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Geração de casos de teste para linguagens com aritimética de ponteiro

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●❊❘❆➬➹❖ ❉❊ ❈❆❙❖❙ ❉❊ ❚❊❙❚❊ P❆❘❆

▲■◆●❯❆●❊◆❙ ❈❖▼ ❆❘■❚■▼➱❚■❈❆ ❉❊

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❋❘❆◆❈■❙❈❖ ❉❊▼❖◆❚■✃ ❉❖❙ ❙❆◆❚❖❙ ❏❯◆■❖❘

●❊❘❆➬➹❖ ❉❊ ❈❆❙❖❙ ❉❊ ❚❊❙❚❊ P❆❘❆

▲■◆●❯❆●❊◆❙ ❈❖▼ ❆❘■❚■▼➱❚■❈❆ ❉❊

P❖◆❚❊■❘❖

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❝♦♠♦ r❡q✲ ✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳

❖r✐❡♥t❛❞♦r✿ ▼❛r✐③❛ ❆♥❞r❛❞❡ ❞❛ ❙✐❧✈❛ ❇✐❣♦♥❤❛

❈♦♦r✐❡♥t❛❞♦r✿ ❋❡r♥❛♥❞♦ ▼❛❣♥♦ ◗✉✐♥tã♦ P❡r❡✐r❛

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❋❘❆◆❈■❙❈❖ ❉❊▼❖◆❚■✃ ❉❖❙ ❙❆◆❚❖❙ ❏❯◆■❖❘

●❊◆❊❘❆❚■❖◆ ❖❋ ❚❊❙❚ ❈❆❙❊❙ ❋❖❘

▲❆◆●❯❆●❊❙ ❲■❚❍ P❖■◆❚❊❘ ❆❘■❚❍▼❊❚■❈❙

❉✐ss❡rt❛t✐♦♥ ♣r❡s❡♥t❡❞ t♦ t❤❡ ●r❛❞✉❛t❡ Pr♦❣r❛♠ ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ♦❢ t❤❡ ❋❡❞✲ ❡r❛❧ ❯♥✐✈❡rs✐t② ♦❢ ▼✐♥❛s ●❡r❛✐s ✐♥ ♣❛rt✐❛❧ ❢✉❧✜❧❧♠❡♥t ♦❢ t❤❡ r❡q✉✐r❡♠❡♥ts ❢♦r t❤❡ ❞❡✲ ❣r❡❡ ♦❢ ▼❛st❡r ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✳

❆❞✈✐s♦r✿ ▼❛r✐③❛ ❆♥❞r❛❞❡ ❞❛ ❙✐❧✈❛ ❇✐❣♦♥❤❛

❈♦✲❆❞✈✐s♦r✿ ❋❡r♥❛♥❞♦ ▼❛❣♥♦ ◗✉✐♥tã♦ P❡r❡✐r❛

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© 2016, Francisco Demontiê dos Santos Júnior

Todos os direitos reservados

Ficha catalográfica elaborada pela Biblioteca do ICEx - UFMG

Santos Júnior, Francisco Demontiê dos.

S237g Generation of test cases for languages with pointer arithmetics. / Francisco Demontiê dos Santos Júnior. — Belo Horizonte, 2016.

xx, 65 f.: il.; 29 cm.

Dissertação (mestrado) - Universidade Federal de Minas Gerais – Departamento de Ciência da Computação. Orientadora: Mariza Andrade da Silva Bigonha

Coorientador: Fernando Magno Quintão Pereira

1. Computação – Teses. 2. Diagrama de fluxo de dados. 3. Compiladores (Programas de computador). 4. Software

– Validação. I. Orientadora. II. Coorientador. III. Título.

