Kind of Godel numbering
in [Math]
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From: JohnF on 21 Dec 2009 15:15 I'm looking to construct a Godel-like kind of numbering to address the following problem. I have objects, each of which is an unordered collection of "containers", with each container in an object labelled by an integer, e.g., object B = { (b1,B1), (b2,B2), (b3,B3),... } with Bi a container labelled by integer bi in object B. The bi's are arbitrary integer labels except that bi must uniquely identify the object Bi it labels. Trouble arises because I want to define multiplication of these objects in the following way. Given B above and a similar C, define BC = { (bi*cj,(BiCj)) | (bi,Bi)\in B & (cj,Cj)\in C } where bi*cj is just integer multiplication, and (BiCj) just means a single container with the combined contents of Bi and Cj in it. The problem is that each bi*cj label for container (BiCj) must identify both original component containers Bi and Cj. That is, while ordering BiCj versus CjBi is unimportant, the bi*cj label must nevertheless identify both i and j as well as which (i or j) belongs to b and which (j or i) to c. Any way to define labels that satisfy these requirements? Thanks, -- John Forkosh ( mailto: j(a)f.com where j=john and f=forkosh )
From: Ilmari Karonen on 21 Dec 2009 18:33 On 2009-12-21, JohnF <john(a)please.see.sig.for.email.com> wrote: > I have objects, each of which is an unordered collection > of "containers", with each container in an object labelled by > an integer, e.g., object B = { (b1,B1), (b2,B2), (b3,B3),... } > with Bi a container labelled by integer bi in object B. > The bi's are arbitrary integer labels except that bi must > uniquely identify the object Bi it labels. > Trouble arises because I want to define multiplication > of these objects in the following way. Given B above and a similar > C, define BC = { (bi*cj,(BiCj)) | (bi,Bi)\in B & (cj,Cj)\in C } > where bi*cj is just integer multiplication, and (BiCj) just means > a single container with the combined contents of Bi and Cj in it. > The problem is that each bi*cj label for container (BiCj) > must identify both original component containers Bi and Cj. > That is, while ordering BiCj versus CjBi is unimportant, > the bi*cj label must nevertheless identify both i and j as well as > which (i or j) belongs to b and which (j or i) to c. > Any way to define labels that satisfy these requirements? If I didn't misunderstand your requirements, wouldn't simply letting all the initial labels be distinct primes work? Then every label would have a unique prime factorization identifying the original containers of whose labels it is a product of. -- Ilmari Karonen To reply by e-mail, please replace ".invalid" with ".net" in address.
From: kunzmilan on 22 Dec 2009 04:35 On Dec 21, 9:15 pm, JohnF <j...(a)please.see.sig.for.email.com> wrote: > I'm looking to construct a Godel-like kind of numbering > to address the following problem. > I have objects, each of which is an unordered collection > of "containers", with each container in an object labelled by > an integer, e.g., object B = { (b1,B1), (b2,B2), (b3,B3),... } > with Bi a container labelled by integer bi in object B. > The bi's are arbitrary integer labels except that bi must > uniquely identify the object Bi it labels. > Trouble arises because I want to define multiplication > of these objects in the following way. Given B above and a similar > C, define BC = { (bi*cj,(BiCj)) | (bi,Bi)\in B & (cj,Cj)\in C } > where bi*cj is just integer multiplication, and (BiCj) just means > a single container with the combined contents of Bi and Cj in it. > The problem is that each bi*cj label for container (BiCj) > must identify both original component containers Bi and Cj. > That is, while ordering BiCj versus CjBi is unimportant, > the bi*cj label must nevertheless identify both i and j as well as > which (i or j) belongs to b and which (j or i) to c. > Any way to define labels that satisfy these requirements? > Thanks, > -- > John Forkosh ( mailto: j...(a)f.com where j=john and f=forkosh ) Try to use a matrix: columns containers, rows objects. Elements ij can be used for a more specific identification of objects, and matrix multiplication for countings. kunzmilan
