheck on our example property above. Note that because
`quickCheck` runs in the `IO` monad, you need to use `ghci` to see the
examples in this module. You can start ghci with the command:
stack ghci QuickCheck.hs
ghci> import Test.QuickCheck
Once you have done that, you should see a prompt that you can use to evaluate
definitions in the `QuickCheck` module. Try checking the property above.
~~~~~{.haskell}
ghci> quickCheck prop_revapp
~~~~~
S
P
O
I
L
E
R
S
P
A
C
E
R
U
N
I
N
G
H
C
I
F
I
R
S
T
What's that ?! Let's run the `prop_revapp` function on the two inputs that
quickCheck identified as counter-examples. (Your counterexamples may differ
from the ones below.)
ghci> prop_revapp [0] [1]
QC has found inputs for which the property function *fails* ie, returns
`False`. Of course, those of you who are paying attention will realize there
was a bug in our property, namely it should be
-}
prop_revapp_ok :: [Int] -> [Int] -> Bool
prop_revapp_ok xs ys = reverse (xs ++ ys) == reverse ys ++ reverse xs
{-
because `reverse` will flip the order of the two parts `xs` and `ys` of
`xs ++ ys`. Now, when we run
~~~~~{.haskell}
ghci> quickCheck prop_revapp_ok
~~~~~
you should see
+++ OK, passed 100 tests.
That is, Haskell generated 100 test inputs and for all of those, the
property held. You can up the stakes a bit by changing the number of tests
you want to run
-}
quickCheckN :: (Testable prop) => Int -> prop -> IO ()
quickCheckN n = QC.quickCheck . QC.withMaxSuccess n
{-
and then ask quickcheck to run more tests.
~~~~~{.haskell}
ghci> quickCheckN 1000 prop_revapp_ok
~~~~~
QuickCheck QuickSort
--------------------
Let's look at a slightly more interesting example. Here is an
implementation of *quicksort* in Haskell. For efficiency, we'll use
the `DList` library so that we can `append` quickly. (Some may quibble that this is
actually the quicksort algorithm because it does not modify the list in
place. But it is a reasonable purely functional analogue.)
-}
qsort :: forall a. Ord a => [a] -> [a]
qsort t = DL.toList (aux t)
where
aux :: [a] -> DL.DList a
aux [] = DL.empty
aux (x : xs) = aux lhs `DL.append` DL.cons x (aux rhs)
where
lhs = [y | y <- xs, y < x] -- this is a "list comprehension"
-- i.e. the list of all elements from
-- xs that are less than x
rhs = [z | z <- xs, z > x]
{-
Really doesn't need much explanation! Let's run it "by hand" on a
few inputs to see what it does. Check out each of these to see what
they produce.
-}
-- >>> [10,9..1]
-- >>> qsort [10,9..1]
-- >>> [2,4..20] ++ [1,3..11]
-- >>> qsort $ [2,4..20] ++ [1,3..11]
{-
Looks good -- let's try to test that the output is in
fact sorted. We need a function that checks that a
list is ordered
-}
isOrdered :: Ord a => [a] -> Bool
isOrdered (x : y : zs) = x <= y && isOrdered (y : zs)
isOrdered [_] = True
isOrdered [] = True
{-
and then we can use the above to write a property saying that the
result of qsort is an ordered list.
-}
prop_qsort_isOrdered :: [Int] -> Bool
prop_qsort_isOrdered xs = isOrdered (qsort xs)
{-
Let's test it!
~~~~~{.haskell}
ghci> quickCheckN 1000 prop_qsort_isOrdered
~~~~~
Conditional Properties
----------------------
Here are several other properties that we
might want. First, repeated `qsorting` should not
change the list. That is,
-}
prop_qsort_idemp :: [Int] -> Bool
prop_qsort_idemp xs = qsort (qsort xs) == qsort xs
{-
Second, the head of the result is the minimum element
of the input
-}
prop_qsort_min :: [Int] -> Bool
prop_qsort_min xs = head (qsort xs) == minimum xs
{-
~~~~~{.haskell}
ghci> quickCheck prop_qsort_min
~~~~~
S
P
O
I
L
E
R
S
P
A
C
E
However, when we run this, we run into a glitch.
