Mastering the Ackermann Function

Yesterday I came in touch with a curious, astonishing mathematical function. It’s called Ackermann function.

Mathematically speaking, it is a well-defined total function. That is, it has defined values for every integer input (= total function), and this value is not ambiguous (every input has one and one only possible output value) (= well-defined).

Speaking about computer science, this function is computable, but it’s not a primitive recursive function. In other words, you can implement an algorithm to express the function using while-loops (= computable), but still you cannot write an equivalent algorithm using only do-loops (= not primitive recursive). I suggest you to try this statement.

The function has this form:

As you can see, it’s form is fairly simple. But, even if it seems simple, its values explode quickly. A(4, 2), for example, has more than 19.000 digits.

Ackermann Function on the complex plane

Representation of the Ackermann Function on the complex plane

What I want to talk about is the possible implementation of such a function by means of a computer algorithm. I will use Python for the next examples, but the description is language-independent.

Recursive approach

The quicker approach to solving such a problem is by implementing a recursive algorithm, since the definition of the function itself is recursive. Let’s try with something like this:

import sys

def A(x, y):
  if x == 0:
    return y+1
    if y == 0:
      return A(x-1, 1)
      return A(x-1, A(x, y-1))

x = int(sys.argv[1])
y = int(sys.argv[2])

print "Result of A(%d, %d) is %d" % (x, y, A(x, y))

You will want to try it calling from command line:

python 4 2

You will face a brutal reality: this algorithm will not end for the input (4, 2), since it quickly reaches the maximum recursion depth provided by Python. We will get better results starting from lower inputs.

You will find that the maximum computable values, for each possible x are:

  • A(1, 997) = 999
  • A(2, 497) = 997
  • A(3, 6) = 509
  • A(4, 0) = 13

For every input (x, y) with x greater than 5 you will get no result, for each y. As you can see, the function quickly diverges in complexity when x = 3, and much more when x = 4.

What we can try to do is jump as much iterations as possible with a simple trick: saving already computed values in order to not compute them anymore. This could simplify the navigation through the recursion graph.

Let’s try this implementation (with some debug information):

import sys

res = {}
jumps = 0
recursions = 0

#------ Start Method -------#
def A(x, y):
  global res, jumps, recursions

  recursions += 1
    jumps += 1
    return res[x][y]
  except Exception, e:
    jumps -= 1
    res[x] = {}

  if x == 0:
    res[x][y] = y+1
    if y == 0:
      res[x][y] = A(x-1, 1)
      res[x][y] = A(x-1, A(x, y-1))

  return res[x][y]
#------ End Method -------#

x = int(sys.argv[1])
y = int(sys.argv[2])

  print "Result of A(%d, %d) is %d" % (x, y, A(x, y))
  print "%d total recursions, %d operations avoided" % (recursions, jumps)
except Exception, e:
  print "Exception occurred after %d recursions" % (recursions)

As you can see, the res variable is loaded with each computed value (at lines 20, 23, 25), and at the beginning of the function we try to return the value of A(x, y) if already computed (at line 14). The jumps variable keeps track of the number of the iterations saved by this implementation, while the recursions variable keeps track of the total number of single calls the the A function.

You can now still try to call the function to see what the maximum computable values are. You will find this:

  • A(1, 995) = 997
  • A(2, 496) = 995
  • A(3, 6) = 509
  • A(4, 0) = 13

We got three results: the maximum computable inputs have decreased ((1, 995) instead of (1,997) for example), the number of jumps increases with increasing values of the inputs, thus the computation time has decreased. Sadly, our target was not to reduce computable time, but reduce recursion depth.

We should try something different, to truly master the Ackermann Function. This something different involves computing the analytic form of the function. This approach will show a completely different world.

Probably I’ll talk about that in a future article. Let me know what you think about it, and submit your different implementations (different languages, different approaches) but still focusing on the recursion optimization.

50.697 thoughts on “Mastering the Ackermann Function

  1. Appreciating the time and effort you put into your site and detailed information you offer.
    It’s good to come across a blog every once in a while that isn’t the same unwanted rehashed information. Great read!
    I’ve bookmarked your site and I’m adding your RSS feeds to my Google

  2. Thank you for the auspicious writeup. It in fact used to be a entertainment account it.
    Glance advanced to more introduced agreeable from you!
    However, how could we keep up a correspondence?

  3. I will right away seize your rss feed as I can not to find your email
    subscription hyperlink or e-newsletter service.
    Do you have any? Please let me recognize in order that I
    may subscribe. Thanks.

  4. Hey there would you mind sharing which blog platform you’re
    working with? I’m going to start my own blog in the near future but I’m having a hard time making a decision between BlogEngine/Wordpress/B2evolution and Drupal.

    The reason I ask is because your design and style seems different then most blogs and I’m looking for something completely unique.

    P.S My apologies for being off-topic but I had to ask!

