Programming Languages - The Lambda Lecture

The lecture begins with an introduction to the ambitious goal of reducing programming language constructs to their most fundamental elements. The speaker aims to demonstrate that complex programming can be achieved using only lambda calculus, a concept developed in the 1930s by Alonzo Church. The journey will involve revisiting basic constructs like numbers, strings, and function applications, and exploring how they can be encoded within the lambda calculus framework. The tool of choice for this demonstration is Dr. Racket, specifically using a lazy evaluation language within it called 'lazy', which is crucial for the proper execution of lambda expressions.

The speaker delves into the concept of lazy evaluation, contrasting it with eager evaluation, and how it impacts the execution of programs. The focus then shifts to the representation of data structures, specifically pairs, using lambda expressions. The idea is to define pairs and the selection of their first and second elements through higher-order functions. This approach is extended to the discussion of conditionals, which are also represented as functions that select between two outcomes based on a boolean condition. The importance of lazy evaluation in preventing the premature evaluation of expressions is highlighted, especially in the context of conditionals.

The lecture continues with an exploration of how to encode fundamental programming constructs using lambda calculus. The speaker discusses the representation of natural numbers, conditionals, and arithmetic operations. The concept of Church numerals is introduced as a lambda representation of numbers, where numbers are represented by functions that apply another function a certain number of times. The ability to define addition and multiplication using these numerals is also explored, emphasizing the power and flexibility of the lambda calculus in capturing complex computations.

In this segment, the speaker focuses on the representation of numbers, particularly zero, in the context of lambda calculus. The discussion revolves around the different numeral systems and how they are abstractions of the concept of numbers. The speaker then introduces the concept of Church numerals for representing natural numbers using lambda expressions. The challenge of defining a 'zero?' function that checks if a numeral represents zero is also presented, highlighting the need for a function that can distinguish between zero and any other number.

The speaker discusses the challenge of representing the operation of subtracting one in the lambda calculus, a problem that even stumped Alonzo Church, the inventor of the lambda calculus. The discussion explores the difficulty of representing the predecessor function in a system that only allows function application. The segment ends with a teaser about the solution provided by Stephen Kleene, a PhD student who helped advance the lambda calculus by solving this problem.

The speaker explains Stephen Kleene's solution to the problem of representing subtraction in lambda calculus. Kleene's approach involved using pairs and a function that, when applied n times, would yield the predecessor of a number. The explanation includes how to construct such a function and how it can be used to iterate towards the zero value. The significance of this solution in demonstrating the computational power of lambda calculus is emphasized.

The lecture concludes with an in-depth discussion on achieving generalized recursion in lambda calculus without explicit recursive calls. The Y combinator, a fixed-point combinator, is introduced as a method to create recursive functions. The speaker demonstrates how the Y combinator can be used to implement factorial and Fibonacci sequences as examples of non-recursive functions that can be made recursive using this combinator. The importance of lazy evaluation in making the Y combinator work is also highlighted.

In the final part of the lecture, the speaker summarizes the journey of reducing programming to its core elements using lambda calculus. The ability to represent any computable function using only function application, functions, and parameters is emphasized. The discussion concludes with the Church-Turing hypothesis, which posits that these different formulations of computation, including lambda calculus and Turing machines, have equivalent power. The reliance on lazy evaluation throughout the lecture is reiterated, and the challenge of creating a Y combinator for eager evaluation is presented as a thought experiment.