Global Execution-Order Refinement

Updated 7 February 2026

Global execution-order refinement algorithms are techniques that convert local execution traces into structured, globally optimized orders, ensuring system correctness and efficiency.
They utilize methods such as LLM-based planning, greedy optimization, Bayesian de-linearization, and SAT-driven refinement to handle complex dependencies and scalability challenges.
Empirical results show these techniques reduce message delays, optimize makespan, and verify event ordering, thereby enhancing performance in distributed, concurrent, and transactional systems.

A global execution-order refinement algorithm is a class of techniques and formal frameworks that seek to construct, analyze, or enforce high-level (global) orderings of actions, events, or agent steps, so as to maximize correctness, efficiency, or expressiveness in distributed, concurrent, or hierarchical systems. Such algorithms often transform local or ad hoc execution traces, schedules, or plans into more structured, globally optimized or explicitly constrained orderings, typically reflected as total or partial orders. The constructions arise in diverse domains including LLM-based agent orchestration, distributed multi-agent optimization, workflow de-linearization, concurrent program verification, time modeling in cyber-physical systems, and speculative transactional memory. Recent advancements integrate probabilistic modeling, communication complexity analysis, and SAT-driven abstraction refinement to handle real-world scalability and correctness requirements.

Global execution-order refinement algorithms formalize execution as sequences or partial orders over action/step sets, subject to external constraints (e.g., tools, communication topology, event-dependency, or user queries). Central elements include:

Plans and Actions: Denote the top-level task, action primitives or skills, global plans $G_t$ as ordered tuples (subgoal–action pairs), history $S_t$ of executed steps, and observations from environment/tool invocations (Chen et al., 23 Apr 2025).
Order Structures: Use permutations (for total order), strict partial orders or DAGs (to encode concurrency and precedence), or scheduling graphs (representing event order and possible interleavings) (Li et al., 2 Feb 2026, Montin et al., 2018, Yin et al., 2017).
Refinement Operator: Refinement is defined as an order-theoretic relation ( $<_r$ ), requiring that a more concrete execution order preserves abstract precedences, splits abstract coincidences only as allowed, and never collapses new precedences not present in the abstraction (Montin et al., 2018).
Optimization Objectives: Balance execution utility—e.g., coverage of subgoals, success rates, or makespan—against complexity (plan size, code length, message count). Constraints include skill executability, plan length, transaction serializability, or explicit tool capabilities (Chen et al., 23 Apr 2025, Konda et al., 2021, Saad et al., 2018).

2. Methodological Instantiations in Diverse Domains

Multiple methodological forms of global execution-order refinement arise, including:

Controllable Global Planning in Agent Systems: An LLM-based agent instantiates a global plan as a sequence of high-level skill–subgoal pairs (e.g., searching, coding, writing), continuously refined by incorporating observations from hierarchical executors and using a replan-then-execute feedback loop. The planner operates via LLM prompts embedding plan structure, constraint, and utility desiderata (Chen et al., 23 Apr 2025).
Distributed Greedy Optimization: In distributed submodular maximization, agents select actions in a globally chosen order to optimize both system value and communication delay, with refinement realized as an ordering (permutation) minimizing accumulated message traversal cost in the agent network (Konda et al., 2021).
Bayesian Inference of Latent Partial Orders: De-linearization approaches like BPOP infer a static strict partial order from noisy sequential traces, producing a globally valid static order that allows more efficient (potentially parallel) execution while preserving observed dependencies (Li et al., 2 Feb 2026).
Scheduling Abstraction Refinement for Verification: Bounded model checking avoids initial combinatorial explosion by omitting precise scheduling constraints, then incrementally refines the allowed event orderings by blocking infeasible interleavings, reconstructing minimal kernel causality explanations and refining CNF formulas (Yin et al., 2017).
Behavioral/System Time Refinement: For concurrency and timing semantics, strictly partial order refinements enable model-driven verification of vertical system design steps, with constructive and checking algorithms based on four precise order-theoretic conditions (Montin et al., 2018).
Transactional Age-Based Commit Enforcement: In TM systems with strict predefined commit order, the execution order is globally refined by runtime interleaving and speculative propagation, enforcing that all commit dependencies align with age order, and enabling concurrency while guaranteeing correctness (Saad et al., 2018).

3. Algorithmic Techniques and Structural Properties

Global execution-order refinement algorithms implement distinctive algorithmic patterns and exhibit characteristic structural properties:

Feedback Loop with Dynamic Suffix Replanning: Algorithms like GoalAct employ a loop that, at each step $t$ , executes the current plan prefix, invokes the skill executor, updates history, and then replans only the suffix using the latest execution context. Prompts enforce preservation of completed steps, pruning/reordering upcoming steps, and length constraints.
Plan Optimization and Skill Complexity Tradeoff: Plan suffix optimization is formalized as

$G^* = \arg\max_G U(G|S_t) - \lambda C(G)$

with $U(G|S_t)$ measuring subgoal utility and $C(G)$ skill-complexity regularization, enforced via LLM prompt instructions rather than direct numerical optimization (Chen et al., 23 Apr 2025).

