Maximal Task Decomposition (MAD)

• MAD is a framework that decomposes complex global tasks into the finest, atomic subtasks while preserving crucial global properties.
• It improves error correction and scalability by enabling modular execution in LLM-based systems, discrete-event multi-agent coordination, and communication-constrained control.
• Rigorous formal definitions and algorithmic methodologies in MAD ensure correctness, optimality, and efficient decomposition across a variety of domains.

Maximal Task Decomposition (MAD) refers to the process of partitioning a global, often complex, task into the finest-grained elementary subtasks such that each resultant subcomponent is as local or atomic as possible, subject to preserving key properties of the original task. Across domains—LLM-based agentic systems, discrete-event multi-agent coordination, and communication-constrained decentralized control—MAD enables error correction, decentralization, and scalable task execution by exploiting extreme modularity.

1. Formal Definitions and Variants

MAD has domain-specific formalizations but universally seeks the finest decomposition compatible with global task achievement. Three canonical instantiations are present in the literature:

• LLM Agentic Execution: Given a deterministic, stepwise task of $s$ elementary actions with transition function $T(x, a) = x'$, MAD sets the decomposition granularity $m = 1$, assigning each move to an independent microagent. Formally, $D_1: x \mapsto$ atomic subtasks; execution proceeds with $$r_{i+1} \sim M(\phi(x_i)),\ a_{i+1} = \psi_a(r_{i+1}),\ x_{i+1} = \psi_x(r_{i+1})$$.
• Discrete-event System Decomposition: Given a global automaton $A_S = (Q, q_0, E, \delta)$, assign to each agent $i$ a finest nontrivial event set $E_i \subseteq E$, producing local automata $P_i(A_S)$. Maximality is achieved when further splitting $E_i$ would violate global-to-local bisimulation equivalence.
• Communication-constrained STL Decomposition: Given a multi-agent global STL task graph and a communication graph $G_c$, the MAD problem seeks a maximal-volume decomposition where each STL predicate defined on multi-hop task edges is rewritten as a conjunction of local predicates on single-hop edges, guaranteeing implication to the global task while maximizing spatial extent.

Maximality, in all cases, implies no strictly finer partition enables successful task realization without violating correctness or communication constraints.

2. Algorithmic Methodologies

LLM-Based MAD

The archetypal algorithm (“generate_solution”) executes each atomic move by (i) spawning microagents, (ii) having each propose an action, and (iii) aggregating proposals through majority voting with a confidence margin $k$. Red-flag filtering removes misformatted samples before voting. The process is recursively applied for each subtask.

Key pseudocode components:

def generate_solution(x0, M, k): A = [] x = x0 for step in range(s): a, x = do_voting(x, M, k) A.append(a) return A def do_voting(x, M, k): V = defaultdict(int) while True: r = get_vote(x, M) a, x_prime = psi_a(r), psi_x(r) V[a] += 1 if V[a] >= k + max(V[a_prime] for a_prime != a): return a, x_prime def get_vote(x, M): while True: r = M(phi(x)) if passes_red_flag_checks(r): return r

Discrete-Event MAD

Hierarchical task decomposition proceeds by repeatedly projecting the global task automaton onto agent event sets while checking strong decomposability (DC1–DC4):

1. For each two-agent split, verify (DC1) mutual independence of private events, (DC2) commutability, (DC3) joint enablement of projected event strings, (DC4) determinism of local projections.
2. Iteratively extract maximal local event sets until decomposition cannot proceed without violating bisimulation equivalence.

Communication-Constrained STL MAD

For decomposing STL predicates requiring multi-hop coordination:

1. Represent local subtask predicates as axis-aligned hyper-rectangles in $\mathbb{R}^n$; parameterize by centers $p$ and edge-lengths $v$.
2. For each multi-hop predicate $\Phi_{ij}$, select a 1-hop path $P_{ij}$ in $G_c$; assign variables $\theta_{rs} = [p_{rs}; v_{rs}]$ along the path.
3. Solve the convex program minimizing $\sum_{(r,s)} \sum_m 1/v_{rs}[m]$, subject to inclusion constraints ensuring that the Minkowski sum of boxes along each path stays within the original predicate’s feasible set, and additional convex constraints for conflict resolution.

3. Error Correction and Correctness Guarantees

LLM Microagent Voting

The elementary move error probability of an LLM microagent after red-flag filtering is $p > 0.5$. The probability that $k$-margin voting yields a correct step is:

$p_{\mathrm{sub}}(k) = \frac{1}{1 + ((1-p)/p)^k}.$

For $S = s$ independent subtasks, the probability of zero errors:

$p_{\mathrm{full}} = [1 + ((1-p)/p)^k]^{-S}.$

To target global reliability $t$, the minimal voting margin grows only logarithmically in $s$:

$$k_{\min} = \left\lceil \frac{\ln( t^{-\frac{1}{s}} - 1 ) }{ \ln( (1-p)/p ) } \right\rceil = \Theta(\ln s).$$

Crucially, cost per atomic subtask scales as $O(s \ln s)$ for MAD ($m=1$), whereas bundling increases cost exponentially in $m$, making coarser decompositions infeasible at large scale.

