gested using tree structures to encode the configuration of a model as a starting point.
The structure described in this chapter is the structure behind the work in Chapter 5. The trees are comprised of nodes which have a label and a weight (and, of course, children). Initially, the labels were simply symbols associated with certain attitudes or semantic data which were taken from a nominated set. This had the advantages of simplicity and clarity, but their very simplicity made the multiplicative operator much more complicated. Much of the ma- chinery associated with labels was really just a messy version of arithmetic in commutative ring-like structure of multinomials. Once labels were identified with multinomials, everything became clear. Using the field of rational multi- nomials may be productive, but this notion has not been seriously explored.
4.2
Conventions and preliminary definitions
The structure is, loosely speaking, a set comprised of disjoint subsets with a corresponding family of additive operators (one to each subset). Each subset is closed under its addition, and there is a multiplicative operator which is defined over the whole set.
Generally, we will use lower case, boldfaced symbols to denote a node (or tree), and upper case, boldfaced symbols to denote sets – particularly sets of nodes. Other symbols (such asx) will typically refer to numbers or multino- mials. Elements of a node, uwill be identified using an appropriate subscript, namely uv, for the node’s value and uP for its label. Initially, the children of a node were thought of as refinements or children of an attitude, so the children of uwere its children, and uC denotes the set of children. We will takeK[A] to be the ring of multinomialsover the elements of a finite set of symbols A. HereKwould usually be some numeric field such asQ,RorC, for example. Definition 4.2.1. GivenAandK[A]and an arbitrary field,K, we define the set,T∗ of finite (acyclic) trees where each node is of the form(uv,uP,uC)where the value,
uv, is a member ofK, it’s label, uP is a member ofK[A]and the set of children, uC,
contains leaf nodes with distinct labels.
Nodes or trees with no children,E=∅, will be calledsimple nodes, simple trees, or leaf nodes, and simple nodes which also have scalar multinomials as their labels may be referred to asscalar nodesorscalar trees. The domain of trees,T∗, is the collection of only those trees with a finite number of nodes. We will make use of a set of special elements inT∗, O=(0, 0,∅), which is an an analogue of zero that we will call thezerotree.
In this work, we will occasionally restrict ourselves to an arbitrary subset of T∗, Once labels were identified with multinomials, everything became clear. We will take ˇTpP to be the set of trees whose root nodes have a label equal to
p
P. The element OpP=(0, pP,∅)will be shown to play the role of the additive identity in the set ˇTpP; we will use the symbol Omore generally to refer to
4.2. CONVENTIONS AND PRELIMINARY DEFINITIONS 73
an appropriate member of the set of additive zero elements unless it creates ambiguity.
Note that the choice to use elements of the ring of multinomialsfor labels is, in a sense, arbitrary: elements of any commutative ring will serve, though we will see that if we use a commutative ring, T∗ and the derived domains are also commutative; here, the ring of multinomialsprovide a simple example which is easily manipulated and printed.Kis taken to beRorQ.
Definition 4.2.2. Two nodes or trees, u, v ∈TˇpPf orsomepP ∈ K[A]are said to be compatibleif they have the same label.
u∼ v ⇐⇒ u∈RvP ⇐⇒ v∈RuP.
Clearly, all the trees inTˇpP are compatible, and we will denote the set of all nodes of the form(0, p
P,∅)with ˇ
O.
When there is no risk of ambiguity, we will use the same symbol to refer to a set,T∗for example, and a vector space based on that set. Compatibility (or lack thereof) is really only pertinent to the addition of trees, in the same way that having the same row- and column-rank is only necessary in matrix addition. There is no such constraint in the pairwise multiplication of trees.
First, the definitions for some basic tools for manipulating these trees. Some of the functions defined below are not used in this chapter, but play a role in the explicit model described in Chapter 5.
Definition 4.2.3. The cardinality of a tree u∈T∗is the number of nodes it contains. We define it formally as ∥u∥⊺=⎧⎪⎪⎨⎪⎪ ⎩ 0 if uv =0 and =C∅ 1+∑e∈u C∥e∥⊺.
Simple nodes are the only nodes which have a cardinality of one, and Ois the only node or tree with a cardinality of zero.
Definition 4.2.4. For any u∈T∗we define the function for
depth(u)=⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪ ⎩
0 if uv =0 and =C∅
1 if uis a simple node
1+max({depth(v)∶ ∀v∈ uC}) otherwise which gives us the depth of the tree.
Definition 4.2.5. We will also define for u∈T∗,
trim(u)=⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪ ⎩ O uv =0 and =C∅ O if uis simple (uv,uP,{trim(e)∶ ∀e∈ uC}∖{Oˇ}) otherwise.
4.2. CONVENTIONS AND PRELIMINARY DEFINITIONS 74
Obviously the depth is an indication of how many levels of nodes the tree pos- sesses. Trimming essentially removes all simple nodes from the tree. A recur- sive application of trimming will be denoted trimk, indicating that the tree u will be trimmedktimes. Note that trimdepth(u)u=0 and depth(trimdepthu−1 u)= 1.
Definition 4.2.6. Theoverlapbetween two trees is defined
overlap(u,v)= ⎧⎪⎪⎪ ⎪⎪ ⎨⎪⎪ ⎪⎪⎪⎩ 0 if u∈ Oˇ or v∈ Oˇ or uP ≠ vP 1+ ∑ e∈uC f∈vC overlap(e, f) otherwise
Clearly two trees, uand v, are compatible if and only ifoverlap(u, v)≠0; they will be said tocompletely overlapif∥u∥⊺=∥v∥⊺=overlap(u,v).
We may also make use of the relative overlap of two nodes, uand v, given by overlapr(u,v)=2 overlap(u, v)
∥u∥⊺+∥v∥⊺ . The relative overlap of Owith itself is not defined.
Definition 4.2.7. Theshadowcast by a tree, v, onto another tree, u, is given by shadow(u,v)=⎧⎪⎪⎨⎪⎪
⎩
(uv, uP,{shadow(r,s)∶ r∈ uC,s∈ vC}∖{Oˇ}) if uP = vP
O otherwise
where uand vare compatible.
This lets us restrict our attention to those parts of a tree which conform to some “template” tree. It is easy to see that overlap(u,v)is precisely∥(∥⊺shadow(u,v)). Definition 4.2.8. We will define a few useful notations for sets derived from sets of nodes inT∗. TakeUandV be such sets and abe a node inT∗; then
L(U)={eP ∶ ∀e∈U} U∣L(V)={f ∈U∶ fP ∈L(V)} U∣L(V)={f ∈U∶ f P ∉L(V)} and aU={au∶ ∀u∈U}
Definition 4.2.9. For convenience, we define analogues of several of the above rela- tions for nodes to implicitly refer to the children of those nodes.
Let uand vbe arbitrary nodes inT∗. Then we define the following L(u)≡L(uC)
u∣L(v)≡ uC∣L(v
C)
u∣L(v)≡ uC∣L(v