    Check out my page :: ไทยสบาย

  5. Hello there, just became aware of your blog through Google, and found that it is truly informative.
    I’m gonna watch out for brussels. I’ll appreciate if you continue this in future.
    A lot of people will be benefited from your writing. Cheers!

  6. Thanks for a marvelous posting! I definitely enjoyed reading it, you are a great author.I will ensure that I bookmark your blog and will come back at some point. I want to encourage continue your great job, have a nice weekend!|

  7. Hmm is anyone else having problems with the pictures on this blog loading? I’m trying to find out if its a problem on my end or if it’s the blog. Any responses would be greatly appreciated.|

  8. Great beat ! I wish to apprentice at the same time as you amend your site, how can i subscribe
    for a blog website? The account helped me a appropriate deal.
    I have been a little bit acquainted of this your broadcast offered vibrant clear

  9. Do you have a spam issue on this website; I also am a blogger,
    and I was curious about your situation; we have developed some nice procedures and we are looking to swap solutions with
    other folks, be sure to shoot me an email if interested.

  10. What Are Steroidal Supplements?

    Increased ranges can thicken your blood and improve your
    risk of coronary heart attack and stroke.
    The androgenic results of AAS could cause or worsen male sample baldness.
    Loss of muscle mass has been intently linked to mortality in these
    illnesses and stopping it can enhance therapeutic outcomes and lengthen lifespan .

    Several circumstances can lead to muscle loss, including AIDS, persistent
    obstructive pulmonary disease , cancer, and kidney and liver disease.

    It is an efficient alternative to steroids for conditioning and building dimension, so it can be used for bulking or during a minimize.
    It is a powerful performance enhancer, growing
    the androgen receptor levels. Trenorol is the choice to Trenbolone, one of many strongest AAS.
    Trenorol may be very environment friendly in constructing power and gaining mass quick, with no water retention. It incorporates only pure elements such as B-Sitosterol,
    Nettle leaf, and Samento Inner Bark extract. While their health dangers range by the sort and amount taken, they are often dangerous and
    trigger unwanted side effects at any dose. While they work well to
    regulate certain illnesses, they can trigger a number of side effects,
    such as elevated blood sugar ranges and weight gain.

    Highly Effective Pure Anabolics: Safe Options To Steroids

    They’re not quite as effective, of course, however that’s part of the commerce-off.
    Every yr tens of millions of people come to for performance enhancement drug data.
    Our highly specialised consultants are deeply skilled with multiple medical levels from a
    few of the high universities on the planet. It accommodates Safflower, Wild Yam, Choline
    Bitartrate and Acetyl-L-Carnitine.
    Basically the unwanted effects experienced are these corresponding to the over-consumption of caffeine.
    But that does not imply that they’re absolutely
    protected for the physique. The gentle unwanted side effects of ANAVAR don’t mean they do not
    exist in any respect. Acne, hairs (face & physique) and oily skin are some of the most common problems produced from its
    use. Together with DIANABOL they are the quickest and
    most harmful anabolic merchandise. His capability to advertise muscle growth and growth also introduced it into the gym and
    elevated it to the highest of world favorites listing. The worst anabolic product relating to the unwanted side effects brought on.
    Major League Baseball announced on Wednesday that New York Mets second baseman, Robinson Cano, has examined positive for performance
    enhancing medication for the secon. For this reason it’s widely
    used through the Bulking Phase so that users can obtain most
    results in muscle gain, without feeling drained or exhausted.

    Muscle restoration time is minimal, leaving room for frequent and super-dynamic workouts.

    Buy Steroids Uk

    For sustaining the aesthetic while new one or newbies amazed by
    the steroids result and begins to abuse them.
    The bodybuilding sport is superb, but due to some individuals,
    the level is getting down. Oli Cooney death due to overdose of
    steroidsHe has brazenly talked concerning the steroid use
    to develop his physique. Death of Legendary Bodybuilder because
    of heavy dose of steroid cycleAndreas Munzer who had a tragic demise.
    The abuse of steroids and overdose mixed with inactive life kill him at the very young age of only 26-years old.

    TRENBOLONE comes with lots of incredibly dangerous unwanted
    effects, which definitely cannot be ignored.
    Reduces the body’s metabolic fee, so energy wants throughout training are minimal.

    Today, along with D-BAL they’re the dominant bodybuilding options.
    Ideal for people who find themselves obese and making an attempt to lose weight fast and effectively.
    It additionally manages to decrease bad cholesterol levels, so selling better muscle progress.
    To a large extent, manages to mimic the motion of the properly-identified anabolic (in fact not at a
    degree of a hundred%), with its excessive efficiency opposing its relative with the
    absence of side effects. Off we go along with crazybulk‘s D-BAL,
    a legal model of the classic anabolic DIANABOL.
    Natural dietary supplements , not causing any unwanted effects
    in any respect. Talking about muscle mass and muscle features normally, an inexpensive factor we should always
    examine is how to get the utmost gains from any training course
    of with the much less effort.

