Anthony on Nostr: The lower order comes up a bit unexpectedly in coevolutionary algorithms (what I ...
The lower order comes up a bit unexpectedly in coevolutionary algorithms (what I studied in my PhD), but really in search and optimization more broadly.
Say you have a notion of "context", and a way of ordering these so that some contexts are larger or more expansive than or "above" others. And let's say in each context, there is a set of things that are "best". I'm being vague because you can instantiate this basic idea pretty broadly. For instance, maybe the contexts are states of information in a search algorithm and "best" refers to the possible solutions that seem best in each information state. Or, maybe the contexts are possible worlds and "best" refers to which propositions are true in each possible world.
Anyway, with that simple setup you can associate to each thing the set of all contexts in which it appears best. This set could be empty or could be very large or anything in between depending on the thing and the context. Then the lower order shows up in a weak preference relationship among all the things: one thing is lower preference than another if for each context in which it appears best there's a larger or equal context in which the other thing seems best. This is exactly the lower order between the sets of contexts in which each thing seems best. If the set of contexts in which one thing seems best is higher up the lower order (😝) than the set of contexts in which the other seems best, then the first thing is weakly preferred to the second.
The intuition in a search setting is that contexts are viewed as states of information. If x and y are possible results, and for every context in which you think x is the best there is always a bigger context--i.e., with more information--in which you think y is best instead, you ought to prefer y to x.
Applied to modal logic, this notion corresponds to validity: if in every world p is true there is an accessible world in which q is true, then "p implies possibly q" is true in every world (valid).
#math #ComputerScience #search #CoevolutionaryAlgorithm #SolutionConcept
Say you have a notion of "context", and a way of ordering these so that some contexts are larger or more expansive than or "above" others. And let's say in each context, there is a set of things that are "best". I'm being vague because you can instantiate this basic idea pretty broadly. For instance, maybe the contexts are states of information in a search algorithm and "best" refers to the possible solutions that seem best in each information state. Or, maybe the contexts are possible worlds and "best" refers to which propositions are true in each possible world.
Anyway, with that simple setup you can associate to each thing the set of all contexts in which it appears best. This set could be empty or could be very large or anything in between depending on the thing and the context. Then the lower order shows up in a weak preference relationship among all the things: one thing is lower preference than another if for each context in which it appears best there's a larger or equal context in which the other thing seems best. This is exactly the lower order between the sets of contexts in which each thing seems best. If the set of contexts in which one thing seems best is higher up the lower order (😝) than the set of contexts in which the other seems best, then the first thing is weakly preferred to the second.
The intuition in a search setting is that contexts are viewed as states of information. If x and y are possible results, and for every context in which you think x is the best there is always a bigger context--i.e., with more information--in which you think y is best instead, you ought to prefer y to x.
Applied to modal logic, this notion corresponds to validity: if in every world p is true there is an accessible world in which q is true, then "p implies possibly q" is true in every world (valid).
#math #ComputerScience #search #CoevolutionaryAlgorithm #SolutionConcept