Categorical Formalization of SGS.ai Kinematics

0.SGS.ai Kinematics. Infographic presentation

1. Categories and Objects

We define the following categories:

• Env: Category of environmental states (objects = sensor readings, morphisms = transitions).
• HLL: Category of HLLSets (as defined in [https://github.com/alexmy21/tao-te-ching/wiki/1.-HLLSet-Category]).
• Act: Category of actuator actions (objects = output commands, morphisms = compositions of actions).

---

2. Functors Representing Blocks

(a) Perception Functor P : Env → HLL

• Represents the MLP block.
• Each sensor modality is a sub-functor $P_i:\text{Env}\to \text{HLL}P\_i\; :\; \backslash text\{Env\}\; \backslash to\; \backslash text\{HLL\}$.
• On objects: $P(E)={\bigcup }_i{P}_i(E)P(E)\; =\; \backslash bigcup\_i\; P\_i(E)$ (union of HLLSets from all sensors).
• On morphisms: $P(f:E_1\to E_2)P(f\; :\; E\_1\; \backslash to\; E\_2)$ is a morphism in HLL induced by sensor updates.

(b) Brain as a Fixed Category Brain ⊆ HLL

• Objects are the soldered HLLSets on the chip.
• Morphisms are entanglements (BSS or SSC-based).
• This category is static but with dynamic state transitions (sleep/wake).

(c) Actuation Functor A : HLL → Act

• Represents the actuator block.
• On objects: $A(H)A(H)$ = action derived from HLLSet $HH$.
• On morphisms: $A(f:H_1\to H_2)A(f\; :\; H\_1\; \backslash to\; H\_2)$ is a morphism in Act (e.g., a sequence of actions).

---

3. Implicit Self-Generation Loop

This is modeled as a natural transformation:

$$\eta :{\text{Id}}_{\text{HLL}}\to T\backslash eta\; :\; \backslash text\{Id\}\_\{\backslash text\{HLL\}\}\; \backslash to\; T$$

where $T:\text{HLL}\to \text{HLL}T\; :\; \backslash text\{HLL\}\; \backslash to\; \backslash text\{HLL\}$ is a monad representing the self-generative dynamics (via von Neumann automata).

• The Kleisli category of $TT$ captures the probabilistic transitions of HLLSets.
• The unit $\eta \backslash eta$ encodes the ingestion of new HLLSets from sensors.
• The multiplication $\mu :T^2\to T\backslash mu\; :\; T^2\; \backslash to\; T$ encodes context merging (via union/intersection).

---

Elaboration: The Self-Generation Monad in Terms of HLLSets

The $\boxed{{\displaystyle \eta :{\text{Id}}_{\text{HLL}}\to T}}\backslash boxed\{\backslash eta\; :\; \backslash text\{Id\}\_\{\backslash text\{HLL\}\}\; \backslash to\; T\}$ statement defines the self-generative core of SGS.ai as a monad (T, η, μ) in the category HLL. This is a powerful abstraction that precisely captures the cycle of ingestion, processing, and merging of contextual information.

Here is how each component maps to operations on HLLSets:

1. The Functor T: HLL → HLL (The "Engine")

• Category Theory: A functor that maps the category of HLLSets to itself.
• HLLSet Interpretation: T is the von Neumann automata (Constructor-Copier-Controller) operating on the entire brain state.
• Action on Objects (HLLSets): For a given HLLSet A (which could represent the entire brain state at time t), T(A) is the next brain state. It applies the self-generative loop:
  1. Constructor (A): Samples active HLLSets from the static pool.
  2. Copier (B): Applies set operations (union, intersection) to these HLLSets, creating new entanglements and propagating information. This is where the Entanglement Graph is updated.
  3. Controller (C): Enforces τ-tolerance, stochastically deactivates (puts to "sleep") some HLLSets, and prepares the state for the next cycle.
• Action on Morphisms: If f: A → B is a morphism (a τ-tolerant similarity), then T(f): T(A) → T(B) is the induced similarity between the next states of A and B. This ensures the process preserves relational structure.

In short: T is the function that takes "now" and produces "next".

