**Theorem 6: Causal IPOMDP's:** *Any infra-POMDP with transition kernel ** which fulfills the niceness conditions and always produces crisp infradistributions, and starting infradistribution **, produces a causal belief function via **.*

So, first up, we can appeal to Theorem 4 on Pseudocausal IPOMDP's, to conclude that all the are well-defined, and that this we defined fulfills all belief function conditions except maybe normalization.

The only things to check are that the resulting is normalized, and that the resulting is causal. The second is far more difficult and makes up the bulk of the proof, so we'll just dispose of the first one.

**T6.1** First, , since it's crisp by assumption, always has for all constants . The same property extends to all the (they're all crisp too) and then from there to by induction. Then we can just use the following argument, where is arbitrary.

The last two inequalities were by mapping constants to the same constant regardless of , and by being normalized since it's an infradistribution. This same proof works if you swap out the 1 with a 0, so we have normalization for .

**T6.2** All that remains is checking the causality condition. We'll be using the alternate rephrasing of causality. We have defined as

and want to derive our rephrasing of causality, that for any and and and family , we have

Accordingly, we get to fix a , and family of functions , and assume

Our proof target is

Applying our reexpression of in terms of and and projection, we get

Undoing the projections on both sides, we get

Unpacking the semidirect product, we get

Now, we can notice something interesting here. By concavity for infradistributions, we have

So, our new proof target becomes

Because if we hit that, we can stitch the inequalities together. The obvious move to show this new result is to apply monotonicity of infradistributions. If the inner function on the left hand side was always above the inner function on the right hand side, we'd have our result by infradistribution monotonicity. So, our new proof target becomes

At this point, now that we've gotten rid of , we can use the full power of crisp infradistributions. Crisp infradistributions are just a set of probability distributions! Remember that the semidirect product, , can be viewed as taking a probability distribution from , and for each , picking a conditional probability distribution from , and all the probability distributions of that form make up .

In particular, the infinitely iterated semidirect product here can be written as a repeated process where, for each history of the form Murphy gets to select a probability distribution over the next observation and state from (as a set of probability distributions) to fill in the next part of the action-observation-state tree. Accordingly every single probability distribution in can be written as interacting with an expanded probabilistic environment of type (which gives Murphy's choice wherever it needs to pick the next observation and state), subject to the constraints that

and

We'll abbreviate this property that always picks stuff from the appropriate set and starts off picking from as .

Our old proof goal of

can be rephrased as

and then rephrased as

And now, by our earlier discussion, we can rephrase this statement in terms of picking expanded probabilistic environments.

Now, on the right side, since the set we're selecting the expanded environments from is the same each time and doesn't depend on , we can freely move the inf outside of the expectation. It is unconditionally true that

So now, our proof goal shifts to

because if that were true, we could pair it up with the always-true inequality to hit our old proof goal. And we'd be able to show this proof goal if we showed the following, more ambitious result.

Note that we're not restricting to compatibility with and , we're trying to show it for *any* expanded probabilistic environment of type . We can rephrase this new proof goal slightly, viewing ourselves as selecting an action and observation history.

Now, for any expanded probabilistic environment , there is a unique corresponding probabilistic environment of type where, for any policy, . This is just the classical result that any POMDP can be viewed as a MDP with a state space consisting of finite histories, it still behaves the same. Using this swap, our new proof goal is:

Where is an arbitrary probabilistic environment. Now, you can take any probabilistic environment and view it as a mixture of deterministic environments . So we can rephrase as, at the start of time, picking a from for some particular . This is our mixture of deterministic environments that makes up . This mixture isn't necessarily unique, but there's always *some* mixture of deterministic enivironments that works to make where is arbitrary. So, our new proof goal becomes:

Since is just a dirac-delta distribution on a particular history, we can substitute that in to get the equivalent

Now we just swap the two expectations.

And then we remember that our starting assumption was

So we've managed to hit our proof target and we're done, this propagates back to show causality for .

