Command injection attacks, continuations, and the Lambek calculus

This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambek's syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of propositional logic without structural rules, and so a forerunner of substructural logics like Linear Logic and Separation Logic.Comment: In Proceedings WoC 2015, arXiv:1606.0583

Abstract Tensor Systems as Monoidal Categories

The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda

A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, then instantiate the abstract setting to sets and relations and to finite dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics

A Faithful Representation of Non-Associative Lambek Grammars in Abstract Categorial Grammars

International audienceThis paper solves a natural but still open question: can Abstract Categorial Grammars (ACGs) respresent usual categorial grammars? Despite their name and their claim to be a unifying framework, up to now there was no faithful representation of usual categorial grammars in ACGs. This paper shows that Non-Associative Lambek grammars as well as their derivations can be defined using ACGs of order two. To conclude, the outcomes of such a representation are discussed

Categorial Grammar

The paper is a review article comparing a number of approaches to natural language syntax and semantics that have been developed using categorial frameworks. It distinguishes two related but distinct varieties of categorial theory, one related to Natural Deduction systems and the axiomatic calculi of Lambek, and another which involves more specialized combinatory operations