[kepler-dev] Re: question on typing processes and higher-order functions a la map

[Bertram Ludaescher]

Hi:

sorry for replying late. 

Here is again the issue (see earlier posting): 
A factors actor can be conceived of as consuming one input token I (an int)
and producing multiple output tokens O_1, ..., O_n, the prime factors
of I:   factors(I) = O_1, ... O_n

One way to assign a type to factors is:
	factors :: int --> <int>
where <a> stands for a stream (or sequence, possibly infinite) of
items of type 'a'.

If such a type is indeed assigned, it leads to an interesting typing
issue, when applying the higher-order function
	map :: (a-->b) --> <a> --> <b>
to factors. According to the obvious type inference, the type of the
"pipelined" version of factors is
	Factors :: <int> --> < <int> >
where Factors is defined as being the same as map(factors).

If we treat <a> just as a list [a], all is well.

On the other hand, on Ptolemy one may look at factors as follows:
	factors :: int --> int
indicating that 

(i) the input and output ports, respectively of factors are int (not
arrray of int or sequence or list of int..), and 
(ii) stating that the token consumption rate/pattern is
	factors :: 1 x  int  --> n x int
(1 input token is consumed an n output tokens are produced)

Indeed a straightforward modeling in Ptolemy using the PN director
(not SDF as I indicated earlier) would now produce
	Factors <10, 20, ...> = <2, 5, 2, 2, 5, ...>
and thus loose the information which factors go with which number.

There are different ways to handle it:
(a) not at all (sometimes this behavior might be desired -- examples
please!). 

(b) use a collection type (array in Java, list, etc) and type
	factors :: int --> [int]
resulting in a process of the form
	Factors :: <int> --> < [int] >
Example
	Factors <10,20,...> = < [2,5], [2,2,5], ... >

(c) use a special blend (a variant of (a)!?) 
	factors :: int --> <int>
that when viewed as a process gives
	Factors <int> --> <int>_; 

where the subscript ";" indicates a punctuated stream:
	Factors <10, 20, ...> = < 2, 5 ; 2, 2, 5 ; ... >
in which special control tokens are inserted into the data streams
that have the purpose preserving the nesting structure of variant (b)
while having the streaming features of variant (a).

Does that make sense? 

I'm interested in a type system than can do all of the above in a neat 
way (and then we add semantic types)

Bertram






>>>>> "SAN" == Stephen Andrew Neuendorffer <neuendor at eecs.berkeley.edu> writes:
SAN> 
SAN> At 01:08 PM 5/25/2004, Bertram Ludaescher wrote:
>> Hi:
>> 
>> Liying Sui and I recently came across the following typing problem:
>> 
>> Consider an actor, say "factors" which computes for a given int I, all
>> the prime factors of I. For example factors(20) = [2,2,5]
>> 
>> Thus, the signature of factors is:
>> 
>> factors :: int --> [int]
>> 
>> Now assume factors is to be applied on a stream <x1, x2, x3, ...> of
>> integers, denoted <int>
>> 
>> It seems tempting to view the *process* Factors that is so created as
>> applying the higher-order map function to factors, i.e.,
>> Factors = map(factors)
>> 
>> There are some interesting typing issues. Let's say map has the
>> following type on streams:
>> 
>> map :: (a-->b) --> <a> --> <b>
>> 
>> That is, map takes a function of type (a-->b) and a stream of a's
>> (denoted <a>) and returns a stream of b's.
>> Therefore the type of the Factors process can be determined to be
>> 
>> Factors :: <int> --> < [int] >
>> 
>> Example: Factors( <4, 6, 10, ... > ) = < [2,2], [2,3], [2,5], ... >
>> 
>> So far so good -- no information is lost.
>> 
>> It seems, however, that in practise sometimes  another process is
>> created:
>> Factors'( <4, 6, 10, ... > ) = < 2,2,2,3,2,5, ... >
>> 
>> Clearly this process Factors' does lose some information (the grouping
>> of result tuples into list of prime factors). While for this specific
>> example, Factors' is not desirable, such a "flattening" behavior seems
>> to be used in practise:
SAN> 
SAN> I'm confused: Are you saying that this is what Ptolemy does, and you don't 
SAN> like
SAN> it, or that Ptolemy does not do this, and you would like it to?
SAN> 
SAN> Could you consider this to be another higher-order function that takes
SAN> expandArray :: <[int]> -> <int>?
SAN> 
SAN> 
>> Let say we change our original function factors to produce not a list
>> of ints, but a stream of them:
>> 
>> factors' :: int --> <int>
>> 
>> This correspond to a token consumption/production pattern of " 1:* "
>> (for each input token, we might get multiple output tokens).
>> 
>> Is it correct that in Ptolemy II using factors' with an SDF director
>> produces a process Factors' on streams that has the signature:
>> 
>> Factors' :: <int> --> <int>
>> 
>> In order to make this behavior type-correct it seems we may have to
>> say that <<int>> = <int>, because we get
>> 
>> map(factors') = Factors'
>> 
>> and the former has the type
>> 
>> map(factors') ::  <int> --> < <int> >
>> 
>> Note that the type of map(factors') is obtained by using the general
>> type of map above:
>> 
>> map :: (a-->b) --> <a> --> <b>
>> 
>> and the applying this to factors' :: int --> <int>
>> (hence a = int and b = <int>)
>> 
>> So if indeed Factors' is of the said type we must accept (whether we
>> like it or not ;-) that <<int>> = <int> (or in general, nesting
>> streams in this way yields a "flat" stream).
>> 
>> Comments??
>> Does Ptolemy II with an SDF director and an actor of type
>> myactor :: a --> <b>
SAN> 
SAN> I don't think this makes sence...  SDF actor functions don't have access to 
SAN> the whole stream...  they have access to a fixed length prefix of the stream.
SAN> 
>> produce a process
>> MYACTOR :: <a> --> <b>
>> which thus can be explained as a "special map" over streams with the
>> type identity <<b>> = <b> ??
SAN> 
SAN> why not another HOF: expandStream :: <<b>> -> <b> ?
SAN> I think that the advantage of expandArray over expandStream is that arrays 
SAN> are generally finite, while
SAN> streams are not, and hence it is more likely that the computation I've 
SAN> specified actually processes the
SAN> data being created...  Note that there are two ways to implement 
SAN> expandStream (does it produce an infinite stream consisting of the first 
SAN> element of each input stream, or does it produce an infinite stream that 
SAN> begins with the first
SAN> infinite input stream, and then never gets to the other ones?)
SAN> 
>> Thanks in advance!
>> 
>> Bertram and Liying
>> 
>> 
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SAN>