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❜✉s❛♥❞♦ ❞♦s ❝❧✐❝❤ês✱ ❛❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ q✉❡ ♠❡ ❞❡✉ ♦ ❞♦♠ ❞❛ ✈✐❞❛✱ s❛ú❞❡ ❡ ♦♣♦rt✉♥✐❞❛❞❡s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r ♦♥❞❡ ❝❤❡❣✉❡✐✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❈❧é❝✐❛ ❡ ❉❡♠♦♥t✐ê✱ q✉❡ ❛❧é♠ ❞❡ s❡r❡♠ ❡①❡♠♣❧♦s ❞❡ ❞❡❞✐❝❛çã♦ ❡ ❝❛rát❡r✱ s❡♠♣r❡ ✜③❡r❛♠ ♦ ♣♦ssí✈❡❧ ❡ ♦ ✐♠♣♦ssí✈❡❧ ♣♦r ♠✐♠ ❡ ♣❡❧❛s ♠✐♥❤❛s ✐r♠ãs✳ ❙❛❝r✐✜❝❛♥❞♦✱ ♠✉✐t❛s ✈❡③❡s✱ ♦ ❝♦♥❢♦rt♦ ♣ró♣r✐♦ ❡♠ ❢❛✈♦r ❞♦ ♥♦ss♦✳ ▼❡ ❛♣♦✐❛♥❞♦ ❡♠ ❝❛❞❛ ❞❡❝✐sã♦ ❡ ♠❡ ❛❝♦❧❤❡♥❞♦ s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐ ❞❡ ❝♦❧♦✳ P♦r ❡ss❡s ❡ ♦✉tr♦s ♠♦t✐✈♦s✱ ❞✐❣♦ ❡ s❡♠♣r❡ ❞✐r❡✐ q✉❡ ❡ss❛ ❝♦♥q✉✐st❛ é✱ t❛♠❜é♠✱ ❞❡❧❡s✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦✱ ✏♣❛✐♥❤♦✑ ❡ ✏♠❛✐♥❤❛✑✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♥♦✐✈❛✱ ■③❛❜❡❧❛✱ q✉❡ ❢❛③ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❛❧❣♦ ♠✉✐t♦ ♠❛✐s ❛❧❡❣r❡✳ ◗✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐♦✉✱ ♣♦r ♠❛✐s ❞♦❧♦r♦s♦ q✉❡ ❢♦ss❡✱ ❡ s✉♣♦rt♦✉ ✷ ❛♥♦s ❞❡ ✉♠❛ r❡❧❛çã♦ ❛ ❞✐stâ♥❝✐❛ ✭q✉❡ ♥ã♦ é ❢á❝✐❧✱ ♠❛s✱ ❣r❛ç❛s ❛ ❡❧❛✱ ❞❡✉ ❝❡rt♦✮✳ ❆❣r❛❞❡ç♦ ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♦❢❡r❡❝❡✉ ❛ ♠ã♦✱ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♠❡ ❢❡③ r✐r✱ ❡♠ q✉❡ s❡ ❛❧❡❣r♦✉ ❝♦♠ ❛s ♠✐♥❤❛s ❝♦♥q✉✐st❛s✱ ♦✉✈✐✉ ♠❡✉s ❞❡s❛❜❛❢♦s✳ ❙❡♠ ❡❧❛✱ ♥❛❞❛ s❡r✐❛ ♣♦ssí✈❡❧✳ ❆❣r❛❞❡ç♦ às ♠✐♥❤❛s ✐r♠ãs✱ ❆♥❛ Pr✐s❝✐❧❧❛ ❡ ❆♥❛ ❈❛r♦❧✐♥❛✱ ♣♦r s❡r❡♠ q✉❡♠ sã♦✳ ❙❡♠♣r❡ ❞✐③❡♥❞♦ q✉❡ s❡♥t❡♠ ♦r❣✉❧❤♦ ❞❡ ♠✐♠ ❡ ♠❡ ❞❛♥❞♦ ❢♦rç❛s ♣❛r❛ ♣❡rs❡❣✉✐r ♠❡✉s ♦❜❥❡t✐✈♦s✳ ▼❡ ❛❧❡❣r❛♥❞♦ s❡♠♣r❡ q✉❡ ❡st❛✈❛♠♦s ❥✉♥t♦s✳ ❱♦❝ês sã♦ ❛s ♠❡❧❤♦r❡s ✐r♠ãs q✉❡ ❛❧❣✉é♠ ♣♦❞❡r✐❛ t❡r✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❝✉♥❤❛❞♦ ❆♥❞❡rs♦♥ ♣♦r s❡♠♣r❡ ♠❡ r❡❝❡❜❡r ❞❡ ❜r❛ç♦s ❛❜❡rt♦s ❡ ♥ã♦ ♠❡❞✐r ❡s❢♦rç♦s ♣❛r❛ ♠❡ ❛❥✉❞❛r q✉❛♥❞♦ ♣ô❞❡✳ ➚s ♠✐♥❤❛s s♦❜r✐♥❤❛s✱ ❆♥❛ ❈❧❛r❛ ❡ ❆♥❛ ❱✐tór✐❛✱ q✉❡ sã♦ ❡①❡♠♣❧♦s ❞❡ ❝❛r✐♥❤♦✱ ❛❧❡❣r✐❛ ❡ q✉❡ s❡♠♣r❡ ♠❡ s✉r♣r❡❡♥❞❡♠✱ s❡❥❛ ❝♦♠ ❛ s❛❣❛❝✐❞❛❞❡ ♦✉ ❝♦♠ ❛s ❡str✐♣✉❧✐❛s✳

