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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 ...
Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of finite memory are ignored.
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 ...
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.
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.
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.
Typed lambda calculi with subtyping are the simply typed lambda calculus with conjunctive types and System F <:. All the systems mentioned so far, with the exception of the untyped lambda calculus, are strongly normalizing: all computations terminate. Therefore, they cannot describe all Turing-computable functions. [4]