But of course! The earlier properties held *for all inputs*
while this property makes no sense if the input list is empty!
This is why thinking about specifications and properties has the
benefit of clarifying the *preconditions* under which a given
piece of code is supposed to work.
In this case we want a *conditional properties* where we only want
the output to satisfy to satisfy the spec *if* the input meets the
precondition that it is non-empty.
-}
prop_qsort_nn_min :: [Int] -> Property
prop_qsort_nn_min xs =
not (null xs) ==> head (qsort xs) == minimum xs
{-
We can write a similar property for the maximum element too.
-}
prop_qsort_nn_max :: [Int] -> Property
prop_qsort_nn_max xs =
undefined
{-
~~~~~{.haskell}
ghci> quickCheckN 100 prop_qsort_nn_min
ghci> quickCheckN 100 prop_qsort_nn_max
~~~~~
This time around, both the properties hold.
Note that now, instead of just being a `Bool` the output
of the function is now a `Property`, a special type built into
the QC library. Similarly the *implies* operator `==>`
is one of many QC combinators that allow the construction
of rich properties.
Testing Against a Model Implementation
--------------------------------------
We could keep writing different properties that capture
various aspects of the desired functionality of `qsort`.
Another approach for validation is to test that our `qsort`
is *behaviorally* identical to a trusted *reference
implementation* which itself may be too inefficient or
otherwise unsuitable for deployment. In this case, let's
use the standard library's `sort` function
-}
prop_qsort_sort :: [Int] -> Bool
prop_qsort_sort xs = qsort xs == List.sort xs
{-
which we can put to the test
~~~~~{.haskell}
ghci> quickCheckN 1000 prop_qsort_sort
~~~~~
S
P
O
I
L
E
R
S
P
A
C
E
Say, what?!
~~~~~{.haskell}
ghci> qsort [-1,-1]
~~~~~
Ugh! So close, and yet ... Can you spot the bug in our code? Here's
a simplified version that uses normal lists instead of DList.
(The bug is not in the DList library.)
~~~~~{.haskell}
qsort [] = []
qsort (x:xs) = qsort lhs ++ [x] ++ qsort rhs
where lhs = [y | y <- xs, y < x]
rhs = [z | z <- xs, z > x]
~~~~~
We're assuming that the *only* occurrence of (the value) `x`
is itself! That is, if there are any *copies* of `x` in the
tail, they will not appear in either `lhs` or `rhs` and hence
they get thrown out of the output.
Is this a bug in the code? What *is* a bug anyway? Perhaps the
fact that all duplicates are eliminated is a *feature*! At any
rate there is an inconsistency between our mental model of how
the code *should* behave as articulated in `prop_qsort_sort`
and the actual behavior of the code itself.
We can rectify matters by stipulating that the `qsort` produces
lists of distinct elements
-}
isDistinct :: Eq a => [a] -> Bool
isDistinct = undefined
prop_qsort_distinct :: [Int] -> Bool
prop_qsort_distinct = isDistinct . qsort
{-
and then, weakening the equivalence to only hold on inputs that
are duplicate-free
-}
prop_qsort_distinct_sort :: [Int] -> Property
prop_qsort_distinct_sort xs =
isDistinct xs ==> qsort xs == List.sort xs
{-
QuickCheck happily checks the modified properties
~~~~~{.haskell}
ghci> quickCheck prop_qsort_distinct
ghci> quickCheck prop_qsort_distinct_sort
~~~~~
The Perils of Conditional Testing
---------------------------------
Well, we managed to *fix* the `qsort` property, but beware! Adding
preconditions leads one down a slippery slope. In fact, if we paid
closer attention to the above runs, we would notice something
~~~~~{.haskell}
ghci> quickCheckN 10000 prop_qsort_distinct_sort
...
(5012 tests; 248 discarded)
...
+++ OK, passed 10000 tests.
