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I vote: yes,however, I regret the speed with which we we push these things through. We should have settled open questions first.
The proposed StandardNotation defines two sets of intervals: IR and *IR. Both sets are not suited as basis for an interval arithmetic standard.
In the middle of page 6 it says:"A (real , closed, nonempty) interval is a 1-dimensional box, i.e., a pair x = [xunderline, xoverline] consisting of two real numbers xunderline and xoverline with xunderline lessorequal xoverline. The set of all intervals is denoted by IR." Trying to understand this, I come to the conclusion: "A real interval is a nonempty, connected, closed and bounded set of real numbers".
Reading the definition of *IR on page 8 I come to the conclusion that *IR is the same as IR* would be. The * is just moved to the left hand side to avoid a conflict with a possible power n.
As basis for an interval arithmetic standard an interval over the real numbers should be defined as a closed and connected set of real numbers. This is a well structured set. (To avoid misunderstandings I mention again that in real analysis a subset of real numbers is called closed if the complement is open). A real interval can be bounded, then it is written as [a, b] with real numbers a,b and a lessorequal b, or unbounded, then it is written as (-oo, a] or [b, +oo) with real numbers a and b, or (-oo, +oo). All these intervals are closed and connected sets of real numbers. With respect to set inclusion as an order relation the set of all such intervals is a complete lattice. Its least element is the empty set and the greatest element is the set (-oo, +oo). The infimum is the intersection and the supremum is the interval (convex) hull. Arithmetic operations in this set are clearly defined as operations for sets of real numbers. The structure {IR, +, -, *, /, setinclusion} is free of exceptions and inclusionisotonely oredered. The infimum in IR and in the power set PR are always the same. In the paper I prepared for the proceedings of the Dagstuhl seminar last year I used the notation (IR) for this set of intervals. In a paper I just prepared for a workshop to be held at Karlsruhe in March this year I just used the (upgraded) notation IR for this set. But I am open for other notations. What about CIR for closed real intervals?
I think the development of a standard for interval arithmetic should begin with the definition of the set of intervals that is considered and a notation for it. I would appreciate it very much if we would invent a catchy name for this space of intervals. This is a particular task for native English speaking colleagues. If we do not specify about what we are talking we will never come to a solution.
I wonder whether we should start another motion to define the basic space of intervals and a notation (and perhaps a name) for it before we continue to develop a standard for interval arithmetic. Clarifying the basis can considerably speed up the standardization process.
I attach my paper for the workshop at Karlsruhe. Best whishes Ulrich Kulisch Ralph Baker Kearfott schrieb:
IEEE P1788 working group members: I must apologize for an additional oversight of mine. In particular, the proposer of Motion 1 (John Pryce) sent an addendum to his rationale as follows: ===== Note added after the discussion period: Various people have pointed out perceived deficiencies in the proposed notation. There has been useful discussion on this. But no one has proposed a formal amendment to the motion. Therefore it goes forward to a vote in its original form. Please note you are not asked to fit into a notational strait-jacket. The motion asks you to take the Notation Paper for what it is, and to "follow" it flexibly, as is appropriate for your application. ===== Please take that into consideration in your voting, which may be by posting to stds-1788@xxxxxxxxxxxxxxxxxx Sincerely, Ralph Baker Kearfott (acting chair, P1788) ------------------------------------------------------------- IEEE P1788 working group members: I hereby officially open the voting period on our first motion. The voting period will continue for three weeks, until 2009/02/13/23:59GMT. I remind you that you should be registered officially at the IEEE web site to vote. (We will help you do this if our vote tabulator determines you are not.) You should send your vote to George Corliss at: george.corliss@xxxxxxxxxxxxx The formal rules are, if you vote "No," you should state the reasons and the changes that, if made to the document, would cause you to vote "Yes." Since this motion is just a position paper, it is governed by 10.1 and 10.5 of our policies and procedures, namely, it will pass by a simple majority. A quorum is determined by 10.4 of our policies and procedures document. I append the actual motion, along with the proposer's Rationale. (Although there was some discussion, no substantive amendments were formally proposed, so the motion upon which we are voting is the one originally stated.) Sincerely, R. Baker Kearfott (acting chair, P1788) =============================================================== Motion P1788/M0001.01_StandardizedNotation Proposer: J D Pryce ===== The P1788 standard will initially use the notation proposed in the paper "Standardized notation in interval analysis" by R.B. Kearfott, M.T. Nakao, A. Neumaier, S.M. Rump, S.P. Shary, and P. van Hentenryck, available at http://www.mat.univie.ac.at/~neum/papers.html This notation will be open to amendment after sufficient experience of using it. The standard will include a copy of the above paper (as possibly amended) in an appendix. ===== ==Rationale== As that paper itself says, interval notation is somewhat fragmented at present. Here is the view of some experts who have thought hard about this issue. We can do great service to interval computation for many years ahead by helping to disseminate their recommended practice, and following it ourselves. Rather than spend preliminary time debating whether we want to amend the proposed standard notation, I think it is more fruitful for us all to accept it as it is for now, and accept the discipline of following its notation for future position papers. In due course, either we are satisfied we can accept it permanently, or some of us are so annoyed by its perceived deficiencies that we have some constructive changes to make. The motion does not say that all position papers SHALL use this notation. I just strongly recommend this, so we get experience of using it. ===== ===============================================================
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