Journal of Functional Programming


Small-step and big-step semantics for call-by-need


a1 Institute of Cybernetics, Tallinn University of Technology, Tallinn, Estonia (e-mail:

a2 Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan (e-mail:


We present natural semantics for acyclic as well as cyclic call-by-need lambda calculi, which are proved equivalent to the reduction semantics given by Ariola and Felleisen (J. Funct. Program., vol. 7, no. 3, 1997). The natural semantics are big-step and use global heaps, where evaluation is suspended and memorized. The reduction semantics are small-step, and evaluation is suspended and memorized locally in let-bindings. Thus two styles of formalization describe the call-by-need strategy from different angles. The natural semantics for the acyclic calculus is revised from the previous presentation by Maraist et al. (J. Funct. Program., vol. 8, no. 3, 1998), and its adequacy is ascribed to its correspondence with the reduction semantics, which has been proved equivalent to call-by-name by Ariola and Felleisen. The natural semantics for the cyclic calculus is inspired by that of Launchbury (1993) and Sestoft (1997), and we state its adequacy using a denotational semantics in the style of Launchbury; adequacy of the reduction semantics for the cyclic calculus is in turn ascribed to its correspondence with the natural semantics.

(Online publication September 07 2009)


This paper is dedicated to the memory of Professor Reiji Nakajima (1947–2008).