Machine Theory of Agentic AI

• Machine Theory of Agentic AI is a framework mapping agent memory architectures to the Chomsky hierarchy, distinguishing between finite, stack-based, and Turing-complete systems.
• It extends automata concepts to probabilistic models, enabling quantitative risk analysis and formal verification of safety properties in stochastic agent systems.
• The theory advocates a right-sizing modular design that minimizes computational overhead while maximizing reliability through rigorous, component-based verification.

Machine Theory of Agentic AI

Agentic AI systems are designed to exhibit goal‐directed, adaptive behaviors by leveraging architectures that integrate perception, memory, planning, reasoning, tool-use, coordination, and governance across a spectrum from single autonomous agents to orchestrated multi-agent collectives. The Machine Theory of Agentic AI formalizes the core structural, computational, and verification principles underlying these systems, drawing exact correspondences between agent memory architectures and the abstract machines of the Chomsky hierarchy, and providing a unifying algebra for analysis, right-sizing, and assurance in safety-critical and high-complexity environments (Koohestani et al., 27 Oct 2025).

1. Automata–Agent Correspondence and Chomsky Hierarchy

At the foundational level, agentic AI architectures can be rigorously classified by the “memory discipline” of their agent core, which determines both computational power and verifiability. This maps agent classes directly onto the Chomsky hierarchy:

• Finite memory: Regular Agents correspond to Finite Automata (FA). These support only a finite set of internal states and no persistent memory beyond current state, making all behavior verifiable and all safety properties decidable in polynomial time.
• Stack memory: Context-Free Agents align with Pushdown Automata (PDA), whose stack-based mechanisms support recursive or hierarchical task decomposition. Partial verifiability holds: configuration reachability is decidable, though general language equivalence is undecidable except in the deterministic case.
• Unbounded memory: Turing-Complete Agents map to Turing Machines (TM), equipped with scratchpad memory or external databases permitting reflection and general program execution. Behavior is undecidable in general—termination, safety and other semantic properties cannot be algorithmically established (Rice’s Theorem).

Formally:

• FA: $M = (Q,\Sigma,\delta,q_0,F)$
• PDA: %%%%1%%%%
• TM: $$T = (Q,\Sigma,\Gamma,\delta,q_0,q_{\text{accept}},q_{\text{reject}})$$

Agent perceptions are tokenized onto a finite alphabet, and their transition/control graphs mirror the automaton's state-plus-memory configuration. Mutual simulation theorems ensure the language of perception–action traces matches the automaton's recognized language, delineating sharp boundaries between verifiability and undecidability (Koohestani et al., 27 Oct 2025).

2. Probabilistic Extensions and Quantitative Risk

LLM-based agents inherently exhibit stochastic state transitions. The automata–agent framework thus extends to probabilistic automata:

• Probabilistic Finite Automaton (PFA): Transition function $\delta_p: Q\times\Sigma\times Q \rightarrow [0,1]$ models the probability of next state given current state and input, guaranteeing $\sum_{q'}\delta_p(q,a,q')=1$.
• For a set $U$ of “unsafe” states, Markov-chain analysis yields the cumulative probability of reaching $U$ from $q_0$ within $n$ steps as

$$\Pr_{q_0}[\exists\,t\leq n:\,q_t\in U] = 1 - \sum_{q\notin U} (\mathbf{P}^n)(q_0,q)$$

where $\mathbf{P}$ is the transition matrix.

This enables quantitative risk analysis for agentic policies and provides a groundwork for formal verification under uncertainty (Koohestani et al., 27 Oct 2025).

3. Formal Verification, Static Analysis, and Right-Sizing

The automata–agent correspondence forms a robust theoretical and practical basis for formal verification:

• FA systems are amenable to standard model checking; properties like reachability, safety, and liveness are efficiently decidable.
• PDA systems allow for pushdown model-checking (e.g., CTL* over pushdown systems), supporting verification of hierarchically structured planning with partial decidability.
• TM systems require incomplete or heuristic verification approaches, including abstraction, runtime monitors, and bounded model-checking. General guarantees are impossible.

Static analysis is facilitated by defining control-flow grammars over the agent’s planning language, constraining it to regular or context-free lattices via prompt or memory management policies. The “right-sizing” doctrine prescribes selecting the weakest automaton class achieving the desired agent competence, minimizing computational overhead and verification burden:

Agent Class	Automaton	Memory Architecture	Decidability
Regular (Reflex)	FA	Finite control, no scratch	All properties decidable
Context-Free (Hierarchical)	PDA	Stack (task-stack)	Some (reachability, DPDA)
Turing-Complete (Reflective)	TM	Unbounded read/write	Properties undecidable

(Koohestani et al., 27 Oct 2025)

4. Architectural Patterns and Modularity

Agentic AI systems are universally realized as modular, closed-loop machines whose core is a series-parallel composition of reliability-critical components. The canonical architecture is:

$$\mathcal{A} = \langle G, P, R, E, M, V, S, T \rangle$$

with:

• $G$: Goal Manager (intent normalization),
• $P$: Planner (plan synthesis),
• $R$: Tool Router (maps plans to tool calls),
• $E$: Executor (with sandbox, ensuring idempotency/transaction semantics),
• $M$: Memory (working, episodic, semantic/vector),
• $V$: Verifiers (plan/schema/safety critics),
• $S$: Safety Monitor (budgets, termination, escalation),
• $T$: Telemetry (audit trail).

