
Multidimensional Sequences With respect to language in general, the grammar for a language is the set of rules that operate on the elements in a sentence. With respect to a programming language, this grammar has been sort of formalised in Lisp with the use of Sexpressions inside the lambda calculus. However, there is no simple grammar or rewriting ruleset for a multidimensional sequence, MDS. In a effort to define a simple grammar for MDS's we're attempting to use rewriting rules for a simple sequences inside of loops. Loops, of course, contain the iteration or recursive constructs inside a computer program. The analogy of comparing loops to geometric polytopes is in the possibility that using loops on rewriting sequences will reduce dimensionality and increase order. Another advantage, maybe a necessity, in using loops, is the setting of initial conditions and adjusting dynamic mutable variables and functions when propagating the computations. The picture below is an illustration of this. The formalism for this method of analysis is contained in the notion of rewriting systems. The ChurchTuring Lambda Calculus also contains the notion of rewriting rules. Finite state machines used in production systems are fundamentally the dynamic elements of rewriting rulesets. The construction of a selfaware kind of machine, I think, depends innately on a feed back system built inside of the recursive loop. This is the idea in 3Lisp, and the recursive tower of Brian Smith. [1, 2] [1] Reflection and Sentantics in Lisp, Brian Cantwell Smith, 1984. [2] The Mystery of the Tower Revealed, Mitchell Wand and Daniel Friedman, 1988.

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