# 15

SubagentsEmbedded AgencyAI
Frontpage
Review

This is the tenth post in the Cartesian frames sequence.

Here, we define a bunch of ways to construct new (additive/multiplicative) (sub/super)-(agents/environments) from a given Cartesian frame. Throughout this post, we will start with a single Cartesian frame over .

We will start by defining operations from taking subsets and partitions of  and . We will then define similar operations from taking subsets and partitions of .

## 1. Definitions from Agents and Environments

1.1. Committing

Definition: Given a subset , let  denote the Cartesian frame , with  given by . Let  denote the Cartesian frame , with  given by .

represents the perspective you get when the agent of  makes a commitment to choose an element of , while  represents the perspective you get when the agent of  makes a commitment to choose an element outside of .

Claim: For all  and . Further,  and  are brothers in .

Proof: That  and  is trivial from the committing definition of additive subagent.

Observe that , where  is given by  if , and  if . Let  be the diagonal, . We clearly have that  is in , where  is the restriction of  to , and that ; so  and  are brothers in

Claim:

Proof: Trivial.

1.2. Assuming

Assuming is the dual operation to committing.

Definition: Given a subset , let  denote the Cartesian frame , with  given by . Let  denote the Cartesian frame , with  given by .

represents the perspective you get when you assume the environment is chosen from , while  represents the perspective you get when you assume the environment is chosen from outside of .

In "Introduction to Cartesian Frames" §3.2 (Examples of Controllables), I noted the counter-intuitive result that agents have no control in worlds where a meteor (or other event) could have prevented their existence:

.

Here, we see that we can use  to recover the more intuitive idea of "control." The subagent modified by the assumption "there's no meteor" can have controllables, even though the original agent has no controllables:

.

Claim: For all  and . Similarly, for all  and .

Proof: Trivial.

Claim: For all  and

Proof: Trivial.

1.3. Externalizing

Note that for the following definitions, when we say " is a partition of ," we mean that  is a set of nonempty subsets of , such that for each , there exists a unique  such that

Definition: Given a partition  of , let  denote the set of all functions  from  to  such that  for all .

Definition: Given a partition  of , let  denote the Cartesian Frame , where  is given by . Let  denote the Cartesian Frame , where  is given by .

We say "externalizing " for  and "externalizing mod " for .

can be thought of the result of the agent of  first factoring its choice into choosing an equivalence class in , and choosing an element of each equivalence class, and then externalizing the part of itself that chooses an equivalence class. I.e., we are drawing a new Cartesian frame which treats the choice of equivalence class as part of the environment, rather than part of the agent.

Similarly,  can be thought of the result of the agent of  factoring as above, then externalizing the part of itself that chooses an element of each equivalence class.

Claim: For all partitions  of  and . Further,  and  are sisters in .

Proof: Let  and .

First, observe that for every , there exists a morphism , with  given by , and  given by . To see that this is a morphism, observe that for all  and ,

Let  be given by , and let , where

Our goal is to show that , and that .

To see , we define  and  as follows.

First,  is given by . We first need to confirm that  is surjective. Given any , we can let  be the set with  and construct a function  by saying , and for each , choosing an , and saying . Observing that , we have that  is surjective and thus has a right inverse.

We choose  to be any right inverse to . Similarly, we define  by , which is clearly surjective, and define  to be any right inverse to . (Indeed,  is bijective as long as  is nonempty.)

Then, for all  and , we have

so  is a morphism. This also gives us that for all  and  we have

so  is a morphism. We know  is homotopic to the identity on , since  is the identity on , and we know that  is homotopic to the identity on , since  is the identity on . Thus, .

To show that , it suffices to show that

and

where  and  are given by

Indeed, we show that if  is nonempty, , and .

If  is nonempty, then the function from  to  given by  is a bijection, since it is clearly surjective, and is injective because  is uniquely defined by . This gives a bijection between  and , and we have that for all , and ,

Similarly, we have a bijection between  and , and for all , and ,

If  is empty, then  is empty, and  is a singleton empty function, so , and we either have  or , depending on whether or not  is empty.

