## Science

Is hard.

Just dumping some links here for now:

Feynman on “cargo cult science”: http://www.lhup.edu/~DSIMANEK/cargocul.htm

Feynman on “what is science”: http://www.fotuva.org/feynman/what_is_science.html

Ioannidis, Why Most Published Research Findings Are False: http://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124

On how subtle it is: http://slatestarcodex.com/2014/04/28/the-control-group-is-out-of-control/

Reinhart, http://www.statisticsdonewrong.com

http://www.nytimes.com/2012/04/17/science/rise-in-scientific-journal-retractions-prompts-calls-for-reform.html?pagewanted=all

## The state of static analysis on Python

Here is a program that fails with UnboundLocalError three out of four times:

"""Module to demonstrate an error.""" import random def zero_or_one(): """Returns either 0 or 1 with equal probability.""" return random.choice([0, 1]) def main(): """Function to demonstrate an error.""" if zero_or_one(): first = 42 for _ in range(zero_or_one()): second = 42 print first, second if __name__ == '__main__': main()

(Condensed from a real bug where, because of additional indentation, a variable assignment happened as the last line inside a loop, instead of the first line after it.)

I know of three tools that are popular: pychecker, pyflakes, and pylint. None of them say a single thing about this program. It is questionable whether ever (and if so, how often) code like the above is what the programmer intended.

The first of these, pychecker, is not on pip, and requires you to download a file from Sourceforge and run “python setup.py install”. Anyway, this is the output from the three programs:

```
```

```
/tmp% pychecker test.py
Processing module test (test.py)...
Warnings...
None
/tmp% pyflakes test.py
/tmp% pylint test.py
No config file found, using default configuration
Report
======
12 statements analysed.
Statistics by type
------------------
+---------+-------+-----------+-----------+------------+---------+
|type |number |old number |difference |%documented |%badname |
+=========+=======+===========+===========+============+=========+
|module |1 |1 |= |100.00 |0.00 |
+---------+-------+-----------+-----------+------------+---------+
|class |0 |0 |= |0 |0 |
+---------+-------+-----------+-----------+------------+---------+
|method |0 |0 |= |0 |0 |
+---------+-------+-----------+-----------+------------+---------+
|function |2 |2 |= |100.00 |0.00 |
+---------+-------+-----------+-----------+------------+---------+
Raw metrics
-----------
+----------+-------+------+---------+-----------+
|type |number |% |previous |difference |
+==========+=======+======+=========+===========+
|code |11 |73.33 |11 |= |
+----------+-------+------+---------+-----------+
|docstring |3 |20.00 |3 |= |
+----------+-------+------+---------+-----------+
|comment |0 |0.00 |0 |= |
+----------+-------+------+---------+-----------+
|empty |1 |6.67 |1 |= |
+----------+-------+------+---------+-----------+
Duplication
-----------
+-------------------------+------+---------+-----------+
| |now |previous |difference |
+=========================+======+=========+===========+
|nb duplicated lines |0 |0 |= |
+-------------------------+------+---------+-----------+
|percent duplicated lines |0.000 |0.000 |= |
+-------------------------+------+---------+-----------+
Messages by category
--------------------
+-----------+-------+---------+-----------+
|type |number |previous |difference |
+===========+=======+=========+===========+
|convention |0 |0 |= |
+-----------+-------+---------+-----------+
|refactor |0 |0 |= |
+-----------+-------+---------+-----------+
|warning |0 |0 |= |
+-----------+-------+---------+-----------+
|error |0 |0 |= |
+-----------+-------+---------+-----------+
Global evaluation
-----------------
Your code has been rated at 10.00/10 (previous run: 10.00/10, +0.00)
/tmp%
```

```
```

## Bhavabhuti on finding a reader

Bhavabhūti, the 8th-century author of the very moving play Uttara-rāma-carita, has in one of his other works these lines, any author’s consolation that even if your work receives not enough praise today, someday the right sort of reader will come along, who will derive great joy or meaning from it.

