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Re: mathematical notation and functional programming
by Henning Thielemann 01 Feb '05

01 Feb '05

On Mon, 31 Jan 2005, Chung-chieh Shan wrote: > (Is Lemming the same person as Henning Thielemann?) Yes. :-) > > For the expression '1+x' I > > conclude by type inference that 'x' must be a variable for a scalar > > value, since '1' is, too. But the expression '1/O(n^2)' has the scalar > > value '1' on the left of '/' and a set of functions at the right side. > > Type inference fails, so my next try is to make the operands compatible > > in a somehow natural way. > > It seems to me that your classification of certain notations as "wrong" > and others as "right" assumes a certain type inference system that > allows adding a number to a placeholder but disallows dividing a > function by a set of functions. Let's see if we share a common interpretation of notation before discussing who uses it consistently and who does not. :-) For me 1, x and + are identifiers. The strings "1", "x" and "+" are not mathematical objects. But '1' denotes a unique mathematical object. Only almost, because depending on the underlying explanation it may be identified with the natural number 1, with the rational number 1, with the real 1, with the complex 1, you will probably need a different representation. Further this representation depends on how you represent natural numbers, reals and so on, e.g. by sets. '+' also denotes a mathematical object, more precisely a function, and again we have ambiguities like that for '1', since '+' might be the natural addition, the rational one or even a mixed one. 'x' denotes no special mathematical object but it may be replaced by one, and the special thing is, that within a certain scope all occurences of 'x' must be replaced with the same mathematical object. Functions are special, in the sense that if I write "1+2" I don't want to consider this as the sequence of three objects denoted by "1", "+", "2" but I want that the function denoted by "+" is applied on the values denoted by "1" and "2". So I imagine a layer of notation and a layer of mathematical objects. Do you share this interpretation? What are consequences of this interpretation? "2+2" and "4" denote the same object. A function has only access to the value (the mathematical object) but not to the generating expression. So how can I define e.g. differentation with respect to a variable? I can't, but I don't miss it because I can define it for functions, and yes Jerzy :-), also for other objects. For me differentiation was introduced by limits. The definition of limits don't need expressions as mathematical objects, the differentation of functions don't need them, too. But later we got used to differentiate expression (e.g. x^2 + y^2 is turned into 2*x * dx + 2*y * dy), but no-one defined what that is. What are the consequences of treating expressions as mathematical objects, too? "2+2" and "4" are different expressions. I guess we would still ask for a value, thus we need a mathematical function which maps expressions to values. I think Mathematicas Evaluate is made for this purpose. If mathematical functions can transform expressions - is it possible that they change my writing? ;-) Indeed, I really like this separation: On the one side expression, on the other side mathematical objects. Simplifications of expressions are nothing I allow a mathematical function. But if I simplify an expression I must assert that the denoted mathematical object remains the same. Differentation of a function is possible if you only know its graph, but a computer algebra system can find an expression for the derivation if you have an expression for the function. There are many functions that can be integrated, but a computer algebra system cannot find an expression without the Integrate function for it. When a pupil says "the function can not be integrated" he means that he couldn't find an expression for the integrated function without the integral sign. When a professor teaching calculus says "the function can not be integrated" he means that there is some divergence. It seems to me that in this case the pupil considers expressions as mathematical objects and the professor does not. > Lambda notation also involves much ambiguity and many implicit > conversions. Following the interpretation above I like to see lambda as a notation instead of an mathematical operator. lambda expressions denote functions. I can replace every occurence of lambda expressions textually. If "\A -> B" occurs I can also write "f" and "where holds \forall A f(A)=B". So lambda notation involves no more ambiguities than the rest of the notation. > > You use 'ask' twice in the second expression. Does this mean that there > > may be two different values for 'ask'? If this is the case your second > > interpretation differs from my second interpretation. > > No -- when I use "ask" I mean the one defined in Control.Monad.Reader. Sorry for my imprecise question :-) What I meant was that 'ask' somehow injects a value into the mathematical operation, I wondered if the values it introduces are equal or not.

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Re: mathematical notation and functional programming
by Chung-chieh Shan 01 Feb '05

