Return to Answer

To build UTM is equivalent to define inside recurrence functions space - which consists of certain relations inside or homomorphic to it (or to define typed lambda calculus). So it seems to be natural to require to embed such "subsystem" inside dynamical system in order to call it as You say Alive. To define full class of recurrence function is enough to have for example Peano Arithmetic inside but it seems to much more than minimal requirement. From Robinson Arithmetic article from Wikipedia:

"Robinson (1950) derived the Q axioms (1)-(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA."

It seems that You question is close related ( in spirit) to my question here: Given is "model". How many theories may it be a model? I was asking for information which theories are satisfied in given model ( given model is model for which theories). It looks like You are asking for "dynamical systems" which allows You to embed model of Universal Turing Machine, or model of lambda calculus, or model of recurrence functions. It is somehow connected in weak manner.

Another interesting matter is any intuition within Your question. What do You mean by "alive" beside that UTM machine is representable within such system? Take into account that from equivalence of several known "models of computation" You cannot follow that this models are the only possible. This is only an example of incomplete induction within mathematic ( Church-Turing Thesis mentioned somewhere here). You ask for exactly UTM machine inside system or You are looking for something "more exotic"? I ask because to use word "alive" is very strange in such context. To live is in known meaning, completely nonequivalent to: "be UTM" or "be simulated by UTN". It is strange to use such word.

Is it the case that UTM may simulate shift space? So in fact if You are able to implement UTM inside You also can shift space? It looks like it is possible at least with TM with two tapes. Then maybe You should look for "dead space" definition and not "alive" one;-)

To build UTM is equivalent to define inside recurrence functions space - which consists of certain relations inside or homomorphic to it (or to define typed lambda calculus). So it seems to be natural to require to embed such "subsystem" inside dynamical system in order to call it as You say Alive. To define full class of recurrence function is enough to have for example Peano Arithmetic inside but it seems to much more than minimal requirement. From Robinson Arithmetic article from Wikipedia:

"Robinson (1950) derived the Q axioms (1)-(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA."

It seems that You question is close related ( in spirit) to my question here: Given is "model". How many theories may it be a model? I was asking for information which theories are satisfied in given model ( given model is model for which theories). It looks like You are asking for "dynamical systems" which allows You to embed model of Universal Turing Machine, or model of lambda calculus, or model of recurrence functions. It is somehow connected in weak manner.

Another interesting matter is any intuition within Your question. What do You mean by "alive" beside that UTM machine is representable within such system? Take into account that from equivalence of several known "models of computation" You cannot follow that this models are the only possible. This is only an example of incomplete induction within mathematic ( Church-Turing Thesis mentioned somewhere here). You ask for exactly UTM machine inside system or You are looking for something "more exotic"? I ask because to use word "alive" is very strange in such context. To live is in known meaning, completely nonequivalent to: "be UTM" or "be simulated by UTN". It is strange to use such word.

Is it the case that UTM may simulate shift space? So in fact if You are able to implement UTM inside You also can shift space? It looks like it is possible at least with TM with two tapes. Then maybe You should look for "dead space" definition and not "alive" one;-)

To build UTM is equivalent to define inside recurrence functions space - which consists of certain relations inside or homomorphic to it (or to define typed lambda calculus). So it seems to be natural to require to embed such "subsystem" inside dynamical system in order to call it as You say Alive. To define full class of recurrence function is enough to have for example Peano Arithmetic inside but it seems to much more than minimal requirement. From Robinson Arithmetic article from Wikipedia:

"Robinson (1950) derived the Q axioms (1)-(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA."

It seems that You question is close related ( in spirit) to my question here: Given is "model". How many theories may it be a model? I was asking for information which theories are satisfied in given model ( given model is model for which theories). It looks like You are asking for "dynamical systems" which allows You to embed model of Universal Turing Machine, or model of lambda calculus, or model of recurrence functions. It is somehow connected in weak manner.

