arXiv subjects

Hermann Wilhelm

Publications and source records attributed to Hermann Wilhelm.

3 recordsLinked to original sources

arXiv2026

Refutation of the Non-Cancelling Intersections Conjecture

The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential "toggle game" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erd\H{o}s--Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \ge 10^{5}$ works.

math.CO↗

arXiv2026

The Non-Cancelling-Intersections Conjecture Fails for Left-Linear Trees

First formulated by Amarilli, Monet, and Suciu (arXiv:2401.16210, 2024), the Non-Cancelling Intersections (NCI) conjecture is an open problem in combinatorics stating that any set union can be constructively built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In the same paper, two orthogonal possible strengthenings are proposed: using only left-linear trees, and using non-trivial intersections only positively or only negatively depending on the sign of their M\"obius value. Here we show that using only left-linear trees, the conjecture is false (independent of the other strengthening). Our argument is non-constructive. We prove the existence of a counterexample, though it is of immense size.

math.CO↗

arXiv2023

A characterization of efficiently compilable constraint languages

A central task in knowledge compilation is to compile a CNF-SAT instance into a succinct representation format that allows efficient operations such as testing satisfiability, counting, or enumerating all solutions. Useful representation formats studied in this area range from ordered binary decision diagrams (OBDDs) to circuits in decomposable negation normal form (DNNFs). While it is known that there exist CNF formulas that require exponential size representations, the situation is less well studied for other types of constraints than Boolean disjunctive clauses. The constraint satisfaction problem (CSP) is a powerful framework that generalizes CNF-SAT by allowing arbitrary sets of constraints over any finite domain. The main goal of our work is to understand for which type of constraints (also called the constraint language) it is possible to efficiently compute representations of polynomial size. We answer this question completely and prove two tight characterizations of efficiently compilable constraint languages, depending on whether target format is structured. We first identify the combinatorial property of ``strong blockwise decomposability'' and show that if a constraint language has this property, we can compute DNNF representations of linear size. For all other constraint languages we construct families of CSP-instances that provably require DNNFs of exponential size. For a subclass of ``strong uniformly blockwise decomposable'' constraint languages we obtain a similar dichotomy for structured DNNFs. In fact, strong (uniform) blockwise decomposability even allows efficient compilation into multi-valued analogs of OBDDs and FBDDs, respectively. Thus, we get complete characterizations for all knowledge compilation classes between O(B)DDs and DNNFs.

cs.LO↗