Related papers: Logic as a distributive law
We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories with additional structure. This is made explicit…
We define a model for linear logic based on two well-known ingredients: games and simulations. This model is interesting in the following respect: while it is obvious that the objects interpreting formulas are games and that everything is…
Formal mathematics and computer science proofs are formalized using Hilbert-Russell-style logical systems which are designed to not admit paradoxes and self-refencing reasoning. These logical systems are natural way to describe and reason…
We describe a class calculus that is expressive enough to describe and improve its own learning process. It can design and debug programs that satisfy given input/output constraints, based on its ontology of previously learned programs. It…
Under the Curry--Howard isomorphism, the syntactic structure of programs can be modeled using birelational Kripke structures equipped with intuitionistic and modal relations. Intuitionistic relations capture scoping through persistence,…
Logical frameworks provide natural and direct ways of specifying and reasoning within deductive systems. The logical framework LF and subsequent developments focus on finitary proof systems, making the formalization of circular proof…
The normal form theorem, proved in R. Laver, On the left distributive law and the freeness of an algebra of elementary embeddings, Advances in Mathematics 91 (1992), 209-231, for the free algebra $\Cal A$ on one generator $x$ satisfying the…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms…
Derivations provide a way of transporting ideas from the calculus of manifolds to algebraic settings where there is no sensible notion of limit. In this paper, we consider derivations in certain monoidal categories, called codifferential…
In this chapter, we explore how (Type-2) computable distributions can be used to give both (algorithmic) sampling and distributional semantics to probabilistic programs with continuous distributions. Towards this end, we sketch an encoding…
We introduce $\mathsf{LEM}$, a type-assignment system for the linear $ \lambda $-calculus that extends second-order $\mathsf{IMLL}_2$, i.e., intuitionistic multiplicative Linear Logic, by means of logical rules that weaken and contract…
This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different…
We have previously introduced role logic as a notation for describing properties of relational structures in shape analysis, databases and knowledge bases. A natural fragment of role logic corresponds to two-variable logic with counting and…
In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the…
In this paper we investigate the Curry-Howard correspondence for constructive modal logic in light of the gap between the proof equivalences enforced by the lambda calculi from the literature and by the recently defined winning strategies…
The fundamental algebraic concepts of quantum mechanics, as expressed by many authors, are reviewed and translated into the framework of the relatively new non-distributive system of Boolean fractions (also called conditional events or…
We advocate the use of de Bruijn's universal abstraction $\lambda^\infty$ for the quantification of schematic variables in the predicative setting and we present a typed $\lambda$-calculus featuring the quantifier $\lambda^\infty$…
To support the understanding of declarative probabilistic programming languages, we introduce a lambda-calculus with a fair binary probabilistic choice that chooses between its arguments with equal probability. The reduction strategy of the…
An intriguing interpretation of the time-evolution of dynamical systems is to view it as a computation that transforms an initial state to a final one. This paradigm has been explored in discrete systems such as cellular automata models,…