# Definition of an Algorithm

• posted

I recall in one of the early computer science classes I took, a professor defined an algorithm in a mathematical definition. She gave a list of properties an algorithm had to have to qualify as an algorithm. She also listed some features that were not required, such as being a computer program.

I recall these features:

1) Output - without output a procedure is pointless.

2) Input - ??? I want to say input is optional. So a set of steps to calculate some number of digits of pi would qualify as an algorithm, in spite of not having inputs.

3) Steps - she used qualifiers that indicated the steps had to be clear and unambiguous.

4) Finite - the steps must come to an end, i.e. at some point the algorithm has to produce the result, it can't be infinite.

I don't recall other qualifications, but I think there is at least one I'm missing. It was not a long list, and, like I've said, I don't think Input was on the list.

The web searches seem to produce some pretty vague, garbage definitions, some even saying it is a computer program, which I'm certain it does not have to be.

Anyone familiar with this?

• posted

Meh, there is not really an accepted formal definition, and to some extent it is a philosophy question. The word algorithm comes from Arabic, but the modern notion of Turing machines is from the 1920s.

There are a bunch of viewpoints on the topic here:

A further take is here:

• posted

If there's no definition, I guess I need to stop using the term.

The Wikipedia article starts off as a poor substitute for what it's supposed to be, with no reference anywhere in the section. This is one of the worst examples I've ever seen, even for Wikipedia. I'm amazed it was ever allowed. Probably a matter of the camel getting its nose under the tent.

But the section that actually tried to define an algorithm (Features of a good algorithm) seems to hit the nail on the head, as far as I can see. It mentions input though, which I'm pretty sure can be a null set without impacting that the process is an algorithm. So factually, input is not required, and not part of the definition.

• posted

If you stopped using a term just because there is no single formal definition, you'd have to stop saying anything at all. After all, there's no definition for the terms "definition" or "term".

It just means that if you are writing something that is supposed to be rigorous, you have to define what /you/ mean by a given term in the context of the discussion.

• posted

Just remember that any such definition will be a "lie-to-children" - an oversimplification that covers what you need at the time, but not the full picture. (That is not a bad thing in any way - it is essential to all learning. And it's the progress from oversimplifications upwards that keeps things fun. If you are not familiar with the term "lie-to-children", I recommend "The Science of the Discworld" that popularised it.)

You could argue that there is always some kind of input. For example, you could say that the digit index or the number of digits is the input to the "calculate pi" algorithm.

(As an aside, I find it fascinating that there is an algorithm to calculate the nth digit of the hexadecimal expansion of pi that does not need to calculate all the preceding digits.)

Is that really true?

If you can accept a "calculate pi" algorithm that does not have an input, then it is not finite.

I remember a programming task from my days at university (doing mathematics and computation), writing a function that calculated the digits of pi. The language in question was a functional programming language similar to Haskell. The end result was a function that returned an infinite list - you could then print out as many digits from that list as you wanted.

So what was the output? What kind of output do you allow from your algorithms? Does the algorithm have to stop and produce and output, or can it produce a stream of output underway? Can the output be an algorithm itself? Is it acceptable (according to your definition) for the output of a "calculate pi" algorithm to be a tuple of a digit of pi and an algorithm to calculate the next digit-algorithm tuple?

Was it "deterministic" ? Not all algorithms are deterministic and repeatable, but it is often a very useful characteristic, and it might have been relevant for the course you were doing at the time.

I think it is unlikely that you'll get a perfect match for your professor's definition, except by luck - because I don't think there really is a single definition. I don't think there is even a single consistent definition for your features "output", "input", "steps", or "finite" (in this context).

That's why Turing went to such effort to define his Turing Machine (and some other mathematicians of the time came up with alternative "universal computers"). If you want a rigorous definition, you have to go back to 0's and 1's and something mathematically precise such as a TM or universal register machine (which is easier to program than a TM). Of course, you are then restricting your algorithms - it no longer includes a recipe for brownies, for example, but it does give you something you can define and reason about.

• posted

Your idea that language is not defined is not valid. They are defined in a dictionary, actually, many, not unlike standards, lol. Terms with jargon meaning need their specific definition in that context, or they have little value.

