Lambda Calculus vs. Turing Machines (Theory of Computation)

The speaker expresses great excitement about discussing the theory of computation, a topic they first encountered as an undergraduate at UCLA. Initially finding the subject challenging, the speaker has since come to appreciate its profound implications. The talk is framed as 'A Tale of Two Towers,' with an emphasis on axiomatic thinking, a formal approach to mathematics that may seem unconventional. The speaker uses the example of basic arithmetic operations to illustrate the concept of axioms and theorems, explaining how operations like multiplication can be defined in terms of addition, thereby establishing a foundation for understanding the fundamental rules of mathematics.

The speaker delves into the concept of axiomatic systems, focusing on the Peano axioms as a minimal set of rules that define natural numbers. They explain the axioms, which include the definition of zero, equality, and the successor function, and how they can be used to construct all of mathematics. The axioms are presented as self-evident truths, and the speaker discusses the idea of syntactic sugar, which simplifies complex expressions without adding new information. The summary also touches on the process of defining addition and multiplication based on these axioms.

The concept of 'axiom towers' is introduced, which represents a foundational set of axioms with additional constructs built on top. The speaker discusses the idea that axioms are not inherently special but can vary in their usefulness. They provide examples of different axiom towers, such as Roman numerals and non-Euclidean geometries, to illustrate the point that some axiomatic systems may be less practical than others. The speaker also emphasizes that axiom towers don't necessarily have to correspond to reality, and that the symbols and rules within them are intrinsically linked.

The speaker describes Alan Turing's work on Turing machines, which serve as a foundational axiom for computation. Turing machines are conceptualized as an infinitely long tape, a head that can read and write, and a set of instructions for computation. The speaker explains that Turing demonstrated the power of these machines by showing that any computable function can be computed by a Turing machine. They also mention the Brainfuck programming language as a virtual implementation of a Turing machine and provide examples of how simple algorithms, like addition, can be implemented using this model.

The speaker explores the concept of Turing completeness, which refers to the ability of a system to perform any computation that can be performed by a Turing machine. They provide examples of diverse systems, such as Conway's Game of Life, Magic: The Gathering, and Microsoft PowerPoint, that have been shown to be Turing complete. The speaker emphasizes the surprising power of simple axiomatic systems to represent complex computations.

The speaker introduces lambda calculus as an alternative axiomatic system for computation, developed by Alonzo Church. They explain the basic constructs of lambda calculus, including variables, function definitions, and function application. The speaker highlights the simplicity and elegance of lambda calculus, which can be used to represent complex computational processes despite its foundational simplicity.

The speaker discusses the Church-Turing thesis, which posits that any function that can be computed by a Turing machine can also be computed by a lambda calculus expression, and vice versa. They explain that although Turing machines and lambda calculus were developed independently, they were found to be equivalent in terms of their computational power. The speaker also touches on the implications of this thesis for the study of computation and the development of programming languages.

The speaker describes the transition from theoretical models of computation to their physical realization in the form of computers. They discuss John von Neumann's contributions to the design of computing machines, which included the concept of memory (RAM) and a central processing unit (CPU) with a control unit and a logic unit. The speaker also explains how the von Neumann model laid the groundwork for the development of higher-level programming languages and the abstraction of machine instructions.

The speaker traces the evolution of programming languages from assembly language to modern languages like C++, highlighting how each new language introduced higher-level constructs that could be reduced to the underlying von Neumann machine instructions. They emphasize that despite the increasing abstraction, all programming languages ultimately boil down to the fundamental axioms of computation as defined by the von Neumann model.

The speaker explores the functional programming paradigm, which has its roots in lambda calculus. They discuss the development of Lisp, a language created by John McCarthy that is capable of being implemented in its own language, and the subsequent rise of other functional programming languages like ML, Haskell, and Elm. The speaker connects the principles of functional programming to the constraints and benefits of the lambda calculus, and how these principles have influenced modern software development practices.

In the final paragraph, the speaker reflects on the relationship between theoretical models of computation and their practical implementation. They discuss the historical context of computing and the factors that contributed to the dominance of the von Neumann model over other models like lambda calculus. The speaker also addresses the difficulty of implementing functional programming concepts in hardware and the implications for the future of computing technology.