Graph models of $\lambda$-calculus at work, and variations

• Authors: Chantal Berline
• Venue: Mathematical Structures in Computer Science
• Year: 2006

This paper surveys what we have learned during the last ten years about the lattice $\lambda \mathcal{T}$ of all $\lambda$-theories (= equational extensions of untyped $\lambda$-calculus), via the sets $\lambda \mathcal{C}$ consisting of the $\lambda$-theories that are representable in a uniform class $\mathcal{C}$ of $\lambda$-models. This includes positive answers to several questions raised in Berline (2000), as well as several independent results, the state of the art on the long-standing open questions concerning the representability of $\lambda _{\beta},\lambda _{\beta\eta}$, $H$ as theories of models, and 22 open problems.We will focus on the class $\mathcal{G}$ of graph models, since almost all the existing semantic proofs on $\lambda \mathcal{T}$ have been, or could be, more easily, obtained via graph models, or slight variations of them. But in this paper we will also give some evidence that, for all uniform classes $\mathcal{C},\mathcal{C}^{\prime}$ of proper $\lambda$-models living in functional semantics, $\lambda \mathcal{C}-\lambda \mathcal{C}^{\prime}$ should have cardinality $2^{\omega }$, provided $ \mathcal{C}$ is not included in $\mathcal{C}^{\prime}.$