Another interesting matter is any intuition within Your question. What do You mean by "alive" beside that UTM machine is representable within such system? Take into account that from equivalence of several known "models of computation" You cannot follow that this models are the only possible. This is only an example of incomplete induction within mathematic ( Church-Turing Thesis mentioned somewhere here). You ask for exactly UTM machine inside system or You are looking for something "more exotic"? I ask because to use word "alive" is very strange in such context. To live is in known meaning, completely nonequivalent to: "be UTM" or "be simulated by UTN". It is strange to use such word.

To build UTM is equivalent to define inside recurrence functions space - which consists of certain relations inside or homomorphic to it (or to define typed lambda calculus). So it seems to be natural to require to embed such "subsystem" inside dynamical system in order to call it as You say Alive. To define full class of recurrence function is enough to have for example Peano Arithmetic inside but it seems to much more than minimal requirement. From Robinson Arithmetic article from Wikipedia:

"Robinson (1950) derived the Q axioms (1)-(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA."

It seems that You question is close related ( in spirit) to my question here: Given is "model". How many theories may it be a model? I was asking for information which theories are satisfied in given model ( given model is model for which theories). It looks like You are asking for "dynamical systems" which allows You to embed model of Universal Turing Machine, or model of lambda calculus, or model of recurrence functions. It is somehow connected in weak manner.

Another interesting matter is any intuition within Your question. What do You mean by "alive" beside that UTM machine is representable within such system? Take into account that from equivalence of several known "models of computation" You cannot follow that this models are the only possible. This is only an example of incomplete induction within mathematic ( Church-Turing Thesis mentioned somewhere here). You ask for exactly UTM machine inside system or You are looking for something "more exotic"? I ask because to use word "alive" is very strange in such context. To live is in known meaning, completely nonequivalent to: "be UTM" or "be simulated by UTN". It is strange to use such word.

Is it the case that UTM may simulate shift space? So in fact if You are able to implement UTM inside You also can shift space? It looks like it is possible at least with TM with two tapes. Then maybe You should look for "dead space" definition and not "alive" one;-)

To build UTM is equivalent to define inside recurrence functions space - which consists of certain relations inside or homomorphic to it (or to define typed lambda calculus). So it seems to be natural to require to embed such "subsystem" inside dynamical system in order to call it as You say Alive. To define full class of recurrence function is enough to have for example Peano Arithmetic inside but it seems to much more than minimal requirement. From Robinson Arithmetic article from Wikipedia:

"Robinson (1950) derived the Q axioms (1)-(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA."

It seems that You question is close related ( in spirit) to my question here: Given is "model". How many theories may it be a model? I was asking for information which theories are satisfied in given model ( given model is model for which theories). It looks like You are asking for "dynamical systems" which allows You to embed model of Universal Turing Machine, or model of lambda calculus, or model of recurrence functions. It is somehow connected in weak manner.

To build UTM is equivalent to define inside recurrence functions space - which consists of certain relations inside or homomorphic to it (or to define typed lambda calculus). So it seems to be natural to require to embed such "subsystem" inside dynamical system in order to call it as You say Alive. To define full class of recurrence function is enough to have for example Peano Arithmetic inside but it seems to much more than minimal requirement. From Robinson Arithmetic article from Wikipedia:

"Robinson (1950) derived the Q axioms (1)-(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA."

It seems that You question is close related ( in spirit) to my question here: Given is "model". How many theories may it be a model? I was asking for information which theories are satisfied in given model ( given model is model for which theories). It looks like You are asking for "dynamical systems" which allows You to embed model of Universal Turing Machine, or model of lambda calculus, or model of recurrence functions. It is somehow connected in weak manner.

Another interesting matter is any intuition within Your question. What do You mean by "alive" beside that UTM machine is representable within such system? Take into account that from equivalence of several known "models of computation" You cannot follow that this models are the only possible. This is only an example of incomplete induction within mathematic ( Church-Turing Thesis mentioned somewhere here). You ask for exactly UTM machine inside system or You are looking for something "more exotic"? I ask because to use word "alive" is very strange in such context. To live is in known meaning, completely nonequivalent to: "be UTM" or "be simulated by UTN". It is strange to use such word.