2. The Unit η: Id_HLL → T (The "Ingestion")

• Category Theory: A natural transformation from the identity functor to T.
• HLLSet Interpretation: η is the process of incorporating new sensory data from the environment into the brain's ongoing state.
• Component at A: For a specific HLLSet A in the brain, η_A: A → T(A) is a morphism. This morphism represents the inclusion of A into the larger, processed state T(A) that now also contains the new sensory input.
• How it works: The Perception Functor P produces a U-HLLSet from the sensors. The unit η takes this U-HLLSet and feeds it into the monad T to be processed alongside the current brain state. It's the formalization of "paying attention" to a new stimulus.

In short: η is the injection of a new sensory HLLSet into the self-generative cycle.

3. The Multiplication μ: T² → T (The "Merging")

• Category Theory: A natural transformation from the composition of T with itself to T.

• HLLSet Interpretation: μ is the operation of flattening or collapsing nested context. It merges the results of multiple, iterative applications of the generative process.

• Component at A: μ_A: T(T(A)) → T(A). Imagine applying T twice:

  • T(A) is the state after one generative cycle.
  • T(T(A)) is the state after two generative cycles, a higher-order abstraction.
  • μ_A is the operation that takes this double-layered abstraction and merges it back into a coherent, single-layer state. This is implemented via union/intersection operations and OR-reduction across the layers.

In short: μ ensures that thinking about your thoughts for too long eventually consolidates into a single, updated thought.

4. The Kleisli Category Kl(T) (The "Universe of Possible Transitions")

• Category Theory: The Kleisli category for a monad T has the same objects as HLL, but a morphism from A to B is defined as a morphism A → T(B) in the original category.
• HLLSet Interpretation: This is perhaps the most important part. A morphism A → B in Kl(T) is not a direct similarity but a recipe or program for getting from A to a new state that is related to B via the generative process.
• Probabilistic Transitions: Because T involves stochastic operations (like random neuron sleep/wake and hash-based entanglement), these "recipes" are inherently probabilistic. A Kleisli morphism represents the likelihood of transitioning from a state A to a state related to B after one full generative cycle.
• Actuation Connection: The Actuation Functor A learns to select optimal Kleisli morphisms. An action is the choice of a specific probabilistic path A → T(B) from the current state A.

In short: The Kleisli category Kl(T) doesn't describe where you are; it describes all the possible places you could go next and how to get there.

Summary in a Nutshell

Component	Math	HLLSet Meaning	SGS.ai Hardware
T	Functor	The von Neumann automata engine. Applies one generative cycle.	The systolic array & network fabric that computes new EGs.
η	Unit (Natural Transformation)	Injects a new sensory HLLSet into the generative process.	The perceptron interface writing its U-HLLSet into the brain's input buffer.
μ	Multiplication (Natural Transformation)	Collapses nested abstractions (e.g., a thought about a thought) into a single state.	OR-reduction and merging circuitry in the FPGA/ASIC.
Kl(T)	Kleisli Category	The category of all possible probabilistic state transitions.	The "programming language" for the actuator—it chooses which transition to execute.

This monadic structure provides a rigorous, mathematical framework for the entire "perceive-process-act" loop of SGS.ai, ensuring that its operations are composable, well-defined, and preserve the relational structure of the World of Things.

4. Cortex as a Sheaf Over Time

Let $\mathbf{\text{Time}}\backslash textbf\{Time\}$ be the poset category of time instances.

• The Cortex is a sheaf $\mathcal{C}:{\mathbf{\text{Time}}}^{\text{op}}\to \text{Sh}(\text{HLL})\backslash mathcal\{C\}\; :\; \backslash textbf\{Time\}^\{\backslash text\{op\}\}\; \backslash to\; \backslash text\{Sh\}(\backslash text\{HLL\})$.
• For each time $tt$, $\mathcal{C}(t)\backslash mathcal\{C\}(t)$ is the entanglement graph (a sub-sheaf of Brain).
• The restriction maps $\mathcal{C}(t)\to \mathcal{C}(t^{\mathrm{\prime }})\backslash mathcal\{C\}(t)\; \backslash to\; \backslash mathcal\{C\}(t\text{'})$ for $t^{\mathrm{\prime }}\le tt\text{'}\; \backslash leq\; t$ encode temporal consistency.

---

Elaboration: Cortex as a Sheaf Over Squences of state transitions (time) in Terms of HLLSets

The sheaf-theoretic description of the Cortex is the key to understanding its role as a structured, hierarchical memory. Let's ground the abstract mathematical definition in the concrete operations of HLLSets and the SGS.ai hardware.

