Termination Analysis of the Untyped λ-Calculus

An algorithm is developed that, given an untyped λ-expression, can certify that its call-by-value evaluation will terminate. It works by an extension of the ”size-change principle” earlier applied to first-order programs. The algorithm is sound (and proven so in this paper) but not complete: some λ-expressions may in fact terminate under call-by-value evaluation, but not be recognised as terminating. The intensional power of size-change termination is reasonably high: It certifies as terminating all primitive recursive programs, and many interesting and useful general recursive algorithms including programs with mutual recursion and parameter exchanges, and Colson’s ”minimum” algorithm. Further, the approach allows free use of the Y combinator, and so can identify as terminating a substantial subset of PCF. The extensional power of size-change termination is the set of functions computable by size-change terminating programs. This lies somewhere between Péter’s multiple recursive functions and the class of ε0-recursive functions.