~~~~~
The bit about some tests being *discarded* is ominous. In effect,
when the property is constructed with the `==>` combinator, QC
discards the randomly generated tests on which the precondition
is false. In the above case QC grinds away on the remainder until
it can meet its target of `10000` valid tests. This is because
the probability of a randomly generated list meeting the precondition
(having distinct elements) is high enough. This may not always be the case.
To see why, let's look at another sorting function.
The following code is (a simplified version of) the `insert` function from the
standard library
-}
insert :: forall a. Ord a => a -> [a] -> [a]
insert x = aux
where
aux :: [a] -> [a]
aux [] = [x]
aux (y : ys)
| x <= y = x : y : ys
| otherwise = y : aux ys
{-
Given an element `x` and a list `xs`, the function walks along `xs`
till it finds the first element greater than `x` and it places `x`
to the left of that element. Thus
-}
-- >>> insert 8 ([1..3] ++ [10..13])
{-
Indeed, the following is the well known [insertion-sort][5] algorithm
-}
isort :: Ord a => [a] -> [a]
isort = foldr List.insert []
{-
We could write our own tests, but why do something a machine can do better?!
-}
prop_isort_sort :: [Int] -> Bool
prop_isort_sort xs = isort xs == List.sort xs
{-
~~~~~{.haskell}
ghci> quickCheckN 1000 prop_isort_sort
~~~~~
Now, the reason that the above works is that the `insert`
routine *preserves* sorted-ness. That is, while of course
the property
-}
prop_insert_ordered' :: Int -> [Int] -> Bool
prop_insert_ordered' x xs = isOrdered (insert x xs)
{-
is bogus,
~~~~~{.haskell}
ghci> quickCheckN 1000 prop_insert_ordered'
~~~~~
the output *is* ordered if the input was ordered to begin with
-}
prop_insert_ordered :: Int -> [Int] -> Property
prop_insert_ordered x xs =
isOrdered xs ==> isOrdered (insert x xs)
{-
Notice that now, the precondition is more *complex* -- the property
requires that the input list be ordered. If we QC the property
~~~~~{.haskell}
ghci> quickCheck prop_insert_ordered
~~~~~
<FILL IN WHAT HAPPENS HERE!>
*Aside* the above example also illustrates the benefit of
writing the property as `p ==> q` instead of using the boolean
operator `||` to write `not p || q`. In the latter case, there is
a flat predicate, and QC doesn't know what the precondition is,
so a property may hold *vacuously*. For example consider the
variant
-}
prop_insert_ordered_vacuous :: Int -> [Int] -> Bool
prop_insert_ordered_vacuous x xs =
not (isOrdered xs) || isOrdered (insert x xs)
{-
QC will happily check it for us
~~~~~{.haskell}
ghci> quickCheckN 1000 prop_insert_ordered_vacuous
~~~~~
Unfortunately, in the above, the tests passed *vacuously*
only because their inputs were *not* ordered, and one
should use `==>` to avoid the false sense of security
delivered by vacuity.
QC provides us with some combinators for guarding against
vacuity by allowing us to investigate the *distribution*
of test cases
~~~~~{.haskell}
QC.label :: String -> Property -> Property
QC.classify :: Bool -> String -> Property -> Property
~~~~~
We may use these to write a property that looks like
-}
prop_insert_ordered_vacuous' :: Int -> [Int] -> Property
prop_insert_ordered_vacuous' x xs =
QC.label lbl $
not (isOrdered xs) || isOrdered (insert x xs)
where
lbl =
(if isOrdered xs then "Ordered, " else "Not Ordered, ")
++ show (length xs)
{-
When we run this, we get a detailed breakdown of the 100 passing tests:
~~~~~{.haskell}
ghci> quickCheck prop_insert_ordered_vacuous'
~~~~~
where in the first four lines, `P% COND, N` means that `P` percent of the
ordered inputs had length `N`, and satisfied the predicate denoted by the
string `COND`.
What percentage of lists were ordered? How long were they? <FILL IN HERE>
Generating Data
===============
Before we start discussing how QC generates data (and how we can help it
generate data meeting some pre-conditions), we must ask ourselves a basic
question: how does QC behave *randomly* in the first place?!
