as d: teet Sasa model theoretic semantics

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????d?? S?µpe?asµat?????a?

• ??as??? µ???d?? ???te???e???t??? S?µas??????a
  (model theoretic semantics)
• (???a?e? ????e?a?, ???t?se?? ??a??p???s?µ?t?ta?)
• ?e???e?t?µata
• ???a?e? a???e?a? a?ap?te?esµat???? ??a µe????
  a???µ? p??t?se?? (2n e?µ??e?e?)
• ? e???t?s? ap? t?? ??a??p???s?µ?t?ta d?µ?????e?
  t?? a????? p??sd????sµ?? t?? µ??t???? e???
  s?????? p??t?se??
• ?e?te?? p??s????s? ?p?de??t????e???t???
  S?µas??????a (proof theoretic semantics)
• ?a?a???? ???? p??t?se?? ap? ??a a????? s????? µe
  t? ???s? ?a????? pa?a????? (inference rules) ?a?
  ??????? a???µ?t?? (logical axioms)ap?de???.
• ????p???se??
• ??p??? ???ssa
• ?a???e? s??µat?sµ?? t?? ???ssa?
• ?????? a???µata p?? e??a? p??t?se?? p?? a???e????
  se ??a ta µ??t??a (ta?t?????e?)
• ?a???e? pa?a????? p?? ?a ?d????? µ??a???? ap?
  ???sµ??e? p??t?se? se ???e? µe µ??? ???t???? t?
  µ??f? t??? (a?e???t?ta ap? t? s?µas??????a)

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?p?de???-????µat?p???s?
????µat??? s??µata ?a?t?????e? p?? pe???????
s?µß??a p??t?se?? ta ?p??a ???s?µ?p?????ta? sa?
µetaß??t??. ??ass??? p??s????s? ???a a???µat???
s??µata ?a? t? Modus Ponens sa? ?a???a?
pa?a?????. (P1) a?(b?a) (P2)
(a?(b?a))?((a?b)?(a?c)) (P3) (?a??b) ?
((?a?b)?a)) Modus Ponens 1) a 2) a ? b 3) b (ap?
t?? 1,2) ???sµ?? ap?de???? ?p?de??? t?? f ap? t?
S st? s?st?µa pa?a????? p, (s?µß??????µe S -p
f), ???eta? µ?? pepe?asµ??? a???????a p??t?se??
?1, ?2, ..., ??, t?t??a ?ste ??f ?a? ???e ?i
e??a? ? p??tas? t?? S, ? a???µat??? s??µa p??
??e? ?p?ste? ??p??e? a?t??atast?se??, ? pa???eta?
µe t? M.P. ap? d?? p??????µe?e? p??t?se?? t??
a???????a?.
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?p?de???-????µat?p???s?
?a??de??µa ?a ap?de???e? ?t? S -p ?r , ?p??
Sp?q, q?r ?p?de??? 1. p?q, p?? e??a?
p??tas? t?? S, 2. q?r, p?? e??a? p??tas? t??
S, 3. (q?r)?(p?(q?r)), a???µa (P1) µe
a?t??atast?se?? a(q?r) ?a? bp. 4. p?(q?r), µe
M.P. p??? st?? 2,3. 5. (p?(q?r))?((p?q)?(p?r)),
a???µa (P2) µe a?t??atast?se?? ap, bq, cr. 6.
(p?q)?(p?r), µe M.P. p??? st?? 4,5. 7. (p?r), µe
M.P. p??? st?? 1,6. Te???µa ????t?ta? (Soundness
Theorem) S -p f ? S f ?? ?e???µa a?t? ?ta?
ap?de????eta? ??? ??a s?st?µa pa?a????? fa?e???e?
?t? ???e p??tas? ? ?p??a ap?de????eta? µe ß?s? t?
s?st?µa pa?a????? e??a? p???µat? a????? st??
s?µas??????a. ??µe t?te ?t? t? s?st?µa ap?de????
(? s?st?µa pa?a?????) e??a? ????. Te???µa
?????t?ta? (Completeness Theorem) S f ? S
-p f ?? ?e???µa a?t? ?ta? ap?de????eta? ??? ??a
s?st?µa pa?a????? fa?e???e? ?t? ???e p??tas? ?
?p??a a???e?e? st?? s?µas??????a ap?de????eta? µe
ß?s? t? s?st?µa pa?a?????. ?e ???a ????a, de?
?p???e? p??tas? p?? ?a a???e?e? st?? s?µas??????a
?a? ?a µ? µp??e? ?a ap?de???e? ap? t? s?st?µa
pa?a?????. ??µe se a?t? t?? pe??pt?s? ?t? t?
s?st?µa ap?de???? (? s?st?µa pa?a?????) e??a?
p???e?.
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???? t?? ?p?fas??
???? t?? ?p?fas?? (Resolution Principle) J.A.
Robinson (1965) ?a???a? pa?a????? p?? µp??e? ?a
???s?µ?p????e? ??a t?? pa?a???? t?? ?e???
p??tas?? se µ?a d?ad??as?a p?? µ????e? µe t?? e??
?t?p? apa????. "?ed?µ???? d?? p??t?se?? C1 ?a?
C2 p?? e??a? se d?a?e??t??? µ??f?, e?? ?p???e?
??a ?e?t??? L1 st?? C1 t? ?p??? e??a? a?t??et? µe
??a ?e?t??? L2 t?? C2, t?te d?µ??????se µ?? ??a
p??tas? C p?? ap?te?e?ta? ap? t?? C1, C2 se
d???e??? af?? ??e?? d?a????e? ta L1 ?a? L2". C
ap?fa???µe?? (resolvent) t?? C1, C2 C1, C2
??????? p??t?se?? t?? C p.?. ?p? ?p?q?r, ?q?s
p????pte? ?p?r?s ???sµ?? ?p?de??? µe t?? a???
t?? ap?fas?? ?p?de??? t?? f ap? t? S st? s?st?µa
pa?a????? p?? ???s?µ?p??e? t?? a??? t?? ap?fas??,
(s?µß??????µe S -r f), ???eta? µ?? pepe?asµ???
a???????a p??t?se?? ?1,?2, ...,??, t?t??a ?ste
??? ?a? ???e ?i e??a? ? p??tas? t?? S'??f'
(?p?? ? t???? ?p?d????e? ?t? ta s????a ?????
µetat?ape? se ?a?????? s??e??t??? µ??f?), ?
pa???eta? µe t?? a??? t?? ap?fas?? ap? d??
p??????µe?e? p??t?se?? t?? a???????a?.
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?p?de??? µe t?? ???? t?? ?p?fas??