ये नाम केचिदिह नः प्रथयन्त्यवज्ञां जानन्ति ते किमपि तान्प्रति नैष यत्नः । उत्पत्स्यते तु मम कोऽपि समानधर्मा कालो ह्ययं निरवधिर्विपुला च पृथ्वी ॥ye nāma kecit iha naḥ prathayanti avajñām

jānanti te kim api tān prati na eṣa yatnaḥ |

utpatsyate tu mama ko api samāna-dharmā

kālo hi ayaṃ niravadhiḥ vipulā ca pṛthvī ||Those who deride or ignore my work —

let them know: my efforts are not for them.

There will come along someone who shares my spirit:

the world is vast, and time endless.

This verse has become a favourite of many. It appears already in the first known anthology of Sanskrit verses (*subhāṣita*-collection), Vidyākara’s *Subhāṣita-ratna-koṣa*. (It’s numbered 1731 (= 50.34) in the edition by Kosambi and Gokhale, and translated by Ingalls.) Ingalls writes and translates (1965):

Of special interest are the verses of Dharmakīrti and Bhavabhūti, two of India’s most original writers, which speak of the scorn and lack of understanding which the writings of those authors found among contemporaries. To such disappointment Dharmakīrti replies with bitterness (1726, 1729), Bhavabhūti with the unreasoning hope of a romantic (1731). If the souls of men could enjoy their posthumous fame the one would now see his works admired even far beyond India, the other would see his romantic hope fulfilled.

Those who scorn me in this world

have doubtless special wisdom,

so my writings are not made for them;

but are rather with the thought that some day will be born,

since time is endless and the world is wide,

one whose nature is the same as mine.

A translation of this verse is also included in A. N. D. Haksar’s *A Treasury of Sanskrit Poetry in English Translation* (2002):

The Proud Poet

Are there any around who mock my verses?

They ought to know I don’t write for them.

Someone somewhere sometime will understand.

Time has no end. The world is big.

— translated by V. N. Misra, L. Nathan and S. Vatsyayan [The Indian Poetic Tradition, 1993]

Andrew Schelling has written of it in *Manoa*, Volume 25, Issue 2, 2013:

Critics scoff

at my work

and declare their contempt—

no doubt they’ve got

their own little wisdom.

I write nothing for them.

But because time is

endless and our planet

vast, I write these

poems for a person

who will one day be born

with my sort of heart.“Criticism is for poets as ornithology is for the birds,” wrote John Cage. Bhavabhūti has scant doubt that future generations will honor his work. The reader who will arise, utpalsyate [sic], is somebody of the same faith, heart, or discipline, samānadharmā.

Just now also found it on the internet, here (2014) (misattributed to Bhartṛhari):

There are those who

treat my work with

studied indifference.

Maybe they know something,

but I’m not writing for them.

Someone will come around

who feels the way I do.

Time, after all, is limitless,

and fortune spreads far.

Finally, in Sadāsvada, written by my friend Mohan with some comments from me, this was included in one of our our very first posts (2012):

In his play Mālatīmādhava, he makes a point that deserves to be the leading light of anyone wishing to do something of value and

~~is~~put off by discouragement. Standing beside the words attributed to Gandhi (“First they ignore you, then they laugh at you, then they fight you, then you win.”) and Teddy Roosevelt (“It is not the critic who counts…”), Bhavabhūti’s confidence in the future stands resplendent:“They who disparage my work should know that it’s not for them that I did it. One day, there will arise someone who will truly know me: this world is vast, and time infinite.”

It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.

[Note on the text: in Vidyākara’s compilation the verse ends with “विपुला च लक्ष्मीः” (vipulā ca lakṣmīḥ) instead of “विपुला च पृथ्वी” (vipulā ca pṛthvī), but the actual source work Mālatī-mādhava has the latter, as do all quotations of this verse elsewhere (e.g. the काव्यप्रकाशः of Mammaṭa, the Sahityadarpana of Viśvanātha Kavirāja, the रसार्णवसुधाकरः of श्रीसिंहभूपाल), and that is what Ingalls uses: “For *lakṣmīḥ*, which utterly destroys the line, read *pṛthvi* with the printed texts of *Māl*.” Actually, most quotations have “utpatsyate ‘sti” in place of “utpatsyate tu”: “either will be born, or already exists…”.]