~~~~~{.haskell}
ghci> quickCheck prop_insert_ordered'
ghci> quickCheck prop_insert_ordered'
~~~~~
Eh? This seems most *impure* -- same inputs yielding two totally different
outputs! How does that happen?
The QC library defines a type
Gen a
of "generators for values of type a".
The impurity of random generation is bottled up inside the 'Gen' type. The
**monad** structure of this type let's us work with this impurity in a
controlled way, but we will get to what that means. For now, note that these generators
are a powerful mechanism for creating random data and that the QuickCheck
library contains multiple ways of constructing generators.
For example, we can construct a generator using the `chooseInt` function
to generate a random number in a given range:
-}
-- | generate an Int between 1 and 10, inclusive
genSmallInt :: Gen Int
genSmallInt = QC.chooseInt (1, 10)
{-
If you have a generator, you can see what it produces with the `sample` operation:
~~~~~{.haskell}
sample :: Show a => Gen a -> IO ()
~~~~~
This function will show you a sample of the values produced by the generator (and you'll get different values each time).
~~~~~~~~~~~~~~~~~~~~~~~~~~~{.haskell}
ghci> sample genSmallInt
3
6
9
8
1
2
4
10
1
7
4
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This operation generates some example values and prints them to stdout.
Alternatively, if you want access to the randomly generated values, the
sample' function will return them to you.
-}
-- >>> QC.sample' genSmallInt
{-
Generator Combinators
---------------------
QC comes loaded with a set of combinators that allow us to create
generators for various data structures.
The first of these combinators is `choose`, which is the generalization of `chooseInt.`
~~~~~{.haskell}
choose :: (System.Random.Random a) => (a, a) -> Gen a
~~~~~
This function takes an *interval* and returns an random element from that interval.
(The typeclass `System.Random.Random` describes types which can be
*sampled*. For example, the following is a randomly chosen set of numbers
between `0` and `3`.
-}
-- >>> QC.sample' $ QC.choose (0, 3)
{-
A second useful combinator is `elements`
~~~~~{.haskell}
elements :: [a] -> Gen a
~~~~~
which returns a generator that produces values drawn from the input list
-}
-- >>> QC.sample' $ QC.elements [10, 20..100]
{-
A third combinator is `oneof`
~~~~~{.haskell}
oneof :: [Gen a] -> Gen a
~~~~~
which allows us to randomly choose between multiple generators
-}
-- >>> QC.sample' $ QC.oneof [QC.elements [10,20,30], QC.choose (0,3)]
{-
a related generator is `listOf`
~~~~~{.haskell}
listOf :: Gen a -> Gen [a]
~~~~~
that gives us random lists, where the elements are generated by the argument generator.
-}
-- >>> QC.sample' (QC.listOf (QC.elements [1,2,3]))
{-
and finally, the above is generalized into the `frequency` combinator
~~~~~{.haskell}
frequency :: [(Int, Gen a)] -> Gen a
~~~~~
which allows us to build weighted combinations of individual generators.
-}
-- >>> QC.sample' $ QC.frequency [(1, QC.elements [1,2]), (5, QC.elements [100,200])]
{-
The Generator *Monad*
---------------------
The parameterized type 'Gen' is an instance of the `Monad` type class, one that we
will become more familiar with later one this semester. What this
means, for today, is that the monadic operations are available
for constructing new generators. Two of these operators come directly from
the Monad class itself:
~~~~~~~{.haskell}
-- part of the class Monad
--
return :: a -> Gen a
(>>=) :: Gen a -> (a -> Gen b) -> Gen b -- pronounced "bind"
~~~~~~~
For the `Gen` type, the `return` operator creates a generator that
returns exactly the same thing every time, the argument that we
supplied to return.
For example, we can create a generator that always returns the value 3.
