Roman Kniazev

Eric Goubault, Roman Kniazev, Jeremy Ledent et al.

In recent years, several authors have been investigating simplicial models, a model of epistemic logic based on higher-dimensional structures called simplicial complexes. In the original formulation, simplicial models were always assumed to be pure, meaning that all worlds have the same dimension. This is equivalent to the standard S5n semantics of epistemic logic, based on Kripke models. By removing the assumption that models must be pure, we can go beyond the usual Kripke semantics and study epistemic logics where the number of agents participating in a world can vary. This approach has been developed in a number of papers, with applications in fault-tolerant distributed computing where processes may crash during the execution of a system. A difficulty that arises is that subtle design choices in the definition of impure simplicial models can result in different axioms of the resulting logic. In this paper, we classify those design choices systematically, and axiomatize the corresponding logics. We illustrate them via distributed computing examples of synchronous systems where processes may crash.

Nathanaël Fijalkow, Arka Ghosh, Roman Kniazev et al.

Partially observable Markov decision processes (POMDPs) are a fundamental model for sequential decision-making under uncertainty. However, many verification and synthesis problems for POMDPs are undecidable or intractable. Most prominently, the seminal result of Madani et al. (2003) states that there is no algorithm that, given a POMDP and a set of target states, can compute the maximal probability of reaching the target states, or even approximate it up to a non-trivial constant. This is in stark contrast to fully observable Markov decision processes (MDPs), where the reachability value can be computed in polynomial time. In this work, we introduce posterior-deterministic POMDPs, a novel class of POMDPs. Our main technical contribution is to show that for posterior-deterministic POMDPs, the maximal probability of reaching a given set of states can be approximated up to arbitrary precision. A POMDP is posterior-deterministic if the next state can be uniquely determined by the current state, the action taken, and the observation received. While the actual state is generally uncertain in POMDPs, the posterior-deterministic property tells us that once the true state is known it remains known forever. This simple and natural definition includes all MDPs and captures classical non-trivial examples such as the Tiger POMDP (Kaelbling et al. 1998), making it one of the largest known classes of POMDPs for which the reachability value can be approximated.

Roman Kniazev, Nathanaël Fijalkow

Do transformers, when trained on sequential reasoning traces, build internal models of the underlying task? And if so, does the structure of those internal representations mirror the structure of the domain? We train an 8-layer transformer on Sudoku solving traces and perform a mechanistic analysis of its internal computation. We establish two results. First, the model builds a substructure world model: it does not represent the board state cell by cell, as a human analyst would expect, but organizes information around the rows, columns, and boxes that Sudoku's constraints act on. Second, we identify a naked-single circuit: a small set of dedicated neurons in the final MLP layer, each individually detecting when exactly one digit remains possible for a specific cell, and reliably promoting that digit. These findings show that the geometry of an emergent world model is shaped by the constraint algebra of the domain, not its surface presentation, and that the resulting decision circuit is sparse, monosemantic, and fully interpretable. More broadly, they demonstrate that mechanistic interpretability tools can recover an end-to-end algorithmic account of how a transformer solves a combinatorial reasoning task.