NonSimplyMatching

# Title: NonSimplyMatching

## 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 0 1 0

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}
}

Perdita Stevens