01 Feb '05

(Is Lemming the same person as Henning Thielemann?) On 2005-01-30T21:24:24+0100, Lemming wrote: > Chung-chieh Shan wrote: > > Wait a minute -- would you also say that "1+x" has no meaning at the > > first glance, because "x" is a variable whereas "1" is an integer, so > > some lifting is called for? > For me 'x' is a place holder for a value. But you can only adds two *numbers* to get a number. It makes no sense to add a number to a placeholder. So it makes no sense to write "1+x", no? > For the expression '1+x' I > conclude by type inference that 'x' must be a variable for a scalar > value, since '1' is, too. But the expression '1/O(n^2)' has the scalar > value '1' on the left of '/' and a set of functions at the right side. > Type inference fails, so my next try is to make the operands compatible > in a somehow natural way. It seems to me that your classification of certain notations as "wrong" and others as "right" assumes a certain type inference system that allows adding a number to a placeholder but disallows dividing a function by a set of functions. > Since mathematical notation invokes many > implicit conversions, it's sometimes no longer unique or obvious what > implicit conversion to use. Many users of O(n^2) seem to consider it as > a placeholder for some expression, where the value of the expression is > bounded by n^2. Lambda notation also involves much ambiguity and many implicit conversions. In particular, "x" can mean the identity function from integers to integers as well as the identity function from booleans to booleans, as well as a function that maps an integer-boolean pair to the integer of the pair, as well as a function that maps an integer-boolean pair to the boolean of the pair, and so on. > > Right; they are control operators in the sense that call/cc is a control > > operator. > So they seem to be operators that work on expressions rather than > values. In this respect they are similar to the lambda operator, aren't > they? Yes. > You use 'ask' twice in the second expression. Does this mean that there > may be two different values for 'ask'? If this is the case your second > interpretation differs from my second interpretation. No -- when I use "ask" I mean the one defined in Control.Monad.Reader. > Church style, System F(-omega), alpha-conversion, lambda calculus, eta > reduction, currying - Where can I find some introduction to them? What > about Haskell Curry? What about Bourbaki? - I have heard they worked > hard to find a unified notation. (I'm sure that other people would have better suggestions, but I rather like Benjamin Pierce's book "Types and programming languages".) Ken -- Edit this signature at http://www.digitas.harvard.edu/cgi-bin/ken/sig War crimes will be prosecuted. War criminals will be punished. And it will be no defense to say, "I was just following orders."' George W. Bush, address to the nation, 2003-03-17 http://www.whitehouse.gov/news/releases/2003/03/20030317-7.html

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RE: UTF-8 BOM, really!? (was: [Haskell-cafe] Re: File path progr amme)
by Bayley, Alistair 31 Jan '05

31 Jan '05

> From: Aaron Denney [mailto:wnoise@ofb.net] > > Better yet would be to have the standard never allow the BOM. > > Since some things can't handle it, on output we should never emit it, > but still must handle it on input. Bah. I don't see how banning it from input would help; as I understand it, it's meant to be ignored anyway, so there's no loss or gain there. Whether or not to emit the BOM on output should be a user choice, surely? > From: Graham Klyne [mailto:GK@ninebynine.org] > > How can it make sense to have a BOM in UTF-8? UTF-8 is a sequence of > octets (bytes); what ordering is there here that can > sensibly be varied? >From http://www.unicode.org/faq/utf_bom.html#BOM : "Q: Where is a BOM useful? A: A BOM is useful at the beginning of files that are typed as text, but for which it is not known whether they are in big or little endian format..." i.e. it helps when you need to guess what the encoding of a given file might be. Alistair. ----------------------------------------- ***************************************************************** Confidentiality Note: The information contained in this message, and any attachments, may contain confidential and/or privileged material. It is intended solely for the person(s) or entity to which it is addressed. Any review, retransmission, dissemination, or taking of any action in reliance upon this information by persons or entities other than the intended recipient(s) is prohibited. If you received this in error, please contact the sender and delete the material from any computer. *****************************************************************

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File paths and internationalization/localization
by Graham Klyne 31 Jan '05

31 Jan '05

I don't offer any answers, but the problems raised on this list concerning file paths using local language characters would appear to have some parallels in the world of URIs. Martin Duerst and others at W3C have been working on the problem of internationalization (I18N) of URIs, and their proposals were recently published as a Proposed Standard RFC [1]. If this area is being considered in the design of Haskell libraries, there may be some lessons there to be copied. #g -- [1] ftp://ftp.rfc-editor.org/in-notes/rfc3987.txt ------------ Graham Klyne For email: http://www.ninebynine.org/#Contact

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ocr'ed version of "The implementation of functional languages"
by Matthew Roberts 31 Jan '05

31 Jan '05

I have just embarked on creating a ocr'ed version of the jpeg images that have been made available for "The implementation of functional languages". However, my ocr software is clunky at best and (not surprisingly) it is slow going. It struch me that someone else might also be hard at work doing this and that there is not point duplicating work. Is anyone else doing this? If so, how can I help? Matt

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JVM-Bridge with Kaffe
by John Goerzen 30 Jan '05

