Rapid Prototyping Formal Systems in MMT: 5 Case Studies

Abstract

Logical frameworks are meta-formalisms in which the syntax and semantics of object logics and related formal systems can be defined. This allows object logics to inherit implementations from the framework including, e.g., parser, type checker, or module system. But if the desired object logic falls outside the comfort zone of the logical framework, these definitions may become cumbersome or infeasible.Therefore, the MMT system abstracts even further than previous frameworks: it assumes no type system or logic at all and allows its kernel algorithms to be customized by almost arbitrary sets of rules. In particular, this allows implementing standard logical frameworks like LF in MMT. But it does so without chaining users to one particular meta-formalism: users can flexibly adapt MMT whenever the object logic demands it.In this paper, we present a series of case studies that do just that, defining increasingly complex object logics in MMT. We use elegant declarative logic definitions wherever possible, but inject entirely new rules into the kernel when necessary. Our experience shows that the MMT approach allows deriving prototype implementations of very diverse formal systems very easily and quickly. PreliminariesThe syntax, type system, and typing algorithms of MMT have been described in detail in [22]. In the sequel, we give a quick example-driven introduction to MMT that builds a prototype implementation of the dependently-typed λ -calculus LF [11].

Building on the Diamonds between Theories: Theory Presentation Combinators

“…Realization Three prototypes exist: a stand-alone version [19,18], one as emacs macros aimed at Agda [4], and another [62] in MMT [60], a framework for developing logics which allows rapid prototyping of formal systems [53]. In MMT, we define the combinators as new theory expressions, defined as untyped symbol declarations in a theory, see Figure 11.…”

Making Isabelle Content Accessible in Knowledge Representation Formats

“…The logical basis of our export is a definition of Pure in the Mmt system. Mmt allows defining a wide variety of logical frameworks, and we use PLF as a starting point, a polymorphic variant of LF [HHP93] that already exists in the Mmt standard library [MR19].…”