Bx Examples Repository
Title: NonSimplyMatching
Version: 0.1
Type: Precise
Overview
This artificial example shows a bx that is matching but not simply matching.
Models
Each model is a boolean: $M = N = \{0,1\}$
Consistency
$R(m,n)$ iff $mn = 0$; that is
| R | 0 | 1 |
|---|---|---|
| 0 | T | T |
| 1 | T | F |
Consistency Restoration
There's only one choice if $R$ is to be correct and hippocratic. Writing Rf, Rb for forward and backward consistency restoration functions:
| Rf | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| Rb | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 0 |
Properties
- correct
- hippocratic
- matching (by bijection $f : 0 \mapsto 1, f : 1 \mapsto 0$)
but not
- simply matching (see discussion)
- undoable.
Discussion
This is Example 8 from Stevens' paper referred to below. In terms of the equivalences $\sim_F$ and $\sim_B$ on each of $M$, $N$ defined there:
Each element of $M$ forms an equivalence class under each equivalence, and dually for $N$. Therefore there is only one choice of transversal, and we identify the elements with the equivalence classes.
Considered as a subset of $M_F \times M_B$, M is the diagonal subset {(0,0),(1,1)} where (0,0) represents 0 and (1,1) represents 1. Similarly for $N$. That is, the coordinate grids for $M$ and $N$ are both, identically:
| 1 | 1 | |
|---|---|---|
| 0 | 0 | |
| 0 | 1 |
We have here the simplest possible example in which there are two distinct elements of $M_F$ (0 and 1) compatible with one element of $N_B$ (0), and only one of those $M_F$ elements (0) is also compatible with a second, distinct element of$N_B$ (1). In other words, both columns of $M$'s coordinate grid are compatible with the 0 row of $N$'s, and the 0 column of $M$'s grid is also compatible with the 1 row of $N$'s.
References
@article{DBLP:journals/eceasst/Stevens12,
author = {Perdita Stevens},
title = {Observations relating to the equivalences induced on model
sets by bidirectional transformations},
journal = {ECEASST},
volume = {49},
year = {2012},
ee = {http://journal.ub.tu-berlin.de/eceasst/article/view/714},
bibsource = {DBLP, http://dblp.uni-trier.de}
}
Author(s)
Perdita Stevens