Let’s say that A is the set of available actions and B is the set of consequences.A→B is then the set of predictions, where a single prediction associates to every possible action a consequence.(A→B)→A is then a choice operator, that selects for each prediction an action to take.
What we have seen so far:
There’s no ‘general’ or ‘natural’ choice operator, that is, every choice operator must be based on at least a partial knowledge of the domain or the codomain;
Unless the possible consequences are trivial, a choice operator will choose the same action for many different predictions, that is a choice operator only uses certain feature of the predictions’ space and is indifferent to anything else [1];
A choice operator defines naturally a ‘preferred outcome’ operator, which is simply the predicted outcome of the chosen action, and is defined by ‘sandwiching’ the choice operator between two predictions. I just thought interleave is a better name than sandwich. It’s of type (A→B)→B.
[1] To show this, let k−1(A) be a partition of A→B and let Rk be the equivalence relation uniquely generated by the partition. Then k(A→B)≡k((A→B)/Rk)
Knowledge that there is an action to select, in the form of having an action in hand, allows the implementation of exactly one chooser: The one that always selects that action.
[1] holds for any function k / partition k−1 between any two sets. The proof you want may be that A→B is an exponential space and therefore usually larger than A.
interleave/sandwich should then take two predictions as parameters. This suggests that we could define a metric on the space of predictions, and then sandwich the chooser between two nearby predictions, to measure its response to inaccurate predictions.
Re: the third point, I think it’s important to differentiate between f(k(f)) and t(k(f)), where t is the true prediction, that is what actually happens when an agent performs the action A.
f(k(f)) is simply the outcome the agent is aiming at, while t(k(f)) is the outcome the agent eventually gets. So maybe it’s more interesting a measure of similarity in B, from which you can compare the two.
Let’s say that A is the set of available actions and B is the set of consequences.A→B is then the set of predictions, where a single prediction associates to every possible action a consequence.(A→B)→A is then a choice operator, that selects for each prediction an action to take.
What we have seen so far:
There’s no ‘general’ or ‘natural’ choice operator, that is, every choice operator must be based on at least a partial knowledge of the domain or the codomain;
Unless the possible consequences are trivial, a choice operator will choose the same action for many different predictions, that is a choice operator only uses certain feature of the predictions’ space and is indifferent to anything else [1];
A choice operator defines naturally a ‘preferred outcome’ operator, which is simply the predicted outcome of the chosen action, and is defined by ‘sandwiching’ the choice operator between two predictions. I just thought interleave is a better name than sandwich. It’s of type (A→B)→B.
[1] To show this, let k−1(A) be a partition of A→B and let Rk be the equivalence relation uniquely generated by the partition. Then k(A→B)≡k((A→B)/Rk)
Knowledge that there is an action to select, in the form of having an action in hand, allows the implementation of exactly one chooser: The one that always selects that action.
[1] holds for any function k / partition k−1 between any two sets. The proof you want may be that A→B is an exponential space and therefore usually larger than A.
interleave/sandwich should then take two predictions as parameters. This suggests that we could define a metric on the space of predictions, and then sandwich the chooser between two nearby predictions, to measure its response to inaccurate predictions.
Re: the third point, I think it’s important to differentiate between f(k(f)) and t(k(f)), where t is the true prediction, that is what actually happens when an agent performs the action A.
f(k(f)) is simply the outcome the agent is aiming at, while t(k(f)) is the outcome the agent eventually gets. So maybe it’s more interesting a measure of similarity in B, from which you can compare the two.