asi-polynomial-operads

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ASI Polynomial Operads Skill

ASI多项式操作子技能

"Pattern runs on matter: The free monad monad as a module over the cofree comonad comonad" — Libkind & Spivak (ACT 2024)

"模式作用于物质：自由单子单子作为余自由余单子余单子上的模块" — Libkind & Spivak（ACT 2024）

1. Polynomial Functors (Spivak)

1. 多项式函子（Spivak）

Core Definition

核心定义

A polynomial functor $p: \text{Set} \to \text{Set}$ is a sum of representables:

$$p \cong \sum_{i \in p(1)} y^{p[i]}$$

Where:

$p(1)$ = set of positions (questions, observations)
$p[i]$ = set of directions at position $i$ (answers, actions)

多项式函子 $p: \text{Set} \to \text{Set}$ 是可表函子的和：

$$p \cong \sum_{i \in p(1)} y^{p[i]}$$

其中：

$p(1)$ = 位置集合（问题、观测）
$p[i]$ = 位置$i$处的方向集合（答案、动作）

Morphisms (Dependent Lenses)

态射（依赖透镜）

A lens $f: p \to q$ is a pair $(f_1, f^\sharp)$:

$$f_1: p(1) \to q(1) \quad \text{(on-positions)}$$ $$f^\sharp_i: q[f_1(i)] \to p[i] \quad \text{(on-directions, contravariant)}$$

透镜 $f: p \to q$ 是一对映射 $(f_1, f^\sharp)$：

$$f_1: p(1) \to q(1) \quad \text{(位置映射)}$$ $$f^\sharp_i: q[f_1(i)] \to p[i] \quad \text{(方向映射，反变)}$$

Hom-set Formula

同态集公式

$$\text{Poly}(p, q) \cong \prod_{i \in p(1)} \sum_{j \in q(1)} p[i]^{q[j]}$$

2. Composition Products

2. 组合积

Substitution ($\triangleleft$) — The Module Action

替换（$\triangleleft$）——模块作用

$$p \triangleleft q \cong \sum_{i \in p(1)} \sum_{\bar{j}: p[i] \to q(1)} y^{\sum_{a \in p[i]} q[\bar{j}(a)]}$$

Interpretation: Substitute $q$ into each "hole" of $p$.

$$p \triangleleft q \cong \sum_{i \in p(1)} \sum_{\bar{j}: p[i] \to q(1)} y^{\sum_{a \in p[i]} q[\bar{j}(a)]}$$

解释： 将$q$代入$p$的每个“空位”中。

Parallel/Dirichlet ($\otimes$)

并行/狄利克雷积（$\otimes$）

$$p \otimes q \cong \sum_{i \in p(1)} \sum_{j \in q(1)} y^{p[i] \times q[j]}$$

Interpretation: Independent parallel execution.

$$p \otimes q \cong \sum_{i \in p(1)} \sum_{j \in q(1)} y^{p[i] \times q[j]}$$

解释： 独立并行执行。

Categorical Product ($\times$)

范畴积（$\times$）

$$p \times q \cong \sum_{i \in p(1)} \sum_{j \in q(1)} y^{p[i] + q[j]}$$

3. Free Monad & Cofree Comonad

3. 自由单子与余自由余单子

Cofree Comonad as Limit

余自由余单子作为极限

The carrier $t_p$ of the cofree comonoid on $p$:

$$t_p = \lim \left( 1 \xleftarrow{!} p \triangleleft 1 \xleftarrow{p \triangleleft !} p^{\triangleleft 2} \triangleleft 1 \leftarrow \cdots \right)$$

基于$p$的余自由余幺半群的载体$t_p$：

$$t_p = \lim \left( 1 \xleftarrow{!} p \triangleleft 1 \xleftarrow{p \triangleleft !} p^{\triangleleft 2} \triangleleft 1 \leftarrow \cdots \right)$$

Trees as Positions

树作为位置

$$t_p \cong \sum_{T \in \text{tree}_p} y^{\text{vtx}(T)}$$

$\text{tree}_p$ = set of $p$-trees (possibly infinite)
$\text{vtx}(T)$ = vertices (rooted paths) of tree $T$

$$t_p \cong \sum_{T \in \text{tree}_p} y^{\text{vtx}(T)}$$

$\text{tree}_p$ = $p$-树集合（可能为无限树）
$\text{vtx}(T)$ = 树$T$的顶点（有根路径）

Comonoid Structure

余幺半群结构

Counit (Extract): $\epsilon_p: t_p \to y$ — picks the root
Comultiplication (Duplicate): $\delta_p: t_p \to t_p \triangleleft t_p$ — path concatenation

