CoLF Logic Programming as Infinitary Proof Exploration

Logical Frameworks such as Automath [de Bruijn, 1968] or LF [Harper et al., 1993] were originally conceived as metalanguages for the specification of foundationally uncommitted deductive systems, yielding generic proof checkers. Their high level of abstraction was soon exploited to also express algorithms over deductive systems such as theorem provers, type-checkers, evaluators, compilers, proof transformers, etc. in the paradigm of computation-as-proof-construction. This has been realized in languages such as $\lambda$-Prolog [Miller et al., 1991] or Elf [Pfenning, 1991] based on backward chaining, and LolliMon [Lopez et al., 2005] or Celf [Schack-Nielsen and Schuermann, 2008], which integrated forward chaining. None of these early frameworks supported the direct expression of infinitary objects or proofs, which are available in the recently developed CoLF$^\omega$ [Chen, 2023]. In this work-in-progress report, we sketch an approach to computation-as-proof-construction over the first-order fragment of CoLF$^\omega$ (called CoLF$^\omega_1$ ) that already includes infinitary objects and proofs. A key idea is the interpretation of logic variables as communication channels and computation as concurrent message-passing. This is realized in a concrete compiler from CoLF$^\omega_1$ to Sax, a proof-theoretically inspired parallel programming language based on the proof-reduction in the semi-axiomatic sequent calculus [DeYoung et al., 2020].