## Printing floating-point numbers

Everyone knows that floating-point numbers, being discrete, cannot possibly exactly represent every real number. In particular, the usual binary (IEEE 754) floating point numbers cannot even exactly store all numbers exactly representable in decimal (e.g. 0.3 or 0.1, which are not dyadic rationals).

But what about *printing* them?

Just because the number stored internally is not 0.1 but the closest approximation to it (say as 0.100000001490116119384765625) doesn’t mean it should be printed as such, when “0.1” means exactly the same number.

This is a solved problem since 1990.

TODO: Write rest of this post.

Bryan O’Sullivan (of Real World Haskell fame):

http://www.serpentine.com/blog/2011/06/29/here-be-dragons-advances-in-problems-you-didnt-even-know-you-had/

Steel & White paper *How to Print Floating-Point Numbers Accurately*: https://lists.nongnu.org/archive/html/gcl-devel/2012-10/pdfkieTlklRzN.pdf

Their retrospective: http://grouper.ieee.org/groups/754/email/pdfq3pavhBfih.pdf

Burger & Dybvig:

http://www.cs.indiana.edu/~dyb/pubs/FP-Printing-PLDI96.pdf

Florian Loitsch:

http://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf

Russ Cox: http://research.swtch.com/ftoa

http://www.ryanjuckett.com/programming/printing-floating-point-numbers/

https://labix.org/python-nicefloat

## Kalidasa’s Deepashikha

(Another example of good vs bad translations from Sanskrit. Previously see here and here.)

One of Kālidāsa’s famous similes is in the following verse from the Raghuvaṃśa, in the context of describing the *svayaṃvara* of Indumatī. The various hopeful suitors of the princess, all kings from different regions, are lined up as she passes them one by one, her friend doing the introductions.

संचारिणी दीपशिखेव रात्रौ यम् यम् व्यतीयाय पतिंवरा सा । नरेन्द्रमार्गाट्ट इव प्रपेदे विवर्णभावम् स स भूमिपालः ॥ ६-६७

saṁcāriṇī dīpa-śikheva rātrau

yam yam vyatīyāya patiṁvarā sā |

narendra-mārga-aṭṭa iva prapede

vivarṇa-bhāvam sa sa bhūmipālaḥ || 6-67

Only today did I discover a decent translation into English. It’s by John Brough (1975/6):

As if a walking lamp-flame in the night

On the king’s highway, flanked with houses tall,

She moved, and lit each prince with hopeful light,

And, passing on, let each to darkness fall.

Every other translation I have seen really falls short. Witness the misunderstandings, and the killing of all feeling.

Here is Ryder (1904), who is usually good:

And every prince rejected while she sought

A husband, darkly frowned, as turrets, bright

One moment with the flame from torches caught,

Frown gloomily again and sink in night.

The idea is there, but requires too much effort to understand.

This is P. de Lacy Johnstone (1902):

Now as the Maid went by, each suitor-King,

Lit for a moment by her dazzling eyes,

Like wayside tower by passing lamp, sank back

In deepest gloom. …

Every king, whom Indumati passed by while choosing her husband, assumed a pale look as the houses on a high way are covered with darkness in the absence of lamps.

Whatsoever king the maiden intent on choosing her husband passed by, like the flame of a moving lamp at night, that same king turned pale, just as a mansion situate on the highway, is shrouded in darkness when left behind (by a moving light).