**Theorem 7: Pseudocausal to Causal Translation:** *The following diagram commutes when the bottom belief function ** is pseudocausal. The upwards arrows are the causal translation procedures, and the top arrows are the usual morphisms between the type signatures. The new belief function ** is causal. Also, if **, and ** is defined as ** for any history containing Nirvana, and ** for any nirvana-free history, then **.*

**T7.0** Preliminaries. Let's restate what the translations are. The notation is that "defends" . defends if it responds to any prefix of which ends in an action with either the next observation in , or Nirvana, and responds to any non-prefixes of with Nirvana.

Also, we'll write if is "compatible" with and . The criterion for this is that , and that any prefix of which ends in an action and contains no Nirvana observations must either be a prefix of , or only deviate from in the last action. An alternate way to state that is that is capable of being produced by a copolicy which defends , interacting with .

With these two notes, we introduce the infrakernels , and .

is total uncertainty over such that defends .

And is total uncertainty over such that is compatible with and .

Now, to briefly recap the three translations.For first-person static translation, we have

Where is the original belief function.

For third-person static translation, we have

Where is the original infradistribution over .

For first-person dynamic translation, the new infrakernel has type signature

Ie, the state space is destinies plus a single Nirvana state, and the observation state is the old space of observations plus a Nirvana observation. is used for the Nirvana observation, and is used for the Nirvana state. The transition dynamics are:

and, if , then

and otherwise,

The initial infradistribution is , the obvious injection of from to .

The obvious way to show this theorem is that we can assume the bottom part of the diagram commutes and are all pseudocausal, and then show that all three paths to produce the same result, and it's causal. Then we just wrap up that last part about the utility functions.

Fortunately, assuming all three paths produce the same belief function, it's trivial to show that it's causal. is an infradistribution since is, and is a crisp infrakernel, so we can just invoke Theorem 6 that crisp infrakernels produce causal belief functions to show causality for . So we only have to check that all three paths produce the same result, and show the result about the utility functions. Rephrasing one of the proofs is one phase, and then there are two more phases showing that the other two phases hit the same target, and the last phase showing our result about the utility functions.

We want to show that

**T7.1** First, let's try to rephrase into a more handy form for future use. Invoking our definition of , we have:

Then, we undo the projection

And unpack the semidirect product

And unpack what means

And then apply the definition of

This will be our final rephrasing of . Our job is to derive this result for the other two paths.

**T7.2** First up, the third-person static translation. We can start with definitions.

And then, we can recall that since and are assumed to commute, we can define in terms of via . Making this substitution, we get

We undo the first pushforward

And then the projection

And unpack the semidirect product

And remove the projection

and unpack the semidirect product and what is

Now, we recall that is all the that defend , to get

At this point we can realize that for any which defends , when interacts with , it makes a that is compatible with and . Further, all compatible with and have an which produces them. If has its nirvana-free prefix not being a prefix of , then the defending which doesn't end early, just sends deviations to Nirvana, is capable of producing it. If looks like but ended early with Nirvana, you can just have a defending which ends with Nirvana at the same time. So we can rewrite this as

And this is exactly the form we got for .

**T7.3** Time for the third and final translation. We want to show that

is what we want. Because the pseudocausal square commutes, we have that , so we can make that substitution to yield

Remove the projection, to get

For this notation, we're using to refer to an element of , the state space. is an element of , a full unrolling. is the projection of this to .

We unpack the semidirect product to yield

and rewrite the injection pushforward

and remove the projection

And unpack the semidirect product along with .

Now we rewrite this as a projection to yield

And do a little bit of reindexing for convenience, swapping out the symbol for , getting

And now, we can ask what is. Basically, the starting state is the destiny , and then interacts with it through the kernel . Due to how is defined, it can do one of three things. First, it could just unroll all the way to completion, if is the sort of history that's compatible with . Second, it could partially unroll and deviate to Nirvana early, even if is compatible with more, because of how is defined. Finally, could deviate from and then would have to respond with Nirvana. And of course, once you're in a Nirvana state, you're in there for good. Hm, you have total uncertainty over histories compatible with and , any of them at all could be made by . So, this projection of the infrakernel unrolling is just . We get

And then we use what does to functions, to get

Which is exactly the form we need. So, all three translation procedures produce identical belief functions.

**T7.4** The only thing which remains to wrap this result up is guaranteeing that, if is 1 if the history contains Nirvana, (or infinity for the type signature), and otherwise, then .

