<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; "><BR><DIV><DIV>On 21 Sep 2007, at 00:36, Chuming Chen wrote:</DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Dear All,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">I am new to Description Logics. I am trying to understand the correct semantics</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">of Description Logics. especially, the changes in semantics.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">I know DL Semantics is defined by interpretations.<SPAN class="Apple-converted-space"> </SPAN>An interpretation I</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">= (Delta^I, .^I), where<SPAN class="Apple-converted-space"> </SPAN>Delta^I is the domain of interpretation (a non-empty set) and</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">.^I is an interpretation function that maps:</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Concept (class) name A to subset of Delta^I, Role (property) name R to</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">a binary relation R over Delta^I, Individual name i to an element of</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Delta^I.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Now let's see an example, if I have concepts "Lawyer" and "Doctor", and</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">role "hasChild",<SPAN class="Apple-converted-space"> </SPAN>John is a "Lawyer" and Mary is "Doctor", John</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">"hasChild" Mary. But later on in my model, Mary gets another degree and</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">becomes "Lawyer" also. Now Mary is both "Lawyer" and "Doctor".<SPAN class="Apple-converted-space"> </SPAN></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>What you have done is you have changed your *interpretation*: you have modified I, so it is no longer I but, say, I'. </DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Do the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">semantics of "Laywer", "Doctor", even "hasChild" change in this case?</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV><DIV>We would say that the *extension* of Lawyer (the set of those elements that are instances of Lawyer) has changed. </DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Because if we treat concept as a subset of domain, adding Mary to</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">"Lawyer" certainly change the set for that concept. <BR></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Indeed. </DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">If role is a subset</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">of pair of elements in the domain, would that be changed too? <BR></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV><DIV>It can certainly be changed in a similar way: for example, we can think of a third interpretation, say I'', where Mary ceases to be a child of John or a fourth one, say I''', where Mary has a child called Foo or ....</DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Can we still think Mary is the same Mary? What are the correct understanding of semantics here?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV><DIV>ok, so you need to distinguish between </DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>- an interpretation (any structure with a non-empty set and the mappings as you have described above)</DIV><DIV>- whether an interpretation *satisfies* an axiom (or a set of axioms or a TBox or an ontology) - such "legal" or "conforming" interpretations are often called *models* of an axiom (or a set of axioms or a TBox or an ontology)</DIV><DIV>- whether an axiom (or a set of axioms or a TBox or an ontology) does have a model, i.e., is *satisfiable* or *consistent*, and </DIV><DIV>- whether an axiom A follows from a set of axioms T (or a TBox or an ontology): this is the case if, in all models of T, the axiom A holds as well. In this case we say that this axiom is *entailed* by T and is often written as T |= A.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>The above 4 points (plus possible some other points) is what we call the semantics of a formalism/logic. </DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Cheers, Uli</DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">I might be missing something obvious here. But mathematically speaking , the set</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">has been changed. Would the semantics be changed also?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Thank you for any comments!</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Chuming Chen</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">---</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">**<SPAN class="Apple-converted-space"> </SPAN>You received this mail via the description logic mailing list; for more<SPAN class="Apple-converted-space"> </SPAN>**</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">**<SPAN class="Apple-converted-space"> </SPAN>information, visit the description logic homepage at <A href="http://dl.kr.org">http://dl.kr.org</A>/.<SPAN class="Apple-converted-space"> </SPAN>**</DIV> </BLOCKQUOTE></DIV><BR><DIV> <SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><DIV>Ulrike Sattler</DIV><DIV><A href="mailto:sattler@cs.man.ac.uk">sattler@cs.man.ac.uk</A></DIV><DIV><A href="http://www.cs.man.ac.uk/~sattler/">http://www.cs.man.ac.uk/~sattler/</A></DIV><BR class="Apple-interchange-newline"></SPAN></SPAN> </DIV><BR></BODY></HTML>