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Topic: What is the difference between probability and "fuzzy logic"? (Read 12221 times) previous topic - next topic

What is the difference between probability and "fuzzy logic"?

What is the difference between probability and "fuzzy logic"?

Re: What is the difference between probability and "fuzzy logic"?

Reply #1 –
Fuzzy logic is fuzzier. Probability probably isn't.

Re: What is the difference between probability and "fuzzy logic"?

Reply #3 –
Fuzzy Wuzzy was a bear.
Fuzzy Wuzzy probably had hair.
Fuzzy Wuzzy probably was not a naked alien.

(see other message thread)

Re: What is the difference between probability and "fuzzy logic"?

Reply #4 –
Amazingly enough, Robert's post is actually a pretty good description of fuzzy logic!

But enough of that now. We wouldn't want anyone to think we're having fun here.

Re: What is the difference between probability and "fuzzy logic"?

Reply #5 –
In order to relate this somewhat to NWC, I am interested
in using some aleatoric methods in music.  I often hear
the term "fuzzy logic" used in descriptions of non-
deterministic situations.  But I don't understand the
difference between using probability and "fuzzy logic".
Since there are so many knowledgeable musicians using
NWC, I thought I would ask here for some guidance.

Thanks!

Re: What is the difference between probability and "fuzzy logic"?

Reply #6 –
I woudn't call myself a knowledgeable "musician," but I do happen to know something about math and logic.

The gist of the difference is that probability is caculated according to established mathematical rules. If A and B are welldefined eents that can be described mathematically as members of a larger set of all possible events, the statement "A is more probable than B" is mathematically true or false, and can be demonstrated by proof. (Let us not get into Goedel here.)

Fuzzy logic allows for user-defined decision-making rules that do not necessarily coincide with the results of mathematical probability. The objective is to get to a decision that perhaps cannot be rigorously proven, or perhaps could be proven (right or wrong) but using more resources or more time than is available.

I sing in a choir. If we are doing an SATB hymn by Bach, i can "guess" my bass part once I hear the beginning. I have not made any extensive study of Bach, or of music theory. It's just that Bach seems to work a certain way. I understand that way, in a "fuzzy logic" sense. Once in a while, I guess the wrong note. But even so, it is rarely dissonant. I cannot say that I am applying the rules of probability. I have no quantitative information about Bach's harmonization whatsoever, and no means to calculate them in real time, even if I had such information. Thus, I am not applying probability.

Incidentally, I cannot do that for Mozart. others claim they can. I cannot even guess notes that will be harmonious, much of the time.

It would be possible to program a computer to choose a "most probable" note to follow a sequence of prior notes, using some sort of mathematical rules. Anyone who knew the rules could duplicate the calculation.

Re: What is the difference between probability and "fuzzy logic"?

Reply #7 –
... but in the context of aleatoric music: The objective of aleatory is to introduce chance. That is, the objective is not to predict the most likely note that follows a sequence of prior notes, but to remove at least some of the determinism.

That is, if it is possible to calculate the "most probable" note following a certain sequence of notes in a certain style of music, aleatoric music would sometimes intentionally choose (or, randomly choose) a less probable note. It's the difference between making a best guess that turns out to be wrong, and deliberately not making the best guess. it is possible to use methods for choosing a note that are not entirely at random, but use certain principles (with a result that nevertheless has an element of unpredictability). That's "fuzzy logic."

One of our choir hymns is "Teach me, O Lord, the way of your statutes, and I will keep them to the end." (There are verses, but that's the refrain.) The sheet music calls for the SATB choir the introduce aleatory at the final word "end." This is done by beginning at defined notes, then having each singer move at will up or down to another note in the key of E major, independently (out of sync) with the others, singing the "e" vowel. This is a capella. At a signal from the choir director, all conclude with the "d" consonant together.

My own spin on it, is that it sounds like the choir got lost, ending with a collective "duh." But what do I know?

