Dyck
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Generating 2-Gray codes for grand Motzkin paths and grand Dyck paths with air pockets in constant amortized time
Announce Type: cross Abstract: A grand Motzkin path with air pockets is a non-empty lattice path in the first and fourth quadrant of $\mathbb{Z}^2$, starting at the origin $(0,0)$, ending on the $x$-axis, and consisting of up-steps $(1, 1)$, horizontal steps $(1, 0)$, down-steps $(1, -k)$ where $k \geq 1$, and with no consecutive down-steps. A {grand Dyck path with air pockets} is a grand Motzkin path with air pockets that uses no horizontal steps. We present the first known 2-Gray codes for...
Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability
Announce Type: new Abstract: Machine learning is increasingly used in mathematical discovery, but in mathematics the desired output is often not a prediction itself, but an explicit construction that can be checked independently. We study this setting through the zeta map on Dyck paths, a classical bijection in the combinatorics of the q,t-Catalan numbers. We train a deliberately small one-layer, one-head encoder-decoder transformer on this map and analyze its learned computation using...
Delacroix's Entry of the Crusaders into Constantinople Restored
Following a meticulous course of conservation treatment, Eugène Delacroix’s Entry of the Crusaders into Constantinople is returning to the Red Rooms, joining a number of other famous works by the artist. By restoring the piece’s colours to their original brilliance, once darkened by yellowed varnish, the conservators’ work has allowed its complex significance to shine through once more. Commissioned in 1838 by King Louis-Philippe I, the Entry of the Crusaders into Constantinople (12 April...
Greedy Grammar Induction with Indirect Negative Evidence
arXiv:2312.15321v3 Announce Type: replace Abstract: This paper proposes a non-lexicalized grammar-induction procedure that separates two tests: recognition of the observed finite presentation, and rejection of short preterminal strings generated by a hypothesis but unsupported by the evidence. The central object is the rule-coverage bound \(\ell^*(G)\): the maximum, over rules in \(G\), of the length of the shortest preterminal string whose derivation uses that rule. This bound induces the...