• ???????µ?? ap?de???? µe t?? a??? t?? ap?fas??
• Te????µe ??a s????? p??t?se?? S ?a? t?? p??tas? f
  p?? ?????µe ?a ap?de????µe. ? a??????µ?? ??e? ??
  e???
• ??µa 1?. ?etat??p??µe ??e? t?? p??t?se?? t?? S se
  ?a?????? s??e??t??? µ??f? (CNF) ?a? d?µ???????µe
  ?ts? t? s????? S' d?a?es?µ?? p??t?se??.
• ??µa 2?. ??t?st??f??µe t?? f (se ?f), t??
  µetat??p??µe se ?a?????? s??e??t??? µ??f? ?a?
  e?????µe t? s????? t?? pa?a??µ???? p??t?se?? F'
  µe t? S' S'' S'?F
• ??µa 3?. ?pa?a?aµß????µe t? pa?a??t? ß?µa ???
  ?t?? ß?e?e? ??p??a a?t?fas? ? ??? ?t?? de? µp??e?
  ?a ???e? ?aµµ?a p???d??
• ??a?????µe t??a?a d?? p??t?se?? ap? t? s????? S''
  t?? d?a??s?µ?? p??t?se?? ?a? efa?µ????µe µeta??
  t??? t?? a??? t?? ap?fas??. ??? ? pa?a??µe??
  p??tas? e??a? ? ?de?a p??tas?, a?t? s?µa??e? ?t?
  ft?saµe se a?t?fas? ?a? staµat?µe, a????? t??
  p??s??t??µe st? s????? t?? d?a??s?µ?? p??t?se??
  ?a? s??e?????µe.

