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Nate, P1788 On 17 Dec 2012, at 08:55, Vincent Lefevre wrote: > On 2012-12-14 20:34:25 -0600, Nathan T. Hayes wrote: >> ...Can you explain this impression more specifically, perhaps with a concrete >> example? I don't understand what you mean. > > Take your example (where you can replace DAC by another decoration). > We are in the case: > > -- the input is nonempty, but the result is empty because of an > intersection operation (as in the case of this particular example). > > What I don't like about giving decorations on the result of set > operations like intersection and union (hull) is that decorations make > sense only when there is a corresponding point function, but once you > have a set operation, you no longer have a point function, at least in > a canonical way. Maybe my remarks aren't relevant to Nate's specific query, but Vincent puts his finger on my main uneasiness with decorations on the result of set operations. I copy, below, one of the examples from the decorations text currently up for vote. This is a VERY common paradigm where set operations are used. A function f(x) is defined piecewise, by different formulae in different regions. To evaluate f over a box xx, - Cut the box in pieces xx_i, by intersecting with each region. - Evaluate f over each piece to get a piece yy_i of result. - Take the union (convexHull) of the yy_i to get final yy. My problem with automatically decorating the result is that it is vacuous. Like automatically certifying a disabled people's care home as competent and caring without actually doing random visits to it (UK readers will know the context), it may be correct a lot of the time, but that's not the point. > Giving decorations in a particular case like intersection and union > might be interesting if existing practice shows that this is useful > and doesn't have significant drawbacks, but you only showed here is > that they are useful for some class of algorithms, about piecewise > functions. But what about all the other uses of intersection and > union? One doesn't know. In the example below it's only the *programmer's knowledge* that f(x) actually is continuous at the region interfaces (here, x=-2 and x=2), that makes it correct to propagate decorations by the simple statement dy = dx Suppose f had been defined so the pieces are continuous, but there is a jump at x=-2 or x=2 or both. Then to decorate it correctly you need to consider whether input interval xx has points in each of the 3 regions, similarly to decorating the "sign" function. Nate may prefer a different interpretation; but like Vincent, I would like decorations to give information about the continuity, etc., of an underlying point function. All the theoretical interval models under consideration (set-based, Kaucher or other) are *inherently* unable to do this automatically for piecewise functions. The only way to achieve it is by individual inspection -- as with care homes. John Pryce
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