What a delicious example!  (I'm taking the liberty of sharing it with 
r-devel, since it raises some good issues.)

You have two questions, presumably:

1 - how could the order of the setMethod calls make a difference in the 
results?
2 - what's causing the infinite loop & how could it be avoided, reliably?

Second question first.  The danger sign is the "vector" method:

setMethod("foo",
           signature(A = "vector"),
           function(A, ...) {
               foo(matrix(A, nrow = 1), ...)
           })

This handles a vector by recalling the generic with a matrix.  But 
"vector" is a superclass of "matrix" (see getClass("matrix")) and you 
DON'T have an explicit method defined for "matrix".  So for sure if 
"vector" was the only method, you would be in immediate trouble.  
Coercing down the inheritance tree is potentially bad news.

Generally, if you're going to take over dispatch by coercing an argument 
& then recalling the generic, the result is simplest if you have an 
exact match for the new call, not relying on inheritance.  After all, if 
you're doing the work to coerce the argument, might as well pick one 
that works right away.   Not a universal rule, but other things being 
equal ...

There's an additional issue with methods for  "matrix" and "array" 
because R never allows 2-way arrays to have class "array", which means 
using array() here with the same dimensions would not have helped.  
Also, matrix and array objects are weird in that they are not basic 
vectors but also have no class attribute, and is.object() is FALSE for them.

More interesting though--how can the order of the setMethod() calls 
matter?  To see that we need to look at the methods list object.  (The 
discussion is simplified to the case that only one argument is involved, 
which doesn't affect the result.)

The MethodsList object has a slot containing a named list of methods, 
with the names being those of the classes that appeared in the setMethod 
calls, in the order that the calls occurred(note!).  All classes are 
essentially equal from the view of the generic function, so there's no 
ordering favoring the "more relevant".

When method dispatch occurs, the code first looks for an exact match to 
the class of the actual argument--that's a quick search in the names of 
the list.

If the direct search fails, the code now looks for an inherited method.  
The key point is that this second search is "greedy"--the first 
inherited method found is used.

NOW it should be clear why the order of the setMethod() calls matters.  
You have two potential inherited methods here for "matrix"; namely, 
"array" and "vector".  WE know that the "array" method is `closer', and 
the R dispatcher could decide this also, if it were willing to look 
through all possible inheritance paths and drop one possibility if a 
better one was found.

It currently doesn't do any further search, and doing so would be a 
modest efficiency hit.  I'm inclined to think the cost would be worth it 
to eliminate unpleasant suprises like this one, but opinions may 
differ.  (Once found, the inherited method is stored directly in the 
list used for the first lookup,  so the hit is only the first time a 
particular signature turns up.)

To make the difference clearer, I added to your example another generic 
"bar" with the same methods as "foo", but with the order of the 
setMethod calls reversed.
By looking at the "methods" slot in the two cases, we can see why the 
bad ("vector") method is selected for bar() but not for foo():

 > names(getMethods("foo")@methods)
[1] "array"  "vector"
 > names(getMethods("bar")@methods)
[1] "vector" "array"

After running foo(1:10) and trying to run bar(1:10):

 > showMethods("foo")

Function "foo":
 A = "array"
 A = "vector"
 A = "integer"
    (inherited from A = "vector")
 A = "matrix"
    (inherited from A = "array")
 > showMethods("bar")

Function "bar":
 A = "vector"
 A = "array"
 A = "integer"
    (inherited from A = "vector")
 A = "matrix"
    (inherited from A = "vector")

But including setMethod("bar", "matrix", ...) in the source code makes 
bar() work fine.



Paul Roebuck wrote:

>Sorry to bother but could you shed some light on this?
>I don't understand why order of setMethod calls makes
>any difference. Since it obviously does, it has shaken
>the foundations of what I thought I understood about
>S4 methods. Even Gabor was surprised...
>
>
>---------- Forwarded message ----------
>Date: Wed, 12 Apr 2006 18:24:46 -0400
>From: Gabor Grothendieck <ggrothendieck@gmail.com>
>To: Paul Roebuck <roebuck@mdanderson.org>
>Cc: R Help Mailing List <r-help@stat.math.ethz.ch>
>Subject: Re: [R] S4 method dispatch matrixOrArray
>
>On 4/12/06, Paul Roebuck <roebuck@mdanderson.org> wrote:
>
>  
>
>>On Wed, 12 Apr 2006, Gabor Grothendieck wrote:
>>
>>    
>>
>>>On 4/12/06, Paul Roebuck <roebuck@mdanderson.org> wrote:
>>>
>>>      
>>>
>>>>I have some code where the primary dispatching is on
>>>>other parameters so I'd like not to have to create a
>>>>set of functions for "matrix" and another duplicate
>>>>set for "array". But the class union technique isn't
>>>>working as implemented below and I don't have my Green
>>>>book with me. How do I fix my infinite recursion problem?
>>>>
>>>>
>>>>##--------------------------------------------------------
>>>>library(methods)
>>>>
>>>>setGeneric("foo",
>>>>          function(A, ...) {
>>>>              cat("generic", match.call()[[1]], "\n")
>>>>              standardGeneric("foo")
>>>>          })
>>>>
>>>>setMethod("foo",
>>>>         signature(A = "vector"),
>>>>         function(A, ...) {
>>>>             callGeneric(matrix(A, nrow = 1), ...)
>>>>         })
>>>>
>>>>setClassUnion("matrixOrArray", c("matrix", "array"))
>>>>
>>>>setMethod("foo",
>>>>         signature(A = "matrixOrArray"),
>>>>         function(A, ...) {
>>>>             cat("A =", A, "\n")
>>>>         })
>>>>
>>>>## Test
>>>>foo(1:4)
>>>>foo(matrix(1:4, 1, 4))
>>>>foo(array(1:4, c(1, 4, 1)))
>>>>        
>>>>
>>>I think its good enough to just define an array method, i.e. you
>>>don't need the matrix method or the matrixOrArray class, and the
>>>vector method can call foo(matrix(A,1), ...) so:
>>>
>>>setGeneric("foo",
>>>          function(A, ...) {
>>>              cat("generic", match.call()[[1]], "\n")
>>>              standardGeneric("foo")
>>>          })
>>>
>>>setMethod("foo",
>>>         signature(A = "array"),
>>>         function(A, ...) {
>>>             cat("A =", A, "\n")
>>>         })
>>>
>>>setMethod("foo",
>>>         signature(A = "vector"),
>>>         function(A, ...) {
>>>             foo(matrix(A, nrow = 1), ...)
>>>         })
>>>      
>>>
>>Something didn't seem right here. That was pretty close
>>to what I had started with, before trying to go the
>>classUnion route. Matter of fact, the vector method can
>>retain use of callGeneric.
>>
>>The solution has to do with the order in which calls to
>>setMethod are made. Adding foo-vector after foo-array
>>works fine; the other way around causes infinite recursion.
>>    
>>
>
>This is surprising.  I would have thought that the
>parent/child relationships determine the order that
>dispatched methods are invoked, not the order that
>the setMethod commands are issued in.
>
>
>  
>

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