Bisimulation in Discrete-Event MAD

Parallel execution of all local projections, under the DC1–DC4 conditions, recovers the original global automaton exactly:

$\left\|_{i=1}^n P_i(A_S) \right\| \equiv A_S$

This ensures the decomposition is lossless at the task specification level.

STL Predicate Inclusion Guarantees

Decomposed boxes (via hyper-rectangles) along paths in $G_c$ are constrained to ensure their sum remains inside the original STL predicate’s feasible set at all vertices, guaranteeing the decentralized controllers’ executions imply satisfaction of the centralized global specification.

4. Empirical Evaluation and Case Studies

LLM Execution Tasks

• Single-step error rates $p$ for LLMs (e.g., GPT-4.1-mini) in the Towers of Hanoi domain remain $\approx 0.997$ even for high-dimensional instances ($D=20$, $s>10^6$).
• A complete run for $D=20$ (over $1,048,575$ moves), with $k=3$ and red-flagging, achieved zero errors, validating predicted convergence rates.
• Cost projections demonstrate that MAD scaling enables the practical use of smaller (and less costly) LLMs to achieve extremely long-horizon, reliable execution when accompanied by robust voting and filtering.

Automaton Decomposition for Multi-Agent Coordination

A three-robot scenario illustrates stepwise maximal decomposition: first agent 2 versus agents 1 and 3, then decomposing the 1,3 block further, all subject to the DC1–DC4 checks. Each robot receives an automaton over its private events that, when composed, precisely realizes the global specification.

STL Task Decomposition

In an 8-agent $\mathbb{R}^2$ formation scenario with a restricted communication graph, MAD yields hyper-rectangle parameterizations for every required local predicate, achieving global formation goals with decentralized controllers. Convex solvers compute optimal parameters in real time (e.g., $0.019$ s for all subtasks), and empirical results show complete specification satisfaction using only local state exchanges.

5. Complexity, Scalability, and Optimality

Computational Complexity

• LLM MAD execution: $O(s \ln s)$ total cost in API calls, assuming per-step costs are constant; no exponential blowup with respect to the task horizon.
• Discrete-event decomposition: Each hierarchical iteration has complexity $$O\bigl(n^2(|E|^2|Q|\kappa + \sum_{a \in E_1\cap E_2}|p_a(L(A_S))|^2)\bigr)$$, where $n$ is the number of agents, $|E|$ the event alphabet, $|Q|$ the state number, and $\kappa$ the maximal loop length.
• Convex STL decomposition: Program size grows linearly with the number of decomposed edges and state dimension. Each convex constraint depends only on a small subset of variables, and off-the-shelf solvers efficiently handle hundreds of parameters.

Scalability

Empirical findings indicate MAD enables million-step or greater task horizons in LLM agentic computation, lossless decentralization in multi-agent automata, and real-time decentralized STL satisfaction for high-dimensional systems.

Optimality

For STL tasks, the MAD solution is globally optimal (maximal-volume boxes) due to convexity. In automata, maximal decomposition is characterized by the strict necessity and sufficiency of the DC1–DC4 conditions.

6. Conflict Handling, Limitations, and Future Directions

Conflict Resolution

In STL-based MAD, maximizing volume can introduce unsatisfiable conjunctions. Four canonical conflict types (overlapping “always” tasks, conflicting “eventually” intervals, cyclic closure failures, mixed temporal cycles) are resolved by imposing further convex constraints, such as vertical containment (box inclusion) or ensuring Minkowski sum intersections are non-empty.

Assumptions and Limitations

• Static communication topologies (for STL MAD).
• Restriction to concave predicates and axis-aligned boxes (STL context); general polytopes or ellipsoids increase program complexity.
• In automata, MAD requires the global task be expressible as a deterministic automaton satisfying the decomposability conditions.
• In LLM agentic MAD, scalability relies on high per-microagent accuracy after filtering; strong task determinism and independence are assumed.

Extensions

Outstanding research directions include time-varying communication networks, richer predicate sets for STL, more general projection techniques for automaton tasks, automated and scalable conflict detection/handling, and the application of finer-grained MAD in non-deterministic or stochastic domains.

7. Cross-Domain Significance and Theoretical Insights

Maximal Task Decomposition fundamentally addresses the reliable, scalable realization of complex global behaviors by insisting on extreme modularity under rigorously formalized correctness constraints. In agentic LLM systems, MAD permits error-corrected, arbitrarily long reasoning chains infeasible for monolithic models. In cooperative multi-agent systems, MAD yields formal local controllers whose joint behavior exactly recovers the intended collective dynamics. In communication-constrained control, MAD generates decentralized tasks maximally aligned with centralized specifications, bounded only by communication structure and predicate geometry.

A plausible implication is that as computational or organizational scale increases, the cost efficiency and error resilience of maximal decompositions will dominate coarser task partitioning approaches under the assumption that suitable corrective or compositional mechanisms exist. MAD thus provides a principled foundation for the synthesis of scalable, provably correct multi-component intelligent systems and novel algorithmic architectures in both artificial-intelligence and control-theoretic domains.