**From:** Tony Simons (*A.Simons@dcs.shef.ac.uk*)

**Date:** Sat 07 Sep 2002 - 13:16:52 BST

**Next message:**Joaquin Miller: "RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP?"**Previous message:**Joaquin Miller: "RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP?"**Maybe in reply to:**Joaquin Miller: "RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP?"

Hi Joaquin, I think this may relate to what theoreticians in algebra and category theory refer to as the "initial algebra" semantics (as opposed to "final algebra" semantics). If I've remembered these the right way, then all semantic elements in an initial algebra are assumed to be distinct unless you can prove, using axioms, that they are equal. In a final algebra, it's the other way around - you assume that elements are potentially equivalent, unless you can prove that they are distinct. (There are many-to-one mappings from elements in initial algebras onto elements in final algebras, which is why I think that things which are distinct in an initial algebra are equivalent in a final algebra. If I have this the wrong way round, someone let me know). It sounds as if this statement is being made about the semantic domain, not the UML model elements. The semantic domain is the underlying mathematical (eg set-theoretic) model on which the concrete syntax of the UML model is based and interpreted. The idea is that the meaning of the UML model is made plain in terms of well-understood math constructs in the semantic model. For example, it is possible to say that the meaning of a standard type declaration x : T is that there exists a T-set of which x is a member. Semanticists write something like: [[ x : T ]] == x' in T' where x, T are syntactic elements from the concrete model and x' is an element and T' is a set in the semantic domain. The double brackets [[ ]] denote "the meaning of" the enclosed expression. This gives a simple set-theoretic semantics for the notion of type. Technically a semantic domain is more than just a set. Domains are constructed to have certain properties, for example, Scott domains are sets whose elements are partially ordered and in which you can construct a complete lattice of elements according to the < less than relation. This kind of domain is useful for interpreting the meaning of things like functional programs, since you can show that recursion will eventually terminate, if the meanings of the original argument and the argument of the recursive call can be shown to be ordered in the lattice (the latter must be "less than" the former). In ordered domains, there is always a "bottom" element that is less than any other element, where algorithms halt. If what you report 2U as saying follows this kind of reasoning, then the consequences for UML models are that you must provide explicit equivalences (in OCL) to assert that model elements are the same, otherwise you must always assume they are distinct. --Tony ========================================================================== > X-Sender: miller%joaquin.net@pop3.joaquin.net > Date: Fri, 06 Sep 2002 12:52:02 -0700 > To: wg@2uworks.org, discussion@2uworks.org > From: Joaquin Miller <miller@joaquin.net> > Subject: RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP? > Cc: pUML <puml-list@cs.york.ac.uk> > Mime-Version: 1.0 > X-Loop: puml-list@cs.york.ac.uk > X-Scanner: exiscan@shef.ac.uk *17nQVm-0001j1-00* http://duncanthrax.net/exiscan/ > > From a 2U draft: > > >A consequence of the semantic domain design principles is that the > >semantic domain should not contain equivalences; i.e. all semantic > >elements denote distinct concepts. > > I'm sorry, but i need to ask what that means. > > Are the semantic elements those in the UML model or those in the semantic > domain, or ...? > > Which are the concepts that the semantic elements denote? > Where are those concepts found? > What do those concepts denote? > Where do we find the meaning of those concepts? > > > > PGP Fingerprint: > CA23 6BCA ACAB 6006 E3C3 0E79 2122 94B4 E5FD 42C3 ========================================================================== Dr Anthony J H Simons a.simons@dcs.shef.ac.uk Senior Lecturer in Computer Science http://www.dcs.shef.ac.uk/~ajhs Director of Teaching Department of Computer Science tel: (+44) 114 22 21838 University of Sheffield dept: (+44) 114 22 21800 Regent Court, 211 Portobello Street fax: (+44) 114 22 21810 SHEFFIELD, S1 4DP univ: (+44) 114 22 22000 United Kingdom ==========================================================================

**Next message:**Joaquin Miller: "RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP?"**Previous message:**Joaquin Miller: "RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP?"**Maybe in reply to:**Joaquin Miller: "RE: [discussion@2uworks.org] RE: [wg@2uworks.org] 3C + 2U = xP?"