Software Theory Change for resilient near-complete Specifications

❙♦❢t✇❛r❡ ❚❤❡♦r② ❈❤❛♥❣❡ ❢♦r r❡s✐❧✐❡♥t ♥❡❛r✲❝♦♠♣❧❡t❡ ❙♣❡❝✐✜❝❛t✐♦♥s ●✐✉s❡♣♣❡ Pr✐♠✐❡r♦ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❋r❛♥❝♦ ❘❛✐♠♦♥❞✐ ▼✐❞❞❧❡s❡① ❯♥✐✈❡rs✐t② ▲♦♥❞♦♥✱ ❯❑ ❆♥t✐❢r❛❣✐❧❡ ✷✵✶✺ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶ ✴ ✶✼ ❈❤❛♥❣❡ ❙♦❢t✇❛r❡ ❝❤❛♥❣❡✿ ❛ ❝r✐t✐❝❛❧ st❡♣ ✐♥ t❤❡ ❧✐❢❡✲❝②❝❧❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ s②st❡♠s✱ ❞✐❝t❛t❡❞ ❜② ◮ ❛r❝❤✐t❡❝t✉r❛❧ ❞❡❣❡♥❡r❛t✐♦♥ ✭✈✐♦❧❛t✐♦♥ ♦r ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❛r❝❤✐t❡❝t✉r❡✮ ◮ ✢❡①✐❜✐❧✐t② r❡q✉✐r❡♠❡♥ts ✭❛❝❝♦♠♠♦❞❛t✐♦♥ ♦❢ ✉♥♣❧❛♥♥❡❞ ♠♦❞✐✜❝❛t✐♦♥s✮ ❘❡s✐❧✐❡♥❝❡✿ ❢✉♥❝t✐♦♥❛❧✐t② ♣r❡s❡r✈❛t✐♦♥ ✉♥❞❡r ❝❤❛♥❣❡s ✐s ♦❢ ♣❛r❛♠♦✉♥t ✐♠♣♦rt❛♥❝❡ t♦ s②st❡♠ ❡✈♦❧✉t✐♦♥✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✷ ✴ ✶✼ ❚②♣❡s ♦❢ ❝❤❛♥❣❡ ❈♦rr❡❝t✐✈❡ ❝❤❛♥❣❡s✿ r❡s✉❧t ❢r♦♠ r❡s♣♦♥s❡ t♦ ❞❡❢❡❝ts ✉♥❞❡r t❡st✐♥❣✳ P❡r❢❡❝t✐✈❡ ❝❤❛♥❣❡s✿ r❡s✉❧t ❢r♦♠ ♥❡✇ ♦r ❝❤❛♥❣❡❞ r❡q✉✐r❡♠❡♥ts✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✸ ✴ ✶✼ ❆ss✉♠♣t✐♦♥s ♦❢ t❤✐s ✇♦r❦ ✶ ✷ ✸ ❙♦❢t✇❛r❡ ❝❤❛♥❣❡ ❛s s❝✐❡♥t✐✜❝ t❤❡♦r② ❝❤❛♥❣❡✱ t♦ ❛❝❝♦✉♥t ❢♦r ❜♦t❤ ♣❡r❢❡❝t✐✈❡ ❛♥❞ ❝♦rr❡❝t✐✈❡ ❝❤❛♥❣❡s❀ ❙♦❢t✇❛r❡ ❡♥❣✐♥❡❡r✐♥❣ ❛s ❜✉✐❧❞✐♥❣ ❛♥❞ ♠❛♥❛❣✐♥❣ ♦❢ t❤❡ ✭♣♦ss✐❜❧② ❧❛r❣❡✮ ❞❡❞✉❝t✐✈❡ ❝❧♦s✉r❡ ♦❢ ❛ s❡t ♦❢ ❧❛✇✲❧✐❦❡ ❡①♣r❡ss✐♦♥s ✭❛①✐♦♠s✮❀ ❱✐♦❧❛t✐♦♥ ♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ✐♥ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❧❡❛❞s t♦ r❡✈✐s✐♦♥ ♦❢ t❤❡ ❡①❛♠✐♥❡❞ ♠♦❞❡❧✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✹ ✴ ✶✼ ▼♦❞❡❧ ♦❢ ❛ ❈♦♠♣✉t❛t✐♦♥❛❧ ❙②st❡♠ ❲❡ ❝♦♥s✐❞❡r ❛ s♦❢t✇❛r❡ ♠♦❞❡❧ S♠ ❛s ❛ ❧✐st ♦❢ ✐♥❢♦r♠❛❧ r❡q✉✐r❡♠❡♥ts✱ ♣r♦❣r❛♠ ♦♣❡r❛t✐♦♥s ❛♥❞ ♦✉t♣✉t st❛t❡s✳ ■❞❡♥t✐✜❡❞ ♣r❡❝♦♥❞✐t✐♦♥s ❣✉❛r❛♥t❡❡ s❛❢❡t② ♦❢ ♣r♦❣r❛♠ ❡①❡❝✉t✐♦♥✳ ■❞❡♥t✐✜❡❞ ♣r❡❝♦♥❞✐t✐♦♥s t❤❛t ✭♠❛②✮ ❧❡❛❞ t♦ ❡①❡❝✉t✐♦♥ ❢❛✐❧✉r❡ ♦❢ ❛♥ ✐♥st❛♥❝❡ ♦❢ ♣ ❛❧❧♦✇ t♦ ❞❡✜♥❡ ❝♦♠♣❧❡t❡♥❡ss✳ ♥❡❛r✲❝♦♠♣❧❡t❡ s♣❡❝✐✜❝❛t✐♦♥✿ ♣♦st❝♦♥❞✐t✐♦♥s ❢♦r s♦♠❡ ❣✐✈❡♥ ✐♥♣✉ts✱ ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ♠❡❛♥s ❢♦r ♦t❤❡r ✐♥♣✉ts ✭❡✳❣✳ ♠❡t❛♠♦r♣❤✐❝ t❡st✐♥❣ ❛♥❞ ❞❡r✐✈❡❞ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ❡①❡❝✉t✐♦♥s✮✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✺ ✴ ✶✼ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❆♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ S ✐s ❛♥ ❛❝t✉❛❧ ✭❢❛✐t❤❢✉❧✮ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✐♥ s♦♠❡ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡❀ ♠ ✐ts ✭♣♦ss✐❜❧② ❛✉t♦♠❛t✐❝❛❧❧② ❡①tr❛❝t❡❞✮ s❡t ♦❢ s♣❡❝✐✜❝❛t✐♦♥s ✐s S ❀ ✐ ■❢ S ❞✐✈❡r❣❡s ❢r♦♠ S ✱ ♣❡r❢❡❝t✐✈❡ ♦r ❝♦rr❡❝t✐✈❡ ❝❤❛♥❣❡s ❛r❡ ❛♣♣❧✐❡❞ t♦ ′ ♣r♦❞✉❝❡ ❛ ♥❡✇ ♠♦❞❡❧ S ❀ ✐ ♠ ♠ r❡q✉✐r❡♠❡♥ts ❡✈♦❧✉t✐♦♥ ❢♦r♠❛❧❧② ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠❛♣♣✐♥❣ S 7→ S ♠ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ′ ♠ ✻ ✴ ✶✼ ❚♦② ❊①❛♠♣❧❡ ❣✐✈❡♥ ❛♥ ❡♠♣t② ❛rr❛② r❡t✉r♥ ❛♥ ❡rr♦r✱ ❛♥❞ ♦t❤❡r✇✐s❡ r❡t✉r♥ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❛rr❛②✳ ❊♥t❡r ❛♥ ❛rr❛② ❛s ✐♥♣✉t❀ ❘❡❛❞ t❤❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ ❛rr❛②✱ st❛rt✐♥❣ ❜② ❝♦✉♥t✐♥❣ t❤❡ ❡♥tr② ❛t ♣♦s✐t✐♦♥ ✵❀ ❋♦r ❡❛❝❤ ❡❧❡♠❡♥t ✇✐t❤ ✈❛❧✉❡ ❣r❡❛t❡r t❤❛♥ ③❡r♦✱ ❛❞❞ ✐t t♦ t❤❡ ❝♦✉♥t❀ ❘❡t✉r♥ t❤❡ ❝♦✉♥t ❛♥❞ st♦♣✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✼ ✴ ✶✼ ❚♦② ❊①❛♠♣❧❡ ♣✉❜❧✐❝ ❝❧❛ss ❚❡st✐♥❣ ④ ♣✉❜❧✐❝ ✐♥t ❝♦✉♥tP♦s✐t✐✈❡ ✭✐♥t❬❪ ①✮ ④ ⑥ ⑥ ✐♥t ❝♦✉♥t ❂ ✵❀ ❢♦r ✭✐♥t ✐❂✵❀ ✐ ❁ ①✳❧❡♥❣t❤❀ ✐✰✰✮ ④ ✐❢ ✭①❬✐❪ ❃❂ ✵✮ ④ ❝♦✉♥t✰✰❀ ⑥ ⑥ r❡t✉r♥ ❝♦✉♥t❀ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✽ ✴ ✶✼ ❚❤❡♦r② ✫ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❙ ♠ = ❈♥(▲(S )) := {φ | ▲(S )  φ} ♠ ♠ ▲ ⊇ ▲′ = ▲(S ) ✐ ∃φ .∃❙ (❙  φ ∧ (❙ 2 φ ∨ ❙  ¬φ ))✳ ✐ ✐ ✐ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✐ ♠ ✐ ♠ ✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✾ ✴ ✶✼ ❈♦♥s❡q✉❡♥❝❡ ❘❡❧❛t✐♦♥ ✭Pr♦♣❡rt✐❡s✮ ✶ φ  φ s❛②s t❤❛t ❛ ♣r♦♣❡rt② s♣❡❝✐✜❝❛t✐♦♥ φ ❤♦❧❞✐♥❣ ❢♦r ❛ s②st❡♠ S ✐♥❞✉❝❡s ♣r♦♣❡rt② s♣❡❝✐✜❝❛t✐♦♥ φ ✐ ❥ ✐ ♠ ❥ ✷ ❙ ⊤ ✸ ❙  (φ✶ → φ✷ ) ❛♥❞ ❙  φ✶ ✱ ✐♠♣❧✐❡s ❙  φ✷ ✹ ❙  φ ✐♠♣❧✐❡s ❙ 2 ¬φ ✺ ◆❡✐t❤❡r ❝♦♠♣❧❡t❡♥❡ss ♥♦r ❝♦♠♣❛❝t♥❡ss ❛r❡ ❛ss✉♠❡❞✳ ♠ ♠ ♠ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ♠ ♠ ♠ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✵ ✴ ✶✼ ❊①♣❛♥s✐♦♥ ❢♦r P❡r❢❡❝t✐✈❡ ❈❤❛♥❣❡ ❙′ ♠ = (❙ )+ φ ♠ ✐ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❧♦❣✐❝❛❧ ❝❧♦s✉r❡ ♦❢ ❙ ❛♥❞ φ ✱ ♠ ✐ (❙ )+ φ = ❈♥(▲(S ) ∪ {φ}) ♠ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ♠ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✶ ✴ ✶✼ ❈♦♥tr❛❝t✐♦♥ ❢♦r ❈♦rr❡❝t✐✈❡ ❈❤❛♥❣❡ (❙ )− φ = ❈♥(▲(S )/φ ) ♠ ♠ ✐ ✐ ❚❤❡ ❝♦♥tr❛❝t✐♦♥ ♦♣❡r❛t♦r ✐s t❤❡♥ ❛ ♠❛♣♣✐♥❣ ❢✉♥❝t✐♦♥✿ ❙ ♠ × ▲′ 7→ ❙ ′ := {(❙ , φ )} 7→ (❙ )− φ ♠ ❢r♦♠ t❤❡ s❡t ♦❢ ❛❧❧ ❝♦♠♣❧❡t❡ t❤❡♦r✐❡s ♦❢ ♠ ❙ ♠ ✐ t♦ ❛ ♥❡✇ ♠♦❞❡❧ ✇❤♦s❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❤❛✈❡ ❧❛♥❣✉❛❣❡s t❤❛t ❞♦ ♥♦t ✐♠♣❧② Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ♠ ✐ φ ✐ ✳ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✷ ✴ ✶✼ ❖r❞❡r✐♥❣ ♦♥ ❡①♣r❡ss✐✈❡♥❡ss ✶ ✷ ❈♦♥tr❛❝t✐♦♥ s❤♦✉❧❞ r❡♠♦✈❡ ✇✐t❤ ❛ ♠✐♥✐♠❛❧ ❧♦ss ♦❢ ❢✉♥❝t✐♦♥❛❧✐t✐❡s❀ ✐♠♣♦s❡ ❛♥ ♦r❞❡r✐♥❣ φ ≤ φ s❛②✐♥❣ t❤❛t φ ✐s ❛t ❧❡❛st ❛s ❡♠❜❡❞❞❡❞ ❛s φ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧✐t✐❡s ♦❢ t❤❡ s②st❡♠❀ ✐ ❥ ❥ ✐ ✸ ❛ ♣r♦♣❡rt② φ ∈ ❙ ✐s s❛❢❡ ✇✐t❤ r❡s♣❡❝t t♦ (❙ )− φ ♠♦❞✉❧♦ <✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ φ ✐s ♥♦t ❛ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥t ✉♥❞❡r < ♦❢ ❛♥② ♠✐♥✐♠❛❧ ❙ ′ t❤❛t ✈❡r✐✜❡s φ✳ ❥ ❥ ♠ ✐ ✐ ♠ ✐ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✸ ✴ ✶✼ ■♥❢❡r❡♥t✐❛❧✲❜❛s❡❞ ❛♣♣r♦❛❝❤ t♦ r❡s✐❧✐❡♥❝❡ ❉❡✜♥✐t✐♦♥ ✭Pr♦♣❡rt② ❘❡s✐❧✐❡♥❝❡✮ ❈♦♥s✐❞❡r ♣r♦♣❡rt② s♣❡❝✐✜❝❛t✐♦♥s φ , , ∈ ❙ ❛♥❞ ❛ r❡❧❡✈❛♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❙ ✳ ❚❤❡♥ φ ✐s ♠♦r❡ r❡s✐❧✐❡♥t t❤❛♥ φ ✇✐t❤ r❡s♣❡❝t t♦ ❙ ′ = (❙ )− φ ✐❢ ✐t ✐s ′ ♥♦t ❛ ♠❛①✐♠❛❧❧② ✈✉❧♥❡r❛❜❧❡ ❡❧❡♠❡♥t ♦❢ s♦♠❡ ♠✐♥✐♠❛❧ ❙ ⊆ ❙ ✐♠♣❧②✐♥❣ φ ✳ ✐ ❥ ❦ ✐ ❦ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ❥ ♠ ♠ ✐ ♠ ♠ ✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✐ ✶✹ ✴ ✶✼ ■♥❢❡r❡♥t✐❛❧✲❜❛s❡❞ ❛♣♣r♦❛❝❤ t♦ r❡s✐❧✐❡♥❝❡ ❉❡✜♥✐t✐♦♥ ✭Pr♦♣❡rt② ❘❡s✐❧✐❡♥❝❡✮ ❈♦♥s✐❞❡r ♣r♦♣❡rt② s♣❡❝✐✜❝❛t✐♦♥s φ , , ∈ ❙ ❛♥❞ ❛ r❡❧❡✈❛♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❙ ✳ ❚❤❡♥ φ ✐s ♠♦r❡ r❡s✐❧✐❡♥t t❤❛♥ φ ✇✐t❤ r❡s♣❡❝t t♦ ❙ ′ = (❙ )− φ ✐❢ ✐t ✐s ′ ♥♦t ❛ ♠❛①✐♠❛❧❧② ✈✉❧♥❡r❛❜❧❡ ❡❧❡♠❡♥t ♦❢ s♦♠❡ ♠✐♥✐♠❛❧ ❙ ⊆ ❙ ✐♠♣❧②✐♥❣ φ ✳ ✐ ❥ ❦ ✐ ♠ ❥ ❦ ♠ ✐ ♠ ♠ ✐ ✐ ❉❡✜♥✐t✐♦♥ ✭❙②st❡♠ ❘❡s✐❧✐❡♥❝❡✮ ❆ s♦❢t✇❛r❡ s②st❡♠ s♣❡❝✐✜❝❛t✐♦♥ ❙ ✐s s❛✐❞ r❡s✐❧✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ❛ ♠ ♣r♦♣❡rt② s♣❡❝✐✜❝❛t✐♦♥ φ ✐❢ t❤❡ ❧❛tt❡r ✐s ♥♦t ❛ ♠❛①✐♠❛❧❧② ✈✉❧♥❡r❛❜❧❡ ❡❧❡♠❡♥t ✐♥ ❛♥② ❙ ′ ♣r❡s❡r✈✐♥❣ ♠✐♥✐♠❛❧ ❢✉♥❝t✐♦♥❛❧✐t✐❡s ♦❢ ❙ ✳ ✐ ♠ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ♠ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✹ ✴ ✶✼ ❙♦❢t✇❛r❡ ❘❡s✐❧✐❡♥❝❡ ❛♥❞ ❊✈♦❧✈❛❜✐❧✐t②✿ Pr♦♣❡rt✐❡s ❆❝❝♦✉♥t❛❜✐❧✐t②✿ ●✐✈❡♥ t❤❡ ♥♦♥✲r❡s✐❧✐❡♥t ♣❛rt ♦❢ t❤❡ s②st❡♠✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❡st❛❜❧✐s❤ ✇❤✐❝❤ ♣r♦♣❡rt✐❡s ✇✐❧❧ ♥♦t ❜❡ ✐♥st❛♥t✐❛t❡❞ ✐♥ ❛♥② s✉❜s②st❡♠✳ ❊✈♦❧✈❛❜✐❧✐t②✿ ●✐✈❡♥ t❤❡ r❡s✐❧✐❡♥t ❝♦r❡ ♦❢ t❤❡ s②st❡♠✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❡st❛❜❧✐s❤ ✇❤✐❝❤ ♣r♦♣❡rt✐❡s ✇✐❧❧ ❜❡ ✐♥st❛♥t✐❛t❡❞ ❜② ❛♥② s✉❜s②st❡♠✳ Pr❡✈✐s✐♦♥✿ ●✐✈❡♥ t❤❡ r❡s✐❧✐❡♥t ❝♦r❡ ♦❢ t❤❡ s②st❡♠✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❦♥♦✇ ✇❤✐❝❤ ♣r♦♣❡rt✐❡s ✇✐❧❧ ❜❡ ✐♥st❛♥t✐❛t❡❞ ❜② ❛♥② ♦❢ ✐ts ❡①t❡♥s✐♦♥s✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✺ ✴ ✶✼ ❋✉t✉r❡ ❘❡s❡❛r❝❤ ❯♣❞❛t❡ ♦♣❡r❛t✐♦♥✿ ✐t ❡①♣❧✐❝✐t❧② ❢♦r♠✉❧❛t❡s ❝❤❛♥❣❡s ✐♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t ❛❞ ❤❡♥❝❡ q✉❛❧✐✜❡s ❛s ❛❞❛♣t✐✈❡ ❝❤❛♥❣❡ ❀ ❆♥t✐❢r❛❣✐❧✐t②✿ ❛ str❡♥❣t❤❡♥✐♥❣ ♦❢ ✐♥❢❡r❡♥t✐❛❧ ❜❛s❡❞ r❡s✐❧✐❡♥❝❡ ❛♥❞ ♣r❡✈✐s✐♦♥✱ t♦ ❛❝❝♦✉♥t ❢♦r ✐♠♣r♦✈❡♠❡♥ts ♦❢ t❤❡ s②st❡♠ ✉♥❞❡r ❝❤❛♥❣❡s Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✻ ✴ ✶✼ ❚❤❛♥❦s ◗✉❡st✐♦♥s ❛♥❞ ❝♦♠♠❡♥ts ✇❡❧❝♦♠❡✳ Pr✐♠✐❡r♦✱ ❘❛✐♠♦♥❞✐ ✸r❞ ❏✉♥❡ ✷✵✶✺ ✶✼ ✴ ✶✼