From: Ron Whitney <RFW@math.ams.org>

Date: Sun, 14 Apr 1996 13:59:25 -0400 (EDT)

To: w3c-math-erb@w3.org

Message-Id: <829504765.932602.RFW@MATH.AMS.ORG>

Date: Sun, 14 Apr 1996 13:59:25 -0400 (EDT)

To: w3c-math-erb@w3.org

Message-Id: <829504765.932602.RFW@MATH.AMS.ORG>

Patrick and I have had some discussions on various aspects of html-math, and a point on which I think there may be some merit in general discussion is that of "semantics", as one may understand the term in relation to mathematical notation. A full definition isn't required, but we all anticipate that html-math will feed into various processors which manipulate the html data in mathematical ways, and I'd like a somewhat clearer notion of how others here understand the "semantics" we're trying to capture. I realize that there may be no one in the group claiming to capture "full" semantics (whatever that could mean), but there is clearly an effort to capture something beyond surface-level notation, and I'm trying to understand more about what that deeper level is. And, as in ordinary conversation, I expect that there will be levels of semantical binding between the "free" level of abstract notation and the final level at which the notation is bound sufficiently for the application at hand. There is a meaning of semantics in connection with, say, mathematical logic, wherein one develops a "modeling" relationship between statements in some formal language and the mathematical structures which carry the same signature as the language. In a formal theory of rings, one might say in this connection that the meaning of "+" is that + corresponds to the addition operator in each of the rings of the theory. So we haven't captured one thing, but many of what we might call the "same" thing. The "meaning" of the + then lies in the abstract similarity of the operation across the class of models, i.e. in the position of + within a formal theory of rings. (Meanings in mathematicians' minds do tend to jump among the various models involved and the formal theories which provide abstract means of unifying many disparate instances. Computer algebra systems, in some sense, always reside at the symbolic, formal level insofar as they deal "only" with symbols and no infinitary objects. Whether human minds truly grasp infinitary objects might be another question, but here I'll call the semantical binding to a single mathematical entity a lower level than the level at which binding is made to a term within a formal theory. It's the theory level which may concern us most insofar as we're attempting to serve computer processing. To further indicate my meaning, I'll say that a computer system deals with complex numbers, for example, as a formal system and not as a full mathematical entity.) More analysis in the case of + would suggest that its use in rings is actually based on its use as the binary operator in a commutative group. By parsing operands and passing a string such as "plus(X,Y)", we let the next system in sequence handle the next level of semantics (e.g. to bind the "plus" to complex addition). In contrast, reading a + at surface level within the notation "k < n+" and analyzing it as a postfix operator to pass "successor(n)" will better prepare whatever's down the road to handle that usage. And generally, we may expect that surface-level html is unbound (free notation) and that inner-html (the 2nd parse level) is still unbound, but closer to binding. (Am I actually saying anything?) A more complicated case lies in the various uses of, say, ^ as a "power" operation. Here we have at least integer powers, rational powers, the exponential function, iteration of maps, and topological powers (i.e. cartesian products), each of which may use the surface-level ^ operator. More significantly, each of these uses might be mapped to the same internal "power" operator for passage to the next level of semantical binding. The unifying force here is that the same surface level notation is used and the later semantical analysers may cater to that (or more rationally, may recognize the surface uses as part of a single formal meaning). The distinguishing force is that these "really are" (say in a computational or other formal sense) different uses of the term (e.g. one wants to say the exponential function *truly* isn't a power at all) and should be distinguished as such. As much as we want to say that it does no harm to distinguish at the first level since analyzers are free to map back to indistinguished state, there is cost associated with making distinctions (where does one stop?). In the discussion to this point, if we consider a computer algebra system, the semantical binding made by the CA system is done either on the basis of some gross environment settings (e.g. the specific commutative group in question may have been set to an additive group of matrices as part of the import-environment) or on the basis of "type" characteristics passed with the formula. There are also situations intermediate between these extremes (of global and completely localized specifications) where the contextual theory may require more involved specification before proper binding can be made. (Let E be a certain elliptic curve and + its natural group operation, then discourse about certain equations.) When should binding be made and how? So in addition to the ambiguities of "power"-like operations, there are situations in which one would only expect to make the binding anyway at a late stage within the CA system and not within any html analysis. No? (I'm generally worried about ad hoc or parameterized operators within a paper and the transmission of their proper characteristics to a computer algebra system. The example above regarding an elliptic curve may not be sufficiently complicated. But then maybe I'm seeing complications where they don't exist.) I'm certainly not claiming that any of this is news to people on this list. I would simply like to hear some further discussion of such matters so that I have a clearer idea of what we envision. Perhaps the Illinois workshop is a better venue than this list for such discussion (with feedback to the list afterward). Patrick has pointed out to me that the "extensibility" of html-math is not entirely clear to either of us. I think we both understand that certain surface level distinctions will be possible for altering the 2nd-level parse state achieved by the html analyzer, but it's unclear to us how varied the paths to binding may be. I'd also be interested to know whether we envision computer algebra systems providing an interactive end to this process, wherein queries help a user get to a state of sufficient semantical binding. Comments will be appreciated. -RonReceived on Sunday, 14 April 1996 14:36:58 UTC

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