67. pati.m varA sA= husband, selector, she – she who has come to select her husband, indumati; rAtrau sa.mcAriNI dIpa shikha iva= in night, moving, lamp’s, [glittering] flame, as with; ya.m ya.m= whom, whom; [bhUmi pAlam= king, whomever]; vyatIyAya= passed by; saH saH bhUmipAlaH= he, he, king – such and such a king; narendra mArga= on king’s, way; aTTa= a turret, or a balustrade; iva= like; vi+varNa bhAva.m= without, colour, aspect – they bore a colourless aspect; prapede= [that king] obtained – that king became colourless, he drew blank.

Princess indumati who came to choose her husband then moved like the glittering flame of a lamp on a king’s way, and whichever prince she left behind was suffused with pallor just like a turret or balustrade on the king’s way will be shrouded in darkness and becomes dim when left behind by the moving light on the king’s way. [6-67]

And this is representative of the average quality of Sanskrit-to-English translations, and how much beauty is lost.

## Some playing with Python

A long time ago, Diophantus (sort of) discussed integer solutions to the equation

(solutions to this equation are called Pythagorean triples).

Centuries later, in 1637, Fermat made a conjecture (now called Fermat’s Last Theorem, not because he uttered it in his dying breath, but because it was the last one to be proved — in ~1995) that

has no positive integer solutions for . In other words, his conjecture was that none of the following equations has a solution:

… and so on. An nth power cannot be partitioned into two nth powers.

About a century later, Euler proved the case of Fermat’s conjecture, but generalized it in a different direction: he conjectured in 1769 that an nth power cannot be partitioned into fewer than n nth powers, namely

has no solutions with . So his conjecture was that (among others) none of the following equations has a solution:

… and so on.

This conjecture stood for about two centuries, until abruptly it was found to be false, by Lander and Parkin who in 1966 simply did a direct search on the fastest (super)computer at the time, and found this counterexample:

(It is still one of only three examples known, according to Wikipedia.)

Now, how might you find this solution on a computer today?

In his wonderful (as always) post at bit-player, Brian Hayes showed the following code:

import itertools as it def four_fifths(n): '''Return smallest positive integers ((a,b,c,d),e) such that a^5 + b^5 + c^5 + d^5 = e^5; if no such tuple exists with e < n, return the string 'Failed'.''' fifths = [x**5 for x in range(n)] combos = it.combinations_with_replacement(range(1,n), 4) while True: try: cc = combos.next() cc_sum = sum([fifths[i] for i in cc]) if cc_sum in fifths: return(cc, fifths.index(cc_sum)) except StopIteration: return('Failed')

to which, if you add (say) `print four_fifths(150)`

and run it, it returns the correct answer fairly quickly: in about 47 seconds on my laptop.

The `if cc_sum in fifths:`

line inside the loop is an cost each time it’s run, so with a simple improvement to the code (using a set instead) and rewriting it a bit, we can write the following full program:

import itertools def find_counterexample(n): fifth_powers = [x**5 for x in range(n)] fifth_powers_set = set(fifth_powers) for xs in itertools.combinations_with_replacement(range(1, n), 4): xs_sum = sum([fifth_powers[i] for i in xs]) if xs_sum in fifth_powers_set: return (xs, fifth_powers.index(xs_sum)) return 'Failed' print find_counterexample(150)

which finishes in about 8.5 seconds.

Great!

But there’s something unsatisfying about this solution, which is that it assumes there’s a solution with all four numbers on the LHS less than 150. After all, changing the function invocation to `find_counterexample(145)`

makes it run a second faster even, but how could we know to do without already knowing the solution? Besides, we don’t have a fixed 8- or 10-second budget; what we’d *really* like is a program that keeps searching till it finds a solution or we abort it (or it runs out of memory or something), with no other fixed termination condition.

The above program used the given “n” as an upper bound to generate the combinations of 4 numbers; is there a way to generate all combinations when we don’t know an upper bound on them?

