This site is supported by donations to The OEIS Foundation.

User:Jon Awbrey/Theme One Program • Work Area

From OeisWiki
Jump to: navigation, search

Work Area

Basic Connectives

  • The node connective joins a number of component cacti to a node:
Cactus Cj Node Connective.jpg
Box Cj Node Connective.jpg
  • The lobe connective joins a number of component cacti to a lobe:
Cactus Cj Lobe Connective.jpg
Box Cj Lobe Connective.jpg

Basic Reductions

  • A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.
Box Cj Node Reduction.jpg
  • A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.
Box Cj Lobe Reduction.jpg

Reduction Rules

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:

Cactus Graph Lobe Rule.jpg
Cactus Graph Spike Rule.jpg

Syntax and Semantics of a Calculus for Propositional Logic

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.


Rooted Node.jpg
Rooted Edge.jpg
Cactus A Big.jpg
Cactus (A) Big.jpg
Cactus ABC Big.jpg
Cactus ((A)(B)(C)) Big.jpg
Cactus (A(B)) Big.jpg

Cactus (A,B) Big.jpg

Cactus ((A,B)) Big.jpg

Cactus (A,B,C) Big.jpg

Cactus ((A),(B),(C)) Big.jpg

Cactus (A,(B,C)) Big.jpg


Cactus (X,(A),(B),(C)) Big.jpg


Existential Interpretation

Table A illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.


Cactus Graph Node Big.jpg
Cactus Graph Spike Big.jpg
Cactus A Big.jpg
Cactus (A) Big.jpg

Cactus ABC Big.jpg

Cactus ((A)(B)(C)) Big.jpg

Cactus (A(B)) Big.jpg

Cactus (A,B) Big.jpg

Cactus ((A,B)) Big.jpg

Cactus (A,B,C) Big.jpg

Cactus ((A),(B),(C)) Big.jpg

Cactus (A,(B),(C)) Big.jpg


Entitative Interpretation

Table B illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.


Cactus Graph Node Big.jpg
Cactus Graph Spike Big.jpg
Cactus A Big.jpg
Cactus (A) Big.jpg

Cactus ABC Big.jpg

Cactus ((A)(B)(C)) Big.jpg

Cactus (A)B Big.jpg

Cactus (A,B) Big.jpg

Cactus ((A,B)) Big.jpg

Cactus (A,B,C) Big.jpg

Cactus ((A,B,C)) Big.jpg

Cactus (((A),B,C)) Big.jpg


Existential and Entitative Interpretations of Cactus Structures




Rooted Node.jpg
Rooted Edge.jpg
Cactus Graph • Node Connective.jpg
Cactus Graph • Lobe Connective.jpg