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

[Edward A Lee]

The issue here is that an actor with an input type of int
and output type of int is not a function with
type signature:

    factors_1 :: int --> int

The factors example is a good one.  It is a function
with type signature:

    factors_2 :: <int> --> <int>

The idea of applying "map" to the first version factors_1
of this to get the second factors_2 doesn't work because the
domain of map is functions, and the first version
is not a function.

It is tempting to try to define a "function" for an
individual firing of an actor, but to do this right,
you need to include in the domain of the function the
state of the actor.  Thus, the factors actor could
be viewed as having two firing functions:

   factors_3 :: (int, int) --> int
   factors_4 :: int --> int

where both functions take the state of the actor (encoded as
an int). The second function requires no input on the input port,
but produces an output on the output port.  An execution of
this actor starts by using factors_3, then switches to
factors_4 until all the factors are produced, then goes back
to factors_3.  You can build factors_2 from this, but you
need an infinite-state automaton to do it.

These and related semantic issues are addressed in the following
paper, which has never been submitted anywhere for publication because
there are still vexing open issues:

http://ptolemy.eecs.berkeley.edu/publications/papers/97/dataflow/

Given this, there is no difficulty with the Ptolemy II
type system.  The key is to recognize that every connection
has type <t>. The type of the port is t, but that is not
the type signature of the actor.

It is possible to have streams of streams, e.g. <<int>>.
A few years ago we developed multidimensional SDF (MDSDF),
implemented in Ptolemy Classic.  To be useful, we usually
need for all but one of the dimensions to be finite,
otherwise there is an intrinsic ambiguity in how to make
progress in an infinite execution.  In fact, MDSDF can
be viewed as a way to get static analysis to ensure
dimensional consistency without the pitfalls of
dependent type systems (specifically, the MDSDF analysis
is decidable, whereas in general, analysis of dependent
types is not... for those who don't know, dependent types
are types that are "indexed" by values, such as int[10].
Steve is right that sometimes the undecidable problem can
be solved, as in compiler optimizations that eliminate array
bounds checking, but the factors example is a good illustration
where no compiler optimization will work...  Any type inference
on this will fail.)

MDSDF is described in the following paper, which was submitted
an astonishing 8 years before it was finally published (so much
for the value of technical journals):

http://ptolemy.eecs.berkeley.edu/publications/papers/02/synchronous/

Edward


At 12:59 PM 5/26/2004 -0700, Bertram Ludaescher wrote:

>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
> >>
> >>
> >> 
> ----------------------------------------------------------------------------
> >> Posted to the ptolemy-hackers mailing list.  Please send administrative
> >> mail for this list to: ptolemy-hackers-request at ptolemy.eecs.berkeley.edu
>SAN>
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------------
Edward A. Lee, Professor
518 Cory Hall, UC Berkeley, Berkeley, CA 94720
phone: 510-642-0455, fax: 510-642-2739
eal at eecs.Berkeley.EDU, http://ptolemy.eecs.berkeley.edu/~eal