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====== Solutions to the Propositions and Inference lab ======

===== 5.1 =====
<code>
% heating_k1 is true if kettle k1 is heating 
heating_k1 <-
   plugged_into_k1_p1 & 
   live_p1 & 
   turned_on_k1 & 
   filled_with_water_k1.

% plugged_into_k1_p1 is true if k1 is plugged into power outlet p1
plugged_into_k1_p1.
plugged_into_k2_p2.

% turned_on_k1 is true if k1 is turned on 
turned_on_k1.
turned_on_k3.

% filled_with_water_k1 is true if k1 is filled with water 
filled_with_water_k1.
</code>

===== 5.2 =====
(a)<code>
pressurized p3 <- on t1 & pressurized p1.
wet sink <- on t3 & pressurized p3.
flow d3 <- wet sink & unplugged sink.
flow d1 <- flow d3.</code>

(b) We would expect the user of the house to be able to determine whether the taps are on and whether the plugs are in or not.

<nowiki>(c)</nowiki> Here is the minimal set of clauses:
<code>
wet floor <- wet sink & plugged sink. 
wet floor <- wet bath & plugged bath.
</code>

(d) The denotation is given in the diagram:
{{ :teaching:is:ailog_shower.png?400 |}}
Here is the simplest axiomatization:
<code>
pressurized hws <- on t4 & pressurized p2. 
pressurized p4 <- pressurized hws. 
pressurized p5 <- pressurized hws.
flow shower <- on t5 & pressurized p5.
wet sink <- on t6 & pressurized p4.
</code>
The file [[http://www.cs.ubc.ca/spider/poole/aibook/figures/ch05/plumbing2.ailog|http://www.cs.ubc.ca/spider/poole/aibook/figures/ch05/plumbing2.ailog]] contains a solution that will run in AILog.

===== 5.3 =====
(a) Give a model of the model of the knowledge base:
The minimal model has { a, c, e, f, g } true and { b, d, h } false. Another model is where all atoms are true.

(b) Give an interpretation that is not a model of the knowledge base:
The interpretation with a false, and all other atoms true is not a model as the ﬁrst clause is false in this interpretation.
The interpretation with all atoms false is not a model as the 7th clause is false in this interpretation.

<nowiki>(c)</nowiki> Give two atoms that are logical consequences of the KB:
Any two of { a, c, e, f, g } .

(d) Give two atoms that are not logical consequences of the KB:
Any two of { b, d, h } .

===== 5.4 =====
(a)
<code>
C={}
C={c,e} (by 4 & 5)
C={c,e,b} (by 3)
C={c,e,b,a} (by 1)
C={c,e,b,a,j} (by 9)
</code>

(b) each of {c,e,b,a,j} is true and the other atoms (including f) are false.

<nowiki>(c)</nowiki> The rightmost branch is the top-down derivation:
<code>
   {yes⇽a}
       |
   {yes⇽b,c}
        / \
{yes⇽d,c} {yes⇽e,c}
    |         |
{yes⇽h,c} {yes⇽c}
              |
          {yes⇽}
</code>