On the Relative Usefulness of Fireballs

In CSL-LICS 2014, Accattoli and Dal Lago showed that there is an implementation of the ordinary (i.e. strong, pure, call-by-name) $\lambda$-calculus into models like RAM machines which is polynomial in the number of $\beta$-steps, answering a long-standing question. The key ingredient was the use of a calculus with useful sharing, a new notion whose complexity was shown to be polynomial, but whose implementation was not explored. This paper, meant to be complementary, studies useful sharing in a call-by-value scenario and from a practical point of view. We introduce the Fireball Calculus, a natural extension of call-by-value to open terms for which the problem is as hard as for the ordinary lambda-calculus. We present three results. First, we adapt the solution of Accattoli and Dal Lago, improving the meta-theory of useful sharing. Then, we refine the picture by introducing the GLAMoUr, a simple abstract machine implementing the Fireball Calculus extended with useful sharing. Its key feature is that usefulness of a step is tested---surprisingly---in constant time. Third, we provide a further optimization that leads to an implementation having only a linear overhead with respect to the number of $\beta$-steps.Comment: Technical report for the LICS 2015 submission with the same titl

Beta-Conversion, Efficiently

Type-checking in dependent type theories relies on conversion, i.e. testing given lambda-terms for equality up to beta-evaluation and alpha-renaming. Computer tools based on the lambda-calculus currently implement conversion by means of algorithms whose complexity has not been identified, and in some cases even subject to an exponential time overhead with respect to the natural cost models (number of evaluation steps and size of input lambda-terms). This dissertation shows that in the pure lambda-calculus it is possible to obtain conversion algorithms with bilinear time complexity when evaluation is carried following evaluation strategies that generalize Call-by-Value to the stronger case required by conversion

Proceedings of the 19th International Workshop on Unification

Proceedings of the 19th international workshop on Unification, held during RDP'2005 in Nara, Japan, on April 22, 2005.UNIF is the main international meeting on unification. Unification is concerned with the problem of identifying given terms, either syntactically or modulo a given logical theory. Syntactic unification is the basic operation of most automated reasoning systems, and unification modulo theories can be used, for instance, to build in special equational theories into theorem provers