Many-sorted logic can reflect formally our intention not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming. Both functional and assertive "parts of speech" in the language of the logic reflect this typeful partitioning of the universe, even on the syntax level: substitution and argument passing can be done only accordingly, respecting the "sorts".
Syntax and semantics of dependent types
In this chapter we fix a particular syntax for a dependently typed calculus and define an abstract notion of model as well as a general interpretation function mapping syntactical objects to entities in a model. This interpretation function is shown to be sound with respect to the syntax.
A modal analysis of staged computation | Journal of the ACM
We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of typed λ-calculi and functional languages. We directly demonstrate the sense in which ...
Theorems for free! | Proceedings of the fourth international conference on Functional programming languages and computer architecture
The polymorphic blame calculus integrates static typing, including universal types, with dynamic typing. The primary challenge with this integration is preserving parametricity: even dynamically-typed code should satisfy it once it has been cast to a ...
Probabilistic compositional semantics, purely
We provide a general framework for the integration of formal semantics with probabilistic reasoning. This framework is conservative, in the sense that it relies only on typed λ-calculus and is thus co
The Lambek Calculus
There is a noticeable revival of categorial grammar these days, as a vehicle for linguistic description. The systems used differ somewhat from the original calculus of Ajdukiewicz and Bar-Hillel, however. In particular, there is a component of rules for ‘type change’ of expressions, making for greater flexibility and elegance. One fundamental system of this kind is the so-called ‘Lambek Calculus’, whose type-change rules show a close analogy with the inference rules of constructive propositional logic. In this paper, we present one calculus of this kind, and survey its theoretical properties as a device in linguistic semantics. Our two main new contributions are a new and complete semantics for this calculus, as well as a modest study of its language-accepting capacity. In this way, we hope to provide a better understanding of the background theory of flexible categorial grammar, in tandem with its descriptive uses.
Finite-Choice Logic Programming | Proceedings of the ACM on Programming Languages
Logic programming, as exemplified by datalog, defines the meaning of a program as its unique smallest model: the deductive closure of its inference rules. However, many problems call for an enumeration of models that vary along some set of choices while ...
Functional Query Languages with Categorical Types
We study three category-theoretic types in the context of functional query languages (typed lambda-calculi extended with additional operations for bulk data processing). The types we study are:
Integrating functional and imperative programming | Proceedings of the 1986 ACM conference on LISP and functional programming
We present a new model, based on monads, for performing input/output in a non-strict, purely functional language. It is composable, extensible, efficient, requires no extensions to the type system, and extends smoothly to incorporate mixed-language ...
The Principal Type-Scheme of an Object in Combinatory Logic
R. Hindley, The Principal Type-Scheme of an Object in Combinatory Logic, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 29-60
Slim Lim: "Concrete syntax matters, actually"
Understanding Understanding: A Pragmatic Framework Motivated by...
Motivated by the rapid ascent of Large Language Models (LLMs) and debates about the extent to which they possess human-level qualities, we propose a framework for testing whether any agent (be it...
Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics)
Introduction to higher order catagorical logic by J. Lambek, March 25, 1988, Cambridge University Press edition, Paperback in English
Categorical logic and type theory
Categorical logic and type theory by Bart Jacobs, 1999, Elsevier Science edition, in English - 1st ed.
Syntax and Semantics of Linear Dependent Types
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax...