arXiv subjects

Dave Goel

Publications and source records attributed to Dave Goel.

4 recordsLinked to original sources

arXiv2026

The Hibi--Li (2016) conjecture is fully resolved

Hibi and Li conjectured that the order polytope $\mathcal{O}(P)$ of a finite poset $P$ has at most as many faces of each dimension as its chain polytope $\mathcal{C}(P)$, and that equality in some dimension $k$ with $1\leq k\leq|P|-1$ forces the two polytopes to be unimodularly equivalent. Fourier proved the inequality in a 2026 preprint. Let $X$ be the five-element poset in which an element lies above two incomparable elements and below two other incomparable elements. Using two results from Fourier's preprint, we show that if an $n$-element poset contains $X$ as an induced subposet, then, for every $k$ with $2\leq k\leq n-1$, its chain polytope has at least $\binom{n-3}{k-2}$ more $k$-dimensional faces than its order polytope, and that for every $n\geq5$ some $n$-element poset containing $X$ attains all these bounds at once. Since Hibi and Li showed that $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent exactly when $P$ does not contain $X$, this proves the equality statement for these values of $k$. The second version of Fourier's preprint gives an independent proof of this statement. For $k=1$, the equality statement is false: the polytopes $\mathcal{O}(X)$ and $\mathcal{C}(X)$ have the same number of edges (Hibi, Li, Sahara and Shikama) but are not unimodularly equivalent. Together with Fourier's inequality, these results settle the conjecture. The proofs given in this paper are formally verified in Lean 4.

math.CO↗

arXiv2026

The Cranston--Li--Su--Wang--Xu (2026) conjecture is true

Let $A$ be a finite abelian group. A nowhere-zero $A$-flow of a graph gives every edge a direction and a non-zero value in $A$, so that at every vertex the sum of the values on the entering edges equals the sum on the leaving edges. Such a flow can be changed along a cycle: going around the cycle, add a fixed non-zero element of $A$ to the value of every edge that points forward and subtract it from the value of every edge that points backward. The Cranston--Li--Su--Wang--Xu (2026) conjecture states that if $A$ has at least four elements, then in every graph in which all degrees are even, any two nowhere-zero $A$-flows are joined by a chain of such changes through nowhere-zero $A$-flows. It also states that, for every integer $k\ge 4$, the same holds for integer flows whose values are between $1$ and $k-1$ in absolute value, where each change adds a non-zero integer. We prove this conjecture. The core of the proof is an induction on the number of edges: if a flow enters a vertex $v$ along an edge $uv$ and leaves $v$ along an edge $vw$ with the same value, the two edges can be replaced by one edge $uw$, and the induction hypothesis applies to the smaller graph. So the main task is to change every flow into a flow with such a pair of edges. For nowhere-zero $A$-flows with $|A|\ge 4$, replacing vertices of degree more than $4$ by triangles reduces the problem to graphs with all degrees at most $4$. For integer flows, a counting argument joins any two nowhere-zero flows with values in the cyclic group $\mathbb{Z}_k$, $k\ge 4$, by changes that add $1$, $-1$, $2$ or $-2$; every such chain gives a chain of integer flows with values between $1$ and $k-1$ in absolute value, and every integer flow of this chain reduces modulo $k$ to a flow of the given chain. All results of this paper are also proved in the Lean 4 proof assistant, from Lean's standard axioms alone.

math.CO↗

arXiv2026

Finite Automata Extraction: Low-data World Model Learning as Programs from Gameplay Video

World models are defined as a compressed spatial and temporal learned representation of an environment. The learned representation is typically a neural network, making transfer of the learned environment dynamics and explainability a challenge. In this paper, we propose an approach, Finite Automata Extraction (FAE), that learns a neuro-symbolic world model from gameplay video represented as programs in a novel domain-specific language (DSL): Retro Coder. Compared to prior world model approaches, FAE learns a more precise model of the environment and more general code than prior DSL-based approaches.

cs.AI↗

arXiv2024

Label-Free Subjective Player Experience Modelling via Let's Play Videos

Player Experience Modelling (PEM) is the study of AI techniques applied to modelling a player's experience within a video game. PEM development can be labour-intensive, requiring expert hand-authoring or specialized data collection. In this work, we propose a novel PEM development approach, approximating player experience from gameplay video. We evaluate this approach predicting affect in the game Angry Birds via a human subject study. We validate that our PEM can strongly correlate with self-reported and sensor measures of affect, demonstrating the potential of this approach.

cs.HC↗