Yes! One of the things I learned from Knuth volume 4 is that if you simply write down each combination in descending order and order them lexicographically, the combinations you get for each upper bound are a prefix of the list of the next bigger one, i.e., for any upper bound, all the combinations form a prefix of the same infinite list, which starts as follows (line breaks for clarity):

1111, 2111, 2211, 2221, 2222, 3111, 3211, 3221, 3222, 3311, 3321, 3322, 3331, 3332, 3333, 4111, ... ... 9541, 9542, 9543, 9544, 9551, ... 9555, 9611, ...

There doesn’t seem to be a library function in Python to generate these though, so we can write our own. If we stare at the above list, we can figure out how to generate the next combination from a given one:

- Walk backwards from the end, till you reach the beginning or find an element that’s less than the previous one.
- Increase that element, set all the following elements to 1s, and continue.

We could write, say, the following code for it:

def all_combinations(r): xs = [1] * r while True: yield xs for i in range(r - 1, 0, -1): if xs[i] < xs[i - 1]: break else: i = 0 xs[i] += 1 xs[i + 1:] = [1] * (r - i - 1)

(The `else`

block on a `for`

loop is an interesting Python feature: it is executed if the loop wasn’t terminated with `break`

.) We could even hard-code the `r=4`

case, as we’ll see later below.

For testing whether a given number is a fifth power, we can no longer simply lookup in a fixed precomputed set. We can do a binary search instead:

def is_fifth_power(n): assert n > 0 lo = 0 hi = n # Invariant: lo^5 < n <= hi^5 while hi - lo > 1: mid = lo + (hi - lo) / 2 if mid ** 5 < n: lo = mid else: hi = mid return hi ** 5 == n

but it turns out that this is slower than one based on looking up in a growing set (as below).

Putting everything together, we can write the following (very C-like) code:

largest_known_fifth_power = (0, 0) known_fifth_powers = set() def is_fifth_power(n): global largest_known_fifth_power while n > largest_known_fifth_power[0]: m = largest_known_fifth_power[1] + 1 m5 = m ** 5 largest_known_fifth_power = (m5, m) known_fifth_powers.add(m5) return n in known_fifth_powers def fournums_with_replacement(): (x0, x1, x2, x3) = (1, 1, 1, 1) while True: yield (x0, x1, x2, x3) if x3 < x2: x3 += 1 continue x3 = 1 if x2 < x1: x2 += 1 continue x2 = 1 if x1 < x0: x1 += 1 continue x1 = 1 x0 += 1 continue if __name__ == '__main__': tried = 0 for get in fournums_with_replacement(): tried += 1 if (tried % 1000000 == 0): print tried, 'Trying:', get rhs = get[0]**5 + get[1]**5 + get[2]**5 + get[3]**5 if is_fifth_power(rhs): print 'Found:', get, rhs break

which is both longer and slower (takes about 20 seconds) than the original program, but at least we have the satisfaction that it doesn’t depend on any externally known upper bound.

I originally started writing this post because I wanted to describe some experiments I did with profiling, but it’s late and I’m sleepy so I’ll just mention it.

python -m cProfile euler_conjecture.py

will print relevant output in the terminal:

26916504 function calls in 26.991 seconds Ordered by: standard name ncalls tottime percall cumtime percall filename:lineno(function) 1 18.555 18.555 26.991 26.991 euler_conjecture.py:1() 13458164 4.145 0.000 4.145 0.000 euler_conjecture.py:12(fournums_with_replacement) 13458163 4.292 0.000 4.292 0.000 euler_conjecture.py:3(is_fifth_power) 175 0.000 0.000 0.000 0.000 {method 'add' of 'set' objects} 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}

Another way to view the same thing is to write the profile output to a file and read it with `cprofilev`

:

python -m cProfile -o euler_profile.out euler_conjecture.py cprofilev euler_profile.out

and visit http://localhost:4000 to view it.

Of course, simply translating this code to C++ makes it run much faster:

#include <array> #include <iostream> #include <map> #include <utility> typedef long long Int; constexpr Int fifth_power(Int x) { return x * x * x * x * x; } std::map<Int, int> known_fifth_powers = {{0, 0}}; bool is_fifth_power(Int n) { while (n > known_fifth_powers.rbegin()->first) { int m = known_fifth_powers.rbegin()->second + 1; known_fifth_powers[fifth_power(m)] = m; } return known_fifth_powers.count(n); } std::array<Int, 4> four_nums() { static std::array<Int, 4> x = {1, 1, 1, 0}; int i = 3; while (i > 0 && x[i] == x[i - 1]) --i; x[i] += 1; while (++i < 4) x[i] = 1; return x; } std::ostream& operator<<(std::ostream& os, std::array<Int, 4> x) { os << "(" << x[0] << ", " << x[1] << ", " << x[2] << ", " << x[3] << ")"; return os; } int main() { while (true) { std::array<Int, 4> get = four_nums(); Int rhs = fifth_power(get[0]) + fifth_power(get[1]) + fifth_power(get[2]) + fifth_power(get[3]); if (is_fifth_power(rhs)) { std::cout << "Found: " << get << " " << known_fifth_powers[rhs] << std::endl; break; } } }

and

```
clang++ -std=c++11 euler_conjecture.cc && time ./a.out
```

runs in 2.43s, or 0.36s if compiled with `-O2`.

But I don’t have a satisfactory answer to how to make our Python program which takes 20 seconds as fast as the 8.5-second known-upper-bound version.

Edit [2015-05-08]: I wrote some benchmarking code to compare all the different “combination” functions.

import itertools # Copied from the Python documentation def itertools_equivalent(iterable, r): pool = tuple(iterable) n = len(pool) if not n and r: return indices = [0] * r yield tuple(pool[i] for i in indices) while True: for i in reversed(range(r)): if indices[i] != n - 1: break else: return indices[i:] = [indices[i] + 1] * (r - i) yield tuple(pool[i] for i in indices) # Above function, specialized to first argument being range(1, n) def itertools_equivalent_specialized(n, r): indices = [1] * r yield indices while True: for i in reversed(range(r)): if indices[i] != n - 1: break else: return indices[i:] = [indices[i] + 1] * (r - i) yield indices # Function to generate all combinations of 4 elements def all_combinations_pythonic(r): xs = [1] * r while True: yield xs for i in range(r - 1, 0, -1): if xs[i] < xs[i - 1]: break else: i = 0 xs[i] += 1 xs[i + 1:] = [1] * (r - i - 1) # Above function, written in a more explicit C-like way def all_combinations_clike(r): xs = [1] * r while True: yield xs i = r - 1 while i > 0 and xs[i] == xs[i - 1]: i -= 1 xs[i] += 1 while i < r - 1: i += 1 xs[i] = 1 # Above two functions, specialized to r = 4, using tuple over list. def fournums(): (x0, x1, x2, x3) = (1, 1, 1, 1) while True: yield (x0, x1, x2, x3) if x3 < x2: x3 += 1 continue x3 = 1 if x2 < x1: x2 += 1 continue x2 = 1 if x1 < x0: x1 += 1 continue x1 = 1 x0 += 1 continue # Benchmarks for all functions defined above (and the library function) def benchmark_itertools(n): for xs in itertools.combinations_with_replacement(range(1, n), 4): if xs[0] >= n: break def benchmark_itertools_try(n): combinations = itertools.combinations_with_replacement(range(1, n), 4) while True: try: xs = combinations.next() if xs[0] >= n: break except StopIteration: return def benchmark_itertools_equivalent(n): for xs in itertools_equivalent(range(1, n), 4): if xs[0] >= n: break def benchmark_itertools_equivalent_specialized(n): for xs in itertools_equivalent_specialized(n, 4): if xs[0] >= n: break def benchmark_all_combinations_pythonic(n): for xs in all_combinations_pythonic(4): if xs[0] >= n: break def benchmark_all_combinations_clike(n): for xs in all_combinations_clike(4): if xs[0] >= n: break def benchmark_fournums(n): for xs in fournums(): if xs[0] >= n: break if __name__ == '__main__': benchmark_itertools(150) benchmark_itertools_try(150) benchmark_itertools_equivalent(150) benchmark_itertools_equivalent_specialized(150) benchmark_all_combinations_pythonic(150) benchmark_all_combinations_clike(150) benchmark_fournums(150)

As you can see, I chose inside the benchmarking function the same statement that would cause `all_combinations`

to terminate, and have no effect for the other combination functions.

