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  1. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ▲❛♥❣✉❛❣❡ ♦✈❡r✈✐❡✇ ❚✐❧❧ ▼♦ss❛❦♦✇s❦✐ ✶ ❖❧✐✈❡r ❑✉t③ ✶ ❋❛❜✐❛♥ ◆❡✉❤❛✉s ✶ ❈❤r✐st♦♣❤ ▲❛♥❣❡ ✷ ▼✐❤❛✐ ❈♦❞❡s❝✉ ✶ ✶ ❯♥✐✈❡rs✐t② ♦❢ ▼❛❣❞❡❜✉r❣ ✷ ❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥ ❖❜❡rs❡♠✐♥❛r✱ ✷✵✶✹✲✶✶✲✶✷ ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✶

  2. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥ ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✷

  3. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❇✐❣ P✐❝t✉r❡ ♦❢ ■♥t❡r♦♣❡r❛❜✐❧✐t② ▼♦❞❡❧✐♥❣ ❙♣❡❝✐✜❝❛t✐♦♥ ❑♥♦✇❧❡❞❣❡ ❡♥❣✐♥❡❡r✐♥❣ ❖❜❥❡❝ts✴❞❛t❛ ❙♦❢t✇❛r❡ ❈♦♥❝❡♣ts✴❞❛t❛ ▼♦❞❡❧s ❙♣❡❝✐✜❝❛t✐♦♥s ❖♥t♦❧♦❣✐❡s ▼❡t❛♠♦❞❡❧s ❙♣❡❝✐✜❝❛t✐♦♥ ❧❛♥❣✉❛❣❡s ❖♥t♦❧♦❣② ❧❛♥❣✉❛❣❡s ❉✐✈❡rs✐t② ❛♥❞ t❤❡ ♥❡❡❞ ❢♦r ✐♥t❡r♦♣❡r❛❜✐❧✐t② ♦❝❝✉r ❛t ❛❧❧ t❤❡s❡ ❧❡✈❡❧s✦ ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✸

  4. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ❖♥t♦❧♦❣✐❡s Class : Person Class : Female Class : Woman EquivalentTo : Person and Female Class : Man EquivalentTo : Person and not Woman ObjectProperty : hasParent ObjectProperty : hasChild InverseOf: hasParent ObjectProperty : hasHusband Class : Mother EquivalentTo : Woman and hasChild some Person Class : Father EquivalentTo : Man and hasChild some Person Class : Parent EquivalentTo : Father or Mother ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✹

  5. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧s Pr♦t♦❝♦❧ st❛t❡ ♠❛❝❤✐♥❡ ❙t❛t❡ ♠❛❝❤✐♥❡ ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✺

  6. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ❙♣❡❝✐✜❝❛t✐♦♥s sort Elem free type List[Elem] ::= [] | __::__(Elem; List[Elem]) pred __elem__ : Elem * List[Elem} preds is_ordered : List[Elem]; permutation : List[Elem] * List[Elem] op sorter : List[Elem]->List[Elem] forall x,y:Elem; L,L1,L2:List[Elem] . not x elem [] . x elem (y :: L) <=> x=y \/ x elem L . is_ordered([]) . is_ordered(x::[]) . is_ordered(x::y::L) <=> x<=y /\ is_ordered(y::L) . permutation(L1,L2) <=> ( forall x:Elem . x elem L1 <=> x elem L2) . is_ordered(sorter(L)) . permutation(L,sorter(L)) ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✻

  7. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ❤❛✈❡ ♦♥t♦❧♦❣✐❡s✱ ♠♦❞❡❧s ❛♥❞ s♣❡❝✐✜❝❛t✐♦♥s ✐♥ ❝♦♠♠♦♥❄ ❢♦r♠❛❧✐s❡❞ ✐♥ s♦♠❡ ❧♦❣✐❝❛❧ s②st❡♠ s✐❣♥❛t✉r❡ ✇✐t❤ ♥♦♥✲❧♦❣✐❝❛❧ s②♠❜♦❧s ✭❞♦♠❛✐♥ ✈♦❝❛❜✉❧❛r②✮ ❛①✐♦♠s ❡①♣r❡ss✐♥❣ t❤❡ ❞♦♠❛✐♥✲s♣❡❝✐✜❝ ❢❛❝ts s❡♠❛♥t✐❝s✿ ❝❧❛ss ♦❢ str✉❝t✉r❡s ✭♠♦❞❡❧s✮ ✐♥t❡r♣r❡t✐♥❣ s✐❣♥❛t✉r❡ s②♠❜♦❧s ✐♥ s♦♠❡ s❡♠❛♥t✐❝ ❞♦♠❛✐♥ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤♦s❡ str✉❝t✉r❡s ✭♠♦❞❡❧s✮ s❛t✐s❢②✐♥❣ t❤❡ ❛①✐♦♠s ❲❡ ❤❡♥❝❡❢♦rt❤ ❝❛❧❧ t❤❡♠ ✏❖▼❙✑✦ ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✼

