<p>Define P as a CSP where X, Y are the variables, domain of both is {1,2,3,4} and conditions in normal form are:</p>

<ol>
<li>node-condition X&lt;4</li>
<li>arc-condition X=Y</li>
</ol>

<p>P is 2-consistent (arc consistent) because for any X value it is possible to find a Y value that fulfills the arc-condition X=Y.</p>

<p>However, P is not 1-consistent (node-consistent) because exist a X value (X=4) that can not fulfill the node condition x&lt;4.</p>

<p>For these reasons, this problem is 2-consistent but not strongly 2-consistent.  </p>

<p>Obviously, it is straightforward to convert this example in a strongly 2-consistent problem, just reducing the domain to {1,2,3}.</p>

CC BY-SA 3.0
2018-02-18T16:42:19.630Z
2018-02-18T19:00:54.843Z
2018-02-18T19:00:54.843Z
pasaba por aqui
<p>From Russell-Norvig:</p>

<blockquote>
  <p>A CSP is strongly k-consistent if it is k-consistent and is also (k − 1)-consistent, (k − 2)-consistent, . . . all the way down to 1-consistent.</p>
</blockquote>

<p>How can a CSP be k-consistent without being (k - 1)-consistent? I can't think of any counter example for this case. Any help would be appreciated.</p>

<linear-algebra>
Does k consistency always imply (k - 1) consistency?
amad-person
# Does k consistency always imply (k - 1) consistency?

(Asked by: username_0 on 2018-02-18)

From Russell-Norvig:

> A CSP is strongly k-consistent if it is k-consistent and is also (k − 1)-consistent, (k − 2)-consistent, . . . all the way down to 1-consistent.

How can a CSP be k-consistent without being (k - 1)-consistent? I can't think of any counter example for this case. Any help would be appreciated.

**Tags:** linear-algebra


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## 1 answers

### Answer by username_1 on 2018-02-18. Score: 2

Define P as a CSP where X, Y are the variables, domain of both is {1,2,3,4} and conditions in normal form are:

1. node-condition X<4
2. arc-condition X=Y

P is 2-consistent (arc consistent) because for any X value it is possible to find a Y value that fulfills the arc-condition X=Y.

However, P is not 1-consistent (node-consistent) because exist a X value (X=4) that can not fulfill the node condition x<4.

For these reasons, this problem is 2-consistent but not strongly 2-consistent.

Obviously, it is straightforward to convert this example in a strongly 2-consistent problem, just reducing the domain to {1,2,3}.


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