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Supplement to The Lambda Calculus

To show that all recursive functions can be represented in the \(\lambda\)-calculus, one reproduces the definition of recursive functions in the \(\lambda\)-calculus. We recall the definition.

Definition The class of is the (smallest) class of functions from natural numbers to natural numbers that contains:

and is closed under the operations of:

: if \(G\) and \(H\) are recursive functions, and if the numeric function \(F\) satisfies the relation:

then \(F\) is a recursive function.

: if \(G\) and \(H\) are recursive functions, and if the numeric function \(F\) satisfies the relations:

: if \(G\) is a recursive function, and if the numeric function \(F\) satisfies the relation

Say that a number-theoretic function \(f\) of arity \(n\) is \(\lambda\)-definable if there exists a \(\lambda\)-term \(F\) with the property that for every natural number \(a\) and for every \(n\)-tuple \((a_1 ,\ldots a_n)\) of natural numbers, we have

\(\lambda\)-definable

To show that the class of recursive functions can be represented in the \(\lambda\)-calculus, one follows its definition; the step-by-step construction of an arbitrary recursive function from the initial functions can be mimicked in the \(\lambda\)-calculus. One shows first that all initial functions can be represented; then one shows that the operations of substitution/composition, definition by primitive recursion, and minimalization can likewise be expressed. We require, first, a representation of natural numbers in the \(\lambda\)-calculus. Representing the initial functions and substitution/composition will then be straightforward, but some work is required to show that definition by primitive recursion and minimalization can be adequately represented. The development that follows is standard (Barendregt, 1984).

We begin with a representation of natural numbers as \(\lambda\)-terms. Define \(\ulcorner n\urcorner\) by recursion as:

Intuitively, to represent primitive recursion, one ought be able to proceed by cases because one has to test whether an argument is zero (‘base case’) or non-zero (‘recursion case’). One naturally searches for an analogue in the \(\lambda\)-calculus for an ‘if-then-else’ term-building operation. The \(\bB\) combinator accomplishes this. That is, \(\bB\) (given our representation of the constant-true and constant-false by the \(\lambda\)-terms \(\bT\) and \(\bF\)) has the property that

for all terms \(x\) and \(y\). We need, further, \(\lambda\)-terms representing the number-theoretic operations of successor and predecessor. In other words, we need \(\lambda\)-terms \(\Succ\) (successor), \(\Pred\) (predecessor), and \(\Zero\) (test for zero) satisfying

If there is a situation that seems destined to reach that point this summer it just might be William Karlsson and the Vegas Golden Knights, because it might be one of the most difficult and complex contract situations out of the entire RFA group.

Karlsson’s 2017-18 season was simply outstanding.

He finished as the NHL’s third-leading goal-scorer (43).

He posted outstanding possession numbers as part of Vegas’ top line between Jonathan Marchessault and Reilly Smith .

He absolutely shattered all of his previous career numbers — not just from any individual season, but his *entire* career numbers. A fact that Vegas Twitter account was happy to point out on Friday.

WILD season

WILD Bill

— Vegas Golden Knights (@GoldenKnights)

This situation — which absolutely nobody saw coming — presents a lot of problems for the team, and Karlsson, when it comes working out a new contract for this season.

When you look at Karlsson’s actual performance in 2017-18 it absolutely justifies a significant pay raise over the $1 million salary he made the past two years.But how much and how long should Vegas be reasonably comfortable committing to at this point?

The problem for Karlsson is that Vegas has every reason to be skeptical that this type of performance is repeatable.His 23.4 percent shooting percentage was not only the best mark in the NHL, it was the third-highest mark of any NHL player over the past 20 years, finishing behind only Mike Ribeiro’s 25.2 mark in 2007-08 and Curtis Glencross’ 23.6 in 2011-12. That number is almost certain to regress this upcoming season, which would mean an obvious decrease in goals.

At this point the Golden Knights still do not know what they have in Karlsson, and signing him to any sort of a long-term contract extension is probably just too much of a risk. That means a short-term bridge deal is almost certainly in the cards. And that is where things could get a little ugly.

Given that Karlsson only has one year of this level of production to his credit the Golden Knights are going to have a pretty compelling case in arbitration, meaning there probably is not much reason for them to come forward with any sort of a sizable contract offer — even on a short-term bridge deal. That likely means that even after scoring 43 goals this past season and helping lead a first-year expansion team to the Stanley Cup Final he probably still will not fully cash in on that production.

One or way or another he is probably going to have to prove it again in 2018-19 before that can happen. If it ever happens. Unfortunately for Karlsson it seems unlikely that another 23 percent shooting percentage and 40-goal season is on the horizon … unless he finds a way to completely shock the hockey world for a second year in a row.

Related: Forty-four players file for salary arbitration

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