-}
genThree :: Gen Int
genThree = return 3
-- >>> QC.sample' genThree
{-
The `(>>=)` operator is a bit more interesting. It takes a generator and a
function that takes a value and returns a generator. It then returns a
generator that applies the function to the value generated by the first
generator. That's a mouthful, so let's look at some examples.
Here's a crazy way to always generate the value five: first generate a three and then
add two to it.
-}
genFive :: Gen Int
genFive = genThree >>= \x -> return (x + 2)
-- >>> QC.sample' genFive
{-
Here's a slightly more interesting generator: we first create an arbitrary
boolean value, then if that value is `True` we generate a three, otherwise we
generate a five.
-}
genThreeOrFive :: Gen Int
genThreeOrFive = QC.choose (False, True) >>= \x -> return (if x then 3 else 5)
-- >>> QC.sample' genThreeOrFive
{-
The next three useful operations are from the library
[Control.Monad](http://hackage.haskell.org/package/base-4.17.1.0/docs/Control-Monad.html).
These are defined in terms of `return` and `(>>=)` above, so they
are available for any type constructor that is an instance of
the Monad class, including `Gen`.
~~~~~~~{.haskell}
liftM :: (a -> b) -> Gen a -> Gen b
liftM2 :: (a -> b -> c) -> Gen a -> Gen b -> Gen c
liftM3 :: (a -> b -> c -> d) -> Gen a -> Gen b -> Gen c -> Gen d
~~~~~~~
The `lift` in these names comes from an analogy: we are taking normal functions
and "lifting" them to work with generators. For example, `liftM` takes any
regular function of type `a -> b` and converts it to be a function of
type `Gen a -> Gen b`.
Note, `liftM` above has another name---`fmap`. That's right, every monad is
also a functor. Furthermore, the infix operator `(<$>)` is yet another name
for `fmap` that can look nice in your definitions.
We will cover what it exactly means for `Gen` to be a monad later on in the
course. However, as we will see, these operations let us put generators
together compositionally.
-}
genPair :: Gen a -> Gen b -> Gen (a, b)
genPair = liftM2 (,) -- a generator for pairs
-- >>> QC.sample' (genPair genThree genFive)
{-
Generator Practice
------------------
Use the operators above to define generators. Make sure that you test them out
to make sure that they are what you want.
-}
genBool :: Gen Bool
genBool = undefined
-- >>> QC.sample' genBool
genTriple :: Gen a -> Gen b -> Gen c -> Gen (a, b, c)
genTriple = undefined
-- >>> QC.sample' (genTriple genBool genThree genFive)
genMaybe :: Gen a -> Gen (Maybe a)
genMaybe ga = undefined
-- >>> QC.sample' (genMaybe genThree)
{-
The Arbitrary Typeclass
-----------------------
To keep track of all these generators, QC defines a typeclass containing types
for which random values can be generated!
~~~~~{.haskell}
class Arbitrary a where
arbitrary :: Gen a
~~~~~
Thus, to have QC work with (ie generate random tests for) values of type
`a` we need only make `a` an instance of `Arbitrary` by defining an
appropriate `arbitrary` function for it. QC defines instances for base
types like `Int` , `Float`, etc
~~~~~{.haskell}
ghci> sample (arbitrary :: Gen Int)
~~~~~
and lifts them to compound types.
~~~~~{.haskell}
instance (Arbitrary a, Arbitrary b, Arbitrary c) => Arbitrary (a,b,c) where
arbitrary = liftM3 (,,) arbitrary arbitrary arbitrary
~~~~~
-}
-- >>> QC.sample' (arbitrary :: Gen (Int,Float,Bool))
-- >>> QC.sample' (arbitrary :: Gen [Int])
{-
However, you'll need to make your own instances of `Arbitrary` for user
defined datatypes. As we'll discuss below, there are two many options in
generation for GHC to make this class automatically derivable. Below, we will
walk through constructing a good generator for the list type as an example of
constructing a good generator for an arbitrary datatype. (This code is just an
example --- if you need to generate a list, you can use the `listOf` function
from the library.) After reading this section, challenge yourself to write
a generator for a `Tree` type.
Generating Trees
----------------
Here's our familiar type for binary trees. Let's generate some
arbitrary values of this type!