余单位（提取）： $\epsilon_p: t_p \to y$ —— 选取根节点
余乘法（复制）： $\delta_p: t_p \to t_p \triangleleft t_p$ —— 路径拼接

Module Action: Pattern Runs On Matter

模块作用：模式作用于物质

$$\Xi_{p,q} : \mathfrak{m}p \otimes \mathfrak{c}q \to \mathfrak{m}(p \otimes q)$$

Where:

$\mathfrak{m}p$ = free monad (Pattern, decision trees, wellfounded)
$\mathfrak{c}q$ = cofree comonad (Matter, behavior trees, non-wellfounded)

Examples:

Pattern	Matter	Runs On
Interview script	Person	Interview
Program	OS	Execution
Voting scheme	Voters	Election
Game rules	Players	Game
Musical score	Performer	Performance

$$\Xi_{p,q} : \mathfrak{m}p \otimes \mathfrak{c}q \to \mathfrak{m}(p \otimes q)$$

其中：

$\mathfrak{m}p$ = 自由单子（模式、决策树、良基）
$\mathfrak{c}q$ = 余自由余单子（物质、行为树、非良基）

示例：

模式	物质	作用场景
面试脚本	面试者	面试
程序	操作系统	执行
投票方案	选民	选举
游戏规则	玩家	游戏
乐谱	演奏者	演出

4. Dynamical Systems (Libkind-Spivak)

4. 动力系统（Libkind-Spivak）

Discrete Dynamical System

离散动力系统

$$f^{upd}: A \times S \to S \quad \text{(update)}$$ $$f^{rdt}: S \to B \quad \text{(readout)}$$

$$f^{upd}: A \times S \to S \quad \text{(更新)}$$ $$f^{rdt}: S \to B \quad \text{(读出)}$$

Continuous Dynamical System

连续动力系统

$$f^{dyn}: A \times S \to TS \quad \text{(dynamics: } \dot{s} = f^{dyn}(a, s) \text{)}$$ $$f^{rdt}: S \to B \quad \text{(readout)}$$

$$f^{dyn}: A \times S \to TS \quad \text{(动力学: } \dot{s} = f^{dyn}(a, s) \text{)}$$ $$f^{rdt}: S \to B \quad \text{(读出)}$$

Wiring Diagram Composition

连线图组合

For $\phi: X \to Y$: $$\phi^{in}: X^{in} \to X^{out} + Y^{in}$$ $$\phi^{out}: Y^{out} \to X^{out}$$

对于$\phi: X \to Y$： $$\phi^{in}: X^{in} \to X^{out} + Y^{in}$$ $$\phi^{out}: Y^{out} \to X^{out}$$

Composed Update

组合更新

$$\bar{f}^{upd}(y, s) := f^{upd}(\phi^{in}(y, f^{rdt}(s)), s)$$ $$\bar{f}^{rdt}(s) := \phi^{out}(f^{rdt}(s))$$

5. Compositional Algorithms (Bumpus)

5. 组合式算法（Bumpus）

Structured Decomposition

结构化分解

$$d: \int G \to \mathbf{K}$$

Where $\int G$ is the Grothendieck construction.

$$d: \int G \to \mathbf{K}$$

其中$\int G$是Grothendieck构造。

Complexity Bound

复杂度边界

$$O\left(\max_{x \in VG} \alpha(dx) + \kappa^{|S|} \kappa^2\right) |EG|$$

Where:

