This section defines some terms and notation that will be used freely in later sections. The list of definitions given below is ordered so that each definition only uses terms that have been defined in previous definitions.

Throughout this manual, when we say "...X1 (ditto, X2)...", we mean that everything we said about X1 applies also to X2. 
A bounded shape.
arrow (edge) 
A directed line.
Two separate nodes.
internal arrow 
An arrow that connects two nodes.
external arrow 
An arrow that exits a node, but does not enter another node. These arrows are NOT draw by Quantum Fog, but it is occasionally useful to speak of them. See Ref. [Tuc95] if interested. A net is considered the same, whether its external arrows are drawn or not.
internal node 
A node that has one or more outgoing internal arrows.
external node 
A node that has no outgoing internal arrows.
A collection of nodes and arrows. In this manual, the arrows of our graphs are all internal ones.
labeled graph 
A graph in which each node has been assigned a name (label).
A closed path of arrows within a graph, with the arrows in the path all pointing either clockwise or counter-clockwise.
acyclic graph 
A graph that contains no cycles.
root node 
A node with no incoming internal arrows. It corresponds to a "wave-function" in Quantum Mechanics.
parent nodes of node X 
Nodes connected to X by arrows that enter X.
state of node X 
A state that X may assume.
state set of node X 
The set of states that X may assume.
degeneracy of node X 
The number of states that X may assume.
input state (parent state) of node X 
An assignment of a state to each of the parent nodes of node X.
node matrix (transition matrix) 
A matrix assigned to a single node. The rows of the matrix are labeled by the states of the node and the columns by its input states.
net (network) 
As in Ref. [Tuc95], we define a net to be an abstract construct comprising a graph and a node matrix for each node of the graph. The graph is labeled and acyclic. We define a Classical Bayesian net  to be a net of the type used in Refs.1-3 —a net such that the entries of its node matrices are non-negative real numbers. We define a Quantum Bayesian net  to be a net of the type used in Ref.[Tuc95]— a net such that the entries of its node matrices are complex numbers. In this manual, unless stated otherwise, when we speak simply of a net, we mean a Quantum Bayesian one.
node amplitude (transition amplitude)   
An entry of a node matrix of a Quantum Bayesian net.
story (graph instantiation)
An assignment of a state to each node of the graph. Stories correspond to what the Quantum Mechanics literature calls Feynman paths. Unlike the term "story", the term "path" feels awkward to us when speaking about a system of more than one particle. Whenever possible, we will use the term "story " instead of "path".
An assignment of a state to each of the external nodes of a graph. An ending is part of a story.
conditional probability P(H|E)
The probability that H occurs, conditioned on  the fact (i.e., assuming that) E occurs. H is often called the hypothesis and E the evidence or condition or data.
active state 
A state is active in a physical experiment if the experimenter has determined by prior measurement (or can determine if he tries) that the state can occur during the experiment. Quantum Fog calculates conditional probability distributions for nodes and bi-nodes, the evidence for the conditional probabilities being which states of each node are active.
active state list (set) of node X   
The set of states of node X that are active.
default active state list   
An active state list that is as full as possible—it contains ALL the states of the node.
evidence case (e-case) 
An assignment of an active state set to each node of the graph.
story amplitude (net amplitude) 
A story 's amplitude is calculated as follows. A story determines a unique transition amplitude Ai for each node. How? For the i'th node, choose as Ai the element of its transition matrix with (row, column) = (its state, its input state). Only one such row-column pair is consistent with the story. The story amplitude is defined as the product A1*A2*...AN, where N is the number of nodes.
possible story 
A story that assigns active states to all nodes of the graph, and that has a non-zero amplitude.
Feynman Integral 
The Feynman Integral FI(e) for the ending e is the sum of the complex amplitudes of all possible stories with ending e. In general, FI(e) depends on which states of each node are active.
total probability of all stories 
The sum over all e of |FI(e)|2, assuming that all node states are active, where FI(e) is the Feynman Integral corresponding to the ending e. This probability should be 1.
deterministic node 
A node for which the input state uniquely determines the state of the node. For each input state, only one state of the node has a non-zero amplitude.
Given a classical probability distribution p1, p2, ..., pD, its Entropy S is the following number :

S = -[p1 log2(p1)   + p2 log2(p2) + ... + pD log2(pD)].

For example, if all pi are equal to 1/D, this gives S=log2(D). D is the number of states or degeneracy of the distribution. If D = 2B (B is referred to as the number of bits), then S = B.

As another example, suppose that one pi equals one and all others equal zero. Then, S=0 (because log1=0,  and xlog(x) tends to 0 as x tends to 0 with x>0.)

In general, one can show that 0<= S <= log2(D) so the two examples just given represent the two extremes.

You can think of S as measuring the spread (dispersion) of the distribution.

TEXT file 
A data file in a special format. Most word processors and spreadsheets allow you to save a file in this format. We recommend that you use TextWrangler, a free application, to deal with all TEXT files associated with Quantum Fog. 
Quantum Fog uses TEXT files for both reading and writing data; specifically, it reads and writes TEXT files with UNIX line endings (TextWrangler allows you to save TEXT files with either UNIX, Mac Classic, or Windows line endings).
TEXT files generated by Quantum Fog should be viewed in a fixed-width font like Monaco or Courier. Otherwise, columns will not line up properly. Alternatively, you can open the file with a spreadsheet application. This way, you can use the powerful features of the spreadsheet to manipulate the data. When reading a TEXT file, spreadsheets will jump to the cell in the next column and the same row when they see a tab, and to the first cell in the next row when they see an end-of-line. TEXT files created by Quantum Fog are formatted with this convention in mind.
Result of pressing the space-bar, RETURN or TAB keys.
[description of something] 
A symbol that should be replaced by whatever is described inside the square brackets. In particular, [white-spaces]  means one or more white-spaces.
Quantum Fog allowed name 
A string of no more than 31 characters and such that none of those characters is the hash character (#) or a white-space.

What Next...

Some of the above definitions may not make sense to you the first time you read them. Don't worry. You'll understand them better as you learn more about Quantum Fog.

Congratulations! You've just endured one of the hardest parts of this manual. And now for a fun part.

We suggest that next you read the section entitled "Quick Tour".

[Table Of Contents]