 
Summary: Principal Typings for Explicit Substitutions Calculi
Daniel Lima Ventura , Mauricio AyalaRinc´on , and Fairouz Kamareddine2
1
Grupo de Teoria da Computa¸c~ao, Departamento de Matem´atica, Universidade de Bras´ilia, Bras´ilia D.F., Brasil
{ayala,ventura}@mat.unb.br
2
School of Mathematical and Computer Sciences, HeriotWatt University, Edinburgh, Scotland
fairouz@macs.hw.ac.uk
Abstract. Having principal typings (for short PT) is an important property of type systems. This
property guarantees the possibility of type deduction which means it is possible to develop a complete
and terminating type inference mechanism. It is wellknown that the simply typed calculus has this
property, but recently, J. Wells has introduced a systemindependent definition of PT which allows to
prove that some type systems do not satisfy PT. The main computational drawback of the calculus
is the implicitness of the notion of substitution, a problem which in the last years gave rise to a number
of extensions of the calculus where the operation of substitution is treated explicitly. Unfortunately,
some of these extensions do not necessarily preserve basic properties of the simply typed calculus
such as preservation of strong normalization. We consider two systems of explicit substitutions ( and
se) and we show that they can be accommodated with an adequate notion of PT. Specifically, our
results can be summarized as follows:
· We introduce PT notions for the simply typed versions of the and the secalculus according to