When run with

`python -m cProfile benchmark_combinations.py`

the results include:

2.817 benchmark_combinations.py:80(benchmark_itertools) 8.583 benchmark_combinations.py:84(benchmark_itertools_try) 126.980 benchmark_combinations.py:93(benchmark_itertools_equivalent) 46.635 benchmark_combinations.py:97(benchmark_itertools_equivalent_specialized) 44.032 benchmark_combinations.py:101(benchmark_all_combinations_pythonic) 18.049 benchmark_combinations.py:105(benchmark_all_combinations_clike) 10.923 benchmark_combinations.py:109(benchmark_fournums)

Lessons:

- Calling
`itertools.combinations_with_replacement`

is by far the fastest, taking about 2.7 seconds. It turns out that it’s written in C, so this would be hard to beat. (Still, writing it in a`try`

block is seriously bad.) - The “equivalent” Python code from the itertools documentation (
`benchmark_itertools_combinations_with_replacment`

) is about 50x slower. - Gets slightly better when specialized to numbers.
- Simply generating
*all*combinations without an upper bound is actually faster. - It can be made even faster by writing it in a more C-like way.
- The tuples version with the loop unrolled manually is rather fast when seen in this light, less than 4x slower than the library version.

## Boolean blindness

*(Just digesting the first page of Google search results.)*

One of the lessons from functional programming is to encode as much information as possible into the types. Almost all programmers understand to some extent that types are helpful: they know not to store everything as `void*` (in C/C++) or as `Object` (in Java). They even know not to use say `double` for all numbers, or `string` for everything (as shell scripts / Tcl do). (This pejoratively called “stringly typed”.)

A corollary, from slide 19 here, is that (when your type system supports richer types) a boolean type is almost always the wrong choice, as it carries too little information.

The name “Boolean blindness” for this seems to have been coined by Dan Licata when taught a course at CMU as a PhD student.

From here (blog post by Robert Harper, his advisor at CMU):

There is no

informationcarried by a Boolean beyond its value, and that’s the rub. As Conor McBride puts it, to make use of a Boolean you have to know itsprovenanceso that you can know what itmeans.

[…]

Keeping track of this information (or attempting to recover it using any number of program analysis techniques) is notoriously difficult. The only thing you can do with a bit is to branch on it, and pretty soon you’re lost in a thicket of if-then-else’s, and you lose track of what’s what.

[…]The problem is computing the bit in the first place. Having done so, you have blinded yourself by reducing the information you have at hand to a bit, and then trying to recover that information later by remembering the provenance of that bit. An illustrative example was considered in my article on equality:

`fun plus x y = if x=Z then y else S(plus (pred x) y)`

Here we’ve crushed the information we have about x down to one bit, then branched on it, then were forced to recover the information we lost to justify the call to pred, which typically cannot recover the

factthat its argument is non-zero and must check for it to be sure. What a mess! Instead, we should write

`fun plus x y = case x of Z => y | S(x') => S(plus x' y)`

No Boolean necessary, and the code is improved as well! In particular, we obtain the predecessor

en passant, and have no need to keep track of the provenance of any bit.

Some commenter there says

To the best of my knowledge, Ted Codd was the first to point out, in his relational model, that there is no place for Boolean data types in entity modeling. It is a basic design principle to avoid characterizing data in terms of Boolean values, since there is usually some other piece of information you are forgetting to model, and once you slam a Boolean into your overall data model, it becomes very hard to version towards a more exact model (information loss).

