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:
“In general I do not draw well with literary men—not that I dislike them but I never know what to say to them after I have praised their last publication.”
—George Gordon Noel Byron (1788–1824)
“Science is a dynamic undertaking directed to lowering the degree of the empiricism involved in solving problems; or, if you prefer, science is a process of fabricating a web of interconnected concepts and conceptual schemes arising from experiments and observations and fruitful of further experiments and observations.”
—James Conant (1893–1978)