From: Don Stockbauer on 22 Dec 2009 05:09 On Dec 22, 3:35 am, kunzmilan <kunzmi...(a)atlas.cz> wrote: > On Dec 21, 9:15 pm, JohnF <j...(a)please.see.sig.for.email.com> wrote: > > > > > I'm looking to construct a Godel-like kind of numbering > > to address the following problem. > > I have objects, each of which is an unordered collection > > of "containers", with each container in an object labelled by > > an integer, e.g., object B = { (b1,B1), (b2,B2), (b3,B3),... } > > with Bi a container labelled by integer bi in object B. > > The bi's are arbitrary integer labels except that bi must > > uniquely identify the object Bi it labels. > > Trouble arises because I want to define multiplication > > of these objects in the following way. Given B above and a similar > > C, define BC = { (bi*cj,(BiCj)) | (bi,Bi)\in B & (cj,Cj)\in C } > > where bi*cj is just integer multiplication, and (BiCj) just means > > a single container with the combined contents of Bi and Cj in it. > > The problem is that each bi*cj label for container (BiCj) > > must identify both original component containers Bi and Cj. > > That is, while ordering BiCj versus CjBi is unimportant, > > the bi*cj label must nevertheless identify both i and j as well as > > which (i or j) belongs to b and which (j or i) to c. > > Any way to define labels that satisfy these requirements? > > Thanks, > > -- > > John Forkosh ( mailto: j...(a)f.com where j=john and f=forkosh ) > > Try to use a matrix: columns containers, rows objects. Elements ij can > be used for a more specific identification of objects, and matrix > multiplication for countings. > kunzmilan Any idiot can number things.
From: JohnF on 22 Dec 2009 10:59
Ilmari Karonen <usenet2(a)vyznev.invalid> wrote: > JohnF <john(a)please.see.sig.for.email.com> wrote: >> I have objects, each of which is an unordered collection >> of "containers", with each container in an object labelled by >> an integer, e.g., object B = { (b1,B1), (b2,B2), (b3,B3),... } >> with Bi a container labelled by integer bi in object B. >> The bi's are arbitrary integer labels except that bi must >> uniquely identify the object Bi it labels. >> Trouble arises because I want to define multiplication >> of these objects in the following way. Given B above and a similar >> C, define BC = { (bi*cj,(BiCj)) | (bi,Bi)\in B & (cj,Cj)\in C } >> where bi*cj is just integer multiplication, and (BiCj) just means >> a single container with the combined contents of Bi and Cj in it. >> The problem is that each bi*cj label for container (BiCj) >> must identify both original component containers Bi and Cj. >> That is, while ordering BiCj versus CjBi is unimportant, >> the bi*cj label must nevertheless identify both i and j as well as >> which (i or j) belongs to b and which (j or i) to c. >> Any way to define labels that satisfy these requirements? > > If I didn't misunderstand your requirements, wouldn't simply letting > all the initial labels be distinct primes work? Then every label > would have a unique prime factorization identifying the original > containers of whose labels it is a product of. Thanks, Ilmari, but if I didn't misunderstand your suggestion, that'll fail the part "which (i or j) belongs to b and which (j or i) to c". As I understand it, you're suggesting B = { (2,B1), (3,B2), (5,B3),... }. If I label C similarly, starting with (2,C1), then a product label of, say, 14=2*7=7*2, would identify either B1C4 or B4C1, but wouldn't resolve the two cases. But maybe you're suggesting labelling C beginning with the first prime not used for B, e.g., (97,C1) if (91,B1) was the last B. That would indeed resolve the two cases. Trouble is I don't, in general, know the "rank" of B. Moreover, instead of BC, maybe I want to calculate DC or EC or FC, etc, for other objects D,E,F, etc. Then I might need different C-representations for each case. -- John Forkosh ( mailto: j(a)f.com where j=john and f=forkosh ) |