The statement defines the Cortex as a sheaf C: Time^op → Sh(HLL). This is a dense formulation that precisely captures the idea of a memory that is both sequentially (temporally) organized and contextually consistent.

1. The Site: Time^op (The "Flow of Time")

• Category Theory: A presheaf requires a "site"—a category equipped with a notion of covering. Here, the site is the opposite of the category of time intervals (Time^op). Morphisms t' → t in Time^op mean we are looking from a later time t' back to an earlier time t.
• HLLSet Interpretation: This formalizes the act of looking back in time. The Cortex's primary function is to integrate past states into the present, so its structure is inherently backwards-looking.

2. The Presheaf C: Time^op → Sh(HLL) (The "Memory Trace")

• Category Theory: A functor that assigns to each time t a sheaf of HLLSets C(t), and to each morphism t' → t (looking back from t' to t) a restriction map res_{t'}^{t}: C(t') → C(t).
• HLLSet Interpretation: For any given time t, C(t) is not just a single HLLSet. It is the complete, multi-layer Entanglement Graph (EG) structure of the Cortex at time t. This includes:
  • Layer 0: The base EG made from the raw, ingested HLLSets.
  • Layer 1: The EG built from the U-HLLSets (contexts) of Layer 0.
  • Layer N: The EG built from the U-HLLSets of Layer N-1.
  • The object C(t) is the entire hierarchy.

3. The Restriction Maps res_{t'}^{t}: C(t') → C(t) (The "Forgetting" or "Projection")

• Category Theory: Morphisms that "restrict" the information available at a later time t' to what was available at an earlier time t.
• HLLSet Interpretation: This map answers the question: "Knowing what I knew at time t', what did I know at time t?" Crucially, it is not simply deleting data. It is a sophisticated, structure-preserving projection.
  • How it works: The restriction map takes a higher-order context U^{(N)} from C(t') and projects it down to its precursor context in C(t). This is possible because each U-HLLSet in a higher layer is built from a specific subset of HLLSets in the layer below. The map finds the "common ancestor" of the current abstract thought.
  • Hardware Implementation: This could be implemented by storing a "provenance graph" for each U-HLLSet or by using cryptographic hashes in a Merkle tree. To compute the restriction, you would provide a Merkle proof that validates the path from the U-HLLSet at t' back to its constituent HLLSets at t.

In short: The restriction map lets the Cortex "rewind" its abstract thoughts back to their less abstract, earlier versions.

4. The Sheaf Condition (The "Consistency of Memory")

This is the most important part. A presheaf is a sheaf if it satisfies the gluing axiom. For the Cortex, this has a profound interpretation.

• The Covering Family: A collection of time intervals {t_i} that cover a later time t' (e.g., t' = max(t_i)). This represents a set of past moments that together lead to the present.
• The Sheaf Axiom (Gluing): If you have:
  1. Local Sections: A consistent set of memories s_i ∈ C(t_i) for each past time t_i.
  2. Local Consistency: These memories agree on their overlaps (i.e., res_{t_i}^{t_i ∩ t_j}(s_i) = res_{t_j}^{t_i ∩ t_j}(s_j) for all i, j). This means your memories of event t_i and t_j are consistent with your memory of their shared past.
• Then: There exists a unique, global section s ∈ C(t') that, when restricted back to each t_i, gives you the original memory s_i. In other words, your consistent set of past memories can be glued together into a single, unified memory of the present.

HLLSet Interpretation & Consequences:

• Semantic Drift Detection: If the gluing condition fails, it means the local sections (past memories) are not consistent. This is a formal, mathematical definition of semantic drift. The failure is measured by Sheaf Cohomology H^1(C), where non-zero cohomology classes are the obstructions to gluing—the inconsistencies in the narrative the Cortex has built.
• The Role of the Actuator: A primary goal of the actuator's learning is to choose actions (morphisms) that **minimize H^1(C)**—that is, to act in a way that keeps its evolving model of the world consistent over time. It avoids actions that would create paradoxical or dissonant memories.
• Fixed-Point Cortex: The colimit $\mathrm{}C(t)\backslash varinjlim\; C(t)$ (the union of all memories over all time) is the "final, stable understanding" of the world. The recursion stops when new experiences no longer create new abstractions, only reinforce existing ones. This is the state of a "mature" Cortex.