Reliability emerges as a product of these components, with discipline enforced via schema-typed interfaces, least-privilege permission, idempotency keys, transactional semantics, explicit safety monitoring, and comprehensive auditing. Assurance loops for verification and safety containment localize faults and reduce emergent error trajectories to deterministic “refuse,” “retry,” or “escalate” outcomes. This generalizes to all major agentic subtypes (tool-using, memory-augmented, planning/self-improving, multi-agent, embodied/web agents) (Nowaczyk, 10 Dec 2025).

5. System-Theoretic and Game-Theoretic Foundations

Agentic AI as a discipline extends beyond automata theory into dynamic system-theoretic and game-theoretic analysis, particularly in adversarial or cyber-resilience contexts (Li et al., 28 Dec 2025). The formal abstraction is: $\langle M, \pi, \mu, \eta \rangle$ where

• $M$: agent memory state,
• $\pi$: reasoning-and-policy map,
• $\mu$: environment sensing,
• $\eta$: adaptation operator.

Multi-agent workflows are explicitly modeled as stochastic games, with equilibrium concepts (Nash, Stackelberg, Markov Perfect Equilibrium) both characterizing and guiding the design of attacker–defender and cooperative agentic strategies. System-level resilience is realized by optimizing autonomy allocation, information flow (conditional independence structure $\mathcal{I}$), and temporal composition (concatenated subgame phases). These principles guarantee that composite workflows are robust to unilateral or adversarial perturbations, as exemplified in formal case studies of pentesting/remediation, deception, and detection (Li et al., 28 Dec 2025).

6. Probabilistic Subagent Structures and Agency Composition

In deep neural models, subagentic structures are mathematically represented as probability distributions over outcomes, with agent welfare given by log-score utility ($U(o)=\log P(o)$). Composition is governed by weighted logarithmic pooling:

$$P_{\text{pool}}(o) \propto \prod_{i=1}^n P_i(o)^{w_i}$$

Logarithmic pooling can be shown to strictly improve every member’s welfare when the outcome set has cardinality $\geq 3$, while linear pooling admits no such unanimity (entropy/concavity arguments). Structural properties such as cloning invariance, continuity, and openness yield a scale-free conceptual basis for latent subagent discovery. This model formalizes phenomena in LLMs such as the induction of antagonistic “Waluigi” personas when benevolent “Luigi” subagents are elicited, and demonstrates optimal alignment strategies through “manifest-then-suppress” protocols, outperforming pure reinforcement (Lee et al., 8 Sep 2025).

7. Unified Formal Modeling and Verification

Domain-agnostic, mechanized model checking of agentic AI is enabled by two foundational primitives:

• Host-agent model ($\mathcal{H}$): Captures agent orchestration, registry, capability mapping, intent resolution, task decomposition, execution, and state (encompassing both tool and agent calls).
• Task-lifecycle model ($\mathcal{L}$): Formalizes sub-task state machines and lifecycle transitions with explicit liveness, safety, completeness, and fairness properties in temporal logic (CTL/LTL).

A finite Kripke structure encodes the composition, allowing the direct application of symbolic model checkers (e.g., NuSMV, Uppaal) to verify properties such as termination, privilege escalation resistance, DAG dependency ordering, and liveness across both Model Context Protocol (MCP) and Agent-to-Agent (A2A) protocol interactions (Allegrini et al., 15 Oct 2025).

8. Implications and Engineering Principles

The Machine Theory of Agentic AI provides a dual foundation:

• Theoretical rigor: By importing automata theory and computability, it establishes sharp verifiability and undecidability frontiers, reducing design risk around agentic unpredictability.
• Engineering methodology: Architectures must favor the minimal computational class necessary, implement modular componentization with typed, least-privilege interfaces, and employ explicit control and auditing mechanisms. This approach maximizes reliability and safety, making agentic AI systems tractable for formal analysis and practical deployment.

These principles apply universally, from single reflex agents through hierarchical planners and fully reflective, Turing-complete orchestrators. The framework supports not only AI safety, but also adaptive robustness under uncertainty, compositional scalability, and transparent governance (Koohestani et al., 27 Oct 2025, Nowaczyk, 10 Dec 2025, Li et al., 28 Dec 2025, Lee et al., 8 Sep 2025, Allegrini et al., 15 Oct 2025).