Much of the engineering and computer science I learned in college was taught like math. Terms were defined, in order to know what is being said. Look at math. It's as much about the definitions as the various rules.

If a term like algorithm is not well defined, then I won't use it in engineering unless I define it first.

• posted

Sorry, I have no idea what you are talking about.

You can argue anything. A procedure to calculate pi without specifying how many digits would not be an algorithm because of the "finite" requirement. It doesn't need an input to set the number of digits if that is built into the procedure.

Not true.

If it has no definite end, it is not an algorithm. That's why an operating system is not typically an algorithm, it has no specific termination unless you have a command to shut down the computer, I suppose.

How did it return an infinite list? You mean it returned as many digits as you specified or waited to have calculated? If you have an infinite storage system, that's a pretty remarkable achievement. But I expect it would be powered by an infinite improbability drive.

If it's not deterministic, it's not an algorithm. Flipping a coin is not an algorithm.

Ok, that explains a lot about software development.

Nothing useful here. Thanks anyway.

• posted

Of course it is input. You change the number and the output changes.

Of course it is true. Unless you specify the limits for an operation which is infinite in nature you have the calculating algorithm running infinitely. No need to go about calculating Pi, divide 1 by 3 using the algorithm taught at primary school using a pencil.

Without giving it much thought I'd say an algorithm is an unambiguous flowchart made up in order to achieve some goal. Anything more than that would take some qualifier to the word "algorithm".

• posted
[snip]

Probably by using a non-strict evaluation order

where a data object can be "unbounded" or "potentially infinite" -- only the parts of the data structure that are later accessed become "real" data held in memory.

You could of course say that the part (function) of the program that produced the potentially infinite list is not an "algorithm" by itself, and becomes an algorithm only when combined with the part of the program that outputs a finite part of the list. But that seems too limited, because it that function is clearly independent of the rest of the program and implements a well-defined computation.

Randomized or probabilistic algorithms are a huge research subject with many practical applications, for example for the approximate solution of hard optimization problems

A very simple example is the "random sample consensus" method (RanSaC), which is often called an "algorithm" although it is not deterministic

• posted

The more relevant term is "lazy evaluation". It is very common in functional programming languages, and allows you to work with infinite lists, infinite mutually recursive functions, and all kinds of fun.

The function returns a function as part of the output.

All sorts of algorithms are non-deterministic. Pretty much any algorithm that is executed in parallel parts is going to have non-deterministic execution, which can result in non-deterministic outputs.

• posted

I wouldn't say there's no definition. I'd say there's no definition that everybody agrees on, so if you are trying to make formal claims about algorithms, you have to be precise. Otherwise, I would say, the different concepts of algorithm have enough shared features that in most informal usages, you can use the term without confusion.

The same thing applies to the term "probability", and we all use that term anyway.

• posted

When you are explaining a complicated subject to someone who does not know about it (and that's typically the case at school or university), you simplify - you are not telling the truth, you are telling a "lie to children". If the professor were to try to tell you all about what she understood by the meaning of "algorithm", it would take the full course instead of a class. You were given a "lie to children" - and that is entirely appropriate.

You appear to be mistaking it for a complete definition, which is less good.

It is most certainly an algorithm - just not a finite algorithm. Your arguments here are circular.

Yes, true - pi is not finite.

It is actually pretty irrelevant whether you consider a way to calculate the first ten digits of pi as a function "calculate_pi_10()" or as "calculate_pi(10)". It doesn't matter if the input is fixed in the algorithm or not.

(Can I assume you have never done any functional programming? You would find it very enlightening - it is much more appropriate for appreciating computation theory than imperative languages like C or Forth, or hardware design languages.)

Again, you are giving a circular argument. Your objection to unending algorithms is merely that they don't fit your remembered lesson that talked about finite algorithms. (An OS would probably be better described as a collection of algorithms rather than a single algorithm.)

No, it returned an infinite list.

I know you are familiar with Python, but I don't know at what level of detail - let me give a comparison here.

When you write "range(1000000)", Python generates a list of a million numbers from 0 to 999,999. This takes time and memory. But if you write "xrange(1000000)", it returns a generator expression, quickly and efficiently. It only actually generates the numbers when it needs to.

