ONT Re: Zeroth Order Theories (ZOT's)
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One of the issues that my pondering weak and weary over
has caused me to burn not a few barrels of midnight oil
over the past elventeen years or so is the relationship
among divers and sundry "styles of inference", by which
I mean particular choices of inference paradigms, rules,
or schemata. The chief breakpoint seems to lie between
information-losing and information-maintaining modes of
inference, also called "implicational" and "equational",
or "projective" and "preservative" brands, respectively.
Since it appears to be mostly the implicational and projective
styles of inference that are more familiar to folks hereabouts,
I will start off this subdiscussion by introducing a number of
risibly simple but reasonably manageable examples of the other
brand of inference, treated as equational reasoning approaches
to problems about satisfying "zeroth order constraints" (ZOC's).
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Applications of a Propositional Calculator:
Constraint Satisfaction Problems.
Jon Awbrey, April 24, 1995.
The Four Houses Puzzle
Constructed on the model of the "Five Houses Puzzle" in [VaH, 132-136].
Problem Statement. Four people with different nationalities live in the
first four houses of a street. They practice four distinct professions,
and each of them has a favorite animal, all of them different. The four
houses are painted different colors. The following facts are known:
| 1. The Englander lives in the first house on the left.
| 2. The doctor lives in the second house.
| 3. The third house is painted red.
| 4. The zebra is a favorite in the fourth house.
| 5. The person in the first house has a dog.
| 6. The Japanese lives in the third house.
| 7. The red house is on the left of the yellow one.
| 8. They breed snails in the house to right of the doctor.
| 9. The Englander lives next to the green house.
| 10. The fox is in the house next to to the diplomat.
| 11. The Spaniard likes zebras.
| 12. The Japanese is a painter.
| 13. The Italian lives in the green house.
| 14. The violinist lives in the yellow house.
| 15. The dog is a pet in the blue house.
| 16. The doctor keeps a fox.
The problem is to find all of the assignments of
features to houses that satisfy these requirements.
Logical Input File: House^4.Log
o---------------------------------------------------------------------o
| |
| eng_1 doc_2 red_3 zeb_4 dog_1 jap_3 |
| |
| (( red_1 yel_2 ),( red_2 yel_3 ),( red_3 yel_4 )) |
| (( doc_1 sna_2 ),( doc_2 sna_3 ),( doc_3 sna_4 )) |
| |
| (( eng_1 gre_2 ), |
| ( eng_2 gre_3 ),( eng_2 gre_1 ), |
| ( eng_3 gre_4 ),( eng_3 gre_2 ), |
| ( eng_4 gre_3 )) |
| |
| (( dip_1 fox_2 ), |
| ( dip_2 fox_3 ),( dip_2 fox_1 ), |
| ( dip_3 fox_4 ),( dip_3 fox_2 ), |
| ( dip_4 fox_3 )) |
| |
| (( spa_1 zeb_1 ),( spa_2 zeb_2 ),( spa_3 zeb_3 ),( spa_4 zeb_4 )) |
| (( jap_1 pai_1 ),( jap_2 pai_2 ),( jap_3 pai_3 ),( jap_4 pai_4 )) |
| (( ita_1 gre_1 ),( ita_2 gre_2 ),( ita_3 gre_3 ),( ita_4 gre_4 )) |
| |
| (( yel_1 vio_1 ),( yel_2 vio_2 ),( yel_3 vio_3 ),( yel_4 vio_4 )) |
| (( blu_1 dog_1 ),( blu_2 dog_2 ),( blu_3 dog_3 ),( blu_4 dog_4 )) |
| |
| (( doc_1 fox_1 ),( doc_2 fox_2 ),( doc_3 fox_3 ),( doc_4 fox_4 )) |
| |
| (( |
| |
| (( eng_1 ),( eng_2 ),( eng_3 ),( eng_4 )) |
| (( spa_1 ),( spa_2 ),( spa_3 ),( spa_4 )) |
| (( jap_1 ),( jap_2 ),( jap_3 ),( jap_4 )) |
| (( ita_1 ),( ita_2 ),( ita_3 ),( ita_4 )) |
| |
| (( eng_1 ),( spa_1 ),( jap_1 ),( ita_1 )) |
| (( eng_2 ),( spa_2 ),( jap_2 ),( ita_2 )) |
| (( eng_3 ),( spa_3 ),( jap_3 ),( ita_3 )) |
| (( eng_4 ),( spa_4 ),( jap_4 ),( ita_4 )) |
| |
| (( gre_1 ),( gre_2 ),( gre_3 ),( gre_4 )) |
| (( red_1 ),( red_2 ),( red_3 ),( red_4 )) |
| (( yel_1 ),( yel_2 ),( yel_3 ),( yel_4 )) |
| (( blu_1 ),( blu_2 ),( blu_3 ),( blu_4 )) |
| |
| (( gre_1 ),( red_1 ),( yel_1 ),( blu_1 )) |
| (( gre_2 ),( red_2 ),( yel_2 ),( blu_2 )) |
| (( gre_3 ),( red_3 ),( yel_3 ),( blu_3 )) |
| (( gre_4 ),( red_4 ),( yel_4 ),( blu_4 )) |
| |
| (( pai_1 ),( pai_2 ),( pai_3 ),( pai_4 )) |
| (( dip_1 ),( dip_2 ),( dip_3 ),( dip_4 )) |
| (( vio_1 ),( vio_2 ),( vio_3 ),( vio_4 )) |
| (( doc_1 ),( doc_2 ),( doc_3 ),( doc_4 )) |
| |
| (( pai_1 ),( dip_1 ),( vio_1 ),( doc_1 )) |
| (( pai_2 ),( dip_2 ),( vio_2 ),( doc_2 )) |
| (( pai_3 ),( dip_3 ),( vio_3 ),( doc_3 )) |
| (( pai_4 ),( dip_4 ),( vio_4 ),( doc_4 )) |
| |
| (( dog_1 ),( dog_2 ),( dog_3 ),( dog_4 )) |
| (( zeb_1 ),( zeb_2 ),( zeb_3 ),( zeb_4 )) |
| (( fox_1 ),( fox_2 ),( fox_3 ),( fox_4 )) |
| (( sna_1 ),( sna_2 ),( sna_3 ),( sna_4 )) |
| |
| (( dog_1 ),( zeb_1 ),( fox_1 ),( sna_1 )) |
| (( dog_2 ),( zeb_2 ),( fox_2 ),( sna_2 )) |
| (( dog_3 ),( zeb_3 ),( fox_3 ),( sna_3 )) |
| (( dog_4 ),( zeb_4 ),( fox_4 ),( sna_4 )) |
| |
| )) |
| |
o---------------------------------------------------------------------o
Sense Outline: House^4.Sen
o-----------------------------o
| eng_1 |
| doc_2 |
| red_3 |
| zeb_4 |
| dog_1 |
| jap_3 |
| yel_4 |
| sna_3 |
| gre_2 |
| dip_1 |
| fox_2 |
| spa_4 |
| pai_3 |
| ita_2 |
| vio_4 |
| blu_1 |
o-----------------------------o
Table 1. Solution to the Four Houses Puzzle
o------------o------------o------------o------------o------------o
| | House 1 | House 2 | House 3 | House 4 |
o------------o------------o------------o------------o------------o
| Nation | England | Italy | Japan | Spain |
| Color | blue | green | red | yellow |
| Profession | diplomat | doctor | painter | violinist |
| Animal | dog | fox | snails | zebra |
o------------o------------o------------o------------o------------o
Have to break here ...
Will explain later ...
Jon Awbrey
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