❆❣r❛❞❡ç♦ ♦s ♠❡✉s t✐♦s ❱❛♥❞❛ ❡ ❘♦❜ér✐♦ ❡ ♠❡✉s ♣r✐♠♦s ▼❛t❤❡✉s ❡ ❏♦❤❛♥♥❛✱ q✉❡ ❢♦r❛♠ ♠✐♥❤❛ s❡❣✉♥❞❛ ❢❛♠í❧✐❛ ♥♦ ✐♥í❝✐♦ ❞❛ ♠✐♥❤❛ ❣r❛❞✉❛çã♦ ✭♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✮ ❡ ❝♦♥t✐♥✉❛♠ s❡♠♣r❡ t♦r❝❡♥❞♦ ♣♦r ♠✐♠✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛♦ ♠❡✉ t✐♦ ❏♦sé ❆ss✐s✱ ❛ ❏✉❧✐❛♥♦ ❡ ❛ ❆❧❢r❡❞♦✱ ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦ ❡♠ ❇❡❧♦ ❍♦r✐③♦♥t❡✱ s❡♥❞♦ s❡♠♣r❡ ♣r❡st❛t✐✈♦s✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s s♦❣r♦s ❱❛♥✐♦ ❡ ❊❧✐③❛❜❡t❡ ❡ ♠❡✉ ❝✉♥❤❛❞♦ ▼❛r❝♦s✱ ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦ ❝♦♠♦ ♣❛rt❡ ❞❛ ❢❛♠í❧✐❛ ❡ t❡r❡♠ s❡♠♣r❡ ♠❡ ❛♣♦✐❛❞♦✳ P♦r ♠❡ ❢❛③❡r❡♠ s❡♥t✐r q✉❡ ❡♠ ▼❛❝❡✐ó ❡✉ t❡♥❤♦ ✉♠❛ t❡r❝❡✐r❛ ❝❛s❛✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛✈ós ❆♥tô♥✐♦ ❇❛t✐st❛ ✭✐♥ ♠❡♠♦r✐❛♠✮✱ ❏❛❝✐r❛✱ ❏♦sé ❆❧❜✐♥♦ ✭✐♥ ♠❡♠♦r✐❛♠✮ ❡ ❊✉♥✐❝❡ ✭✐♥ ♠❡♠♦r✐❛♠✮✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s q✉❡ s❡❣✉✐rã♦ ❝♦♠✐❣♦

