PAL+: a lambda-free logical framework

A lambda-free logical framework takes parameterisation and definitions as the basic notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL+, is a logical framework for specification and implementation of type theories, such as Martin-Löf's type theory or UTT. As in Martin-Löf's logical framework (Nordström et al., 1990), computational rules can be introduced and are used to give meanings to the declared constants. However, PAL+ only allows one to talk about the concepts that are intuitively in the object type theories: types and their objects, and families of types and families of objects of types. In particular, in PAL+, one cannot directly represent families of families of entities, which could be done in other logical frameworks by means of lambda abstraction. PAL+ is in the spirit of de Bruijn's PAL+ for Automath (de Bruijn, 1980). Compared with PAL, PAL+ allows one to represent parametric concepts such as families of types and families of non-parametric objects, which can be used by themselves as totalities as well as when they are fully instantiated. Such parametric objects are represented by local definitions (let-expressions). We claim that PAL+ is a correct meta-language for specifying type theories (e.g., dependent type theories), as it has the advantage of exactly capturing the intuitive concepts in object type theories, and that its implementation reflects the actual use of type theories in practice. We shall study the meta-theory of PAL+ by developing its typed operational semantics and showing that it has nice meta-theoretic properties.