ActPC-Chem: Adaptive Algorithmic Chemistry

• ActPC-Chem is a computational framework that employs discrete active predictive coding and metagraph rewrite rules to drive goal-guided, adaptive behavior in AI systems.
• It utilizes error-driven, reward-modulated updates to continuously refine rule patterns, balancing instrumental and epistemic rewards for robust prediction.
• The framework integrates symbolic, subsymbolic, and probabilistic reasoning to support complex applications such as transformer-like sequence modeling and computational chemistry.

ActPC-Chem is a computational framework designed for goal-guided artificial intelligence founded on Discrete Active Predictive Coding (ActPC) operating over an algorithmic chemistry of metagraph rewrite rules. It integrates symbolic, subsymbolic, and probabilistic reasoning, serving as a cognitive kernel for advanced architectures such as OpenCog Hyperon and PRIMUS. Central to ActPC-Chem is the self-organization and refinement of rule patterns by prediction errors, instrumental and epistemic reward, and semantic constraints, enabling adaptive, logic-consistent behavior in complex algorithmic and chemical domains (Goertzel, 2024).

1. Discrete Active Predictive Coding: Principles and Formalism

ActPC in ActPC-Chem replaces standard backpropagation and continuous activation updates with discrete structures—sets of rewrite rules over a metagraph—optimized using local, information-theoretic prediction errors. At each timestep $t$, the agent maintains a metagraph $G_t$, partitioned into input and output subgraphs ($G_t^{\rm in}$, $G_t^{\rm out}$), and a rule set $R_t = \{ r_1, ..., r_N \}$ with stochastic application probabilities $\{p_i\}$. Applying rules yields a predicted output subgraph $\hat G_t^{\rm out} = \Gamma(R_t, G_t^{\rm in})$, where $\Gamma$ denotes the rule-application engine.

Prediction error is quantified as a Kullback–Leibler divergence

$$e_t = D_{KL}(q_t \| p_t) = \sum_m q_t(m)[\ln q_t(m) - \ln p_t(m)],$$

where $p_t(m)$ is the predicted pattern distribution and $q_t(m)$ is the observed distribution. Rule learning occurs via local search—replacing rules with neighborhood candidates that minimize $D_{KL}(q_t \| p_t)$—or by optimizing a differentiable loss over rule configurations using a Wasserstein natural-gradient step in probability space: $$\xi_{k+1} = \xi_k - h G(\xi_k)^{-1} \nabla_\xi F(p(\xi_k)).$$ Here, $\xi$ parameterizes distributions over rules and $G(\xi)$ is the Fisher information metric tensor defined on the probability simplex.

2. Algorithmic Chemistry and Metagraph Rewrite Rule Dynamics

In ActPC-Chem, “algorithmic chemistry” refers to representing all data, models, and rules as a single, potentially nested, directed-labeled metagraph $G=(V,E)$. Subgraphs serve as patterns $P$, and rules take the form $r: P^{\rm(in)} \Rightarrow P^{\rm(out)}$. Pattern-matching leverages subgraph isomorphism with semantic type constraints.

Rule weights $p_i$ determine stochastic application order when multiple candidates match. Rule selection and application proceed iteratively, halting on exhaustiveness or upon reaching a depth limit. Critically, because rules are themselves graphs, ActPC-Chem permits self-modifying architectures, including rules that rewrite other rules. Semantic filters enforce type-safety and external predicate consistency during pattern matching, ensuring domain-coherence.

3. Reward Mechanisms, Error-Driven Adaptation, and Rule Refinement

Adaptation in ActPC-Chem is governed by two core, reward-driven update streams:

• Instrumental (extrinsic) reward penalizes prediction error: $r_t^{\rm int} = -e_t$.
• Epistemic (intrinsic) reward encourages model exploration: $r_t^{\rm ep} = \sum_m q_t(m)\ln[1/p_t(m)]$.

A scalarized reward $$r_t = \alpha_{\rm int} r_t^{\rm int} + \alpha_{\rm ep} r_t^{\rm ep}$$ controls the rule-refinement process, with candidate modifications accepted if $r_t$ increases. Both local search and Wasserstein natural-gradient schemes use these composite rewards to refine stochastic rule application. All structural modifications are filtered through semantic constraints, maintaining type integrity and domain logic.