In Haskell, you can write :

numbers = 1 : [x + 1 | x <- numbers]

This would be like "numbers = [1] + [x + 1 for x in numbers]" in Python, if such recursive statements were allowed.

This is an infinite list - it behaves in all ways as an unending list of integers.

Non-deterministic does not mean random.

Software development is a pretty big field.

I can't imagine anything in this thread being "useful" - surely it is just an interesting discussion?

• posted

There are lots of what are called "randomized algorithms" which involve flipping coins. They are used in practice all the time. An example might be a Monte Carlo method for computing an integral. Whether every problem solvable by a randomized algorithm can be solved equally efficiently by a non-randomized one is a big unsolved problem in CS theory, called "P vs BPP". If you've heard of the more famous P vs NP problem, it's sort of similar.

There aren't necessarily termination guarantees either. Tons of practical cryptography programs somewhere contain a routine to generate, say, a 100 bit prime number. The method (that I would informally say is an algorithm) is:

1. Generate a random 1000 bit number by flipping coins. 2. Check whether the resulting number is prime. (There are known deterministic methods for this, but in practice one often uses a probabilistic test for this step too). 3. If yes, you are done. Otherwise, go back to step 1 and try again.

You can make a good probabilistic estimate of how long the above method will take, but obviously there is no definite upper bound, since you can keep generating composite numbers at step 1.

Almost everything in real-world engineering depends on randomness too. According to statistical thermodynamics, the air molecules in your room are moving around at random, and there is a small chance that all of them could coincidentally travel to one small corner of the room. Your vacuum cleaner's engineering depends on that not happening, and in practice it doesn't. Semiconductors depend on the statistical properties of charge carriers, and all that. Randomness is everywhere, don't be afraid of it.

It's a finite data structure that represents an infinite list, just like a sequence of 32 ones and zeros is a data structure that represents an integer in a certain range. In Haskell you can say "x = [1,2..]" and that means x represents the infinite list 1,2,3,4,5..... If you try to print x, you will see 1,2,3,4,5... spewing onto the terminal similar to if you ran "1 begin dup . 1+ again" in Forth. More normally you would say "print (take 5 x)" to print the first 5 elements of x, and get 1,2,3,4,5.

There's no infinite storage system, it's more like a symbolic algebra system that has a symbol representing the exact number pi. If you actually try to print pi, it will start printing 3.14159... using more and more computation and storage. But if you say "print cos(pi)", it will print -1 exactly, that sort of thing. There's no infinite storage, just some behind the scenes shortcuts.

• posted

If it's part of the procedure, that's not input by definition.

Why bother with that? Just define it as 3 and be done!

Must be nice to be the guy who makes the definitions.

• posted

Why would it have to be an algorithm?

Yeah, people often misuse terms.

• posted

I believe electrical engineers frequently use the Dirac delta function, whose value is infinite at x=0 and zero everywhere else. When you convolve it with another function, you get back the function that you convolved with. But by standard mathematical formalism, there is no such function as the Dirac delta. That didn't stop physicists and engineers from using it anyway, starting around the 1920s(?).

A rigorous mathematical formalism for such "functions" was not developed until the 1950s, called "generalized functions" or "Schwartz distributions". I'd be very surprised if any engineers ever study that. They just use the Dirac delta without worrying too much about mathematical rigor, and as long as they don't go too crazy, it acts the way they expect it to.

Engineering practice by others has not been so careful. You might find a "definition" of the Dirac delta in an engineering book, but if you look closely at the mathematical details, it will be hand-wavy or have gaps. They want mostly to agree on the features that they rely on, and to not worry about the corner cases that they don't use.

Here's a definitional problem for you. Let's say you flip a coin and then cover it with your hand, so you don't see how it came up. What is the probability that it came up heads? One notion is that the probability is obviously 50%, assuming it was a fair coin. A conflicting notion is that it is obviously not 50%: the coin has already been flipped and has landed, so it is either definitely heads or else it is definitely tails. That is, the probability is either 100% or 0%, and you simply don't know which.

Those two notions of probability are claled Bayesianism and frequentism. Philosophers have been arguing about them for 100s of years and they still haven't stopped. Mathematicians deal with this by treating probability as a topic in analysis involving measure spaces of yada yada. But a philosopher might say instead that probability is a sub-topic in the part of philosophy that deals with the nature of knowledge, and then the Bayesian view is simply a scheme for saying how strongly you believe a given proposition.

