A meta-probabilistic-programming language for bisimulation of probabilistic and non-well-founded type systems
This work addresses the problem of enabling AGIs to learn reasoning methods, though it appears incremental as it builds on existing frameworks like cubical type theory.
The authors introduced a formal meta-language for probabilistic programming that can express both programs and their type systems, enabling AGIs to learn appropriate reasoning methods alongside knowledge. They demonstrated bisimulations for pure type systems and probabilistic dependent type systems, with an implementation in a Guarded Cubical Type Theory type checker.
We introduce a formal meta-language for probabilistic programming, capable of expressing both programs and the type systems in which they are embedded. We are motivated here by the desire to allow an AGI to learn not only relevant knowledge (programs/proofs), but also appropriate ways of reasoning (logics/type systems). We draw on the frameworks of cubical type theory and dependent typed metagraphs to formalize our approach. In doing so, we show that specific constructions within the meta-language can be related via bisimulation (implying path equivalence) to the type systems they correspond. This allows our approach to provide a convenient means of deriving synthetic denotational semantics for various type systems. Particularly, we derive bisimulations for pure type systems (PTS), and probabilistic dependent type systems (PDTS). We discuss further the relationship of PTS to non-well-founded set theory, and demonstrate the feasibility of our approach with an implementation of a bisimulation proof in a Guarded Cubical Type Theory type checker.