# Notation

Potentially idiosyncratic notation used throughout the rest of the pages.

## Sets

When the integer $$X$$ appears in a context where a set is expected, it is taken to mean $$\{ 0, \ldots, X-1 \}$$. F.ex. $$A \times 2 = A \times \{0,1\}$$.

When an index $$i \in \{ 0, 1 \}$$ appears primed, $$i^\prime$$ it is taken to mean $$i$$'s complement in $$\{ 0, 1 \}$$.

## Variables

A variable $$x$$ inhabits some set $$X$$. Its valuation is denoted $$?x$$. The valuation is assumed to change in discrete steps, and the trace of the variable is the sequence $$\{ ?x^{t} \ldots \}$$,with $$t \in \N$$. If the superscript is omitted $$?x$$ typically refers to the latest valuation of the trace. $$?x \in X$$ is taken to mean a variable $$x$$ in $$X$$.

If a variable $$?p \in P$$, is a tuple, $$?(x,y) \in X \times Y$$ the same notation is used on the components of the tuple, ie $$?p^t$$, $$?(x,y)^t$$ and $$(?x^t,?y^t)$$ all convey the same meaning.

## Higher order functions

A higher order function is a function, $$f \colon A \to G$$, where $$g \in G$$ are themselves functions, $$g \colon B \to C$$. It is denoted here with $$f \colon A \to B \to C$$, ie $$\to$$ is right-associative in this context.

## Graphs

The junction of a directed graph, $$G = (V,E)$$, is a set of incident edges, $$y \subseteq E$$. Two edges, $$e_0 = (v_{00},v_{01})$$, $$e_1 = (v_{10},v_{11})$$, are said to be incident if source or target is equal, ie $$(v_{00} = v_{10} \lor v_{01} = v_{11})$$. The junctions, $$Y = \{ y \ldots \}$$, are found by taking the transitive closure of this incidence relation on $$E$$. Note that $$\bigcup \limits_{y \in Y} = E$$ and $$y_0 \neq y_1 \implies y_0 \cap y_1 = \emptyset$$.

$$V(y)$$ denotes vertices incident to $$e \in y$$, $$V^0(y)$$ denotes sources in $$e \in y$$, and $$V^1(y)$$ denotes targets. $$y^0(v)$$ denotes the junction containing the edges whose target is $$v$$, and $$y^1(v)$$ denotes the junction containing the edges whose source is $$v$$.

Lets clarify with an example, the graph $$G_0$$:

\begin{equation*} \begin{gathered} G_0 = (V_0,E_0) \\ V_0 = \{ v_i \mid i \in 4 \}\\ E_0 = \{ (v_0,v_2),(v_1,v_2),(v_2,v_3) \} \\ \end{gathered} \end{equation*}

Or, as a diagram:

Here are some sample results for $$G_0$$ using the definitions above:

\begin{equation*} \begin{aligned} Y &= \{ \{e_0,e_1 \}, \{e_2\} \} \\ V^0(y_0) &= \{ v_0, v_2 \} \\ V^1(y_1) &= \{ v_3 \} \\ y^0(v_2) &= y_0 \\ y^0(v_0) &= \emptyset \end{aligned} \end{equation*}