Changes

A painted cactus, as a rooted graph, has a distinguished node that is called its ''root''.  By starting from the root and working recursively, the rest of its structure can be described in the following fashion.

A painted cactus, as a rooted graph, has a distinguished node that is called its ''root''.  By starting from the root and working recursively, the rest of its structure can be described in the following fashion.

Each ''node'' of a PARC consists of a graphical ''point'' or ''vertex'' plus a finite sequence of ''attachments'', described in relative terms as the attachments ''at'' or ''to'' that node.  An empty sequence of attachments defines the ''empty node''.  Otherwise, each attachment is one of three kinds:  a blank, a paint, or a type of PARC that is called a ''lobe''.

Each "node" of a PARC consists of a graphical "point" or "vertex" plus

a finite sequence of "attachments", described in relative terms as the

attachments "at" or "to" that node.  An empty sequence of attachments

defines the "empty node".  Otherwise, each attachment is one of three

kinds:  a blank, a paint, or a type of PARC that is called a "lobe".

Each "lobe" of a PARC consists of a directed graphical "cycle" plus a

Each ''lobe'' of a PARC consists of a directed graphical ''cycle'' plus a finite sequence of ''accoutrements'', described in relative terms as the accoutrements ''of'' or ''on'' that lobe.  Recalling the circumstance that every lobe that comes under consideration comes already attached to a particular node, exactly one vertex of the corresponding cycle is the vertex that comes from that very node.  The remaining vertices of the cycle have their definitions filled out according to the accoutrements of the lobe in question.  An empty sequence of accoutrements is taken to be tantamount to a sequence that contains a single empty node as its unique accoutrement, and either one of these ways of approaching it can be regarded as defining a graphical structure that is called a ''needle'' or a ''terminal edge''.  Otherwise, each accoutrement of a lobe is itself an arbitrary PARC.

finite sequence of "accoutrements", described in relative terms as the

accoutrements "of" or "on" that lobe.  Recalling the circumstance that

every lobe that comes under consideration comes already attached to a

particular node, exactly one vertex of the corresponding cycle is the

vertex that comes from that very node.  The remaining vertices of the

cycle have their definitions filled out according to the accoutrements

of the lobe in question.  An empty sequence of accoutrements is taken

to be tantamount to a sequence that contains a single empty node as its

unique accoutrement, and either one of these ways of approaching it can

be regarded as defining a graphical structure that is called a "needle"

or a "terminal edge".  Otherwise, each accoutrement of a lobe is itself

an arbitrary PARC.

Although this definition of a lobe in terms of its intrinsic structural

Although this definition of a lobe in terms of its intrinsic structural components is logically sufficient, it is also useful to characterize the structure of a lobe in comparative terms, that is, to view the structure that typifies a lobe in relation to the structures of other PARC's and to mark the inclusion of this special type within the general run of PARC's. This approach to the question of types results in a form of description that appears to be a bit more analytic, at least, in mnemonic or prima facie terms, if not ultimately more revealing.  Working in this vein, a ''lobe'' can be characterized as a special type of PARC that is called an ''unpainted root plant'' (UR-plant).

components is logically sufficient, it is also useful to characterize the

structure of a lobe in comparative terms, that is, to view the structure

that typifies a lobe in relation to the structures of other PARC's and to

mark the inclusion of this special type within the general run of PARC's.

This approach to the question of types results in a form of description

that appears to be a bit more analytic, at least, in mnemonic or prima

facie terms, if not ultimately more revealing.  Working in this vein,

a "lobe" can be characterized as a special type of PARC that is called

an "unpainted root plant" (UR-plant).

An "UR-plant" is a PARC of a simpler sort, at least, with respect to the

An ''UR-plant'' is a PARC of a simpler sort, at least, with respect to the recursive ordering of structures that is being followed here.  As a type, it is defined by the presence of two properties, that of being ''planted'' and that of having an ''unpainted root''.  These are defined as follows:

recursive ordering of structures that is being followed here.  As a type,

it is defined by the presence of two properties, that of being "planted"

and that of having an "unpainted root".  These are defined as follows:

1.  A PARC is "planted" if its list of attachments has just one PARC.

# A PARC is ''planted'' if its list of attachments has just one PARC.

# A PARC is ''UR'' if its list of attachments has no blanks or paints.

In short, an UR-planted PARC has a single PARC as its only attachment, and since this attachment is prevented from being a blank or a paint, the single attachment at its root has to be another sort of structure, that which we call a ''lobe''.

 

In short, an UR-planted PARC has a single PARC as its only attachment,

and since this attachment is prevented from being a blank or a paint,

the single attachment at its root has to be another sort of structure,

that which we call a "lobe".

To express the description of a PARC in terms of its nodes, each node

To express the description of a PARC in terms of its nodes, each node

can be specified in the fashion of a functional expression, letting a

can be specified in the fashion of a functional expression, letting a