Related papers: Adjunction As Substitution: An Algebraic Formulati…
This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The…
We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague…
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…
Recently, unsupervised parsing of syntactic trees has gained considerable attention. A prototypical approach to such unsupervised parsing employs reinforcement learning and auto-encoders. However, no mechanism ensures that the learnt model…
We consider a language together with the subword relation, the cover relation, and regular predicates. For such structures, we consider the extension of first-order logic by threshold- and modulo-counting quantifiers. Depending on the…
The meaning of a sentence is a function of the relations that hold between its words. We instantiate this relational view of semantics in a series of neural models based on variants of relation networks (RNs) which represent a set of…
An algebraic structure with two constants and one ternary operation, which is not completely commutative, is put forward to accommodate ternary Boolean algebras. When the ternary operation is interpreted as Church's conditioned disjunction,…
Our goal is to define an algebraic language for reasoning about non-deterministic computations. Towards this goal, we introduce an algebra of string-to-string transductions. Specifically, it is an algebra of partial functions on words over…
This paper describes an alignment-based model for interpreting natural language instructions in context. We approach instruction following as a search over plans, scoring sequences of actions conditioned on structured observations of text…
In codeswitching contexts, the language of a syntactic head determines the distribution of its complements. Mahootian 1993 derives this generalization by representing heads as the anchors of elementary trees in a lexicalized TAG. However,…
The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional…
We study the question of whether a given regular language of finite trees can be defined in first-order logic. We develop an algebraic approach to address this question and we use it to derive several necessary and sufficient conditions for…
Techniques in which words are represented as vectors have proved useful in many applications in computational linguistics, however there is currently no general semantic formalism for representing meaning in terms of vectors. We present a…
We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
We propose two new dependent type systems. The first, is a dependent graded/linear type system where a graded dependent type system is connected via modal operators to a linear type system in the style of Linear/Non-linear logic. We then…
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…
These notes present the essentials of first- and second-order monadic logics on strings with introductory purposes. We discuss Monadic First-Order logic and show that it is strictly less expressive than Finite-State Automata, in that it…
We investigate the new, Turing-complete class of layered systems, whose lefthand sides of rules can only be overlapped at a multiset of disjoint or equal positions. Layered systems define a natural notion of rank for terms: the maximal…
We define a coherent adjunction in a strict $3$-category and we use string diagrams to show that any adjunction can be extended to a coherent adjunction in an essentially unique way.