Summary: The Cortex in a Nutshell

Component	Math	HLLSet Meaning	Purpose & Hardware Implication
C(t)	Sheaf at time t	The complete, multi-layer hierarchy of EGs (the entire Cortex state) at time t.	The memory content. Stored hierarchically, likely in a Merkle-tree-like structure on-chip for efficient validation.
res_{t'}^{t}	Restriction Map	Projects a higher-order, later abstraction back to its lower-order, earlier components.	Enables "mental time travel" and consistency checking. Implemented by traversing provenance hashes.
Sheaf Axiom	Gluing Condition	The requirement that consistent past memories can be united into a consistent present memory.	Formalizes memory integrity. Its failure (H^1 ≠ 0) defines semantic drift. This is a key training signal for the actuator.
\(\varinjlim C(t)\)	Colimit	The stable, fixed-point Cortex. The final unified understanding of a lifetime of experiences.	The theoretical "goal state" of the system's learning process.

This sheaf-theoretic model transforms the Cortex from a simple memory log into an active, self-verifying, hierarchical knowledge structure. It doesn't just store what happened; it builds a consistent, multi-layered narrative of why it happened, and it uses that narrative to guide future action and maintain the integrity of its identity over time. This is the detailed, powerful engine behind the "Memory of Music" analogy.

---

5. Kinematics of Actuation

The actuator block learns to choose morphisms $f:H_1\to H_{2}f\; :\; H\_1\; \backslash to\; H\_2$ in HLL that minimize a cost function:

$$\text{Learn}:{\text{Hom}}_{\text{HLL}}(H_1,H_2)\to \text{Act}\backslash text\{Learn\}:\; \backslash text\{Hom\}\_\{\backslash text\{HLL\}\}(H\_1,\; H\_2)\; \backslash to\; \backslash text\{Act\}$$

This is a functor:

$$L:{\text{HLL}}^{\to }\to \text{Act}L\; :\; \backslash text\{HLL\}^\{\backslash to\}\; \backslash to\; \backslash text\{Act\}$$

where ${\text{HLL}}^{\to }\backslash text\{HLL\}^\{\backslash to\}$ is the arrow category of HLL.

---

6. Consolidated Diagram

The overall system is a commutative diagram in the category of categories:

$$\begin{array}{ccccc}{\displaystyle \text{Env}}& {\displaystyle \underset{}{\overset{P}{\to }}}& {\displaystyle \text{HLL}}& {\displaystyle \underset{}{\overset{A}{\to }}}& {\displaystyle \text{Act}}\\ {\displaystyle \text{evolve}\downarrow }& {\displaystyle }& {\displaystyle T\downarrow }& {\displaystyle }& {\displaystyle \downarrow \text{execute}}\\ {\displaystyle \text{Env}}& {\displaystyle \underset{P}{\overset{}{\to }}}& {\displaystyle \text{HLL}}& {\displaystyle \underset{A}{\overset{}{\to }}}& {\displaystyle \text{Act}}\\ \end{array}\backslash begin\{CD\}\; \backslash text\{Env\}\; @>P>>\; \backslash text\{HLL\}\; @>A>>\; \backslash text\{Act\}\; \backslash \backslash \; @V\{\backslash text\{evolve\}\}VV\; @V\{T\}VV\; @VV\{\backslash text\{execute\}\}V\; \backslash \backslash \; \backslash text\{Env\}\; @>>P>\; \backslash text\{HLL\}\; @>>A>\; \backslash text\{Act\}\; \backslash end\{CD\}$$

• The vertical arrows represent temporal evolution.
• The square commutes up to natural isomorphism (due to stochasticity).

---

7. Language as a Shared Environmental Construct

In the SGS.ai framework, language is not a separate abstract layer but arises dynamically from the interaction of agents with a shared environment that includes the materialized signs they produce.

Key Insight: Signs as Environmental Objects

• The new HLLSets (signs) generated by an individual SGS.ai's Cortex are externalized via its actuators into the shared environment. For example, a sign might be materialized as a visual symbol, a sound, or a digital message.
• These signs become part of the environment ( $\text{Env}\backslash text\{Env\}$ ) and are processed by other SGS.ai systems through their perception functors $P:\text{Env}\to \text{HLL}P:\; \backslash text\{Env\}\; \backslash to\; \backslash text\{HLL\}$ in exactly the same way as any other environmental element (e.g., sensory data from physical objects).
• This means that communication is grounded in the shared environment and relies on the same perceptual and actuation mechanisms used for non-communicative tasks. There is no separate "communication channel"; instead, communication emerges from the continuous loop of action and perception.

