Computability of Agentic Systems

TL;DR

This work provides a formal methodology for classifying and understanding the fundamental capabilities of agentic systems, and shows that reference-augmented (Turing-complete) systems can be exponentially more efficient at simulating complex graphs than their non-augmented (context-free) counterparts.

Abstract

This paper introduces the Quest Graph, a formal framework for analyzing the capabilities of agentic systems with finite context. We define abstractions that model common reasoning techniques and establish their computational power: the base Quest Graph is equivalent to an unrestricted Turing machine; the forward-only Finite Quest Decision Process (FQDP), despite its wide use, is only equivalent to a pushdown automaton (context-free); and the Reference-Augmented QDP (RQDP) regains Turing completeness only when stateful queries are allowed. Since computability affects efficiency, we then analyze the theoretical efficiency of each model by simulating task dependencies in computation graphs. We show that this computational hierarchy translates to concrete performance trade-offs: reference-augmented (Turing-complete) systems can be exponentially more efficient at simulating complex graphs than their non-augmented (context-free) counterparts. This work provides a formal methodology for classifying and understanding the fundamental capabilities of agentic systems.

Figures (8)

• Figure 1: Illustrative Quest Graphs. Top Left: A single-node graph. Top Right: A graph representing a multi-hop comparison task. Bottom: A 2-level hierarchical graph rollout for an RL problem. The focus node is the one with a filled circle. $\pi$ represents a parent mark (defined in Section \ref{['section:qdp']}).
• Figure 2: Evolution of a Quest Graph for a multi-hop question answering task. Steps 1-2: "Discover node" actions generate new nodes for sub-quests. Step 3: A "Respond then move focus" action returns empty update to the main node but shifts focus to a sub-quest. Steps 4-6: Subsequent "Respond then move focus" actions provide answers to the sub-quests and, finally, to the main quest node.
• Figure 3: An RQDP simulating the recursive computation of a Fibonacci number, $F(n) = F(n-1) + F(n-2)$. Top: The execution trace (rollout), where the agent explores the dependency graph depth-first, recursively computing the $F(n-1)$ term while retrieving the pre-computed value for the $F(n-2)$ term. For termination, we assume an omitted input is discovered at each $F(n-1)$ node to count down to the base case. Bottom: The corresponding reference graph, where edges are defined by the reference-generating function, $\tau$. The label in each node (e.g., $F(5)$) represents its reference; the corresponding node in the reference graph stores the cached response.
• Figure 4: Computation graph (left) and its max dependency version (right) with equal number of nodes.
• Figure 5: A node of degree five (left) and its k-ary tree of proxy nodes for $C=2$ (right). The numbers represent the proxy nodes and their potential increase in the response size. The red edges are the added proxy edges.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• proof : Proof Sketch
• Theorem 3
• Theorem 4
• proof : Proof Sketch
• Lemma 5
• proof : Proof Sketch
• Corollary 6
• Corollary 7