Given a finite set Ω of cardinality n, find a computable upper bound on the largest finite factored set model that is combinatorially different from all smaller finite factored set models. (We say that two FFS models are combinatorially different if they say the same thing about the emptiness of all boolean combinations of histories and conditional histories of partitions of Ω.) (Such an upper bound must exist because there are only finitely many combinatorially distinct FFS models, but a computable upper bound, would tell us that temporal inference is computable.)
Prove the fundamental theorem for finite dimensional factored sets. (Seems likely combinatorial-ish, but I am not sure.)
Figure out how to write a program to do efficient temporal inference on small examples. (I suspect this requires a bunch of combinatorial insights. By default this is very intractable, but we might be able to use math to make it easier.)
Axiomatize complete consistent orthogonality databases (consistent means consistent with some model, complete means has an opinion on every possible conditional orthogonality) (To start, is it the case that compositional semigraphoid axioms already work?)
If by “pure” you mean “not related to history/orthogonality/time,” then no, the structure is simple, and I don’t have much to ask about it.
Right, the structure is quite simple. The only thing that came to mind about finite factored sets as combinatorial objects was studying the L-function of the number of them, which surely has some nice Euler product. Maybe you can write it as a product of standard zeta functions or something?
Are there any interesting pure combinatorics problems about finite factored sets that you’re interested in?
Given a finite set Ω of cardinality n, find a computable upper bound on the largest finite factored set model that is combinatorially different from all smaller finite factored set models. (We say that two FFS models are combinatorially different if they say the same thing about the emptiness of all boolean combinations of histories and conditional histories of partitions of Ω.) (Such an upper bound must exist because there are only finitely many combinatorially distinct FFS models, but a computable upper bound, would tell us that temporal inference is computable.)
Prove the fundamental theorem for finite dimensional factored sets. (Seems likely combinatorial-ish, but I am not sure.)
Figure out how to write a program to do efficient temporal inference on small examples. (I suspect this requires a bunch of combinatorial insights. By default this is very intractable, but we might be able to use math to make it easier.)
Axiomatize complete consistent orthogonality databases (consistent means consistent with some model, complete means has an opinion on every possible conditional orthogonality) (To start, is it the case that compositional semigraphoid axioms already work?)
If by “pure” you mean “not related to history/orthogonality/time,” then no, the structure is simple, and I don’t have much to ask about it.
Right, the structure is quite simple. The only thing that came to mind about finite factored sets as combinatorial objects was studying the L-function of the number of them, which surely has some nice Euler product. Maybe you can write it as a product of standard zeta functions or something?