🤖 AI Summary
This work addresses the challenges of modeling causality and temporal ordering in recursive agent execution traces by proposing a formal framework called the Causal-Temporal Event Graph (CTEG). CTEG represents strict causal-temporal relationships through a rooted tree annotated with timestamps and event types, and constructs recursive execution levels via the least fixed point of a monotonic operator. By integrating causal semantics, recursive structure, and temporal sequencing for the first time, CTEG enables decentralized, compositional trace construction, preserves well-formedness under partial failures, and supports tamper-evident verification through compatibility with Merkle trees. Theoretical analysis shows that, under reasonable conditions, the execution hierarchy converges at the first level, yielding a compositional, verifiable, and fault-tolerant model for agent trajectories.
📝 Abstract
We introduce causal-temporal event graphs (CTEGs) as a formal model for fully resolved recursive agent execution records under single-parenthood causal semantics. We formalise direct event emissions and recursive subagent invocations as extension procedures on generic typed temporal graphs and show that the recursive closure $\mathscr{E}_\infty$ of the induced maximal dynamics starting from single causal roots consists entirely of finite sequences of CTEGs. A CTEG is a rooted arborescence whose nodes carry timestamps and event types, subject to the constraint that timestamps be strictly increasing along causal paths. We realise $\mathscr{E}_\infty$ as the increasing union of a recursive hierarchy $\mathscr{E}_0 \subseteq \mathscr{E}_1 \subseteq \cdots$ of agent execution levels parametrised by recursion depth, which is recognised as the ascending Kleene chain of a monotone operator $\varphi$ admitting $\mathscr{E}_\infty$ as its least fixed point. Although the introduction of the full hierarchy is natural, stabilisation occurs already at $\mathscr{E}_1$ if one insists that the internal construction of a subagent execution trace be a delegated and opaque computational unit. The CTEG formalism supports compositional construction of globally well-formed execution traces from local agent behaviour without centralised coordination, preserves well-formedness under partial execution failure, and admits a natural relational database encoding. The arborescent structure of CTEGs is further compatible with cryptographic Merkle tree commitments for tamper-evident session verification.