Model-checking ATL under Imperfect Information and Perfect Recall Semantics is Undecidable

We propose a formal proof of the undecidability of the model checking problem for alternating- time temporal logic under imperfect information and perfect recall semantics. This problem was announced to be undecidable according to a personal communication on multi-player games with imperfect information, but no formal proof was ever published. Our proof is based on a direct reduction from the non-halting problem for Turing machines.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of Alternating‑time Temporal Logic (ATL) by providing a rigorous proof that model checking ATL under imperfect information combined with perfect recall semantics is undecidable. While ATL was introduced to reason about the strategic abilities of agents in multi‑agent systems, its model‑checking problem has been thoroughly studied for four semantic variants: perfect information/perfect recall, perfect information/imperfect recall, imperfect information/imperfect recall, and imperfect information/perfect recall. The last case, however, had only been claimed to be undecidable based on informal arguments and personal communications, with no formal reduction published.

The authors fill this gap by reducing the non‑halting problem for deterministic Turing machines to ATL model checking. Given a deterministic Turing machine M = (Q, Σ, q₀, B, δ), they construct a concurrent game structure (CGS) G with three agents {1,2,3}. The CGS’s states encode the tape cells, the left border, and auxiliary control nodes (e.g., cell separators, blank‑cell generators). Crucially, the agents’ observations are defined by equivalence relations ∼₁ and ∼₂ that make many distinct states indistinguishable, thereby enforcing imperfect information. Strategies are required to be perfect‑recall: a strategy may depend on the entire history of visited states, but must assign the same action to any two histories that are ∼‑equivalent for the corresponding agent.

The core of the construction is a “horizontal” encoding of Turing‑machine configurations into the levels of a computation tree generated by the CGS. Even‑numbered levels represent the contents of tape cells together with the current state, while odd‑numbered levels contain auxiliary nodes that implement the transition step. For a transition a₁ q a₂ a₃ → a₁ a′₂ q′ a₃, the tree is extended over two levels: first the agents choose the action tuple (q, q′, R) (or (q, q′, L) for a left move), which is forced to be the same in all ∼‑equivalent histories; then the resulting state reflects the updated configuration. Agent 3 plays a purely environmental role, providing the necessary “synchronization” nodes (e.g., s_gen, s_tr) that allow the two “simulating” agents to coordinate without direct communication.

The ATL formula used for the reduction is ⟪{1,2}⟫ ◻ ok, where the atomic proposition ok holds in every “regular” configuration node and is absent precisely when the simulated Turing machine reaches a halting configuration (i.e., a state where δ is undefined). The authors prove two directions:

1. If M never halts on the empty input, then the coalition {1,2} has a perfect‑recall strategy that keeps the system forever in states labelled ok; consequently (G, s_init) ⊧_iR ⟪{1,2}⟫ ◻ ok.

2. If M halts, any strategy for {1,2} eventually forces the system into a node without ok, violating the □‑condition; thus the formula is false.

Since the non‑halting problem is undecidable, the ATL model‑checking problem under imperfect information and perfect recall is likewise undecidable. The authors also note that the strategies employed are primitively recursive, emphasizing that the source of undecidability is the agents’ inability to distinguish certain states, not the computational power of the strategies themselves.

Beyond the main result, the paper discusses how the same construction works for alternative ATL semantics (e.g., de dicto strategies) and situates the finding within the broader landscape of ATL research. It highlights that while ATL model checking becomes decidable under memoryless strategies or perfect information, the combination of imperfect information with perfect recall pushes the problem into the realm of non‑computable. The authors suggest future work on identifying subclasses of imperfect‑information CGSs where decidability might be recovered, possibly by restricting the structure of the observation equivalence relations or limiting the depth of recall.

In summary, the paper delivers the first formal, self‑contained proof that ATL model checking with imperfect information and perfect recall is undecidable, closing a notable open question and clarifying the precise role of information asymmetry in the complexity of strategic reasoning.