An example from Hacker News (on an unrelated post):

Basically, the idea is that when you branch on a conditional, information is gained. This information may be represented in the type system and used by the compiler to verify safety, or it can be ignored. If it is ignored, the language is said to have “boolean blindness”.

Example:`if (ptr == NULL) { ... a ... } else { ... b ... }`

In branch a and branch b, different invariants about ptr hold. But the language/compiler are not verifying any of these invariants.

Instead, consider:

`data Maybe a = Nothing | Just a`

This defines a type “Maybe”, parameterized by a type variable “a”, and it defines two “data constructors” that can make a Maybe value: “Nothing” which contains nothing, and “Just” which contains a value of type “a” inside it.

This is known as a “sum” type, because the possible values of the type are the sum of all data constructor values.

We could still use this sum data-type in a boolean-blind way:`if isJust mx then .. use fromJust mx .. -- UNSAFE! else .. can't use content of mx ..`

However, using pattern-matching, we can use it in a safe way. Assume “mx” is of type “Maybe a”:

`case mx of Nothing -> ... There is no value of type "a" in our scope Just x -> ... "x" of type "a" is now available in our scope!`

So when we branch on the two possible cases for the “mx” type, we gain new type information that gets put into our scope.

“Maybe” is of course a simple example, but this is also applicable for any sum type at all.

Another from notes of someone called HXA7241:

A nullable pointer has two ‘parts’: null, and all-other-values. These can be got at with a simple if-else conditional:

`if p is not null then ... else ... end`

. And there is still nothing wrong with that. The problem arrives when you want to handle the parts – and you lack a good type system. What do you do with the non-null pointer value? You just have to put it back into a pointer again – anullablepointer: which is what you started with. So where has what you did with the test been captured? Nowhere. When you handle that intended part elsewhere you have to do the test again and again.

A Reddit discussion: Don Stewart says

Not strongly typed,

richly typed. Where evidence isn’t thrown away. Agda is about the closest thing to this we have.

### Relation to “object-oriented programming”

There’s also a good (and unintentionally revealing) example there by user `munificent`. Consider Harper’s example, which in more explicit Haskell could be:

```
data Num = Zero | Succ Num
plus :: Num -> Num -> Num
plus Zero y = y
plus (Succ x) y = Succ (plus x y)
```

We might write it in Java as the following:

```
```

```
interface Num {}
class Zero implements Num {}
class Succ implements Num {
public Succ(Num pred) {
this.pred = pred;
}
public final Num pred;
}
Num plus(Num x, Num y) {
if (x instanceof Succ) { // (1)
Succ xAsSucc = (Succ)x; // (2)
return new Succ(plus(xAsSucc.pred, y));
} else {
return y;
}
}
```

```
```

Here instanceof returns a boolean (comment (1)), but doesn’t carry with it any information about what that boolean represents (namely that x is an instance of Succ) so when we get to the next line (2) we’re forced to do an unsafe cast. As programmers, we know it’s safe from context, but the compiler doesn’t.

There *is* a way in Java of avoiding this situation (where the programmer has context the compiler doesn’t):

```
interface Num {
Num plus(Num other);
}
class Zero implements Num {
public Num plus(Num other) {
return other;
}
}
class Succ implements Num {
public Succ(Num pred) {
this.pred = pred;
}
public Num plus(Num other) {
return new Succ(pred.plus(y));
}
public final Num pred;
}
```

But see what has happened (by user `aaronia`):

There’s a rub though — in your first code snippet, “plus” was written as a non-member function; anyone can write it as a stand alone library. It was modular. But, as you demonstrated, it had the redundant type check.

However, your second code snippet has lost this modularity. You had to add new methods to the Num data type.

I think this is the pitfall of Object-Oriented Programming (which is like “Pants-Oriented Clothing”): objects can talk about themselves, but it’s harder to talk about objects.

If you want to write a function that does similar things for a bunch of types, you have to write similar function definitions in all those different classes. These function definitions do not stay together, so there is no way of knowing or ensuring that they are similar.