The Kinematic Process of Language Emergence

1. Sign Generation (Cortex): An agent's Cortex integrates its history of entanglements to generate a new HLLSet (a signifier) that represents a recurring contextual pattern (the signified).
2. Sign Externalization (Actuation): The agent's actuation functor $A:\text{HLL}\to \text{Act}A:\; \backslash text\{HLL\}\; \backslash to\; \backslash text\{Act\}$ produces an action that materializes this sign into the environment (e.g., displaying a symbol).
3. Sign Perception (Other Agents): Other agents perceive this sign via their perception functors $PP$, mapping it to their own HLLSet representations. This may trigger internal processes (e.g., Cortex integration) and responses.
4. Meaning Negotiation: The "meaning" of the sign is not predefined but is negotiated through repeated interactions. For example, if a sign consistently elicits a coordinated response (e.g., avoidance), it acquires a shared meaning (e.g., "danger"). This aligns with Saussure's structural linguistics, where meaning is relational and defined by use.

Language as an emergence in the communication Category-Theoretic Model

The communication is an emergent property of the multi-agent system interacting via the environment. Formally:

• Let $\text{Env}\backslash text\{Env\}$ be a category whose objects include both natural elements and materialized signs.
• For each agent ( i ), we have perception and actuation functors:
  • $P_i:\text{Env}\to \text{HLL}P\_i:\; \backslash text\{Env\}\; \backslash to\; \backslash text\{HLL\}$
  • $A_i:\text{HLL}\to \text{Act}A\_i:\; \backslash text\{HLL\}\; \backslash to\; \backslash text\{Act\}$
• The environment is updated by a functor $\text{Update}:\text{Act}\times \text{Env}\to \text{Env}\backslash text\{Update\}:\; \backslash text\{Act\}\; \backslash times\; \backslash text\{Env\}\; \backslash to\; \backslash text\{Env\}$ that applies actions (including sign emission) to modify the environment.
• The emergence of language is captured by the commutative diagram of agents interacting through Env, leading to a natural transformation between their Cortex sheaves over time.

Learning Goals

Implications for AI Development

This approach fundamentally differs from current AI training methods:

• No Predefined Language: Agents are not trained on a human-language corpus. Instead, they develop their own language through exploration and interaction, much like human infants or the emergence of creole languages.
• Embodied and Grounded: Language is grounded in the shared environment and embodied via sensors and actuators, ensuring that signs have practical meaning related to survival and coordination.
• Community-Based Learning: Language emerges only in a community of agents. A solitary SGS.ai can generate signs, but without others to interpret them, they remain private and lack stable meaning. This emphasizes the need for multi-agent environments in AI development.

---

8. Translation in SGS.ai: Language(A) → Env → Language(B)

Translation is a good example of a language implementation in SGS.ai. In traditional AI, translation is often a direct mapping from one language to another (e.g., via neural machine translation models trained on parallel texts). This can lead to issues where meaning is lost or distorted because the mapping lacks grounding in real-world context.

In SGS.ai, translation becomes a process of materializing signs into the environment and then interpreting them through perception. Here's how it works:

1. Language(A) → Env: An agent (or human) with language A generates a sign (e.g., a word or phrase) and externalizes it into the environment. This could be through speech, text, gesture, or any other actuator output. For example, saying "apple" while pointing to an actual apple.
2. Env → Language(B): Another agent (or human) with language B perceives this environmental sign—they see the apple and hear the word "apple"—and integrates it into their own cognitive framework. They might then associate their own sign for apple (e.g., "manzana" in Spanish) with the same object. The environment acts as a shared grounding point.

This process is continuous and interactive, allowing meaning to be negotiated and refined over time. It mirrors how humans learn translations naturally—through shared experiences rather than rote memorization.