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???? t?? ap?fas?? st?? ???tas?a?? ??????
?a??de??µa G?a t? s????? Sp ? q, ?p ? q, p ?
?q, ?p ? ?q . ?a ap?de???e? ?t? t? S e??a? µ?
??a??p???s?µ? s????? p??t?se??. (1) p ? q (2)
?p ? q (3) p ? ?q (4) ?p ? ?q pa?????µe ta
pa?a??t? ap?fa???µe?a (5) q ap? (1) ?a?
(2) (6) ?q ap? (3) ?a? (4) (7) ? ap?
(5) ?a? (6) S??µat??? a?apa??stas?
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???? t?? ap?fas?? st?? ???tas?a?? ??????
?a??de??µa 1. Ta pa?a???????s? t? ????, e?? ???
?????µ? t??e??as? ?a? de? e?µa? apas????µ????. 2.
Ta ????? t? ????, e?? ??? video ?a? e?µa?
apas????µ????. 3. Ta pa?a???????s? t? ???? e?? t?
??? ????e?. 4. ??? ?????µ? t??e??as?. 5. ???
video. p"Ta pa?a???????s? t? ????" q"???
?????µ? t??e??as?" r"e?µa? apas????µ????"
s"?a ????? t? ????" t"??? video" S p ? q
??r, s? t ? r, p ? s, q, t. (CNF) S' p ?
?q ? r, s ? ?t ? ?r, p ? ?s, q, t. T????µe
?a ap?de????µe t?? p, ??a pa?????µe t? ?p (?d? se
CNF) S''S??p p ? ?q ? r, s ? ?t ? ?r,
p ? ?s, q, t, ?p 1. p ? ?q ? r, (p??tas? t??
S') 2. ?p, (p??tas? t?? S) 3. ?q ? r, (a??? t??
ap?fas?? st?? 1,2) 4. s ? ?t ? ?r, (p??tas? t??
S') 5. p ? ?s, (p??tas? t?? S') 6. p ? ?t ?
?r, (a??? t?? ap?fas?? st?? 4,5) 7. ?t ? ?r,
(a??? t?? ap?fas?? st?? 2,6) 8. t, (p??tas? t??
S') 9. ?r, (a??? t?? ap?fas?? st?? 7,8) 10. q,
(p??tas? t?? S') 11. r, (a??? t?? ap?fas?? st??
3,10) 12. ? , (a??? t?? ap?fas?? st?? 9,11)
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????eµat??? S??e??t??? ?a?????? ???f?
???sµ?? ??? p??tas? e??a? se ????eµat???
?a?????? ???f? (Prenex conjunctive normal form)
a? ??e? t?? e??? d??ta?? (Q1 X1) (Q2 X2) ...
(Qn Xn) M ?p?? ta Q1, Q2, ..., Qn e??a?
p?s?de??te?, ?a? M µ?a p??tas? p?? de? pe????e?
p?s?de??te?. ?etas??µat?sµ?? ap? wff se
p???eµat??? ?a?????? s??e??t??? µ??f? 1)
??t??at?stas? t?? f?? µe f??? ?a? t?? f?? µe
(f???)?(?f??). 2) ??t??at?stas? t?? ?(?f) µe f,
?(f??) µe ?f???, ?(f??) µe ?f???,? (??)f µe
(??)(?f), ?(??)f µe (??)(?f) ??? ?t?? ???e
eµf???s? e??? ? p????e?ta? µ??? e??? at?µ??. 3)
?et???µ????µe t?? µetaß??t?? ?ts? ?ste ???e
p?s?de??t?? ?a pe????e? µ?a µ??ad???
µetaß??t?. 4) ??t??at?stas? t?? (??) f?? µe (??)
(f??) ?a? t?? f?(??)? µe (??)(f??) ?a? ?e???? t??
(Q?)f???? µe (Q?)(f????) ?a? t?? f ??
(Q(?)??) µe (Qx)(f????) ??? ?t?? ???? ??
p?s?de??te? ß??s???ta? st?? a??? t??
p??tas??. 5) ??t??at?stas? t?? (f??)?? µe
(f??)?(???) ?a? f?(???) µe (f??)?(f??) ??? ?t?? ?
p??tas? ?a µetat?ape? st?? ep???µ?t? µ??f?
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F??se?? (clauses)
??a s????? ap? f??se?? ap?te?e?ta? ap?
d?a?e??t???? p??t?se?? p?? ?e?????ta? ?t?
ß??s???ta? se s??e???. ?? f??se?? de? pe???????
p?s?de??te?, a??? ??? ???e µetaß??t? t???
e???e?ta? ??a? ?a??????? p?s?de??t?? st?? a???
t?? ???e f??s??. ?etas??µat?sµ?? ap? p???eµat???
?a?????? s??e??t??? µ??f? se f??se?? ?st? ? (Q1
X1) (Q2 X2) ... (Qn Xn) M se Prenex-CNFµ??f? 1)
?etas??µat????µe t?? M se ?a?????? s??e??t???
µ??f? (e?? de? e??a? ?d?). 2) G?? ???e ?pa???a??
p?s?de??t? Qi (d??. Qi ?) t?? ?p???? de?
p???????ta? ?a??????? p?s?de??te?, a?t??a??st??µe
t?? a?t?st???? µetaß??t? µe µ?? sta?e??
d?af??et??? ap? t?? ???e? sta?e??? t?? p??tas??
M. 3) G?? ???e ?pa???a?? p?s?de??t? Qi, (d??. Qi
?), t?? ?p???? p???????ta? ?? (?Xs1),...,(?Xsn)
?a??????? p?s?de??te?, a?t??a??st??µe t??
a?t?st???? µetaß??t? µe µ?? s????t?s?
f(Xs1,..,Xsn) ?p?? t? s??a?t?s?a?? s?µß??? f
e??a? d?af??et??? ap? ta ???a s??a?t?s?a??
s?µß??a t?? p??tas?? M. 4) ?fa????µe t???
?a???????? p?s?de??te?.
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