30 Jan '05

Hello, I am attempting to package up the JVM bridge for Debian. As such, I am limited to the purely Free JVM's available within Debian. They include Kaffe, SableVM, GCJ/GIJ, and the Jikes compiler available for most of the above. I could not convince jvm-bridge's Native/configure to even see a gcj environment. Kaffe worked (after updating the include path to /usr/lib/kaffe/include). However, when I tried to start the compile: g++ -DHAVE_CONFIG_H -I. -I. -I. -I/usr/lib/kaffe/include/ -DVM_KAFFE -O3 -W -Wall -fno-rtti -g -O2 -MT JVMBridge.lo -MD -MP -MF .deps/JVMBridge.Tpo -c JVMBridge.cpp -fPIC -DPIC -o .libs/JVMBridge.o JVMBridge.cpp: In function `JNIEnv* JVMBridge_AttachCurrentThread(JavaVM*)': JVMBridge.cpp:57: error: invalid conversion from `JNIEnv**' to `void**' JVMBridge.cpp: In function `jboolean* JVMBridge_GetBooleanArrayRegion(JNIEnv*, _jobject*, int, int)': JVMBridge.cpp:239: warning: unused variable `jboolean foo' JVMBridge.cpp: In function `_jobject* JVMBridge_DefineClass(JNIEnv*, const char*, _jobject*, const jbyte*, int)': JVMBridge.cpp:1187: warning: ignoring #pragma unused JVMBridge.cpp:1188: error: no matching function for call to `_Jv_JNIEnv:: DefineClass(_jobject*&, const jbyte*&, jsize&)' /usr/lib/kaffe/include/jni.h:498: error: candidates are: _jclass* _Jv_JNIEnv::DefineClass(const char*, _jobject*, const jbyte*, int) make[2]: *** [JVMBridge.lo] Error 1 This is jvm-bridge 0.3 and kaffe 1.1.4 (Debian version 1.1.4PRECVS6-1). Thanks, -- John Goerzen Author, Foundations of Python Network Programming http://www.amazon.com/exec/obidos/tg/detail/-/1590593715

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Looking for these libraries...
by John Goerzen 30 Jan '05

30 Jan '05

Hi, I'm looking for libraries / interfaces to these systems from Haskell: LDAP ncurses zlib (the one in darcs doesn't suit my needs) bz2lib Thanks, John

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RE: [Haskell-cafe] File path programme
by Simon Marlow 30 Jan '05

30 Jan '05

On 24 January 2005 22:11, Ben Rudiak-Gould wrote: > Please, let's not ship this with the hierarchical libraries. It's not > ready for prime time. We decided to provide this library for purely pragmatic reasons. It's better than nothing. http://www.haskell.org/pipermail/libraries/2004-October/002605.html Doing a reasonable job on System.FilePath, even if it isn't perfect, will help prevent lots of applications from falling into common traps, and help make more code portable. Cheers, Simon

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field record update syntax
by S. Alexander Jacobson 29 Jan '05

29 Jan '05

I have a lot of code of the form foo {bar = fn $ bar foo} Is there a more concise syntax? I am thinking the record equivalent of C's foo+=5... I imagine there is some operator that does this e.g. foo {bar =* fn} But I don't know what it is... -Alex- ______________________________________________________________ S. Alexander Jacobson tel:917-770-6565 http://alexjacobson.com

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Re: [Haskell-cafe] mathematical notation and functional programming
by Michael Matsko 29 Jan '05

29 Jan '05

Also, Walter Noll of Carnegie Mellon Univ. wrote a book, "Finite-Dimensional Spaces" in 1987 which basically presented undergraduate math in a notationally and conceptually unified manner. Some of the notation and terminology was strange, but consistent. Mike Matsko ----- Original Message ----- From: Fritz Ruehr <fruehr(a)willamette.edu> Date: Friday, January 28, 2005 3:10 pm Subject: Re: [Haskell-cafe] mathematical notation and functional programming > Well, I don't know about modern works which might appeal to > knowledge > of FP languages, but there is a well-known, 2-volume work by Cajori: > > Cajori, F., A History of Mathematical Notations, The Open > Court > Publishing Company, Chicago, 1929 (Available from Dover). > > I know it through Ken Iverson (may he rest in peace), the creator > of > APL. (Dr. Iverson's own notations were not to everyone's taste, but > I > think they were a bigger influence on Backus and the recent wave of > FP > than is generally acknowledged.) > > APL *did* have "implicit maps and zipWiths" in the sense that > scalar > functions would be automatically extended to vectors (and similarly > for > higher dimensions). I think my PhD advisor, Satish Thatte, did some > work on extending this sort of "notational abuse" to Hindley-Milner > systems, but I don't have the citations at hand. > > OK then, googling on Cajori yields this quote from a math history > site: > "He almost single-handedly created the history of mathematics as > an > academic subject in the United States > and, particularly with his book on the history of mathematical > notation, he is still one of the most quoted > historians of mathematics today." > > More googling on "mathematical notation" reveals that there *are* > people concerned about these issues, Steven Wolfram being an > easily-recognized example (he refers to Cajori's work). > > -- Fritz > > On Jan 27, 2005, at 12:14 PM, Henning Thielemann wrote: > > > I wonder if > > mathematical notation is subject of a mathematical branch and > whether> there are papers about this topic, e.g. how one can > improve common > > mathematical notation with the knowledge of functional languages. > > _______________________________________________ > Haskell-Cafe mailing list > Haskell-Cafe(a)haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe >

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