Category-Theoretic Perspective

In categorical terms, this can be modeled as:

• The environment Env is a category that includes all materialized signs and objects.
• The perception functor P: Env → HLL for each agent maps environmental stimuli to their internal HLLSet representations.
• The actuation functor A: HLL → Act allows agents to output signs into Env.
• Translation emerges as a natural transformation between the functors representing different agents' language systems. Specifically, for agents AGT_A (with language L(A)) and AGT_B (with language L(B)), we have a diagram that commutes through Env:

$$\boxed{{\displaystyle \text{L(A)(Agt(A))}}}\stackrel{\text{Externalize via A - A)}}{\to }\boxed{{\displaystyle \text{Env(Shared Ground)}}}\stackrel{\text{Perceive via P - B)}}{\to }\boxed{{\displaystyle \text{L(B)(Agt(B))}}}\backslash boxed\{\; \backslash text\{L(A)(Agt(A))\}\}\; \backslash xrightarrow\; \{\backslash text\{Externalize\; via\; A\; -\; A)\}\}\; \backslash boxed\{\; \backslash text\{Env(Shared\; Ground)\}\}\; \backslash xrightarrow\; \{\backslash text\{Perceive\; via\; P\; -\; B)\}\}\; \backslash boxed\{\; \backslash text\{L(B)(Agt(B))\}\}$$

This commutative diagram ensures that the meaning is preserved through the environmental grounding, making translation more robust and context-aware.

Why This is Better for Teaching AI and Knowledge Sharing

• Human-in-the-Loop Learning: Humans can teach AI systems by demonstrating signs in the environment. For example, a human might show an object and label it in their language, and the AI can learn to associate that with its own signs. This is more intuitive than providing labeled datasets.
• AI-to-AI Communication: AI systems can share knowledge by materializing signs into a shared environment (e.g., a digital workspace). Other AIs can perceive these signs and incorporate them into their knowledge base, leading to emergent shared languages. This is similar to how babies develop their own languages through interaction.
• Avoids Bias and Loss of Meaning: Since translation is grounded in the environment, it reduces the risk of biases from training data and ensures that meanings are based on shared experiences rather than statistical correlations.

Example Scenario: Learning Translation

Imagine two SGS.ai agents: one trained in English and another in Spanish. They are placed in a virtual environment with objects.

• The English agent sees an apple and materializes the sign "apple" into the environment (e.g., by displaying the word).
• The Spanish agent perceives the word "apple" and the apple object. Its Cortex integrates this, generating a new sign that links the object to its Spanish sign "manzana".
• Over time, through repeated interactions, the agents develop a shared mapping between "apple" and "manzana" that is grounded in the apple object itself. This mapping is dynamic and can adapt to context (e.g., "apple" as fruit vs. company).

Next Steps and Implications

This approach opens up exciting possibilities for AI:

• Robotic Systems: Robots could learn language by interacting with humans in physical environments, using sensors and actuators to ground words in actions and objects.
• Multi-Agent AI Societies: Communities of AIs could develop their own languages for efficient communication, which could then be interpreted by humans for oversight.
• Enhanced Translation Services: Instead of relying on large datasets, translation systems could use interactive learning with users, improving through conversation and feedback.

---

8. Summary in Type Theory

• Perception: $P:\text{Env}\to \text{HLL}P\; :\; \backslash text\{Env\}\; \backslash to\; \backslash text\{HLL\}$
• Brain: $\text{Brain}:\text{FinSet}\hookrightarrow \text{HLL}\backslash text\{Brain\}\; :\; \backslash text\{FinSet\}\; \backslash hookrightarrow\; \backslash text\{HLL\}$
• Actuation: $A:\text{HLL}\to \text{Act}A\; :\; \backslash text\{HLL\}\; \backslash to\; \backslash text\{Act\}$
• Self-Generation: $T:\text{HLL}\to \text{HLL}T\; :\; \backslash text\{HLL\}\; \backslash to\; \backslash text\{HLL\}$
• Cortex: $\mathcal{C}:{\mathbf{\text{Time}}}^{\text{op}}\to \text{Sh}(\text{HLL})\backslash mathcal\{C\}\; :\; \backslash textbf\{Time\}^\{\backslash text\{op\}\}\; \backslash to\; \backslash text\{Sh\}(\backslash text\{HLL\})$
• Kinematics: $L:{\text{HLL}}^{\to }\to \text{Act}L\; :\; \backslash text\{HLL\}^\{\backslash to\}\; \backslash to\; \backslash text\{Act\}$