Frontier Scheduling and Partial Order Compilation: Bayesian de-linearization compiles the inferred strict partial order $\hat h$ into a runtime executor which tracks the current frontier and dispatches all feasible (unblocked) actions in parallel, obviating the need for repeated local LLM steps except for fallback (Li et al., 2 Feb 2026).
Graph-Based Counterexample Validation and Refinement: In program verification, event-order graphs with deduced causality and “no-write” rules iteratively propagate constraints to detect cycles (infeasibility) and extract minimal blocking sets—refining the abstraction efficiently with minimal CNF increments (Yin et al., 2017).
Order Consistency and Preservation Laws: Strict partial order refinement is characterized by four implications over concrete/abstract precedences and coincidences, and has been mechanized in Agda for behavioral model preservation and verified compositional reasoning (Montin et al., 2018).
Transaction Dependency Graphs: Refined commit order enforcement is ensured by tracking write-read/write-write/read-write conflicts and only allowing dependency edges to proceed along the predetermined global age ordering, with robust correctness guarantees (strict serializability) (Saad et al., 2018).

4. Performance, Scalability, and Complexity

Empirical and theoretical evaluations demonstrate substantial impact on efficiency, scalability, and correctness:

LLM Agent Performance: On LegalAgentBench, GoalAct’s global refinement boosts success by 12.22% over baselines; particularly for higher-hop tasks (+20%) and writing-intensive tasks (+12.74%) (Chen et al., 23 Apr 2025).
Distributed Communication Cost: In distributed submodular optimization, DFS-based orderings achieve $O(n)$ message delays versus $\Theta(n^2)$ in worst-case random orderings, with DFS schedules within 2–4 hops of optimal (Konda et al., 2021).
Bayesian Trace De-linearization Gains: BPOP achieves edge-F1 of 0.91 on scientific workflows vs. 0.43 (Heuristic Miner), and compiled execution reduces LLM token usage from 17K–32K to near zero in full coverage scenarios (Li et al., 2 Feb 2026).
Concurrent Program Verification: Constraint-based refinement slashes formula size to 1/8–1/1200 of monolithic encodings, with near-instant per-refinement iterations and 5–10x solver speedups; complete on large SV-COMP suites (Yin et al., 2017).
Transactional Throughput: Age-order-preserving TM algorithms attain 4.3x–16.5x speedups over single-threaded and STMLite, with strict correctness by construction (Saad et al., 2018).

5. Implementation Heuristics, Gating, and Safety Measures

Refinement algorithms are stabilized further by domain-specific control heuristics and abstraction:

Skill Complexity Caps and Failure Gating: GoalAct uses coding complexity thresholds (downgrading high-complexity coding to searching), failure caps (reverting skills after repeated errors), and branch-pruning based on utility scores embedded directly into the global planner’s LLM prompt (Chen et al., 23 Apr 2025).
Message Size and Scheduling Robustness: Distributed greedy orderings account for message payload growth and avoid random order’s inefficiency by careful walk construction—balancing memory cost against communication delay (Konda et al., 2021).
Safe Thresholding under Uncertainty: In Bayesian order inference, missing edges are safety-biased (additional, not missing, constraints), with edge threshold parameters empirically tuned for optimal feasibility and success (Li et al., 2 Feb 2026).
Minimal Blocking Clause Extraction: Instead of bloating the status formula, concurrent verification refines only on the essential causality kernels responsible for infeasibility, minimizing overfitting and keeping SAT increments tiny (Yin et al., 2017).

6. Theoretical Guarantees and Preservation Properties

Execution-order refinement frameworks are characterized by formal correctness, optimality and preservation properties:

Approximation Guarantees: Distributed greedy execution schedules retain $1/2$-approximation for submodular maximization, regardless of agent ordering, as long as each agent learns the prior choices; performance guarantees degrade only with increased communication hops (Konda et al., 2021).
Soundness and Completeness: Scheduling abstraction approaches are both sound and (up to bounded unwinding) complete: every feasible path is preserved and all infeasible schedules can be incrementally blocked, converging to conclusive safety or a concrete bug (Yin et al., 2017).
Order-Preserving Refinement: Vertical execution-order refinement is itself a partial order and preserves behavioral composition operators (union, subclocking), enabling compositional system design verified both axiomatically and in mechanized proof environments (Montin et al., 2018).
Transaction Graph Acyclicity: Strict serializability in ordered TM protocols follows from enforced acyclicity in the dependency graph, strictly aligned with global age order (Saad et al., 2018).

7. Cross-Domain Significance and Future Directions

Global execution-order refinement unifies planning, distributed coordination, concurrent verification, and workflow compilation under an order-centric paradigm. It enables agents—human or artificial, distributed or local—to evolve from stepwise, myopic strategies to globally coherent, efficient, and robust execution schemes. Practical implications include SOTA performance in agent benchmarks, provably scalable distributed algorithms, and tractable verification and synthesis of complex multi-component systems. A plausible implication is continuous refinement toward tighter integration of learning-based, formal, and distributed methods for execution-order optimization across emerging AI-infused and concurrent systems (Chen et al., 23 Apr 2025, Konda et al., 2021, Li et al., 2 Feb 2026, Yin et al., 2017, Montin et al., 2018, Saad et al., 2018).