This is the first post in a sequence on Cartesian frames, a new way of modeling agency that has recently shaped my thinking a lot.
Traditional models of agency have some problems, like:
They treat the "agent" and "environment" as primitives with a simple, stable input-output relation. (See "Embedded Agency.")
They assume a particular way of carving up the world into variables, and don't allow for switching between different carvings or different levels of description.
Cartesian frames are a way to add a first-person perspective (with choices, uncertainty, etc.) on top of a third-person "here is the set of all possible worlds," in such a way that many of these problems either disappear or become easier to address.
The idea of Cartesian frames is that we take as our basic building block a binary function which combines a choice from the agent with a choice from the environment to produce a world history.
We don't think of the agent as having inputs and outputs, and we don't assume that the agent is an object persisting over time. Instead, we only think about a set of possible choices of the agent, a set of possible environments, and a function that encodes what happens when we combine these two.
This basic object is called a Cartesian frame. As with dualistic agents, we are given a way to separate out an “agent” from an “environment." But rather than being a basic feature of the world, this is a “frame” — a particular way of conceptually carving up the world.
We will use the combinatorial properties of a given Cartesian frame to derive versions of inputs, outputs and time. One goal here is that by making these notions derived rather than basic, we can make them more amenable to approximation and thus less dependent on exactly how one draws the Cartesian boundary. Cartesian frames also make it much more natural to think about the world at multiple levels of description, and to model agents as having subagents.
Mathematically, Cartesian frames are exactly Chu spaces. I give them a new name because of my specific interpretation about agency, which also highlights different mathematical questions.
Using Chu spaces, we can express many different relationships between Cartesian frames. For example, given two agents, we could talk about their sum (⊕), which can choose from any of the choices available to either agent, or we could talk about their tensor (⊗), which can accomplish anything that the two agents could accomplish together as a team.
Cartesian frames also have duals (−∗) which you can get by swapping the agent with the environment, and ⊕ and ⊗ have De Morgan duals (& and ⅋ respectively), which represent taking a sum or tensor of the environments. The category also has an internal hom, ⊸, where C⊸D can be thought of as "D with a C-shaped hole in it." These operations are very directly analogous to those used in linear logic.
1. Definition
Let W be a set of possible worlds. A Cartesian frame C over W is a triple C=(A,E,⋅), where A represents a set of possible ways the agent can be, E represents a set of possible ways the environment can be, and ⋅:A×E→W is an evaluation function that returns a possible world given an element of A and an element of E.
We will refer to A as the agent, the elements of A as possible agents, E as the environment, the elements of E as possible environments, W as the world, and elements of W as possible worlds.
This is the first post in a sequence on Cartesian frames, a new way of modeling agency that has recently shaped my thinking a lot.
Traditional models of agency have some problems, like:
Cartesian frames are a way to add a first-person perspective (with choices, uncertainty, etc.) on top of a third-person "here is the set of all possible worlds," in such a way that many of these problems either disappear or become easier to address.
The idea of Cartesian frames is that we take as our basic building block a binary function which combines a choice from the agent with a choice from the environment to produce a world history.
We don't think of the agent as having inputs and outputs, and we don't assume that the agent is an object persisting over time. Instead, we only think about a set of possible choices of the agent, a set of possible environments, and a function that encodes what happens when we combine these two.
This basic object is called a Cartesian frame. As with dualistic agents, we are given a way to separate out an “agent” from an “environment." But rather than being a basic feature of the world, this is a “frame” — a particular way of conceptually carving up the world.
We will use the combinatorial properties of a given Cartesian frame to derive versions of inputs, outputs and time. One goal here is that by making these notions derived rather than basic, we can make them more amenable to approximation and thus less dependent on exactly how one draws the Cartesian boundary. Cartesian frames also make it much more natural to think about the world at multiple levels of description, and to model agents as having subagents.
Mathematically, Cartesian frames are exactly Chu spaces. I give them a new name because of my specific interpretation about agency, which also highlights different mathematical questions.
Using Chu spaces, we can express many different relationships between Cartesian frames. For example, given two agents, we could talk about their sum (⊕), which can choose from any of the choices available to either agent, or we could talk about their tensor (⊗), which can accomplish anything that the two agents could accomplish together as a team.
Cartesian frames also have duals (−∗) which you can get by swapping the agent with the environment, and ⊕ and ⊗ have De Morgan duals (& and ⅋ respectively), which represent taking a sum or tensor of the environments. The category also has an internal hom, ⊸, where C⊸D can be thought of as "D with a C-shaped hole in it." These operations are very directly analogous to those used in linear logic.
1. Definition
Let W be a set of possible worlds. A Cartesian frame C over W is a triple C=(A,E,⋅), where A represents a set of possible ways the agent can be, E represents a set of possible ways the environment can be, and ⋅:A×E→W is an evaluation function that returns a possible world given an element of A and an element of E.
We will refer to A as the agent, the elements of A as possible agents, E as the environment, the elements of E as possible environments, W as the world, and elements of W as possible worlds.
Definition: A Cartesian frame