4. Symbolic Integration: AIRIS and PLN

The algorithmic chemistry substrate accommodates higher-order symbolic engines:

• AIRIS (causal rule inference): On observing $(s,a) \rightarrow s'$ transitions that violate current model expectations, AIRIS proposes new causal rules using Bayesian updates:

$P(r_i | D) \propto P(D | r_i) P(r_i)$

for observed data $D$. High-confidence rules are inserted into the metagraph with low initial probability and tested via subsequent predictive errors.

• PLN (probabilistic logical abstraction): PLN induces higher-level abstraction rules by exploring clusters of co-occurring or confidently applied rewrite rules. E.g., abstraction from specific food-related rules to generalized shape-based rules, with probabilities assigned as:

$$P(\text{Food} \mid \text{Round} \lor \text{Oval}) \approx \frac{P(\text{Round} \land \text{Food}) + P(\text{Oval} \land \text{Food})}{P(\text{Round} \lor \text{Oval})}.$$

PLN introduces uncertain-implication rules back into the system, which are then validated by the core ActPC reward mechanisms.

This layered approach intertwines fine-grained causal patching (AIRIS) and coarse-grained abstraction (PLN) for robust, context-aware rule evolution.

5. Continuous–Discrete Fusion via Predictive Coding Networks

To support noisy sensory input and fine-grained motor control, ActPC-Chem incorporates continuous predictive-coding neural networks (e.g., Neural Generative Coding). Each layer $\ell$ maintains state units $z^\ell \in \mathbb{R}^{n_\ell}$, computes predictions $\hat z^\ell = W^\ell \phi(z^{\ell-1}) + U^\ell m_t$, and tracks error units $e^\ell = z^\ell - \hat z^\ell$. Inference dynamics and synaptic updates are local and Hebbian-like: $$\tau \dot z^\ell = -\gamma z^\ell - e^\ell + (W^{\ell+1})^T e^{\ell+1}, \qquad \Delta W^\ell = \eta (e^\ell [\phi(z^{\ell-1})]^T) - \lambda W^\ell.$$

Symbolic metagraph patterns are embedded into continuous codes for input to the network, and discrete outputs decoded from higher-level continuous states. Bidirectional error signaling connects discrete rule adjustment and continuous context vector updates.

6. Transformer-Like Sequence Modeling Without Backpropagation

ActPC-Chem generalizes to transformer-like, next-token prediction entirely via rewrite rules and ActPC updates:

• Working Memory (WM): Encodes recent tokens/features as a symbolic subgraph.
• Long-Term Memory (LTM): Stores the suite of rewrite rules, including those generated by AIRIS, PLN, and ActPC learning.
• At each step, attention-like rule matching retrieves applicable rules based on WM context. Feedforward-like application of those rules generates candidate outputs. Distributions over next tokens are formed by aggregating rule outputs:

$$p(w_{t+1} \mid w_{1:t}) = \sum_{r \in R} p(r \mid \mathrm{WM}) 1_{\mathrm{out}(r)=w_{t+1}}.$$

• The actual next token is observed, and a symbolic prediction error is computed and used to update rule probabilities via discrete natural-gradient steps.
• AIRIS and PLN layers continuously propose/validate causal and abstraction rules, yielding hierarchical organization analogous to stacking layers in deep transformers; lower layers focus on local patterns, upper layers on larger-scale abstractions.

7. Prospects and Applicability to Computational Chemistry AI

A plausible implication is that ActPC-Chem's self-referential “algorithmic chemistry” unifies data, models, and learning dynamics in a framework blending discrete, continuous, and symbolic computation. Its error-driven, reward-modulated adaptive process is hypothesis to provide a “cognitive kernel” suited for highly flexible, multi-modal, and logically robust AI systems (Goertzel, 2024). Design patterns such as multi-agent orchestration, modular tool interfaces, and dynamic method selection—pioneered in frameworks like ChemGraph (Pham et al., 3 Jun 2025)—can directly inform ActPC-Chem implementations, enabling autonomous, GPU-accelerated, multi-scale workflows that adaptively combine ML and ab initio methods, and support end-to-end scientific automation from natural language through high-performance simulation.

By leveraging structured task decomposition, real-time error correction, and integrated symbolic reasoning, ActPC-Chem delineates a path toward scalable, logically consistent, and adaptable AI infrastructures for scientific computing and artificial general intelligence.