    Horse Steroids Are Merely Steroids:

    While not as frequent, AAS can be utilized
    in these populations to help protect muscle mass
    . Many users on this category also make the most of a strategy called “stacking,” which is a slang time period for mixing a number of types of AAS.

    Some athletes also include other artificial hormones, such as growth hormone and insulin.
    Post questions about bulking, getting lean, healthy consuming, weight loss, etc.
    Post questions on weight coaching as it pertains to muscle
    constructing. This discussion board is guided by MrRippedZilla backed by legitimate
    scientific data and case studies. While many philosophies are
    typical of bodybuilders and powerlifters, there’s actually surprisingly little evidence to help such distinctions.

    Discover, debate and share the truth between myths and facts.
    Most of them have a lack of knowledge about steroid cycle, in order that they run in their own method and ended
    with the drastic aspect-results even, death.

    How Steroids Are Taken

    For many trainers, it is one of favourite merchandise, primarily because
    reduces muscle tissue damage caused by hard workouts, reducing also the metabolic rate.
    Next on the list is another anabolic steroid, the TRENBOLONE.
    Dating back to 1955 created by CIBA aiming solely on the anabolic enhancement of physique.
    DIANABOL might be essentially the most well-known anabolic steroid so far.

    On the opposite, the Best Legal Steroid Alternatives recommended
    current no side effects and in very uncommon circumstances cause some insignificant and much less harmful to well being
    unwanted side effects of temporary character.
    But however you can maximize the muscle gain, even by minimizing the effort in the gym, just by using the
    right dietary supplements. The outcomes are milder and primarily
    relate to burning saved fats and muscle strengthening and ribbing.

    Testosterone may also construct up the mineral density
    in your bones, allowing you to develop the capability
    for bigger muscle tissue and extra energy.

    They have been twice as more likely to abuse other body-shaping substances similar
    to amphetamines, anabolic steroids, and muscle-constructing dietary supplements in the course of the season.

    Use this steroid whenever you’re looking for a approach to get more power, enhance your vascularity, and jumpstart
    your stamina and endurance.

    There are legal steroids that mimic the pure effects of testosterone in your physique, and contribute to large muscle features.

    The feminine athletes who didn’t obtain the ATHENA coaching had been three times more prone to begin using slimming
    capsules through the sports season.

    Week Pattern Cycle: Moderate To Excessive Dosage

    In these sports activities, muscle strength, measurement, and energy directly relate to general performance.
    Though traditionally regarded as a male hormone, ladies additionally produce
    testosterone but in much smaller quantities. It serves
    a number of features for ladies, primarily promoting bone density and
    a wholesome libido . They affect numerous parts of your
    physique, similar to your muscle tissue, hair follicles, bones, liver, kidneys,
    and reproductive and nervous systems. For some, but not all,
    social strain such as media influence, peer affect, and sport or social norms additionally emerged as an necessary driver behind anabolic steroid use.

    For this cause, they’re reserved only for reasonable
    to extreme inflammatory situations . This is as a result of many steroids are produced in illegal labs that don’t observe the identical procedures as industrial labs.
    Lower HDL and better LDL ranges may enhance coronary heart disease risk.
    These blood markers play an necessary function in oxygen delivery throughout your body.

    The abusing of testosterone result in the disturbance
    of androgenic and estrogenic stage of the physique.
    In Thailand, the hormones and Performance enhancing medicine can be found at cheaper charges.

    If you’re looking to seriously bulk up and remodel your physique,
    then you definitely want the easiest natural steroid various
    in the marketplace. Our analysis staff has combed by way
    of the scientific literature to find out the most effective natural steroids you will get.
    Natural steroids supply most of the identical
    benefits, however with out these downsides.

    my website :: what are The two main types of steroids

  11. Heya! I just wanted to ask if you ever have any trouble with hackers?

    My last blog (wordpress) was hacked and I ended up losing many months of hard work due to no back up.
    Do you have any methods to prevent hackers?

  12. Great beat ! I wish to apprentice while you amend your website, how could i subscribe for a blog website?
    The account helped me a acceptable deal. I had been tiny bit acquainted of this your broadcast offered bright clear idea

  13. I’ve been surfing on-line more than 3 hours today, yet I
    never discovered any fascinating article like yours.
    It’s beautiful worth enough for me. In my opinion, if all website owners and bloggers made excellent content material as you did, the web will likely be a lot more helpful than ever before.

  14. Definitely believe that which you said. Your favorite reason appeared to
    be on the net the easiest thing to be aware of. I
    say to you, I certainly get annoyed while people think about worries
    that they just don’t know about. You managed to hit the nail upon the top and also defined out the whole thing without having side effect
    , people could take a signal. Will likely be back to get more.


  15. An interesting discussion is definitely worth comment. I
    believe that you need to write more on this subject matter, it may not be a taboo matter but typically people don’t talk about these
    subjects. To the next! Many thanks!!

Rispondi a Total Marketing Services Lille Annulla risposta

Il tuo indirizzo email non sarà pubblicato. I campi obbligatori sono contrassegnati *