Open breandan opened 4 years ago
An investigation pursuing the connection between GF(2) and automatic differentiation has been undertaken. Preliminary findings indicate that finite fields possess the remarkable ability to both recognize and parse context-free languages, with suspected extensions to LCFRS [Cohen et al. (2016)] and context-sensitive grammars [Tanaka and Fu (1978)]. These findings are noteworthy as they unlock a nearly-optimal compilation pathway from graph computation to Boolean circuits. To our surprise, rootfinding in GF(2) has adjacent applications to robust-parsing, error-correcting codes, type inference and sketch-based program synthesis. Further evidence is needed to support the postulated connection [Considine, 2019] between differential and algebraic geometry unifying derivatives of continuous functions [Leibniz, 1675], derivatives of regular languages [Brzozowski (1964), Antimirov (1996)] and derivatives of algebraic data types [McBride (2011)], however our findings indicate that automatic differentiation has a meaningful interpretation in discrete-valued spaces (contrary to verbal objections by illustrious professors who shall remain unnamed) and lends evidence to support the claim that binary matrix multiplication is a universal programming language conjectured by Considine (2020).
Kotlin∇ tries to build a mathematically sound type system capturing the relationship between functions and fields. This design should be described more formally, but in short there are two approaches:
Fun<X>: Field<X>
whereX
is a member of some algebra over the reals, equipped with the usual arithmetic operators. When applying an operator, it is evaluated eagerly and you get backX
immediately. This is a shallow eDSL, evaluated eagerly.Fun<X>: Field<Fun<X>>
. Instead of returningX
, operators returnFun<X>
by default, which can be evaluated by usinginvoke(...): X
(X
andFun<X>
are both fields).The second representation comes from finite field theory, which has important implications for expression parsing and language design. It would be useful to understand this connection more deeply.
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