Cartesian Frames Definitions

post by Rob Bensinger (RobbBB) · 2020-11-08T12:44:34.509Z · LW · GW · 0 comments

Contents

  1. Small Cartesian Frames
  2. Binary Operations
  3. Frames, Morphisms, and Equivalence Relations
  4. Functors
  5. Subagents
    5.1. Additive and Multiplicative Subagents
    5.2. Ways to Construct Subagents, Sub-Environments, etc.
  6. Controllables and Observables
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This is a list of the main definitions from Scott Garrabrant's Cartesian Frames [? · GW] sequence. (I'll update it as more posts come out.)

 

1. Small Cartesian Frames

Let $W=\{w_0,w_1\}$ for the matrix visualizations below. Let $C$ be an arbitrary Cartesian frame.

	visualization	definition	notes
$0$	$\begin{array}{cc}& \begin{array}{c}e\end{array}\\ \begin{array}{c}\end{array}& \left(\begin{array}{c}\end{array}\right)\end{array}$	$0=\left(\right\{\},\{e\},\cdot )$, where $\text{Agent}\left(0\right)$ is empty, $\text{Env}\left(0\right)=\left\{e\right\}$ is any singleton set, and $\text{Eval}\left(0\right)$ is trivial.	${\perp }_{\left\{\right\}}$. Initial. Identity of sum ($\oplus$).
$\top$	$\begin{array}{cc}& \begin{array}{}\end{array}\\ \begin{array}{c}a\end{array}& \left(\begin{array}{c}\end{array}\right)\end{array}$	$\top =\left(\right\{a\},\{\},\cdot )$, where $\text{Agent}\left(\top \right)$ is any singleton set, $\text{Env}\left(\top \right)$ is empty, and $\text{Eval}\left(\top \right)$ is trivial.	$1_{\left\{\right\}}$. Terminal. $\top ◃C$. Identity of product ($\&$).
$1$	$\begin{array}{cc}& \begin{array}{cc}{w}_0& w_1\end{array}\\ \begin{array}{c}a\end{array}& \left(\begin{array}{cc}{w}_0& w_1\end{array}\right)\end{array}$	$1_S=\left(\right\{a\},S,\star )$, where $S\subseteq W$ and $a\star s=s$ for all $s\in S$. $1$ is the frame $1_W$.	Identity of tensor ($\otimes$).
$\perp$	$\begin{array}{cc}& \begin{array}{c}e\end{array}\\ \begin{array}{c}{w}_0\\ w_1\end{array}& \left(\begin{array}{c}{w}_0\\ w_1\end{array}\right)\end{array}$	${\perp }_S=(S,\{e\},\star )$, where $S\subseteq W$ and $s\star e=s$ for all $s\in S$. $\perp$ is the frame ${\perp }_W$.	$C◃\perp$. Identity of par ($⅋$).
$\text{null}$	$\begin{array}{cc}& \begin{array}{}\end{array}\\ \begin{array}{c}\end{array}& \left(\begin{array}{c}\end{array}\right)\end{array}$	$\text{null}=\left(\right\{\},\{\},\cdot )$, with empty agent, environment, and evaluation function.

 

2. Binary Operations

Sum. For Cartesian frames $C=(A,E,\cdot )$ and $D=(B,F,\star )$ over $W$, $C\oplus D$ is the Cartesian frame $(A\bigsqcup B,E\times F,\diamond )$, where $a\diamond (e,f)=a\cdot e$ if $a\in A$, and $a\diamond (e,f)=a\star f$ if $a\in B$.

Product. For Cartesian frames $C=(A,E,\cdot )$ and $D=(B,F,\star )$ over $W$, $C\&D$ is the Cartesian frame $(A\times B,E\bigsqcup F,\diamond )$, where $(a,b)\diamond e=a\cdot e$ if $e\in E$, and $(a,b)\diamond e=b\star e$ if $e\in F$.

Tensor. Let $C=(A,E,\cdot )$ and $D=(B,F,\star )$ be Cartesian frames over $W$. The tensor product of $C$ and $D$, written $C\otimes D$, is given by $C\otimes D=(A\times B,\text{hom}(C,D^{\ast }),\diamond )$, where $\text{hom}(C,D^{\ast })$ is the set of morphisms $(g,h):C\to D^{\ast }$ (i.e., the set of all pairs $(g:A\to F,h:B\to E)$ such that $b\star g\left(a\right)=a\cdot h\left(b\right)$ for all $a\in A$, $b\in B$), and $\diamond$ is given by $(a,b)\diamond (g,h)=b\star g\left(a\right)=a\cdot h\left(b\right)$.

