On Writing Op enMath Content Dictionaries James H. Davenp ort Dept. Mathematical Sciences University of Bath Bath BA2 7AY England [email protected] Abstract by which Op enMath achieves its goal of b eing an \an ex- tensible framework for exchanging semantically-ri ch repre- This pap er is based on various discussion with the Op en- sentations of mathematical ob jects". So the motivation for Math Consortium, and recently at the UniversityofWest- writing a CD is that there is some semantics that one wishes ern Ontario. All errors are the author's. Many helpful sug- to exchange. The consequences of this motivation are the gestions have b een made, particularly by Dr. Dewar and following. Dr. Naylor. 1. There must b e a \new" piece of semantic information This pap er outlines some of the issues that a ect au- to convey. thors of Op enMath Content Dictionarie s, and their asso ci- ated Small Typ e System [4] les. 2. It must b e p ossible to write down informally the \com- mented mathematical prop erties" or CMPs the se- mantics that the author of the CD intends to convey. 1 Intro duction It should also b e p ossible to write Formal Mathemat- This pap er addresses the following questions ab out the writ- ical Prop erties FMPs, but this is not necessary, and ing of Op enMath Content Dictionaries CDs, and asso ci- is clearly imp ossible for all of mathematics. It maybe ated Small Typ e System STS [4] les. that some of the new items can b e de ned in terms of others, even if not everything can b e de ned formally. 1. Why write a Content Dictionary? 3. There must b e a motivation for wishing to convey it. 2. What should I lo ok at b efore/while writing a Content One such motivation would b e that two algebra systems Dictionary? wished to communicate ob jects with these semantics, but this is far from the only motivation. Databases 3. What should I b ear in mind while writing a Content might contain these ob jects consider the data de- Dictionary? scrib ed in [2], or wemay wish to search in pap ers containing such items in formulae. 4. What is approval for a Content Dictionary, and howdo I get a Content Dictionary approved? 3 An example: multisets 1.1 Content Dictionary Groups MathML [12] de nes the concept of set, and says that it can 1 A Content Dictionary Group is a le normally with exten- haveatype attribute of normal or multiset. Op enMath sion .cdg which sp eci es a grouping of CDs for a logical has a set1.ocd and corresp onding set1.sts available as purp ose. For example, the MathML group is a group of [9] which de ne the semantics for ordinary sets. Though CDs whose symb ols are equivalent to the symb ols of Con- they do not explicitly say so, it is the case that A [ A = A. 2 tent MathML [12, App endix C]. Since Op enMath do es not have the same sort of attribute A CD can b e in more than one CD group. For example, concept that MathML has, we need a way to enco de multi- setname1.ocd is in the MathML CD group, since it contains sets in Op enMath. symb ols Z etc. which are in content MathML. It is also in The following p ossibili ti es exist. the setname.cdg group, which also contains setname2.ocd, 1. Add a multiset constructor to set1.ocd which has various set names, suchasP, which are not in content MathML. 2. De ne a new CD probably called multiset1 with the same op erators as set1 except that set would proba- 2 Why write a CD? bly b e called multiset but di erent semantics. 3. De ne a new CD probably called multiset1 with op- A Content Dictionary is a fundamental concept of Op en- erators suchasmultiset-union etc. Math. The symb ols contained in a CD form the mechanism 1 Presumably the default, though this is not made explicit. Supp orted by the Op enMath Esprit Pro ject 24.696. 2 For go o d reasons: it's only p ossible in XML to have xed at- tributes of the MathML kind, whichwould con ict with the extensi- bility of Op enMath. 1 The rst has some drawbacks. 