General Observations
Given the standard semantics, the simply typed lambda calculus is strongly normalizing: that is, well-typed terms always reduce to a value, i.e., a abstraction. This is because recursion is not allowed by the typing rules: it is impossible to find types for fixed-point combinators and the looping term . Recursion can be added to the language by either having a special operator of type or adding general recursive types, though both eliminate strong normalization.
Since it is strongly normalizing, it is decidable whether or not a simply typed lambda calculus program halts: it does! We can therefore conclude that the language is not Turing complete.
Read more about this topic: Simply Typed Lambda Calculus
Famous quotes containing the words general and/or observations:
“At Hayes’ General Store, west of the cemetery, hangs an old army rifle, used by a discouraged Civil War veteran to end his earthly troubles. The grocer took the rifle as payment ‘on account.’”
—Administration for the State of Con, U.S. public relief program (1935-1943)
“The truth is, the Science of Nature has been already too long made only a work of the brain and the fancy: It is now high time that it should return to the plainness and soundness of observations on material and obvious things.”
—Robert Hooke (1635–1703)