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A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely…
Unreplicated two-level factorial designs are often used in screening experiments to determine which factors out of a large plausible set are active. A theorem regarding the generalized word count pattern is stated and proved for…
In combinatorics on words, the well-studied factor complexity function $\rho_{\infw{x}}$ of a sequence $\infw{x}$ over a finite alphabet counts, for every nonnegative integer $n$, the number of distinct length-$n$ factors of $\infw{x}$. In…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
The Pascal matrix, $P$, is an upper diagonal matrix whose entries are the binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies the beautiful relation $P=\exp(H)$ in which $H$ has the numbers 1, 2, 3, etc. on its…
There are two main approaches to the distributed representation of words: low-dimensional deep learning embeddings and high-dimensional distributional models, in which each dimension corresponds to a context word. In this paper, we combine…
Continuous word representations learned separately on distinct languages can be aligned so that their words become comparable in a common space. Existing works typically solve a least-square regression problem to learn a rotation aligning a…
A gapped repeat (respectively, palindrome) occurring in a word $w$ is a factor $uvu$ (respectively, $u^Rvu$) of $w$. In such a repeat (palindrome) $u$ is called the arm of the repeat (respectively, palindrome), while $v$ is called the gap.…
Consider a growing system of random walks on the 3,2-alternating tree, where generations of nodes alternate between having two and three children. Any time a particle lands on a node which has not been visited previously, a new particle is…
Recursive distinctioning (RD) is a name coined by Joel Isaacson in his original patent document describing how fundamental patterns of process arise from the systematic application of operations of distinction and description upon…
It is natural for probabilistic programs to use conditionals to express alternative substructures in models, and loops (recursion) to express repeated substructures in models. Thus, probabilistic programs with conditionals and recursion…
Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…
We introduce data structures answering queries concerning the occurrences of patterns from a given dictionary $\mathcal{D}$ in fragments of a given string $T$ of length $n$. The dictionary is internal in the sense that each pattern in…
In the theory of answer set programming, two groups of rules are called strongly equivalent if, informally speaking, they have the same meaning in any context. The relationship between strong equivalence and the propositional logic of…
Grammar induction is the task of learning a grammar from a set of examples. Recently, neural networks have been shown to be powerful learning machines that can identify patterns in streams of data. In this work we investigate their…
A problem of reconstructing words from their subwords involves determining the minimum amount of information needed, such as multisets of scattered subwords of a specific length or the frequency of scattered subwords from a given set, in…
Homonyms are words with identical spelling but distinct meanings, which pose challenges for many generative models. When a homonym appears in a prompt, diffusion models may generate multiple senses of the word simultaneously, which is known…
Mechanistic interpretability aims to understand how neural networks generalize beyond their training data by reverse-engineering their internal structures. We introduce patterning as the dual problem: given a desired form of generalization,…
The block reversal of a word $w$, denoted by $\mathtt{BR}(w)$, is a generalization of the concept of the reversal of a word, obtained by concatenating the blocks of the word in the reverse order. We characterize non-binary and binary words…
Math word problem (MWP) solving aims to understand the descriptive math problem and calculate the result, for which previous efforts are mostly devoted to upgrade different technical modules. This paper brings a different perspective of…