5 What to Bear in Mind The semantics are not the same, so one needs to say The key things to b ear in mind while writing a CD are the something like \If A is a set rather than a multiset following. then A [ A = A". Lest this seem trivial, consider the fact that 1. Op enMath is ab out semantics, rather than the elegance of rendering. There are many alternativeways to de- A [ B =AnB[A\B[BnA termine how something is rendered, but this is the job is true for sets but false for multisets: if A = of MathML [12], particularly its presentation mo de, f1; 1; 2g and B = f1; 2; 2g, the the left-hand side is rather than of Op enMath. f1; 1; 1; 2; 2; 2g, but the right-hand side is f1; 1; 2; 2g. One example will illustrate this p oint. A colleague com- mented as follows. \It o ccurred to me whilst writing It is then imp ossible to distinguish b etween the op er- that some Op enMath phrases like`a; b; c 2 Z' seem to ation of union on sets and multisets. Some texts use o ccur quite frequently in mathematics. At the moment di erent symb ols for the two, thus allowing a text to the only way to enco de this in Op enMath is the follow- write \A [ A = A, but A t A 6= A". It is also p ossible ing: that an algorithm might wish to consider b oth. Hence the choice lies b etween the second and third. This is <OMA> a matter of preference, but it seems that the general Op en- <OMS cd="logic1" name="and"/> 3 Math convention has b een to cho ose the second rather than <OMA> the third. In the current context, one can see that <OMS cd="set1" name="in"/> <OMS name="multiset-union" cd="multiset1"/> <OMV name="a"/> <OMS cd="setname1" name="Z"/> seems somewhat redundant compared with </OMA> <OMS name="union" cd="multiset1"/> <OMA> <OMS cd="set1" name="in"/> Examples of a CD and the asso ciated STS le constructed <OMV name="b"/> on this principle are given in the full version [5], also [9]. <OMS cd="setname1" name="Z"/> </OMA> 4 What to Lo ok at <OMA> <OMS cd="set1" name="in"/> The key do cument is the formal Op enMath standard [8]. <OMV name="c"/> Clearly this do cument is imp ortant, as is the description of <OMS cd="setname1" name="Z"/> the Small Typ e System [4]. Existing CDs, particularly the 4 </OMA> draft ocial ones at [9], are a useful source of layout , but </OMA> also of examples of how things can b e describ ed. In par- ticular, when lo oking at writing multiset1.ocd, the author Surely it would b e much neater if instead 'in' was made lo oked carefully at set1.ocd.However, the reader will no- n + 1-ary, and we could say something like: tice that not all the examples were blindly copied: some have b een changed to examples more appropriate for the case at <OMA> hand. <OMS cd="set1" name="in"/> In the eld in whichyou are working, there may b e some <OMV name="a"/> standard reference works, e.g. [1] in the area of sp ecial func- <OMV name="b"/> tions. These should certainly b e consulted, but it should b e <OMV name="c"/> b orne in mind that suchworks, however famous, may not b e <OMS cd="setname1" name="Z"/> complete see [3] for examples. Equally, there may b e stan- </OMA> dard software systems, and it would make sense to lo ok at them rst. However, one should not exp ect uniformity here. :::". The following table shows what happ ens on an apparently simple example: the de nition of arccot1. We note, however, that no \new" semantics were in- [1] 1st printing 3=4 inconsistent volved: indeed the fact that an existing representation [1] 9th printing =4 in Op enMath was quoted essentially proves this. This [6] 5th edition ? inconsistent led to the analysis in Table 1, and a general principl e: [13] 30th edition 3=4 inconsistent MathML Content and Op enMath will aim for sim- Maple V release 5 3=4 plicity in the core language at the cost of requiring a Axiom 2.1 3=4 more complex translation into \go o d" written mathe- Mathematica [11] =4 matics. Reduce 3.4.1 =4 in oating p oint At this p oint, the reader maywell ob ject that this prin- Matlab 5.3.0 =4 in oating p oint ciple has not always b een followed in the most basic Matlab 5.3.0 3=4 symb olic to olb ox Op enMath Content Dictionaries . For example, wehave 3 For example, the op erator in arith2 is called times not not and in,sowhy notin, since the justi cation for re- commutat iv e-t im es .However, this is not an invariable rule, since the placing prop osed CD for multi-valued inverse functions uses Log for the multi- valued equivalent of log, since this is the common mathematical con- <OMA> vention.