Related papers: Bilingual analogical proportions via hedges
An alternative proof of the completeness of relational algebra with respect to allowed formulas of first-order logic is presented. The proof relies on the well-known embedding of relational algebra into cylindric algebra, which makes it…
In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…
In this paper, first we show that $(\g,[\cdot,\cdot],\alpha)$ is a hom-Lie algebra if and only if $(\Lambda \g^*,\alpha^*,d)$ is an $(\alpha^*,\alpha^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie…
Order-sorted algebras and many sorted algebras exist in a long history with many different implementations and applications. A lot of language specifications have been defined in order-sorted algebra frameworks such as the language…
These are lecture notes on the algebraic approach to regular languages. The classical algebraic approach is for finite words; it uses semigroups instead of automata. However, the algebraic approach can be extended to structures beyond…
The field of AI alignment aims to steer AI systems toward human goals, preferences, and ethical principles. Its contributions have been instrumental for improving the output quality, safety, and trustworthiness of today's AI models. This…
Recent work on vector-based compositional natural language semantics has proposed the use of density matrices to model lexical ambiguity and (graded) entailment (e.g. Piedeleu et al 2015, Bankova et al 2019, Sadrzadeh et al 2018). Ambiguous…
This research started with an algebra for reasoning about rely/guarantee concurrency for a shared memory model. The approach taken led to a more abstract algebra of atomic steps, in which atomic steps synchronise (rather than interleave)…
Analogical reasoning has been a principal focus of various waves of AI research. Analogy is particularly challenging for machines because it requires relational structures to be represented such that they can be flexibly applied across…
Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the…
We propose a new formalism for specifying and reasoning about problems that involve heterogeneous "pieces of information" -- large collections of data, decision procedures of any kind and complexity and connections between them. The essence…
Rhetoric, both spoken and written, involves not only content but also style. One common stylistic tool is $\textit{parallelism}$: the juxtaposition of phrases which have the same sequence of linguistic ($\textit{e.g.}$, phonological,…
In the category of Hom-Leibniz algebras we introduce the notion of representation as adequate coefficients to construct the chain complex to compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of…
Math is constructed by people for people: just as natural language corpora reflect not just propositions but the communicative goals of language users, the math data that models are trained on reflects not just idealized mathematical…
In the rapidly evolving field of Large Language Models (LLMs), there is a critical need to thoroughly analyze their capabilities and risks. Central to our investigation are two novel elements. Firstly, it is the innovative parallels between…
Similarity is a core notion that is used in psychology and two branches of linguistics: theoretical and computational. The similarity datasets that come from the two fields differ in design: psychological datasets are focused around a…
Word embeddings generated by neural network methods such as word2vec (W2V) are well known to exhibit seemingly linear behaviour, e.g. the embeddings of analogy "woman is to queen as man is to king" approximately describe a parallelogram.…
Capturing the similarities between human language units is crucial for explaining how humans associate different objects, and therefore its computation has received extensive attention, research, and applications. With the ever-increasing…
We introduce the notion of pre-weight structure on a triangulated category and study the corresponding pseudo-identities. We propose the notion of canonical derived equivalence between algebras that are not necessarily flat, which is…
One of the major open problems in automata and logic is the following: is there an algorithm which inputs a regular tree language and decides if the language can be defined in first-order logic? The goal of this paper is to present this…