**Author:** Martin Hyland (University of Cambridge)

**Reference:** arXiv preprint arXiv:1211.5762v1 [math.CT], 2012.

**Comments:** For those reading this blog who aren't familiar with Google Scholar Updates, it might help to describe the format of this tool, because it is a little different from how search tools are usually set up. Instead of being ranked by relevance, my Updates page's 42 results, all presumably judged as crossing some threshold of relevancy, are ranked *chronologically*, so the most recently archived papers are at the top. The oldest paper is 326 days old; I do not know for sure if they will start dropping off the list, but I suspect they will once they get to a year old. I should say that the 42 results are from my list of 'top' recommendations, which I am using for this blog - there is also a less selective list of 'all' recommendations that currently has 99 results. In weeks where no new papers have been added I will go down the list till I get to my first unblogged paper; there hasn't been a lot of activity lately, so this week's paper is currently fifth.

The subject of this paper is the

λ (lambda) calculus, an area to which Böhm made important contributions. The λ-calculus is a mathematical formalism developed in the 1930s by

Alonzo Church as a 'theory of functions'. The two basic constructions are

*application* - given a function

**f** and expression

**e**, we get a new expression

**f(e)** (usually written without brackets), and

*abstraction* - given an expression

**e** which may mention a fresh variable

**x**, the function

**λx.e** will evaluate by binding whatever argument it's given to

**x** throughout

**e**. For example,

**λx.x** is the identity function that returns whatever it's given unchanged. Finally, raw λ-calculus expressions are considered to be equal if they differ only by certain variable renamings, or they can be simplified in the same way by evaluating them as functions. I should also note that the rhetorical distinction I'm making between 'functions' and 'expressions' is a bit misleading - in the pure λ-calculus, everything is a function, and functions happily take other functions as inputs and produce them as outputs.

Remarkably, the simple system I have sketched can be used to compute any function (on natural numbers) that it is possible to compute! This claim is called the

Church–Turing thesis (where 'Turing' refers to the

Turing machines of

Alan Turing). So the λ-calculus is important and useful to people like me whose job it is to ponder the nature of computation all day.

It's one thing to define a formal system of rules and procedures, but it seems a bit like a pointless game if we don't have a precise understanding of what they all *mean*. After all, why these rules and not some others? This is a question of *semantics*, and in the case of the λ-calculus this question has, in the words of this paper, "proved vexed". The simplest interpretation - that λ-expressions correlate to ordinary functions - turns out not to work. Computer scientists have had 80-odd years to work on this problem, so of course there are now a number of answers on the market; this paper adds a new one, and shows that it is mathematically natural and yields some of the earlier approaches as special cases.

The semantics for the λ-calculus given by Hyland are

*category theoretic* (an approach I discussed generally

here). The basic concept is that of an

'operad', although it could also be given in the more familiar (to me, at least) guise of

Lawvere theories. The general idea is that for every natural number

**n** we have a set

**L(n)** of all the λ-expressions with at most n free variables. This is the setting for ordinary equational logic (see

here). To get to the λ-calculus we must also have functions

**L(n+1)→L(n)** and

**L(n)→L(n+1)**, where the first corresponds to adding an abstraction, and the second to applying an expression to a fresh variable. These functions are related because doing the first, then the second, leaves you back where you started -

**e = (λx.e)x**. Doing the second, then the first, need not leave you back where you started, but if we added the requirement that it did - that

**f = λx.(fx)** - this is exactly

η-equivalence, a common condition to add on to the basic λ-calculus.

This is attractive stuff if you are a computer science and category theory nerd - and I am! The rest of the paper is dedicated to showing that this construction, called a 'semi-closed structure' is indeed a good way to interpret the λ-calculus, by relating it to the syntax, and to some other notions of semantics. There's a lot in this paper and I couldn't absorb it all in one week - the 'Taylor Fibration' material went a bit over my head - but I still greatly enjoyed it.