Re: What is the difference between probability and "fuzzy logic"?

Reply #8 –
Another possible aspect to "fuzzy logic" is the idea of learning. For instance, you could write a computer program that initially assigned an equal probability to all possible note motions. It would watch you compose (or perhaps analyse existing scores) and "discover" that most of the time, if you're playing a ti and fa, you will follow it with a do and mi. It therefore increases the probability for this pairing, and if asked to compose "in your style," will follow ti/fa intervals with do/mi more frequently than others. So in the realm of software development, where the term "fuzzy logic" originated, there's a strong connection with AI (artificial intelligence).

This is perhaps one reason why I find the fractal model of automatic music composition much more satisfying than pure aleatory. Fractals tend to have their own sort of fuzzy logic "built-in", due to their self-similar nature. (Or perhaps it would be more accurate to call it "patterned randomness" or something like that.) Artifacts will have a similar, but not identical, structure at different scales. Whether this simulates "real" music more realistically than other approaches is open to debate, but I did find it a useful starting point for "avant-garde" musical compositions on numerous occasions.
 

Re: What is the difference between probability and "fuzzy logic"?

Reply #9 –
Greetings,

Thank you Robert and Fred for your in-depth answers;
but I am still confused - perhaps it is the term
'probability' that I don't understand.

Robert, you wrote:

"If we are doing an SATB hymn by Bach, I can "guess" my bass part once I hear the beginning. I have not made any extensive study of Bach, or of music theory. It's just that Bach seems to work a certain way. I understand that way, in a "fuzzy logic" sense. Once in a while, I guess the wrong note. But even so, it is rarely dissonant. I cannot say that I am applying the rules of probability. I have no quantitative information about Bach's harmonization whatsoever, and no means to calculate them in real time, even if I had such information. Thus, I am not applying probability."

Why does this not mean that you are unconsciously using
probability?  I assume you have heard enough hymns by
Bach that your mind has made some observations about
the 'probability' of one note following another in a given
context.

Fred, you wrote:

"Another possible aspect to "fuzzy logic" is the idea of learning. For instance, you could write a computer program that initially assigned an equal probability to all possible note motions. It would watch you compose (or perhaps analyze existing scores) and "discover" that most of the time, if you're playing a ti and fa, you will follow it with a do and mi. It therefore increases the probability for this pairing, and if asked to compose "in your style," will follow ti/fa intervals with do/mi more frequently than others. So in the realm of software development, where the term "fuzzy logic" originated, there's a strong connection with AI (artificial intelligence)."

Again, why is this considered "fuzzy logic" and not
probability?  It seems that this program would be using
probability as the criteria for making decisions.

Thanks again, and I apologize for dragging-out this
off-topic.

Re: What is the difference between probability and "fuzzy logic"?

Reply #10 –
I was trying to be precise, regarding usage of the term "probability." When I guess a Bach note (and even if I have the sheet music in front of me, I often guess) I cannot quantify the likelihood of being correct. That is, I cannot say "the next note has a 65% chance of being A#, 22% chance of being E, and 13% chance of being any other note.

Presumably, it is possible for a computer to extensively study Bach, and quantify his music so that such a prbability predication can be made. But that's not what I'm doing.

When i guess notes, I am not really being aleatoric. That is, I am not intentionally using chance to choose a note that differs from whatever Bach wrote. I am merely mistaken. If I were trying to be aleatoric, I would say to myself, "I know perfectly well that Bach wrote A# here, because I have the sheet music. But before I sing that note, I'll flip a coin. If it comes up heads, I'll sing E instead of A#." The odds don't have to be 50-50. This concept could be extended to choosing notes at random without being different from an established standard. But I don't think that's the intent of aleatoric music.

