Changes

We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.

We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.

The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules ''q''₁₃₉, ''q''₁₇₅, ''q''₁₈₇, ''q''₂₀₇.

The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules <math>q_{139}, q_{175}, q_{187}, q_{207}.\!</math>

The function ''q''₁₃₉ : '''B'''³ &rarr; '''B''' and its fiber <nowiki>[|</nowiki> ''q''₁₃₉ <nowiki>|]</nowiki> &sube; '''B'''³ appeared to be key to many structures in this setting, and so I singled them out under the new names of ''syll'' : '''B'''³ &rarr; '''B''' and ''Syll'' &sube; '''B'''³, respectively.

The function <math>q_{139} : \mathbb{B}^3 \to \mathbb{B}</math> and its fiber <math>[| q_{139} |] \subseteq \mathbb{B}^3</math> appeared to be key to many structures in this setting, and so I singled them out under the new names of <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B}</math> and <math>\operatorname{Syll} \subseteq \mathbb{B}^3,</math> respectively.

Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations".  The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.

Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations".  The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.