-}
data Tree a = Empty | Branch a (Tree a) (Tree a) deriving (Show, Foldable)
{-
Here's our first generator. It uses the `liftM3` combinator above
to generate an arbitrary tree. It type checks, but that is the
only good thing about this code.
-}
genTree1 :: (Arbitrary a) => Gen (Tree a)
genTree1 = liftM3 Branch arbitrary genTree1 genTree1
{-
Only run this if you have a lot of time to kill!
~~~~~~~~~~~{.haskell}
ghci> QC.sample' (genTree1 :: Gen (Tree Int))
~~~~~~~~~~~
Can you spot a problem in the above?
<FILL IN HERE>
Let's try again,
-}
genTree2 :: forall a. (Arbitrary a) => Gen (Tree a)
genTree2 =
QC.oneof
[ return Empty,
liftM3 Branch arbitrary genTree2 genTree2
]
{-
Let's take a look at how big the trees are that we are generating. Because
we derived the `Foldable` class for our `Tree` type above, the `length`
function will tell us how many values are stored in the generated trees.
Refresh this value a few times to see the distribution of tree sizes that
our generator is producing.
-}
-- >>> map length <$> QC.sample' (genTree2 :: Gen (Tree Int))
{-
This is not bad, but there is still something undesirable.
What is wrong with this output?
<FILL IN HERE>
This version fixes that problem. We only choose `Empty` one third of the time.
-}
genTree3 :: forall a. (Arbitrary a) => Gen (Tree a)
genTree3 =
QC.frequency
[ (1, return Empty),
(2, liftM3 Branch arbitrary genTree3 genTree3)
]
{-
But, if you try it out, you'll find that this generator is rather slow.
In fact, I was never patient enough to let it finish.
-}
-- >>> map length <$> QC.sample' (genTree3 :: Gen (Tree Int))
{-
Now `genTree3` has the opposite problem --- it generates a lot of big
trees (more than 4 or 5 values) but not so many short ones. But finding bugs
with small data is a lot faster than finding bugs with large data.
So, two last tweaks. We let quickcheck determine what frequency to use, and we
decrease the frequency of `Branch` with each recursive call. For the former, we
rely on the following function from QC library.
sized :: (Int -> Gen a) -> Gen a
This function is higher-order; it takes a generator with a size parameter
(i.e. the Int) and uses it to develop a new generator by progressively
increasing this size.
For the latter, when we define this "size-aware" function, we cut the size in
half for each recursive call.
(Note: to give a type annotation for the local definition `gen`,
we have to bring the type variable `a` into scope with the `forall`
keyword.)
-}
genTree :: forall a. (Arbitrary a) => Gen (Tree a)
genTree = QC.sized gen
where
gen :: Int -> Gen (Tree a)
gen n =
QC.frequency
[ (1, return Empty),
(n, liftM3 Branch arbitrary (gen (n `div` 2)) (gen (n `div` 2)))
]
{-
Now look at that distribution! Not too small, not too big, not too many empty trees.
-}
-- >>> map length <$> QC.sample' (genTree :: Gen (Tree Int))
-- [0,3,1,4,4,8,8,9,19,8,11]
{-
I encourage you to look at the implementation of `genTree4` closely. This use
of `frequency` and `sized` is particularly important to controlling the
generation of arbitrary tree-structured data.
Shrinking
---------
When properties fail, QuickCheck provides a counterexample. But sometimes
this counterexample could be rather complex and not much use in finding
your bug.
As an example, consider this buggy function that adds together all of the
values stored in a tree of integers. I've added an "optimization" to this
function to make it super fast.
-}
treeSum :: Tree Int -> Int
treeSum = aux
where
aux Empty = 0
aux (Branch x l r) = if x == 0 then 0 else aux l + x + aux r
{-
Can you see the bug? The special case of 0 would be great if we were
multiplying the values in the tree but computes the wrong answer for addition.
It turns out that, because I derived `Foldable` above, there is already
an overloaded function `sum` that I can use to sum up the tree values.