Thus, , so  and  are sisters in

1.4. Internalizing

Definition: Given a partition  of , let  denote the Cartesian Frame , where  is given by . Let   denote the Cartesian Frame , where  is given by

We say "internalizing " for  and "internalizing mod " for .

Claim: For all partitions  of  and . Similarly, for all partitions  of  and

Proof: Trivial.

Claim: For all partitions  of  and .

Proof: This follows from the fact that  and , and the fact that multiplicative subagent is equivalent to multiplicative sub-environment.

## 2. Definitions from Worlds

The above definitions are dependent on subsets and partitions of a given  and , and thus do not represent a single operation that can be applied to an arbitrary Cartesian frame over . We will now use the above eight definitions to define another eight operations that instead use subsets and partitions of .

Once we have the following definitions in hand, our future references to committing, assuming, externalizing, and internalizing will use the definitions from worlds unless noted otherwise.

2.1. Committing

Definition: Given a set , we define  and , where  is given by .

Claim: For all  and . Further,  and  are brothers in .

Proof: Trivial.

Unlike before, it is not necessarily the case that . This is because there might be rows that contains both elements of  and elements of .

2.2. Assuming

Definition: Given , we define  and , where  is given by

Claim: For all  and  and .

Proof: Trivial.

Claim: For all  and .

Proof: Trivial.

2.3. Externalizing

Definition: Given a partition  of , let  send each element  to the part that contains it. We define  and , where .

Claim: For all partitions  of  and . Further,  and  are sisters in .

Proof: Trivial.

2.4. Internalizing

Definition: Given a partition  of , let  send each element  to the part that contains it. We define  and , where .

Claim: For all partitions  of  and

Proof: Trivial.

Claim: For all partitions  of , and

Proof: Trivial.

## 3. Basic Properties

3.1. Biextensional Equivalence

Committing and assuming are well-defined up to biextensional equivalence.

Claim: If  are Cartesian frames over , then for any subset ,, and .

Proof: Let , and let  and  compose to something homotopic to the identity in both orders. Let  be defined by .

Observe that if , then for all , so . Similarly, if , then . Thus, if we let  be given by , we get morphisms , which are clearly morphisms, since they are restrictions of our original morphisms .

Since  and  compose to something homotopic to the identity in both orders,  is a morphism, so  is a morphism, so  and  compose to something homotopic to the identity in both orders. Thus .

Similarly, if , then there exists an  such that . But then

, so . Similarly, if , then . Thus, if we let  be given by , we get morphisms , which (similarly to above) compose to something homotopic to the identity in both orders. Thus, .

We know that  and , because

and

Externalizing and internalizing are also well-defined up to biextensional equivalence.

Claim: If  are Cartesian frames over , then for all partitions  of , and .

Proof: Let , and let  and  compose to something homotopic to the identity in both orders. Let  be a partition of W, and let  send each element  to the part that contains it. Let  be the partition of  defined by

Let , send each element of  to its part in , so . Since  is surjective, it has a right inverse. Let  be any choice of right inverse to . This gives a pair of functions  given by .

We start by showing that  and  are inverses, and thus bijections between  and . We do this by showing that , and that  , and thus we will have

which is the identity on .

To see that , observe that for all , we have that for all , so, . Thus, .

To see that , first observe that for all , we have that , and thus, for all ,

Thus, . Thus, we have

This also gives us functions , by . To ehow that these functions are well-defined, we need to show that for all ,   is in fact in , by showing that for all ,  , or equivalently that  is the identity on . Since , we already have that  is the identity of . Thus, we have that

is the identity on .

We are now ready to demonstrate that .

Let , and define

by , while  is given by .

To see that  is a morphism, observe that for all , and  we have

To see that  is homotopic to the identity, we show that

is a morphism. Indeed, for all  and ,

Thus,

Similarly, let , and define

by , and  is given by .

To see that  is a morphism, observe that for all , and , we have

Clearly,  is homotopic to the identity, since  is the identity on . Thus, .

We know that  and , because

and

3.2. Committing and Assuming Can Be Defined Using Lollipop and Tensor

Claim:  and .

Proof: Let , and let

Let , where , and

Let