We start by proving statement (ii). We now prove statement (iii). The last constraint is trivially satisfied. This can be easily shown by induction. 's constraint remains equal when Let's pick such a branching Moreover, observe that every edge in B is tight.

We introduce the best order score search (BOSS) and grow-shrink trees (GSTs) for learning directed acyclic graphs (DAGs) in this paradigm.

We refer the readers to ( Peters et al., 2017) for more detailed graphical terminology. We base our proof mostly on ( Kirsch, 2019). The first statement follows directly from the first theorem in ( Haviland, 1936). Without loss of generality, we reorder the variables according to reversed topological ordering, i.e. a Follows directly from Lemma 1. Lemma 4. Recall condition 2) in Causal de Finetti states that 8 i, 8 n 2 N: X The first equality holds by well-defindedness. The fourth equality follow from well-definedness.

Fulkerson's algorithm, we develop a polynomial-time algorithm for computing the

The Directed Acyclic Graph (DAG) task model for real-time scheduling finds its primary practical target in Robot Operating System 2 (ROS 2). However, ROS 2's publish/subscribe API leaves DAG precedence constraints unenforced: a callback may publish mid-execution, and multi-input callbacks let developers choose topic-matching policies. Thus preserving DAG semantics relies on conventions; once violated, the model collapses. We propose the Function-as-Subtask (FasS) API, which expresses each subtask as a function whose arguments/return values are the subtask's incoming/outgoing edges. By minimizing description freedom, DAG semantics is guaranteed at the API rather than by programmer discipline. We implement a DAG-native scheduler using FasS on a Rust-based experimental kernel and evaluate its semantic fidelity, and we outline design guidelines for applying FasS to Linux Linux sched_ext.

With the increased use of learning algorithms as a means to optimize agents' objectives in unknown

We start by proving statement (ii). We now prove statement (iii). The last constraint is trivially satisfied. This can be easily shown by induction. 's constraint remains equal when Let's pick such a branching Moreover, observe that every edge in B is tight.