$G$ = shape graph of decomposition
$S$ = feedback vertex set
$\kappa$ = max local solution space size
$\alpha(c)$ = time to compute sheaf on object $c$

$$O\left(\max_{x \in VG} \alpha(dx) + \kappa^{|S|} \kappa^2\right) |EG|$$

其中：

$G$ = 分解的形状图
$S$ = 反馈顶点集
$\kappa$ = 最大局部解空间大小
$\alpha(c)$ = 在对象$c$上计算层的时间

Tree-Shaped Bound

树形边界

For tree-shaped decompositions ($|S| = 0$): $$O(\kappa^2) |EG|$$

对于树形分解（$|S| = 0$）： $$O(\kappa^2) |EG|$$

6. Cohomological Obstructions (Bumpus)

6. 上同调障碍（Bumpus）

Čech Cohomology

切赫上同调

$$H^n(X, \mathcal{U}, F) := \ker(\delta^n) / \text{im}(\delta^{n-1})$$

Global Existence Constraint

全局存在性约束

$$FX \neq \emptyset \iff H^0(X, \mathfrak{M}F) = 0$$

Interpretation: A problem has a solution iff the zeroth cohomology of its model-collecting presheaf is trivial.

$$FX \neq \emptyset \iff H^0(X, \mathfrak{M}F) = 0$$

解释： 问题有解当且仅当其模型收集预层的0阶上同调是平凡的。

GF(3) Connection

GF(3)关联

While Bumpus uses $\mathbb{Z}[S]$ (free Abelianization), the methods generalize to:

$\text{Vect}(\mathbb{F}_3)$ — vector spaces over GF(3)
Balanced ternary conservation = cohomological constraint

尽管Bumpus使用$\mathbb{Z}[S]$（自由阿贝尔化），该方法可推广至：

$\text{Vect}(\mathbb{F}_3)$ —— GF(3)上的向量空间
平衡三进制守恒 = 上同调约束

7. Spined Categories (Bumpus)

7. 带刺范畴（Bumpus）

Definition

定义

A spined category $(\mathcal{C}, \Omega, \mathfrak{P})$:

$\Omega: \mathbb{N}_{=} \to \mathcal{C}$ — the spine functor
$\mathfrak{P}$ — proxy pushout operation

带刺范畴$(\mathcal{C}, \Omega, \mathfrak{P})$：

$\Omega: \mathbb{N}_{=} \to \mathcal{C}$ —— 刺函子
$\mathfrak{P}$ —— 代理推出操作

Proxy Pushout

代理推出

For span $G \xleftarrow{g} \Omega_n \xrightarrow{h} H$: $$G \xrightarrow{\mathfrak{P}(g,h)_g} \mathfrak{P}(g,h) \xleftarrow{\mathfrak{P}(g,h)_h} H$$

对于跨度$G \xleftarrow{g} \Omega_n \xrightarrow{h} H$： $$G \xrightarrow{\mathfrak{P}(g,h)_g} \mathfrak{P}(g,h) \xleftarrow{\mathfrak{P}(g,h)_h} H$$

Chordal Objects (Recursive)

弦对象（递归定义）

Smallest set $S$ where:

$\Omega_n \in S$ for all $n$
$\mathfrak{P}(a,b) \in S$ for $A, B \in S$ and arrows to $\Omega_n$

最小集合$S$满足：

对所有$n$，$\Omega_n \in S$
若$A, B \in S$且存在指向$\Omega_n$的态射，则$\mathfrak{P}(a,b) \in S$

Width/Triangulation

宽度/三角化

$$\Delta[X] = \min { \text{width}(\delta) \mid \delta: X \hookrightarrow H \text{ pseudo-chordal}}$$

$$\Delta[X] = \min { \text{width}(\delta) \mid \delta: X \hookrightarrow H \text{ 伪弦} }$$

8. Open Games (Hedges)

8. 开放博弈（Hedges）

Parametrised Lens (Arena)

参数化透镜（竞技场）

ParaLens p q x s y r = (get, put)
  get : p → x → y        -- forward
  put : p → x → r → (s, q)  -- backward

The 6 wires:

```
x
```
= observed states (from past)
```
y
```
= output states (to future)
```
r
```
= utilities received (from future)
```
s
```
= back-propagated utilities (to past)
```
p
```
= strategies (parameters)
```
q
```
= rewards (co-parameters)

ParaLens p q x s y r = (get, put)
  get : p → x → y        -- forward
  put : p → x → r → (s, q)  -- backward

6条连线：

```
x
```
= 观测状态（来自过去）
```
y
```
= 输出状态（去往未来）
```
r
```
= 接收的效用（来自未来）
```
s
```
= 反向传播的效用（去往过去）
```
p
```
= 策略（参数）
```
q
```
= 奖励（余参数）