What I'm saying here is that probability, like algorithm, is another one of those terms that doesn't have a single, universally accepted, rigorous definition, but that doesn't stop it from being important anyway.

• posted

Dictionaries do not /define/ words, they /describe/ words. They say how words are used, and what people generally mean when they use the word. That is viewing the issue from the opposite side - use first, then definition. For technical terms in technical fields, you define the term first, then use it.

Yes.

But often the definition is not universally accepted - that's why in technical papers, standards documents, etc., important terms are defined in the document. They say exactly what you mean by the term in that context.

So you can use the word "algorithm" loosely, and people usually know approximately what you mean. But if you want to be rigorous and rely on properties that are not always agreed upon, you must define you /you/ will use the term in the particular context.

Sure. But a lot of these things are simplifications, and sometimes they will be the particular choice of definition that suits the teacher or class. It applies to maths, it applies to engineering, computer science, and any other technical topic.

You probably first learned that "integration" meant "finding the area under a curve". Then you later learn about complex analysis and integration in circles around poles of complex functions - the "definition" of integration that you learned now makes no sense whatsoever. (It is also meaningless for the Dirac delta function, since Paul brought that up.)

When you learned about "algorithms" in your early computer science classes, you similarly got a simplistic view. Since my university course was more theoretical than yours (AFAIUI), I was probably given a more advanced and nuanced definition of "algorithm" - and I learned that there is no single definition. You can keep learning, keep investigating, keep thinking, and come up with new ways to define such terms, new subclasses or variants of "algorithm", new characteristics that can be useful or interesting. It is all "right" - the only thing that is /wrong/ is to claim that /your/ definition is /the/ definition.

If you need it to be well defined in the context of what you are doing, then yes, you need to define what you mean by it. If it doesn't need to be rigorously defined, then you don't need to do so.

So if you are writing "This is the documentation for the algorithm used to control the speed of the motor" for some embedded software, then there is absolutely no need to define the term "algorithm". On the other hand, if you are writing a paper titled "Proof that there is no algorithm that can determine the halting state of all possible Turing Machines", then you absolutely /do/ need to define the term.

• posted

By /your/ definition only.

In Python (for familiarity), you could have :

mult = lambda x, y : x * y mult3 = lambda x : mult(3, x) mult3_5 = lambda : mult3(5)

It makes no difference to the theory or the meaning of "algorithm" whether the input is part of the function or not. (Of course it makes a major difference in practical use, but not for the theory.)

This is a discussion, not an argument. You brought up a rather theoretical topic, and people have been discussing it beyond your current level of familiarity. The sensible reaction is to see what you can learn from this, or to accept that it is going beyond what you know about and you are dropping out. This is all very theoretical, with no practical application (that I can imagine) for the kind of work we comp.arch.embedded denizens do in real life - either you find these things interesting and fun to thing about, or you don't. (I hope you /do/ find it interesting - but I do understand that mathematics and computational theory is an unusual hobby, even for professional software and hardware developers.)

You perhaps made the first post thinking there is a nice, simple and clear definition of "algorithm" and you'd get a nice, simple and clear answer to which clause you'd forgotten from that long-ago lecture. The list on Wikipedia of the dozens of mathematicians' definitions of "algorithm" through the years should have been a clue that things aren't that simple.

Define 1 / 3 to equal 3? ": 1/3 3 ;" might be acceptable in Forth, but not in mathematics!

We all get to make such definitions. The question is whether other people agree with them or not. Dimiter's suggestion here covers the lowest common denominator that I expect most people would agree with - even if it is also common to include other features (such as your own suggestions of input, output and finiteness).

• posted

Actually dictionaries REFLECT usage of language rather than defining it. The meanings of words often differ with locality and often change over time.

Unfortunately, many dictionaries are "abridged": they contain current and popular definitions - not necessarily ALL definitions - and pronunciation keys, but generally nothing of the origins or historical usages of the words.

To get the history of a word you need either to find an an unabridged print dictionary, or in an online search you must include the term "etymology" (and then hope the information is not behind a pay wall).