Par. Let $C=(A,E,\cdot )$ and $D=(B,F,\star )$ be Cartesian frames over $W$. $C⅋D=\left(\text{hom}\right(C^{\ast },D),E\times F,\diamond )$, where $(g,h)\diamond (e,f)=g\left(e\right)\star f=h\left(f\right)\cdot e$.

Lollipop. Given two Cartesian frames over $W$, $C=(A,E,\cdot )$ and $D=(B,F,\star )$, we let $C⊸D$ denote the Cartesian frame $C⊸D=\left(\text{hom}\right(C,D),A\times F,\diamond )$, where $\diamond$ is given by $(g,h)\diamond (a,f)=g\left(a\right)\star f=a\cdot h\left(f\right)$.

 

3. Frames, Morphisms, and Equivalence Relations

Cartesian frame. A Cartesian frame $C$ over a set $W$ is a triple $(A,E,\cdot )$, where $A$ and $E$ are sets and $\cdot :A\times E\to W$. If $C=(A,E,\cdot )$ is a Cartesian frame over $W$, we say $\text{Agent}\left(C\right)=A$, $\text{Env}\left(C\right)=E$, $\text{World}\left(C\right)=W$, and $\text{Eval}\left(C\right)=\cdot$.

Environment subset. Given a Cartesian frame $C=(A,E,\cdot )$ over $W$, and a subset $S$ of $W$, let $E_S$ denote the subset $\{e\in E|\forall a\in A,e\cdot a\in S\}$.

Cartesian frame image. $\text{Image}\left(C\right)=\{w\in W|\exists a\in A,\exists e\in E\text{s.t.}a\cdot e=w\}$.

Chu category. $\text{Chu}\left(W\right)$ is the category whose objects are Cartesian frames over $W$, whose morphisms from $C=(A,E,\cdot )$ to $D=(B,F,\star )$ are pairs of functions $(g:A\to B,h:F\to E)$, such that $a\cdot h\left(f\right)=g\left(a\right)\star f$ for all $a\in A$ and $f\in F$, and whose composition of morphisms is given by $(g_1,h_1)\circ (g_0,h_0)=(g_1\circ g_0,h_0\circ h_1)$.

Isomorphism. A morphism $(g,h):C\to D$ is an isomorphism if both $g$ and $h$ are bijective. If there is an isomorphism between $C$ and $D$, we say $C\cong D$.

Homotopic. Two morphisms $(g_0,h_0),(g_1,h_1):C\to D$ with the same source and target are called homotopic if $(g_0,h_1)$ is also a morphism.

Homotopy equivalence / biextensional equivalence. $C$ is homotopy equivalent (or biextensionally equivalent) to $D$, written $C\simeq D$, if there exists a pair of morphisms $\varphi :C\to D$ and $\psi :D\to C$ such that $\psi \circ \varphi$ is homotopic to the identity on $C$ and $\varphi \circ \psi$ is homotopic to the identity on $D$.

Sub-sum. Let $C=(A,E,\cdot )$, and let $D=(B,F,\star )$. A sub-sum of C and D is a Cartesian frame of the form $(A\bigsqcup B,X,\diamond )$, where $X\subseteq \text{Env}(C\oplus D)$ and $\diamond$ is $\text{Eval}(C\oplus D)$ restricted to $(A\bigsqcup B)\times X$, such that $C\simeq (A,X,{\diamond }_C)$ and $D\simeq (B,X,{\diamond }_D)$, where ${\diamond }_C$ is $\diamond$ restricted to $A\times X$ and ${\diamond }_D$ is $\diamond$ restricted to $B\times X$. Let $C⊞D$ denote the set of all sub-sums of $C$ and $D$.

Sub-tensor. Let $C=(A,E,\cdot )$, and let $D=(B,F,\star )$. A sub-tensor of $C$ and $D$ is a Cartesian frame of the form $(A\times B,X,\bullet )$, where $X\subseteq \text{Env}(C\otimes D)$ and $\bullet$ is $\text{Eval}(C\otimes D)$ restricted to $(A\times B)\times X$, such that $C\simeq (A,B\times X,{\bullet }_C)$ and $D\simeq (B,A\times X,{\bullet }_D)$, where ${\bullet }_C$ and ${\bullet }_D$ are given by $a{\bullet }_C(b,x)=(a,b)\bullet x$ and $b{\bullet }_D(a,x)=(a,b)\bullet x$. Let $C⊠D$ denote the set of all sub-tensors of $C$ and $D$.