  8. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥✿ ❉✐✈❡rs✐t② ♦❢ ❖♣❡r❛t✐♦♥s ♦♥ ❛♥❞ ❘❡❧❛t✐♦♥s ❛♠♦♥❣ ❖▼❙ ❱❛r✐♦✉s ♦♣❡r❛t✐♦♥s ❛♥❞ r❡❧❛t✐♦♥s ♦♥ ❖▼❙ ❛r❡ ✐♥ ✉s❡✿ str✉❝t✉r✐♥❣✿ ✉♥✐♦♥✱ tr❛♥s❧❛t✐♦♥✱ ❤✐❞✐♥❣✱ ✳ ✳ ✳ r❡✜♥❡♠❡♥t ♠❛t❝❤✐♥❣ ❛♥❞ ❛❧✐❣♥♠❡♥t ♦❢ ♠❛♥② ❖▼❙ ❝♦✈❡r✐♥❣ ♦♥❡ ❞♦♠❛✐♥ ♠♦❞✉❧❡ ❡①tr❛❝t✐♦♥ ❣❡t r❡❧❡✈❛♥t ✐♥❢♦r♠❛t✐♦♥ ♦✉t ♦❢ ❧❛r❣❡ ❖▼❙ ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧ ✐♥ ❛♥ ❡①♣r❡ss✐✈❡ ❧❛♥❣✉❛❣❡✱ r❡❛s♦♥ ❢❛st ✐♥ ❛ ❧✐❣❤t✇❡✐❣❤t ♦♥❡ ♦♥t♦❧♦❣②✲❜❛s❡❞ ❞❛t❛❜❛s❡ ❛❝❝❡ss✴❞❛t❛ ♠❛♥❛❣❡♠❡♥t ❞✐str✐❜✉t❡❞ ❖▼❙ ❜r✐❞❣❡s ❜❡t✇❡❡♥ ❞✐✛❡r❡♥t ♠♦❞❡❧❧✐♥❣s ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✽

  9. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ❖♥t♦■❖♣ ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✾

  10. ▼♦t✐✈❛t✐♦♥ ❖♥t♦■❖♣ ❉❖▲ ❋♦❝✉s❡❞ ❖▼❙ ❖▼❙ ▲✐❜r❛r✐❡s ❈♦♥❝❧✉s✐♦♥ ◆❡❡❞ ❢♦r ❛ ❯♥✐❢②✐♥❣ ▼❡t❛ ▲❛♥❣✉❛❣❡ ◆♦t ②❡t ❛♥♦t❤❡r ❖▼❙ ❧❛♥❣✉❛❣❡✱ ❜✉t ❛ ♠❡t❛ ❧❛♥❣✉❛❣❡ ❝♦✈❡r✐♥❣ ❞✐✈❡rs✐t② ♦❢ ❖▼❙ ❧❛♥❣✉❛❣❡s tr❛♥s❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ ❞✐✈❡rs✐t② ♦❢ ♦♣❡r❛t✐♦♥s ♦♥ ❛♥❞ r❡❧❛t✐♦♥s ❛♠♦♥❣ ❖▼❙ ❈✉rr❡♥t st❛♥❞❛r❞s ❧✐❦❡ t❤❡ ❖❲▲ ❆P■ ♦r t❤❡ ❛❧✐❣♠❡♥t ❆P■ ♦♥❧② ❝♦✈❡r ♣❛rts ♦❢ t❤✐s ❚❤❡ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ■♥t❡❣r❛t✐♦♥ ❛♥❞ ■♥t❡r♦♣❡r❛❜✐❧✐t② ✭❖♥t♦■❖♣✮ ✐♥✐t✐❛t✐✈❡ ❛❞❞r❡ss❡s t❤✐s ▼♦ss❛❦♦✇s❦✐ ❉✐str✐❜✉t❡❞ ❖♥t♦❧♦❣②✱ ▼♦❞❡❧✐♥❣ ❛♥❞ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡ ✭❉❖▲✮ ✷✵✶✹✲✶✶✲✶✷ ✶✵

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