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

Abstract

Abstract. A fully-automated algorithm is developed able to show that evaluation of a given untyped λ-expression will terminate under CBV (call-by-value). The "size-change principle" from first-order programs is extended to arbitrary untyped λ-expressions in two steps. The first step suffices to show CBV termination of a single, stand-alone λ-expression. The second suffices to show CBV termination of any member of a regular set of λ-expressions, defined by a tree grammar. (A simple example is a minimum function, when applied to arbitrary Church numerals.) The algorithm is sound and proven so in this paper. The Halting Problem's undecidability implies that any sound algorithm is necessarily incomplete: some λ-expressions may in fact terminate under CBV evaluation, but not be recognised as terminating.The intensional power of the termination algorithm is reasonably high. It certifies as terminating many interesting and useful general recursive algorithms including programs with mutual recursion and parameter exchanges, and Colson's "minimum" algorithm. Further, our type-free approach allows use of the Y combinator, and so can identify as terminating a substantial subset of PCF.

Analysing the complexity of functional programs: higher-order meets first-order

“…the recent survey of Midtgaard [39]. Closest to our work, control flow analysis has been successfully employed in termination analysis, for brevity we mention only [25,34,43]. By Jones and Bohr [34] a strict, higherorder language is studied, and control flow analysis facilitates the construction of size-change graphs needed in the analysis.…”

“…Closest to our work, control flow analysis has been successfully employed in termination analysis, for brevity we mention only [25,34,43]. By Jones and Bohr [34] a strict, higherorder language is studied, and control flow analysis facilitates the construction of size-change graphs needed in the analysis. Based on earlier work by Panitz and Schmidt-Schauß [43], Giesl et al [25] study termination of Haskell through so-called termination or symbolic execution graphs, which under the hood corresponds to a careful study of the control flow in Haskell programs.…”

Analysing the complexity of functional programs: higher-order meets first-order

“…the recent survey of Midtgaard [38]. Closest to our work, control flow analysis has been successfully employed in termination analysis, for brevity we mention only [24,33,42]. By Jones and Bohr [33] a strict, higher-order language is studied, and control flow analysis facilitates the construction of size-change graphs needed in the analysis.…”

“…Closest to our work, control flow analysis has been successfully employed in termination analysis, for brevity we mention only [24,33,42]. By Jones and Bohr [33] a strict, higher-order language is studied, and control flow analysis facilitates the construction of size-change graphs needed in the analysis. Based on earlier work by Panitz and Schmidt-Schauß [42], Giesl et al [24] study termination of Haskell through so-called termination or symbolic execution graphs, which under the hood corresponds to a careful study of the control flow in Haskell programs.…”

Shape Analysis in the Absence of Pointers and Structure

“…Anodization, using factored sets of singleton and non-singleton bindings, is most closely related to the Balakrishnan and Reps's recency abstraction [2], except that anodization works on bindings instead of addresses, and anodization is not restricted to a most-recent allocation policy. Superficially, one might also term Jones and Bohr's work on termination analysis of the untyped λ-calculus via size-change as another kind of shape analysis for higher-order programs [14].…”