 

4. Functors

Functions between worlds. Given a Cartesian frame $C=(A,E,\cdot )$ over $W$, and a function $f:W\to V$, let $f^{\circ }\left(C\right)$ denote the Cartesian frame over $V$, $f^{\circ }\left(C\right)=(A,E,\star )$, where $a\star e=f(a\cdot e)$.

Dual. Let ${-}^{\ast }:\text{Chu}\left(W\right)\to \text{Chu}(W{)}^{\text{op}}$ be the functor given by $(A,E,\cdot {)}^{\ast }=(E,A,\star )$, where $e\star a=a\cdot e$, and $(g,h{)}^{\ast }=(h,g)$.

Functor (from functions between worlds). Given two sets $W$ and and $V$, and a function $p:W\to V$, let $p^{\circ }:\text{Chu}\left(W\right)\to \text{Chu}\left(V\right)$ denote the functor that sends the object $(A,E,\cdot )\in \text{Chu}\left(W\right)$ to the object $(A,E,\star )\in \text{Chu}\left(V\right)$, where $a\star e=p(a\cdot e)$, and sends the morphism $(g,h)$ to the morphism with the same underlying functions, $(g,h)$.

Functor (from Cartesian frames). Let $C=(V,E,\cdot )$ be a Cartesian frame over $W$, with $\text{Agent}\left(C\right)=V$. Then $C^{\circ }:\text{Chu}\left(V\right)\to \text{Chu}\left(W\right)$ is the functor that sends $(B,F,\star )$ to $(B,F\times E,\diamond )$, where $b\diamond (f,e)=(b\star f)\cdot e$, and sends the morphism $(g,h)$ to $(g,h^{\prime })$, where $h^{\prime }(f,e)=\left(h\right(f),e)$.

 

5. Subagents

Subagent (categorical definition). Let $C$ and $D$ be Cartesian frames over $W$. We say that $C$ is a subagent of $D$, written $C◃D$, if for every morphism $\varphi :C\to \perp$ there exists a pair of morphisms ${\varphi }_0:C\to D$ and ${\varphi }_1:D\to \perp$ such that $\varphi ={\varphi }_1\circ {\varphi }_0$.

Subagent (currying definition). Let $C$ and $D$ be Cartesian frames over $W$. We say that $C◃D$ if there exists a Cartesian frame $Z$ over $\text{Agent}\left(D\right)$ such that $C\simeq D^{\circ }\left(Z\right)$.

Subagent (covering definition). Let $C=(A,E,\cdot )$ and $D=(B,F,\star )$ be Cartesian frames over $W$. We say that $C◃D$ if for all $e\in E$, there exists an $f\in F$ and a $(g,h):C\to D$ such that $e=h\left(f\right)$.

Sub-environment. We say $C$ is a sub-environment of $D$, written $C{◃}^{\ast }D$, if $D^{\ast }◃C^{\ast }$.

 

5.1. Additive and Multiplicative Subagents

Additive subagent (sub-sum definition). $C$ is an additive subagent of $D$, written $C{◃}_{+}D$, if there exists a $C^{\prime }$ and a $D^{\prime }\simeq D$ with $D^{\prime }\in C⊞C^{\prime }$.

Additive subagent (brother definition). $C^{\prime }$ is called a brother to $C$ in $D$ if $D\simeq D^{\prime }$ for some $D^{\prime }\in C⊞C^{\prime }$. We say $C{◃}_{+}D$ if $C$ has a brother in $D$.

Additive subagent (committing definition). Given Cartesian frames $C$ and $D$ over $W$, we say $C{◃}_{+}D$ if there exist three sets $X$, $Y$, and $Z$, with $X\subseteq Y$, and a function $f:Y\times Z\to W$ such that $C\simeq (X,Z,\diamond )$ and $D\simeq (Y,Z,\bullet )$, where $\diamond$ and $\bullet$ are given by $x\diamond z=f(x,z)$ and $y\bullet z=f(y,z)$.

Additive subagent (currying definition). We say $C{◃}_{+}D$ if there exists a Cartesian frame $M$ over $\text{Agent}\left(D\right)$ with $|\text{Env}\left(M\right)|=1$, such that $C\simeq D^{\circ }\left(M\right)$.