Sequential Composition

顺序组合

haskell

(MkLens get put) >>>> (MkLens get' put') =
  MkLens
    (\(p, p') x -> get' p' (get p x))           -- compose forward
    (\(p, p') x t ->
      let (r, q') = put' p' (get p x) t         -- future first
          (s, q) = put p x r                     -- then past
      in (s, (q, q')))

Key insight: Backward pass = constraint propagation / abduction.

haskell

(MkLens get put) >>>> (MkLens get' put') =
  MkLens
    (\(p, p') x -> get' p' (get p x))           -- compose forward
    (\(p, p') x t ->
      let (r, q') = put' p' (get p x) t         -- future first
          (s, q) = put p x r                     -- then past
      in (s, (q, q')))

核心洞见： 反向传播 = 约束传播 / 溯因推理。

Equilibrium

均衡

$$E_G(x, k) := \varepsilon_G(x; A_G; k)$$

Where $\varepsilon = \bigotimes_{p \in P} \varepsilon_p$ is the joint selection function.

$$E_G(x, k) := \varepsilon_G(x; A_G; k)$$

其中$\varepsilon = \bigotimes_{p \in P} \varepsilon_p$是联合选择函数。

9. Integration: DiscoHy Operads

9. 整合：DiscoHy操作子

The 7 Operad Network

7操作子网络

Operad	Trit	Description
Little Disks (E₂)	+1	Non-overlapping disk configurations
Cubes (E_∞)	-1	Infinite-dimensional parallelism
Cactus	-1	Trees with cycles (self-modification)
Thread	0	Linear continuations + DuckDB
Gravity	-1	Moduli M_{0,n} with involutions
Modular	+1	All genera, runtime polymorphism
Swiss-Cheese	+1	Open/closed for forward-only learning

GF(3) Total: $(+1) + (-1) + (-1) + (0) + (-1) + (+1) + (+1) = 0$ ✓

操作子	三元值	描述
Little Disks (E₂)	+1	不重叠圆盘配置
Cubes (E_∞)	-1	无限维并行性
Cactus	-1	带环的树（自修改）
Thread	0	线性延续 + DuckDB
Gravity	-1	带对合的模空间M_{0,n}
Modular	+1	所有亏格，运行时多态
Swiss-Cheese	+1	面向前向学习的开放/封闭结构

GF(3)总和： $(+1) + (-1) + (-1) + (0) + (-1) + (+1) + (+1) = 0$ ✓

Libkind-Spivak Dynamical Operads

Libkind-Spivak动力操作子

Operad	Trit	Type
Directed (⊳)	+1	Output → Input wiring
Undirected (○)	-1	Interface matching via pullback
Machines	0	State machines with dynamics
Dynamical	+1	Open ODEs

操作子	三元值	类型
Directed (⊳)	+1	输出→输入连线
Undirected (○)	-1	通过拉回实现接口匹配
Machines	0	带动力学的状态机
Dynamical	+1	开放常微分方程

10. General Intelligence Requirements (Swan/Hedges)

10. 通用智能需求（Swan/Hedges）

From "Road to General Intelligence":

来自《通用智能之路》：

Value Proposition

价值主张

General intelligence must:

Perform work on command — respond to dynamic goal changes
Scale to real-world concerns
Respect safety constraints
Be explainable and auditable

通用智能必须：

按需执行任务 —— 响应动态目标变化
扩展至现实场景
遵守安全约束
可解释、可审计

Structural Causal Model

结构因果模型

$$X_i = f_i(\text{PA}_i, U_i), \quad i = 1, \ldots, n$$

Where:

$\text{PA}_i$ = parent nodes
$U_i$ = exogenous noise (jointly independent)

$$X_i = f_i(\text{PA}_i, U_i), \quad i = 1, \ldots, n$$

其中：

$\text{PA}_i$ = 父节点
$U_i$ = 外生噪声（联合独立）

Ladder of Causality

因果阶梯

Observational — statistical learning
Interventional — setting variables despite natural processes
Counterfactual — inferences from alternate histories

观测层 —— 统计学习
干预层 —— 突破自然过程设置变量
反事实层 —— 基于替代历史的推理

Lens-Based Abduction

透镜式溯因

Component	Role
get (forward)	Induction / forward inference
put (backward)	Abduction / constraint propagation
Selection function	Attention mechanism
Equilibrium checking	Reflective reasoning

