syntax and encoding in LF In this section we introduce a logical framework with a cumulative hierarchy of predicative universes (in this way we don't need to distinguish between contexts and signatures ). Our main motivation for introducing a logical framework is to have precise and concise descriptions of (the well-formed expressions of) languages and translations: languages are described by signatures and translations by signature realizations. 3.1 The logical framework LF The logical framework is give by a set of inference rules for deriving judgements of the following forms: `, i.e. is a context ` A : Type i , i.e. A is a type (in the i-th universe) in context ` M : A, i.e. M is a term of type A in context where M and A range over pseudo-terms described by the following BNF: identiers x 2 Id ::= an innite set pseudo-terms A; M 2 Exp ::= x j Type i j x : A 1 :A 2 j x : A:M j M 1 M 2 empty ; ` ext ` A : Type i ; x : A ` x 62 DV() type-2 ` ` Type i : Type i+1 i...