Re: Objects and Relations
Date: Wed, 31 Jan 2007 15:24:23 GMT
Message-ID: <Hw2wh.390$R71.6757_at_ursa-nb00s0.nbnet.nb.ca>
Marshall wrote:
> On Jan 30, 6:32 am, "David BL" <davi..._at_iinet.net.au> wrote:
>
>>On Jan 30, 6:33 pm, "Marshall" <marshall.spi..._at_gmail.com> wrote: >>
>
>
> Okay. Going to try to recreate my earlier post. Will not be
> as good but hope to get the basic sense across, along
> with some actual references to stuff.
>
>
>
>>>>Consider the following relation >> >>>> charsInString(StringId, Index, Char) :- StringId has Char at >>>>Index >> >>>This isn't right as is, though; you're putting every string in >>>a single table. Coincidental cohesion; grouping things based >>>on a superficial type commonality. Are you going to put all >>>your ints in a single table also? I would expect not. >> >>I'm glad to see that you don't like the idea of putting all characters >>of all strings in a single table. >> >>I believe a relational database should treat strings as an ADT to be >>used for domains for attributes.
>
>
> Yes, I'm aware of that perspective, and some people I really
> respect subscribe to it. However I don't; if you encapsulate
> a string as an ADT then you lose the ability to use the RA
> on the characters, which I believe I've already demonstrated
> has some utility.
>
> I also have an argument that encapsulation itself is merely
> a crutch to make up for the fact that a language lacks
> declarative integrity constraints.
>
>
>
>>>Instead, let's imagine we're in a nested relational system,
>>>and strings are a relation type {index:int, c:char} I will use
>>>the name S to denote the string at hand; sort of the analog
>>>of "this" in an OO language.
>>
>>Well, I was assuming a flat relational system. If you are going to
>>allow nested relations then I agree that things are rather
>>different. In fact I believe this is a bit of a cop out. Please
>>correct me if I'm wrong.
>
>
> Not 100% sure I understand, but it sounds like you're saying
> nested relations are a cop-out. If so I disagree. Again, this
> is a theory group, so we do not have to restrict ourselves
> to discussing existing implementations or SQL or whatever.
> As a logical model nested relations have a long history of
> study, and they do solve some problems. My personal model
> includes nested relations.
NF^2 nested relations? Or relation-valued attributes?
>>At the outermost nesting level you have >>chosen to deal with strings opaquely - as a domain for attributes. In >>other words at that level of abstraction you have chosen not to deal >>with strings relationally.
>
> Yes however I consider this a non-issue. The advantage of
> recursive structures is that one can deal with each level
> at a time, and not have to address them all at once.
And I would expect a String type to permit multiple possible representations. If one can represent a String as an array, one already represents them as a relation given the operations to convert back and forth. So why not have a relation-valued representation?
/amplification
>>You don't seem to appreciate the fact that the onus of proof often >>depends on the nature of the claim rather that who is making it. It >>would obviously be difficult to construct a formal proof that there >>doesn't exist elegant and useful string processing algorithms using >>RA, whereas it is comparatively easy to show examples if they in fact >>exist. All I said is that I can't think of any and perhaps there >>aren't any. I make no apologies for having rather less experience in >>RM than you do. If you look carefully at my post you will find that I >>never even claim that there are no such examples. I only raise the >>question.
>
> I don't believe I have made any claims as to where the anus of
> proof lies.
Intentional? ;> LOL
>>I note that this is the second time you have expected proofs of non- >>existence of examples (cf our discussions about fine grained mutative >>operations in a previous thread). Do you agree that this is often >>impossible?
>
> I don't believe that I have expected proofs. Let me change
> my earlier phrase "prove a point" to "make a point" so that
> we avoid any potentially-ambiguous use of the word "prove."
>
> Some of what we're discussing are design questions and
> they are not subject to proof. (In fact, surprisingly little *is*
> subject to proof.)
Indeed. The more we make subject to proof, the better, however.
>>>>Won't you almost always >>>>join/select for the simple purpose of looking up a character in a >>>>given string at a given index position? >> >>>You say that like it's a bad thing. Join is the primary tool >>>of the RA, so yes, we will join. And select. We'll use the >>>algebra, just like an OO programmer would use loops >>>and method calls. >> >>Of course. What I mean is that the access pattern on the relation is >>almost always the same, and no different to the functionality provided >>by the ADT. >> >>This contrasts with relations like Employee which join on different >>attributes at different times, revealing the power of RA for adhoc >>query.
