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Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
This illustrates one reason why relatively powerful non-Turing-complete languages are rare: the more powerful the language is initially, the more complex are the tasks to which it is applied and the sooner its lack of completeness becomes perceived as a drawback, encouraging its extension until it is Turing-complete. The untyped lambda calculus ...
The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable." [54]: 101 The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another.
Upon hearing of Church's work, Turing enrolled at Princeton later that year under Church for a Ph.D. [23] Church and Turing then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative "mechanical processes for computation ...
As shown by Alan Turing and Alonzo Church, the λ-calculus is strong enough to describe all mechanically computable functions (see Church–Turing thesis). [1] [2] [3] Lambda-calculus is thus effectively a programming language, from which other languages can be built.
Unlike the untyped lambda calculus, the simply typed lambda calculus is not Turing complete. All programs in the simply typed lambda calculus halt. For the untyped lambda calculus, there are programs that do not halt, and moreover there is no general decision procedure that can determine whether a program halts.
The Church–Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. [dubious – discuss] In the untyped lambda calculus the only primitive data type is the function.
General models of computation equivalent to a Turing machine (see Church–Turing thesis) include: Lambda calculus A computation consists of an initial lambda expression (or two if you want to separate the function and its input) plus a finite sequence of lambda terms, each deduced from the preceding term by one application of beta reduction.