$μλεδ$-Calculus: A Self Optimizing Language that Seems to Exhibit Paradoxical Transfinite Cognitive Capabilities

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 $\mu\lambda$ 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) $\epsilon$-expressions to model syntactic to semantic expansion as a general model of macros; 2) $\delta$-functional expressions as a minimal model of input and output. We provide detailed step-by-step construction of our language interpreter, compiler and optimizer.