>
> Joining on single-attribute relations is a practical and useful
> thing to do; it doesn't indicate any problem, any more than
> a function that takes only a single argument indicates a
> problem.
>
> Join of two one-attribute relations on their single attributes is
> set intersection, which I hope you will agree is a meaningful
> and useful operator.
Or cross-product which I hope everyone will agree is a meaningul and useful operation too.
>>>indexOf(int ch, int fromIndex) >> >>>The source to java.lang.String can be downloaded as part of the JDK; >>>I will refer to it here. (But not excerpt it, much as I would like to >>>for >>>comparison, because it is copyrighted.) >> >>>Here is the relational implementation in pseudo-SQL: >> >>> select min(index) from S where c = ch and index >= fromIndex >> >>>The body of the Java implementation is 37 lines long, and >>>contains 8 if statements and 2 for loops, and can return >>>from any one of five places. >> >>I shall look at this Java code when I get the time. It seems overly >>complicated. >> >>In my C++ code I use a StringRange class that represents an aliased >>range of characters using a pair of pointers, allowing me to write >> >> while(s && *s != ch) ++s; >> >>to scan through characters in range s not equal to ch.
>
> I am aware of the StringRange techinque but not familiar with
> it. I am not immediately clear what the above code does; in
> a C idiom (as opposed to C++) it appears merely to advance
> the (uninitialized) s pointer. Even if it's a complete solution, you
> omit the code from the StringRange class, which has to be
> accounted for to do a fair comparison with my code.
I wonder how the above loop terminates if ch does not occur in s. Then again, if the increment operation can make s evaluate to false at the end of the string, I suppose it is a fair operation to assume just as join is a fair operation to assume.
>>> select min(S1.index) from S S1 where >>> (select min(S2.c = str.c) from S S2, str >>> where S2.index-S1.index = str.index) and >>> S1.index >= fromIndex
>
> [corrected from original]
>
>>>"Select the smallest S.index where >>> every character in S' offset by S.index equals >>> every character in str, >>> that is greater than or equal to fromIndex" >> >>>The body of the java implementation calls an internal helper function >>>that is 33 lines, containing 6 ifs, 2 for loops, 1 while loop, and >>>that >>>can return from any one of 4 different places. >> >>Thinking off the top of my head, in C++ I would write something based >>along the lines of >> >>while(s && str.size() <= s.size() && s.prefix(str.size()) != str) ++s;
>
>
> Again, I note that you're not accounting for the size() and prefix()
> methods;
> you have to include them in the total count.
I tend to disagree with that. You do not account for the physical implementation of join or min.
My code didn't use any
> function abstraction; what you see is the whole deal. Otherwise
> we both just wrap our implementations in a function and call it a
> day. My implementation used only min(), subtraction, and comparison.
I don't think it unfair to assume standard library-supplied code in a comparison, but I agree the other extreme of wrapping everything in a function makes no sense either.
I note that David's algorithm is necessarily at least O(n*m) where n is the number of characters in s and m is the number of characters in str--squared if the .size function scans for null-termination.
Your solution is potentially O(n) depending more on the skill and investment of the compiler writers and less on your skill and investment.
(Unless someone can point to a C++ compiler that has implemented algorithmic replacement.)
>>>>If you want to deal with strings declaratively (not procedurally) then >>>>use a functional programming or logic programming language. RM seems >>>>a poor fit. >> >>>Does it still seem so after reading the above? >> >>Yes!
>
> Okay. What would you find to be a convincing demonstration? If nothing
> else I claim my substring find is quite declarative; as much so as any
> logic program. (In fact, one could make an argument that mine *is*
> a logic program.)
>
> It is in general nearly impossible to separate out one's familiarity
> with
> a particular computational model from how "intuitive" code in that
> model appears. Again, Haskell on first contact appeared ridiculously
> opaque, but now I find it generally very readable.
>
> The Stroustroup quote:
>
> http://www.research.att.com/~bs/bs_faq.html#compare
>
> The only way to get around the familiarity issue is with hard
> metrics of expressiveness, and these are few and far between.
> The one that stands out is Felleisen's "On the Expressive Power
> of Programming Languages." I have it on my to-do list to
> master Felleisen expressiveness and use it to compare a
> relational language.
>
> Until we do that, or something like it, pretty much all we're
> doing with comparing different computational models is
> expressing how familiar its idioms are to us.
