**Authors:** Christoph Berkholz, Matthäus Micun, Igor Razgon Tree Decision Diagrams (TDDs) are a data structure recently introduced by Capelli et al. (SAT 2026). They are structured along a vtree and the size of their canonical form lies between Ordered Binary Decision Diagrams (OBDDs) and deterministic structured DNNF circuits (d-SDNNFs). While the succinctness gap between TDD and d-SDNNF is exponential, only a quasipolynomial separation between OBDD and TDD has been shown and it was left as open question whether this is optimal. We answer this question affirmatively by showing that every TDD can be transformed to an equivalent OBDD of quasipolynomial size. Although this might be seen as a weakness, our second result shows that TDDs share another desirable property with OBDDs that is not known to hold for d-SDNNF: Given a TDD and another target vtree, it is possible to construct the minimal and canonical TDD respecting the new vtree in time polynomial in the input and output. As a result we also obtain that the equivalence test between TDDs over different vtrees can be done in polynomial time.

An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. OBDDs are also known as special cases of oblivious read-once branching programs in the field of complexity theory. Since OBDDs have many nice properties as data structures, they have been extensively studied for decades in both theoretical and practical fields, such as VLSI design, formal verification, machine learning, and combinatorial problems. Arguably, the most crucial problem in using OBDDs is that they may vary exponentially in size depending on their variable ordering (i.e., the order in which the variable are to read) when they represent the same function. Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDD for a given function. Hence, numerous studies have sought heuristics to find an optimal variable ordering. From practical as well as theoretical points of view, it is also important to seek algorithms that output optimal solutions with lower (exponential) time complexity than trivial brute-force algorithms do. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity $O^\ast(3^n)$, where $n$ is the number of variables of the function, which is much better than the trivial brute-force bound $O^\ast(n!2^n)$. This paper shows that a further speedup is possible with quantum computers by demonstrating the existence of a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in $O^\ast(2.77286^n)$ time and space with an exponentially small error. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs, which provide compact representations of sparse sets and are often used in the field of discrete optimization and enumeration.

**Authors:** Alexis de Colnet, Alfons Laarman, Joon Hyung Lee We prove the existence of two thresholds regarding the compilability of random 2-CNF formulas to OBDDs. The formulas are drawn from $\mathcal{F}_2(n,δn)$, the uniform distribution over all 2-CNFs with $δn$ clauses and $n$ variables, with $δ\geq 0$ a constant. We show that, with high probability, the random 2-CNF admits OBDDs of size polynomial in $n$ if $0 \leq δ< 1/2$ or if $δ> 1$. On the other hand, for $1/2 < δ< 1$, with high probability, the random $2$-CNF admits only OBDDs of size exponential in $n$. It is no coincidence that the two ``compilability thresholds'' are $δ= 1/2$ and $δ= 1$. Both are known thresholds for other CNF properties, namely, $δ= 1$ is the satisfiability threshold for 2-CNF while $δ= 1/2$ is the treewidth threshold, i.e., the point where the treewidth of the primal graph jumps from constant to linear in $n$ with high probability.

**Authors:** Matthäus Micun, Christoph Berkholz Since their introduction by Atserias, Kolaitis, and Vardi in 2004, proof systems where each line is represented by an ordered binary decision diagram (OBDD) have been intensively studied as they allow to compactly represent Boolean functions. We extend this line of work by considering representation formats that can be even more succinct than OBDDs and have gained a lot of attention in the area of knowledge compilation: sentential decision diagrams (SDDs) and deterministic structured DNNF circuits (d-SDNNFs). We show that both variants can provide strictly smaller refutations of unsatisfiable CNFs than their OBDD counterparts. Furthermore, we investigate the relative strength of these systems depending on which of the three fundamental derivation rules join, reordering, and weakening are allowed. Here we obtain several separations and identify interesting open problems. To streamline our proofs we establish a sat-to-unsat lifting theorem that might be of independent interest: it turns satisfiable CNFs that are hard to represent by SDDs and d-SDNNFs into unsatisfiable CNFs that are hard to refute in the corresponding proof system.

**Authors:** Ryoma Onaka, Kengo Nakamura, Masaaki Nishino, Norihito Yasuda A knowledge compilation map analyzes tractable operations in Boolean function representations and compares their succinctness. This enables the selection of appropriate representations for different applications. In the knowledge compilation map, all representation classes are subsets of the negation normal form (NNF). However, Boolean functions may be better expressed by a representation that is different from that of the NNF subsets. In this study, we treat tensor trains as Boolean function representations and analyze their succinctness and tractability. Our study is the first to evaluate the expressiveness of a tensor decomposition method using criteria from knowledge compilation literature. Our main results demonstrate that tensor trains are more succinct than ordered binary decision diagrams (OBDDs) and support the same polytime operations as OBDDs. Our study broadens their application by providing a theoretical link between tensor decomposition and existing NNF subsets.

**Authors:** Michael Blondin, Michaël Cadilhac, Xin-Yi Cui, Philipp Czerner, Javier Esparza, Jakob Schulz Ordered binary decision diagrams (OBDDs) are a fundamental data structure for the manipulation of Boolean functions, with strong applications to finite-state symbolic model checking. OBDDs allow for efficient algorithms using top-down dynamic programming. From an automata-theoretic perspective, OBDDs essentially are minimal deterministic finite automata recognizing languages whose words have a fixed length (the arity of the Boolean function). We introduce weakly acyclic diagrams (WADs), a generalization of OBDDs that maintains their algorithmic advantages, but can also represent infinite languages. We develop the theory of WADs and show that they can be used for symbolic model checking of various models of infinite-state systems.