# Topological nominal semantics

- 14:00 20th April 2018 ( week 0, Trinity Term 2018 )Lecture Theatre B

My longest ever paper was called "Representation and duality of the

untyped lambda-calculus" at nearly 91 pages.

Though too long, the paper does something unusual and interesting: we

build a topological semantics for the lambda-calculus, in which

lambda-calculus terms are given semantics as open sets in a

topological space, and lambda-abstraction becomes both a logical

quantifier and a corresponding operation on open sets.

A similar programme has also been carried out for first-order logic.

Taken together, these two papers show how to extend Stone duality from

(for example) Boolean algebras, to algebras for first-order logic and

the lambda-calculus. We can do this because we use nominal semantics

to handle the meaning of variables in first-order logic and the

lambda-calculus.

This talk will not be an exhaustive technical account. Instead, I

will try to give an accessible overview of Stone-style duality theory,

and how and why new possibilities are opened up when looked at from

the point of view of nominal semantics. I will outline what

topological semantics look like in a nominal universe, and explain why

this is new and interesting.

Interested readers can view the two papers here:

"Representation and duality of the untyped lambda-calculus"

http://www.gabbay.org.uk/papers.html#repdul

APAL, 2017

"Semantics out of context"

http://www.gabbay.org.uk/papers.html#semooc

Journal of the ACM, 2016