λ-Calculus: The Other Turing Machine

The early 1930s were bad years for the worldwide economy, but great years for what would eventually be called Computer Science. In 1932, Alonzo Church at Princeton described his λ-calculus as a formal system for mathematical logic,and in 1935 argued that any function on the natural numbers that can be effectively computed, can be computed with his calculus [4]. Independently in 1935, as a master’s student at Cambridge, Alan Turing was developing his machine model of computation. In 1936 he too argued that his model could compute all computable functions on the natural numbers, and showed that his machine and the λ-calculus are equivalent [6]. The fact that two such different models of computation calculate the same functions was solid evidence that they both represented an inherent class of computable functions. From this arose the so-called Church-Turing thesis, which states, roughly, that any function on the natural numbers can be effectively computed if and only if it can be computed with the λ-calculus, or equivalently, the Turing machine. Although the Church-Turing thesis by itself is one of the most important ideas in Computer Science, the influence of Church and Turing’s models go far beyond the thesis itself. The Turing machine has become the core of all complexity theory [5]. In the early 1960’s Juris Hartmanis and Richard Stearns initiated the study of the complexity of computation. In 1967 Manuel Blum developed his axiomatic theory of complexity, which although independent of any particular machine, was considered in the context of a spectrum of Turing machines (e.g., one tape, two tape, or two stack). This was followed in 1971 by Stephen Cooke and Leonid Levin who showed a complete problem for NP (satisfiability) and introducing the question of whether P = NP. Since then a huge number of complexity classes have been isolated and related, including PCP, the class of Probabilistically Checkable Proofs. All this work has been defined in terms of minor variants of the Turing machine. In the field of Algorithms, where analysis needs to be more precise, other models have been developed, such at the Random Access Machine (RAM), that are closer to physical computers while remaining relatively abstract and hence widely applicable. However even these machines were heavily influenced by the Turing machine, and most of the complexity classes defined for the Turing machine carry over to the RAM. Whereas the machine models became the foundation of complexity and algorithmic theory, it was Church’s λ-calculus that became the foundation for the theory of programming languages [3]. This came about largely under the influence of Dana Scott and Christopher Strachey who, at the suggestion of Roger Penrose, developed denotational semantics, a topologically influenced account of higher-order computations acting on infinite data objects such as functions and streams, that are inherent in Church’s formalism. Scott and Strachey’s work meshed with Church’s work on classical type theory as a foundation for mathematics, with L.E.J. Brouwer’s program of constructive foundations that led to Per Martin-Löf’s development of Intuitionistic Type Theory, and with N.G. de Bruijn’s AUTOMATH language for expressing machine-checked proof, all of which were similarly founded on the λ-calculus. The result is an integrated theory of computation and deduction, known as the Propositions as Types principle, that consolidates programs with proofs, and types with propositions. This principle lies at the heart of most programming language theory and many program verification and proof checking systems. Robin Milner’s work on the LCF prover in the 1970’s led to the emergence of functional programming, based directly on the λ-calculus, and interactive proof development, based on Scott’s logic of computable functions arising from his program of denota-