Additive subagent (categorical definition). We say $C{◃}_{+}D$ if there exists a single morphism ${\varphi }_0:C\to D$ such that for every morphism $\varphi :C\to \perp$ there exists a morphism ${\varphi }_1:D\to \perp$ such that $\varphi$ is homotopic to ${\varphi }_1\circ {\varphi }_0$ .

Multiplicative subagent (sub-tensor definition). $C$ is a multiplicative subagent of $D$, written $C{◃}_{\times }D$, if there exists a $C^{\prime }$ and $D^{\prime }\simeq D$ with $D^{\prime }\in C⊠C^{\prime }$.

Multiplicative subagent (sister definition).  $C^{\prime }$ is called a sister to $C$ in $D$ if $D\simeq D^{\prime }$ for some $D^{\prime }\in C⊠C^{\prime }$. We say $C{◃}_{\times }D$ if $C$ has a sister in $D$.

Multiplicative subagent (externalizing definition). Given Cartesian frames $C$ and $D$ over $W$, we say $C{◃}_{\times }D$ if there exist three sets $X$, $Y$, and $Z$, and a function $f:X\times Y\times Z\to W$ such that $C\simeq (X,Y\times Z,\diamond )$ and $D\simeq (X\times Y,Z,\bullet )$, where $\diamond$ and $\bullet$ are given by $x\diamond (y,z)=f(x,y,z)$ and $(x,y)\bullet z=f(x,y,z)$.

Multiplicative subagent (currying definition). We say $C{◃}_{\times }D$ if there exists a Cartesian frame $M$ over $\text{Agent}\left(D\right)$ with $\text{Image}\left(M\right)=\text{Agent}\left(D\right)$, such that $C\simeq D^{\circ }\left(M\right)$.

Multiplicative subagent (categorical definition). We say $C{◃}_{\times }D$ if for every morphism $\varphi :C\to \perp$, there exist morphisms ${\varphi }_0:C\to D$ and ${\varphi }_1:D\to \perp$ such that $\varphi \cong {\varphi }_1\circ {\varphi }_0$, and for every morphism $\psi :1\to D$, there exist morphisms ${\psi }_0:1\to C$ and ${\psi }_1:C\to D$ such that $\psi \cong {\psi }_1\circ {\psi }_0$.

Multiplicative subagent (sub-environment definition). We say $C{◃}_{\times }D$ if $C◃D$ and $C{◃}^{\ast }D$. Equivalently, we say $C{◃}_{\times }D$ if $C◃D$ and $D^{\ast }◃C^{\ast }$.

Additive sub-environment. We say $C$ is an additive sub-environment of $D$, written $C{◃}_{+}^{\ast }D$, if $D^{\ast }{◃}_{+}{C}^{\ast }$.

Multiplicative sub-environment. We say $C$ is an multiplicative sub-environment of $D$, written $C{◃}_{\times }^{\ast }D$, if $D^{\ast }{◃}_{\times }{C}^{\ast }$.

 

5.2. Ways to Construct Subagents, Sub-Environments, etc.

Committing. Given a set $S\subseteq W$ and a frame $C=(A,E,\cdot )$ over $W$, we define ${\text{Commit}}_S\left(C\right)={\text{Commit}}^B\left(C\right)$ and ${\text{Commit}}_{\setminus S}\left(C\right)={\text{Commit}}^{\setminus B}\left(C\right)$, where $B\subseteq A$ is given by $B=\{a\in A|\forall e\in E,a\cdot e\in S\}$.

Assuming. Given a set $S\subseteq W$ and a frame $C=(A,E,\cdot )$ over $W$, we define ${\text{Assume}}_S\left(C\right)={\text{Assume}}^F\left(C\right)$ and ${\text{Assume}}_{\setminus S}\left(C\right)={\text{Assume}}^{\setminus F}\left(C\right)$, where $F\subseteq E$ is given by $F=\{e\in E|\forall a\in A,a\cdot e\in S\}.$

Externalizing. Given a partition $V$ of $W$, let $v:W\to V$ send each element $w\in W$ to the part that contains it. Given a frame $C=(A,E,\cdot )$ over $W$, we define ${\text{External}}_V\left(C\right)={\text{External}}^B\left(C\right)$ and ${\text{External}}_{/V}\left(C\right)={\text{External}}^{/B}\left(C\right)$, where $B=\left\{\right\{a^{\prime }\in A|\forall e\in E,v(a^{\prime }\cdot e)=v(a\cdot e)\}|a\in A\}$.

