Theme One Program • Exposition 8

Transformation Rules and Equivalence Classes

The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming cacti among themselves and partitioning the space of cacti into formal equivalence classes.  The transformation rules and equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

  • A reduction is an equivalence transformation which applies in the direction of decreasing graphical complexity.
  • A basic reduction is a reduction which applies to a basic connective, either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows.

  • A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.

Node Reduction

  • A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.

Lobe Reduction

That is roughly the gist of the rules.  More formal definitions can wait for the day when we need to explain their use to a computer.


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This entry was posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

1 Response to Theme One Program • Exposition 8

  1. Pingback: Survey of Theme One Program • 4 | Inquiry Into Inquiry

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