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#41
 
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Re: What is an automorphism of a database instance? -
01-09-2008
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On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." |
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R
f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)
this is general enough to cover both domain value permutations and
column/relation renamings.
Exersize 1. Show that a mapping of database instance in the example of
the rpost message is automorphism in algebraic sence.
#42
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On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." |
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R
f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)
this is general enough to cover both domain value permutations and
column/relation renamings.
Exersize 1. Show that a mapping of database instance in the example of
the rpost message is automorphism in algebraic sence.
#43
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On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." |
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R
f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)
this is general enough to cover both domain value permutations and
column/relation renamings.
Exersize 1. Show that a mapping of database instance in the example of
the rpost message is automorphism in algebraic sence.
#44
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On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." |
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R
f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)
this is general enough to cover both domain value permutations and
column/relation renamings.
Exersize 1. Show that a mapping of database instance in the example of
the rpost message is automorphism in algebraic sence.
#45
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On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." |
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R
f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)
this is general enough to cover both domain value permutations and
column/relation renamings.
Exersize 1. Show that a mapping of database instance in the example of
the rpost message is automorphism in algebraic sence.
#46
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On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." I suggest defining automorphism of database instance (where "database instance" is understood to be a set of relations) algebraically as a mapping f such that for any relations Q and R f(Q /\ R ) = f(Q) /\ f(R) f(Q \/ R ) = f(Q) \/ f(R) this is general enough to cover both domain value permutations and column/relation renamings. |
Btw. didn't you mean "homomorphism" rather than "automorphism"?
-- Jan Hidders
#47
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On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." I suggest defining automorphism of database instance (where "database instance" is understood to be a set of relations) algebraically as a mapping f such that for any relations Q and R f(Q /\ R ) = f(Q) /\ f(R) f(Q \/ R ) = f(Q) \/ f(R) this is general enough to cover both domain value permutations and column/relation renamings. |
Btw. didn't you mean "homomorphism" rather than "automorphism"?
-- Jan Hidders
#48
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On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." I suggest defining automorphism of database instance (where "database instance" is understood to be a set of relations) algebraically as a mapping f such that for any relations Q and R f(Q /\ R ) = f(Q) /\ f(R) f(Q \/ R ) = f(Q) \/ f(R) this is general enough to cover both domain value permutations and column/relation renamings. |
Btw. didn't you mean "homomorphism" rather than "automorphism"?
-- Jan Hidders
#49
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On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." I suggest defining automorphism of database instance (where "database instance" is understood to be a set of relations) algebraically as a mapping f such that for any relations Q and R f(Q /\ R ) = f(Q) /\ f(R) f(Q \/ R ) = f(Q) \/ f(R) this is general enough to cover both domain value permutations and column/relation renamings. |
Btw. didn't you mean "homomorphism" rather than "automorphism"?
-- Jan Hidders
#50
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On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said: On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote: I need help in understanding what is an automorphism of a database instanc e. The following definition is from the book Relational Database Theory by Atzeni and De Antonellis: Definition: An automorphism of a database instance r is a partial function * * * * h : D --> D where D is the domain of the database r such that 1) the partial function h is a permutation of the active domain D_r, and 2) when we extend its definition to tuples, relations, and database instances, we obtain a function on instances that is the identity on r, namely * * * * h(r) = r. I can understand 1), but I cannot understand 2). In mathematics, an automorphism is a 1-1 mapping that preserves the structure of an underlying set. *For example, if in some set whose members x, y and z obeys * * * * z = x + y, then we expect an automorphism f on that set to also obey * * * * f(z) = f(x) + f(y). So, the structure of "addition" is preserved. Now, back to relational database theory, what "structure" is being preserved by 2)? *Can someone explain the formalization in 2) more carefully? I only just saw your posting so I wondered if you still needed help with this. Thanks for the follow-up. *The notion is still somewhat ambiguous in my mind. *I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism. What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values. The difficulty is in formalizing the term "essentially" and "same information." I suggest defining automorphism of database instance (where "database instance" is understood to be a set of relations) algebraically as a mapping f such that for any relations Q and R f(Q /\ R ) = f(Q) /\ f(R) f(Q \/ R ) = f(Q) \/ f(R) this is general enough to cover both domain value permutations and column/relation renamings. |
Btw. didn't you mean "homomorphism" rather than "automorphism"?
-- Jan Hidders
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