 
Summary: Deep isomorphism of modal derivations
and terms
Sergei N. Artemov
June, 1999
Abstract
The traditional CurryHoward isomorphism given a derivation of F
from assumptions in intuitionistic modal logic constructs the term
t(~x) such that the sequent ~x : ) t(~x) : F is derivable in the corresponding
calculus of typed modal terms. Technically speaking this isomorphism
is a realization by terms of the outermost modalities in some (not all)
derivable modal sequents 2 ) 2F .
In this talk we describe a system of terms (in the combinatory terms
form) LPi suÆcient for a uniform realization of all modalities (not just the
outermost ones) in any derivable modal sequent. The expressive power of
LPi surpasses the one of the usual modal calculus. LPi also extends
the natural provability semantics of CurryHoward terms under which t : F
can be interpreted as \t is a proof of F in a given suÆciently rich system,
e.g. Heyting Arithmetic HA".
1 Introduction
The traditional CurryHoward isomorphism given a derivation ) F constructs
