Stefan O'Rear wrote:
On Sat, Aug 11, 2007 at 03:00:04PM -0400, Ronald Guida wrote:
The question remains: "What is special about Monad or ArrowApply, compared to Arrow?" or "What is more general about Arrow, compared to Monad or ArrowApply?"
If all you have is an Arrow, then you must make up your mind what you're going to do ahead of time.
ArrowChoice gives you the ability to make basic "yes or no" decisions at run time.
ArrowApply gives you arbitrary computed jumps.
OK, so I thought this through. To summarize the classes of arrows: 1. Arrow is a device for sequencing side-effects in a fixed order.
If all you have is an Arrow, then you must make up your mind what you're going to do ahead of time.
2. ArrowChoice is a device for sequencing side-effects in a fixed order with options. Some effects can be selected at run-time, but (1) the available choices are fixed at compile-time and (2) options have to be selected before running the arrow computation.
ArrowChoice gives you the ability to make basic "yes or no" decisions at run time. BUT, you have to make these decisions before you run the arrow computation.
3. ArrowApply is a device for sequencing side-effects, such that functions can dynamically choose side-effects at run-time based on intermediate results of arrow computations.
ArrowApply gives you arbitrary computed jumps.
--- We know that ArrowApply is equivalent to Monad.
Imagine trying to write an interpreter for a toy language with I/O, and IO is a plain Arrow and not a Monad. You can read input and parse it, but you can't actually do IO because the IO you need to do, depends on the input you read - precisely what Arrow forbids!
Here's a toy language, described by a regular expression: 0(10)*110 I want to read characters, one at a time, and eventually decide to "Accept" or "Reject" a string. Let me try to understand my options. * With a simple Arrow, I can create a fixed sequence of "read" operations, and I can't act on the results (i.e. by choosing whether or not to keep reading) at run-time. * With ArrowChoice, I can create a set of alternative paths for "read" operations, and I can choose a path at run-time, but I have to choose a fixed path /before/ I get to see any results. * With ArrowApply, I can "read" one character and act on that character to choose what actions to perform next. I can read characters until I can render an "Accept" or "Reject" decision. Clearly, I need ArrowApply (or Monad) to get the job done. In conclusion: "What is special about Monad or ArrowApply, compared to Arrow?" Arrow lets me sequence side-effects in a fixed order. Monad lets me dynamically choose side effects at run-time based on intermediate results of previous side-effects. -- Ron