Spontaneous jazz improv is, to some extent, aleatoric. There is a genre of music being played, and the live musicians have some idea of what the others are doing. But occasionally, an instrument takes the lead and strays off. The remaining instruments soon find a consistent accompaniment. But I don't think that's what John Cage meant by aleatoric music; he intended "pure chance" rather then "do what I feel would work."

You could call my coin-flip singing "fuzzy logic," in the sense that the result does not necessarily coincide with the logical (printed) note. However, outside the scope of music, I have seen "fuzzy logic" used to mean a method of making a decision based on imperfect knowledge of input, or incomplete logical processing of input. The idea is to quickly come to a result that is likely to be correct, rather than wait for a result that is certain to be correct.

To put it another way: "Fuzzy logic" strives to be correct, but tolerates some error as a practical matter. "Aleatory" deliberately introduces error, even when the true result (printed music) is known beforehand. "Probability" is a mathematical term.

Re: What is the difference between probability and "fuzzy logic"?

Reply #11 –
This doesn't really answer your question, but I feel that it is a good example of what I think you all are talking about.  There is a piece written in 1956-7 by a super-computer (one that was super for the time at least :), even though my personal PC is about 1000+ times faster hehe).
It is called the Illiac Suite, and had the basic rules of modal counterpoint plugged into the ILLIAC computer, and then it 'composed' the music for string quartet using those rules.  It does so 'randomly' based on the given rules.  You might want to check it out, it might give you some ideas.

~Adam White
~Colorado State University
~Music Composition and Percussion Performance

Re: What is the difference between probability and "fuzzy logic"?

Reply #12 –
Getting back to your original question - part of the reason why I originally thought it was a joke (and therefore replied with a joke) is that "fuzzy logic" and "probability" are not in the same set of objects, like "cat" and "dog." It's more like "cat" and "bad breath." My cat can (and does) have bad breath, but I cannot directly compare the concepts "cat" and "bad breath."

Similarly, fuzzy logic uses probability, but can't be compared to it. It can even be said that it manipulates it, so in a sense it's a higher concept. In the example I gave, a fuzzy-logic program will at first assume equal probability to all note motions. As it discovers that ti/fa is followed by do/mi more often than can be expected by equal probability, it "weights" the probability of that note motion outcome in the future.

Similarly, fractal motion is unpredictable in a strict sense, but in another sense it's "less unpredictable" than a perfectly random motion. So there are degrees of predictabilities (or, if you will, probabilities), ranging from the purely and completely random Cage-style aleatory, to the highly structured, almost perfectly predictable work of some of the Baroque composers.

It's telling that Robert used Bach for his illustration. It would be meaningless to try to apply fuzzy logic to "best-guess" what the next note is in a Cage composition.

Re: What is the difference between probability and "fuzzy logic"?

Reply #13 –
'What is the difference between probability and "fuzzy logic"?'

Probability is used to guage the likelihood that some particular, well-defined state is or will be the case, under ignorance or chance.  For instance: a fair coin has a 50% probability of coming up heads.  Note that: 1. we do know ahead of time the outcome due to chance and 2. there are only two, clearly-defined states: "heads" and "tails".

Fuzzy membership indicates the degree to which an individual case or circumstance belongs to a fuzzy set.  Here, the thing being measured is a result of the inherently imprecise nature of the definition of the fuzzy set, not ignorance or chance.  Here's a scenario: Bob is in a house with two adjacent rooms: the kitchen and the dining room.  In many cases, Bob status within the set of things "in the kitchen" is completely plain: he's either "in the kitchen" or "not in the kitchen".  What about when Bob stands in the doorway?  He is "partially in the kitchen".  Quantifying this partial state yields a fuzzy set membership.  With only his little toe in the dining room, we might say Bob is 0.99 "in the kitchen", for instance.

No event (like a coin toss) will resolve Bob to being completely "in the kitchen" or "not in the kitchen", as long as he's standing in that doorway.  Fuzzy sets are based on imprecise definitions of sets, not randomness.

-Will Dwinnell
http://will.dwinnell.com