Call-by-value Termination in the Untyped lambda-calculus

A fully-automated algorithm is developed able to show that evaluation of agiven untyped lambda-expression will terminate under CBV (call-by-value). The``size-change principle'' from first-order programs is extended to arbitraryuntyped lambda-expressions in two steps. The first step suffices to show CBVtermination of a single, stand-alone lambda;-expression. The second suffices toshow CBV termination of any member of a regular set of lambda-expressions,defined by a tree grammar. (A simple example is a minimum function, whenapplied to arbitrary Church numerals.) The algorithm is sound and proven so inthis paper. The Halting Problem's undecidability implies that any soundalgorithm is necessarily incomplete: some lambda-expressions may in factterminate under CBV evaluation, but not be recognised as terminating. The intensional power of the termination algorithm is reasonably high. Itcertifies as terminating many interesting and useful general recursivealgorithms including programs with mutual recursion and parameter exchanges,and Colson's ``minimum'' algorithm. Further, our type-free approach allows useof the Y combinator, and so can identify as terminating a substantial subset ofPCF