
## W.R.T. F¬P.

Proof. It is clear that the mbrs of up, u¬p, fp, f¬p, f 0 p have the same upper left corner point.

We now consider the lower right corner points of M(up) and M(u¬p). The constraints p(up, fp) and up O f¬p restrict the lower right corner point of M(up) to the shaded part in Figure 6(a). Similarly, the constraints p(u¬p, f¬p) and u¬p O f 0 p restrict the lower right corner point of M(u¬p) to the shaded part in Figure 6(b). By adding constraint fp O u¬p, the possible area of the lower right corner point of M(u¬p) is further restricted to the shaded part in Figure 6(c).

Therefore, if up and u¬p are both vertical or both horizontal, then the lower right corner point of M(up) must be in M(u¬p), which implies M(up) ⊂ M(u¬p) as their upper left corner points are the same. By imposing p(up, u¬p) (which needs two extra auxiliary variables), M(up) and M(u¬p) are necessarily partially overlapping, i.e., they can not be vertical or horizontal at the same time.

The mutual exclusion of vertically and horizontally instantiations of dual variables up, u¬p corresponds to the mutual exclusion of the truth values of p and its negation ¬p.

For each propositional variable pi ∈ V = {p1, p2, · · · , pn}, we introduce a pair of dual spatial variables ui ≡ upi and u¬i ≡ u¬pi
, (17)
and three frame spatial variables

$$f_{i}\equiv f_{p_{i}},f_{-i}\equiv f_{\neg p_{i}}\;\mathrm{and}\;f_{i}^{0}\equiv f_{p_{i}}^{0},$$
$\left(18\right)^2$
$\mathbf{a}\cdot\mathbf{a}\in\mathbb{R}^{n}$
, (18)
and construct, as described above in Definition 5, a basic CDC network Npi over spatial variables {ui, u¬i, fi, f¬i, f 0 i }.

In order to fix the relative direction between two frame spatial variables of different propositional variables (cf. Figure 7), we introduce a set of reference spatial variables.

Precisely, let

$$\begin{array}{c}{{V_{r e f}=\{w_{r e f},f_{r e f},f_{-r e f},f_{r e f}^{0}\},}}\\ {{{}}}\\ {{{\mathcal{N}}_{r e f}=\{w_{r e f}\;O\;f_{r e f}\;O\;f_{-r e f}\;O\;f_{r e f}^{0},}}\\ {{f_{r e f}^{0}\;S\!:\!O\;f_{-r e f}\;S\!:\!O\;f_{r e f}\;S\!:\!O\;w_{r e f}\},}}\end{array}$$
$$(20)$$

where the shorthand, say, x S : O y S : O z denotes that x S : O y and y S : O z.

Furthermore, we require

$$\begin{array}{l l}{{\parallel(f_{1},f_{r e f}),\parallel(f_{-1},\ f_{-r e f}),\parallel(f_{1}^{0},f_{r e f}^{0}),}}\\ {{\parallel(f_{i+1},f_{i}),\parallel(f_{-(i+1)},f_{-i}),\parallel(f_{i+1}^{0},f_{i}^{0},),}}&{{\qquad(1\leq i<n)}}\end{array}$$
$$(19)$$
$$\begin{array}{l}{(21)}\\ {(22)}\end{array}$$
$1\,\!$ 3. 