Internalizing. Given a partition $V$ of $W$, let $v:W\to V$ send each element $w\in W$ to the part that contains it. Given a frame $C=(A,E,\cdot )$ over $W$, we define ${\text{Internal}}_V\left(C\right)={\text{Internal}}^F\left(C\right)$ and ${\text{Internal}}_{/V}\left(C\right)={\text{Internal}}^{/F}\left(C\right)$, where $F=\left\{\right\{e^{\prime }\in E|\forall a\in a,v(a\cdot e^{\prime })=v(a\cdot e)\}|e\in E\}$.

 

6. Controllables and Observables

Ensurables (categorical definition). $\text{Ensure}\left(C\right)$ is the set of all $S\subseteq W$ such that there exists a morphism $\varphi :1_S\to C$.

Preventables (categorical definition). $\text{Prevent}\left(C\right)$ is the set of all $S\subseteq W$ such that there exists a morphism ${\varphi }_1:1_{W\setminus S}\to C$.

Controllables (categorical definition). Let $2_S$ denote the Cartesian frame $1_S\oplus 1_{W\setminus S}$. $\text{Ctrl}\left(C\right)$ is the set of all $S\subseteq W$ such that there exists a morphism $\varphi :2_S\to C$.

Observables (original categorical definition). $\text{Obs}\left(C\right)$ is the set of all $S\subseteq W$ such that there exist $C_0$ and $C_1$ with $\text{Image}\left(C_0\right)\subseteq S$ and $\text{Image}\left(C_1\right)\subseteq W\setminus S$ such that $C\simeq C_0\&C_1$.

Observables (definition from subsets). We say that a finite partition $V$ of $W$ is observable in a frame $C$ over $W$ if for all parts $S_i\in V$, $S_i\in \text{Obs}\left(C\right)$. We let ${\text{Obs}}^{\prime }\left(C\right)$ denote the set of all finite partitions of $W$ that are observable in $C$.

Observables (conditional policies definition): We say that a finite partition $V$ of $W$ is observable in a frame $C=(A,E,\cdot )$ over $W$ if for all functions $f:V\to A$, there exists an element $a_f\in A$ such that for all $e\in E$, $f\left(v\right(a_f\cdot e\left)\right)\cdot e=a_f\cdot e$, where  $v:W\to V$ is the function that sends each element of $W$ to its part in $V$.

Observables (non-constructive additive definition): We say that a finite partition $V=\{S_1,\dots ,S_n\}$ of $W$ is observable in a frame $C$ over $W$ if there exist frames $C_1,\cdots C_n$ over $W$, with $C_i◃{\perp }_{S_i}$ such that $C\simeq C_1\&\dots \&C_n$.

Observables (constructive additive definition): We say that a finite partition $V=\{S_1,\dots ,S_n\}$ of $W$ is observable in a frame $C$ over $W$ if $C\simeq {\text{Assume}}_{S_1}\left(C\right)\&\dots \&{\text{Assume}}_{S_n}\left(C\right)$.

Powerless outside of a subset: Given a frame $C=(A,E,\cdot )$ over $W$ and a subset $S$ of $W$, we say that $C$'s agent is powerless outside $S$ if for all $e\in E$ and all $a_0,a_1\in A$, if $a_0\cdot e\notin S$, then $a_0\cdot e=a_1\cdot e$.

Observables (non-constructive multiplicative definition): We say that a finite partition $V=\{S_1,\dots ,S_n\}$ of $W$ is observable in a frame $C$ over $W$ if $C\simeq C_1\otimes \cdots \otimes C_n$, where each $C_i$'s agent is powerless outside $S_i$.

Observables (constructive multiplicative definition):  We say that a finite partition $V=\{S_1,\dots ,S_n\}$ of $W$ is observable in a frame $C$ over $W$ if $C\simeq C_1\otimes \cdots \otimes C_n$, where $C_i={\text{Assume}}_{S_i}\left(C\right)\&1_{T_i}$, where $T_i=\left(W\setminus S_i\right)\cap \text{Image}\left(C\right)$.

Observables (non-constructive internalizing-externalizing definition): We say that a finite partition $V$ of $W$ is observable in a frame $C=(A,E,\cdot )$ over $W$ if either $A=\left\{\right\}$ or $C$ is biextensionally equivalent to something in the image of ${\text{External}}_V\circ {\text{Internal}}_V$.

Observables (constructive internalizing-externalizing definition): We say that a finite partition $V$ of $W$ is observable in a frame $C=(A,E,\cdot )$ over $W$ if either $A=\left\{\right\}$ or $C\simeq {\text{External}}_V\left({\text{Internal}}_V\right(C\left)\right)$.

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