$μλεδ$-Calculus: A Self Optimizing Language that Seems to Exhibit Paradoxical Transfinite Cognitive Capabilities
Ronie Salgado
TL;DR
This work tackles the challenge of paradoxical reasoning within formal computation by introducing a Wittgenstein-style graph-based framework and a self-optimizing $\mu$-calculus built on a restricted Sea of Nodes that encodes data dependencies. It presents an end-to-end pipeline: a standard Lambda calculus interpreter, an optimizing DAG-based compiler, and limit-testing methods that reveal how fixed points can be managed via unification, enabling transfinite-like reduction. The authors further extend the calculus with $\epsilon$-macros and $\delta$-IO functionals to model macros and external interactions, framing these extensions as moves toward open computation beyond traditional closed Turing systems. They discuss rich mathematical, physical, and philosophical implications, including termination properties, potential causal paradoxes, fractal self-similarity, and thought experiments about super-intelligent cognition, with future work aimed at broader IR expressiveness and practical applications in robotics and VR.
Abstract
Formal mathematics and computer science proofs are formalized using Hilbert-Russell-style logical systems which are designed to not admit paradoxes and self-refencing reasoning. These logical systems are natural way to describe and reason syntactic about tree-like data structures. We found that Wittgenstein-style logic is an alternate system whose propositional elements are directed graphs (points and arrows) capable of performing paraconsistent self-referencing reasoning without exploding. Imperative programming language are typically compiled and optimized with SSA-based graphs whose most general representation is the Sea of Node. By restricting the Sea of Nodes to only the data dependencies nodes, we attempted to stablish syntactic-semantic correspondences with the Lambda-calculus optimization. Surprisingly, when we tested our optimizer of the lambda calculus we performed a natural extension onto the $μλ$ which is always terminating. This always terminating algorithm is an actual paradox whose resulting graphs are geometrical fractals, which seem to be isomorphic to original source program. These fractal structures looks like a perfect compressor of a program, which seem to resemble an actual physical black-hole with a naked singularity. In addition to these surprising results, we propose two additional extensions to the calculus to model the cognitive process of self-aware beings: 1) $ε$-expressions to model syntactic to semantic expansion as a general model of macros; 2) $δ$-functional expressions as a minimal model of input and output. We provide detailed step-by-step construction of our language interpreter, compiler and optimizer.