组件	作用
get（正向）	归纳 / 正向推理
put（反向）	溯因 / 约束传播
选择函数	注意力机制
均衡检查	反思性推理

11. Commands

11. 命令

bash

undefined

bash

undefined

Run polynomial functor demo

just poly-functor-demo

Test free monad / cofree comonad pairing

just monad-test

Run DiscoHy operads

python3 src/operads/relational_operad_interleave.py

Run Libkind-Spivak dynamical systems

python3 src/operads/libkind_spivak_dynamics.py

Check GF(3) conservation

just gf3-verify

undefined

just gf3-verify

undefined

12. File Locations

12. 文件位置

lib/
├── free_monad.rb              # Pattern (decision trees)
├── cofree_comonad.rb          # Matter (behavior trees)
├── runs_on.rb                 # Module action implementation
└── discohy.hy                 # Hy operad implementations

src/music_topos/
├── free_monad.clj             # Clojure Pattern
├── cofree_comonad.clj         # Clojure Matter
├── runs_on.clj                # Module action
└── operads/
    ├── relational_operad_interleave.py
    ├── libkind_spivak_dynamics.py
    └── infinity_operads.py

scripts/
├── discohy_operad_1_little_disks.py
├── discohy_operad_2_cubes.py
├── discohy_operad_3_cactus.py
├── discohy_operad_4_thread.py
├── discohy_operad_5_gravity.lisp
├── discohy_operad_6_modular.bb
└── discohy_operad_7_swiss_cheese.py

lib/
├── free_monad.rb              # Pattern (decision trees)
├── cofree_comonad.rb          # Matter (behavior trees)
├── runs_on.rb                 # Module action implementation
└── discohy.hy                 # Hy operad implementations

src/music_topos/
├── free_monad.clj             # Clojure Pattern
├── cofree_comonad.clj         # Clojure Matter
├── runs_on.clj                # Module action
└── operads/
    ├── relational_operad_interleave.py
    ├── libkind_spivak_dynamics.py
    └── infinity_operads.py

scripts/
├── discohy_operad_1_little_disks.py
├── discohy_operad_2_cubes.py
├── discohy_operad_3_cactus.py
├── discohy_operad_4_thread.py
├── discohy_operad_5_gravity.lisp
├── discohy_operad_6_modular.bb
└── discohy_operad_7_swiss_cheese.py

13. References

13. 参考文献

Spivak, D.I. — Polynomial Functors: A General Theory of Interaction (2022)
Libkind, S. & Spivak, D.I. — Pattern Runs on Matter (ACT 2024)
Spivak, D.I. — Dynamical Systems and Sheaves (2019)
Bumpus, B.M. — Compositional Algorithms on Compositional Data (2024)
Bumpus, B.M. — Spined Categories (2023)
Bumpus, B.M. — Cohomology Obstructions (2024)
Swan, J. & Hedges, J. et al. — The Road to General Intelligence (Springer 2022)
Hedges, J. — Open Games with Agency (2023)

Spivak, D.I. — Polynomial Functors: A General Theory of Interaction (2022)
Libkind, S. & Spivak, D.I. — Pattern Runs on Matter (ACT 2024)
Spivak, D.I. — Dynamical Systems and Sheaves (2019)
Bumpus, B.M. — Compositional Algorithms on Compositional Data (2024)
Bumpus, B.M. — Spined Categories (2023)
Bumpus, B.M. — Cohomology Obstructions (2024)
Swan, J. & Hedges, J. et al. — The Road to General Intelligence (Springer 2022)
Hedges, J. — Open Games with Agency (2023)

14. See Also

14. 相关链接

```
acsets
```
— Algebraic databases (schema category)
```
discohy-streams
```
— 7 operad variants with GF(3) balance
```
triad-interleave
```
— Balanced ternary scheduling
```
world-hopping
```
— Badiou triangle navigation
```
open-games
```
— Bidirectional transformations

```
acsets
```
— 代数数据库（模式范畴）
```
discohy-streams
```
— 符合GF(3)平衡的7种操作子变体
```
triad-interleave
```
— 平衡三进制调度
```
world-hopping
```
— Badiou三角导航
```
open-games
```
— 双向变换