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Since language models are used to model a wide variety of languages, it is natural to ask whether the neural architectures used for the task have inductive biases towards modeling particular types of languages. Investigation of these biases…
We study the complexity of solving two-player infinite duration games played on a fixed finite graph, where the control of a node is not predetermined but rather assigned randomly. In classic random-turn games, control of each node is…
A major target of linguistics and cognitive science has been to understand what class of learning systems can acquire the key structures of natural language. Until recently, the computational requirements of language have been used to argue…
Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the…
Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group $A$, a move consists of picking some nonzero element $a \in A$. The game then continues with the quotient group $A/ \langle…
As speakers turn their thoughts into sentences, they maintain a balance between the complexity of words and syntax. However, it is unclear whether this syntax-lexicon tradeoff is unique to the spoken language production that is under the…
When flipping a fair coin, let $W = L_1L_2...L_N$ with $L_i\in\{H,T\}$ be a binary word of length $N=2$ or $N=3$. In this paper, we establish second- and third-order linear recurrence relations and their generating functions to discuss the…
Wordle is a single-player word-guessing game where the goal is to discover a secret word $w$ that has been chosen from a dictionary $D$. In order to discover $w$, the player can make at most $\ell$ guesses, which must also be words from…
In a simple pattern matching problem one has a pattern $w$ and a text $t$, which are words over a finite alphabet $\Sigma$. One may ask whether $w$ occurs in $t$, and if so, where? More generally, we may have a set $P$ of patterns and a set…
Game theory is the standard tool used to model strategic interactions in evolutionary biology and social science. Traditional game theory studies the equilibria of simple games. But is traditional game theory applicable if the game is…
We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game…
We consider a modification of the voter model in which a set of interacting elements (agents) can be in either of two equivalent states (A or B) or in a third additional mixed AB state. The model is motivated by studies of language…
Word ordering is a constrained language generation task taking unordered words as input. Existing work uses linear models and neural networks for the task, yet pre-trained language models have not been studied in word ordering, let alone…
We show that the Word Problem in finitely generated subgroups of $\textsf{GL}_d(\mathbb{Z})$ can be solved in linear average-case complexity. This is done under the bit-complexity model, which accounts for the fact that large integers are…
Given a regular language $L$, we study the language of words $\mathsf{D}(L)$, that distinguish between pairs of different left-quotients of $L$. We characterize this distinguishability operation, show that its iteration has always a fixed…
A neat question involving coin flips surfaced on $\Bbb X$, and generated an intensive `storm' of `social mathematics'. In a sequence of flips of a fair coin, Alice wins a point at each appearance of two consecutive heads, and Bob wins a…
We present a simple new method for proving that languages are not regular. We prove the correctness of the method, illustrate the ease of using the method on well-known examples of nonregular languages, and prove two additional theorems on…
Let $a$, $b$, and $n$ be integers with $0<a<b<n$. In a certain two-player probabilistic chip-collecting game, Alice tosses a coin to determine whether she collects $a$ chips or $b$ chips. If Alice collects $a$ chips, then Bob collects $b$…
Consider a situation with $n$ agents or players where some of the players form a coalition with a certain collective objective. Simple games are used to model systems that can decide whether coalitions are successful (winning) or not…
In this paper, we consider block languages, namely sets of words having the same length, and we propose a new representation for these languages. In particular, given an alphabet of size $k$ and a length $\ell$, a block language can be…