>
>
>
>>I found your post very informative and interesting and shows me that >>RA is more powerful than I imagined - when combined with aggregation. >>I was expecting that substring testing would require transitive >>closure.
>
> As an aside, please note that TC is not a showstopper for a relational
> language. If we are speaking of a language that is equivalent in
> computational power to first order logic, (such as basic SQL) then
> we are speaking of a language that is not Turing complete. However
> if we imagine a general purpose programming language that was
> nonetheless relational through and through, we would of course be
> discussing a Turing complete language and it would be able
> to express TC.
>
> I note that recursive functions are Turing complete, and that
> functions
> are a kind of relation, and we can use join on any kind of relation,
> therefore we can employ join with functions. Being able to do
> so expands the computational power of join to the "goes to 11"
> of Turing, at the cost of the loss of the strong normalization
> property
> that basic RA has, which as a very smart guy once said to me
> "is overrated anyway."
>
>
>
>>I admit the RM approach is a little foreign to me. >>However, I have a remaining objection, and perhaps this could be >>overthrown as well. >> >>The objection is this: I find it difficult to imagine a system being >>able to turn those select statements into fast efficient >>implementations on strings represented as arrays of characters. >>Presumably there is an integrity constraint that shows that a string >>has characters at all index positions from the start to the end of the >>string. Somehow the system sees this integrity constraint, realises >>that strings can be implemented as arrays and then is able to >>translate RA expressions into efficient algorithms. This all seems >>rather speculative to me. >> >>Can you answer this question for me? In the following >> >> select min(index) from S where c = ch and index >= fromIndex >> >>How does the system realise that it can avoid finding all characters >>satisfying the where clause, and then finding the minimum index? ie >>instead it should merely scan the array for the left most character >>matching ch starting at fromIndex.
>
> Well, I can't put the pieces together in a nice package but I can
> enumerate them. In short, the system has to understand order
> theory.
>
> I don't think it is the integrity constraint that matters so much as
> that there is a total order on int, the type of index. Furthermore
> the system understands the relationship between the comparators
> (<, >=, etc.) and this order, and it understands the relationship
> between < and min(). Knowing this it can know that it doesn't
> have to examine any values for index that are <= fromIndex,
> and it knows once it has found a single matching value, it doesn't
> need to examine any values larger than that. Couple this with
> the fact that if the system is using a naive array implementation
> of the string, it will also know that the index values are laid
> out *in order* in memory. (The fact that these values aren't
> reified is irrelevant; they are still ordered. We can consider
> the (index, char) tuple as being sequential, with the index
> taking up 0 bytes and the char sizeof(char) bytes.) Oh, and
> that the optimizer will know how to take advantage of joining
> on an attribute that is in order in memory. IIUC these are all
> things that optimizers are capable of doing in today's products.
Indeed. Considering the physical representation might omit any representation of the index other than in the relative physical position of the characters, the compiler would absolutely have to understand the relationship between index and physical location.
> The thing that I used to think was harder was figuring out
> early termination of aggregates. That is, if you are aggregating
> multiplication across a set of integers, you can stop doing any
> actual multiplication if you hit a zero. I found the key to this
> in some recently published Fortress documents; you can
> stop any time you hit a fixed point of the folded function.
> (So the system has to know fixed points.)
>
> Actually the Fortress documents are quite interesting in
> how they handle both catamorphisms and how they handle
> implicit parallelization. I note these are two things that
> are *quite* difficult to optimize in C++ because of its
> Mr. Pouty Pants memory model, where you pretty
> much have to assume everything is aliased to everything
> else unless you have an affidavit from the Postmaster General.
> I also note that these kinds of optimizations are a cakewalk
> for a relational language, and that the relative importance
> of implicit parallelization vs. standard C-style optimizations
> is strongly increasing over time with multiple CPU cores.
> Finally, I note that I am really starting to overuse the
> phrase "I note."
>
> Upon rereading the last paragraph it appears my coherence
> level is dropping faster than a frat boy's pants at Mardi Gras,
> so I'd better sign off now. Sorry for any incoherence.
The Dragon Book mentions how algorithmic replacement can achieve far greater optimizations than the sum of all the optimizations it describes and notes that nobody has yet figured out how to do algorithmic replacement. Declarative languages expressing needs at a higher level change algorithmic replacement into a matter of algorithmic choice, which is far far easier to automate.
/amplification Received on Wed Jan 31 2007 - 16:24:23 CET