What They Don't Tell You About Type Checking

Typed lambda calculus is not Turing-complete.

There. I said it.

More specifically, simply typed lambda calculus is not Turing-complete, and neither are any variants that are both strongly normalizing and have decidable type-checking. This is because programs that the type-checker verifies are guaranteed to compute a result. If such a type-checker allowed a Turing-complete set of programs, it would be a solution to the halting problem!

Really, I should have put two and two together earlier on this one. I suppose this is what comes of picking lambda calculus up by reading diverse things in diverse places rather than learning it from one authoritative source.

What this indicates for me is that, at least in many cases, the point of allowing more and more sophisticated type systems is to get closer to a Turing-complete system. That is why people add things like parametric polymorphism, dependent types, and kinds. When we add these to typed lambda calculus, it doesn't just get "more expressive" in the sense that a high-level programming language is more expressive than machine code; it is literally able to do things that a simpler type system could not.

This doesn't mean that strongly typed programming languages are not Turing-complete. Typically the type-checkers for these will not guarantee that the program contains no infinite loops. So, one must be careful to figure out exactly what one is dealing with.

## Sunday, June 14, 2009

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