We rephrased as

We can notice something interesting. If is the sort of history that is capable of making, ie , then all of the are going to end up in Nirvana except itself, because all the are like "follow , any deviations of from go to Nirvana, and also Nirvana can show up early". In this case, since any history ending with Nirvana is maximum utility, and otherwise is copied, we have . However, if is incompatible with , then no matter what, a history which follows is going to hit Nirvana and be assigned maximum utility. So, we can rewrite

as

Which can be written as an update, yielding

And then, we remember that regardless of and , for pseudocausal belief functions, which is, we have that , so the update assigns higher expectation values than does. Also, . Thus, the infinimum is attained by , and we get

and we're done.

**Theorem 8: Acausal to Pseudocausal Translation:** *The following diagram commutes when the bottom belief function ** is acausal. The upwards arrows are the pseudocausal translation procedures, and the top arrows are the usual morphisms between the type signatures. The new belief function ** will be pseudocausal.*

**T8.0** First, preliminaries. The two translation procedures are, to go from an infradistribution over policy-tagged destinies , to an infradistribution over destinies , we just project.

And, to go from an acausal belief function to a pseudocausal one , we do the translation

We assume that and commute, and try to show that these translation procedures result in a and which are pseudocausal and commute with each other. So, our proof goals are that

and

Once we've done that, it's easy to get that is pseudocausal. This occurs because the projection of any infradistribution is an infradistribution, so we can conclude that is an infradistribution over . Also, back in the Pseudocausal Commutative Square theorem, we were able to derive that *any* infradistribution over , when translated to a , makes a pseudocausal belief function, no special properties were used in that except being an infradistribution.

**T8.1** Let's address the first one, trying to show that

We'll take the left side, plug in some function, and keep unpacking and rearranging until we get the right side with some function plugged in. First up is applying definitions.

by how was defined. Now, we start unpacking. First, the projection.

Then unpack the semidirect product and , for

Then unpack the update, to get

Then pack up the

And pack up the semidirect product

And since for the original acausal belief function, we can rewrite this as

And pack this up in a projection.

And then, since we defined , we can pack this up as

And then pack up the update

So, that's one direction done.

**T8.2** Now for the second result, showing that

Again, we'll take the more complicated form, and write it as the first one, for arbitrary functions .bWe begin with

and then unpack what is to get

Undo the projection

Unpack the semidirect product and .

Remove the projection

Unpack the semidirect product and again.

Let's fold the two infs into one inf, this doesn't change anything, just makes things a bit more concise.

and unpack the update.

Now, we can note something interesting here. If is selected to be equal to , that inner function is just , because is only supported over histories compatible with . If is unequal to , that inner function is with 1's for some inputs, possibly. So, our choice of is optimal for minimizing when its equal to , so we can rewrite as

and write this as

And pack up as a semidirect product

and then because , we substitute to get

and then write as a projection.

Now, this is just , yielding

and we're done.

**Proposition 2:** *The two formulations of x-pseudocausality are equivalent for crisp acausal belief functions.*

One formulation of x-pseudocausality is that

where

The second formulation is,

This proof will be slightly informal, it can be fully formalized without too many conceptual difficulties. We need to establish some basic probability theory results to work on this. The proof outline is we establish/recap four probability-theory/inframeasure-theory results, and then show the two implication directions of the iff statement.

**P2.1** Our first claim is that, given any two probability distributions and , you can shatter them into measures , and such that and , and . What this decomposition is basically doing is finding the largest chunk of measure that and agree on (that's ), and then subtracting that out from and yields you your and . These measures have no overlap, because if there was overlap, you could put that overlap into the chunk of measure. And, the mass of the "unique to " piece must equal the mass of the "unique to " piece because they must add up to probability distributions, and this quantity is exactly the total variation distance of .

**P2.2** Our second claim is that

Why is this? Well, there's an alternate formulation of total variation distance where

As intuition for this, we can pick a function U which is 1 on the support of , and 0 elsewhere. In such a case, we'd get

and then remember that and have disjoint supports, so we get

And any other function in the range wouldn't do as well at attaining different expectation values on and . So that's where the identity comes from. If we try to do the optimal separation but instead restrict to be in , we'd get

and any other utility function in