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♣❡❧♦ r❡st♦ ❞❛ ✈✐❞❛✳ ❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♣r✐♠♦ ❙❛✉❧♦ ✭✐♥ ♠❡♠♦r✐❛♠✮✱ q✉❡✱ ❛❧é♠ ❞❡ t❡r s✐❞♦ ♣❛r❛ ♠✐♠ ✉♠ ❡①❡♠♣❧♦ ❞❡ s✐♠♣❧✐❝✐❞❛❞❡✱ ❛❧❡❣r✐❛✱ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡ ♣r♦✜ss✐♦♥❛❧✱ ♠❡ ❛♣r❡s❡♥t♦✉ ❛ ❝♦♠♣✉t❛çã♦ ❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ♥♦s ♠❡✉s ❡st✉❞♦s✳ ❊♠ ♥♦♠❡ ❞❡❧❡s✱ ❛❣r❛❞❡ç♦ ❛ t♦❞❛ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✳

❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❋r❛♥❦❧✐♥ ❘❛♠❛❧❤♦✱ q✉❡ ♠❡ ❛♣r❡s❡♥t♦✉ ❛ àr❡❛ ❞❡ ❝♦♠♣✐✲ ❧❛❞♦r❡s✱ ♠❡ ♦r✐❡♥t♦✉ ♥♦ ♠❡✉ ♣r✐♠❡✐r♦ ♣r♦❥❡t♦ ❞❡ ♣❡sq✉✐s❛ ❡ ❢♦✐ ❝♦♠♣r❡❡♥s✐✈♦ ❡ ❛♣♦✐❛❞♦r q✉❛♥❞♦ ❞❡❝✐❞✐ ❢❛③❡r ♦ ♠❡str❛❞♦ ❡♠ ♦✉tr❛ ✉♥✐✈❡rs✐❞❛❞❡✳ ➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❛ ♣r♦❢❡s✲ s♦r❛ ▼❛r✐③❛✱ q✉❡ s❡♠♣r❡ ❢♦✐ ❝✉✐❞❛❞♦s❛✱ ❝♦♠♣r❡❡♥s✐✈❛ ❡ ♣❛❝✐❡♥t❡✱ ♠❛s t❛♠❜é♠ ❛ss❡rt✐✈❛ s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐✳ ❆♦ ♠❡✉ ❝♦✲♦r✐❡♥t❛❞♦r✱ ♦ ♣r♦❢❡ss♦r ❋❡r♥❛♥❞♦✱ q✉❡ ♠❡ ❡♥s✐♥♦✉✱ ♠❡ ❛❝♦♠♣❛♥❤♦✉ ❞❡ ♣❡rt♦ ❡ ❢♦✐ s❡♠♣r❡ ♣r❡st❛t✐✈♦✳ ❆❧é♠ ❞✐ss♦✱ ❛❣r❛❞❡ç♦✲♦✱ t❛♠❜é♠✱ ♣♦r t❡r s❡♠♣r❡ ❜✉s❝❛❞♦ ✉♠❛ r❡❧❛çã♦ q✉❡ ✈❛✐ ❛❧é♠ ❞❛ ♣r♦✜ss✐♦♥❛❧✱ ❢❛③❡♥❞♦ ❞♦ ♠❡✉ ♠❡str❛❞♦ ❛❧❣♦ ♠✉✐t♦ ♠❛✐s ❞✐✈❡rt✐❞♦✳

❆❣r❛❞❡ç♦ ❛ ❉♦✉❣❧❛s✱ ❍❡♥r✐q✉❡✱ ❏✉♥✐♦✱ Pér✐❝❧❡s✱ ❘✉❜❡♥s ❡ ❱✐❝t♦r✱ q✉❡✱ ❛❧é♠ ❞❡ ♠❡ ❛❥✉❞❛r❡♠ ❛♦ ❧♦♥❣♦ ❞❛ ♠✐♥❤❛ ♣❡sq✉✐s❛✱ ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♥❛ ♠✐♥❤❛ ❛❞❛♣t❛çã♦ ❛ ❇❡❧♦ ❍♦r✐③♦♥t❡✳ ❉✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s✱ ♣✉❞❡ ❝♦♥❤❡❝ê✲❧♦s ♠❛✐s ❞❡ ♣❡rt♦✱ t❡r ❜♦♥s ♠♦♠❡♥✲ t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✱ ❞❡s❛❜❛❢❛r✱ ❡s❝✉t❛r✳ ❱♦❝ês ❢♦r❛♠ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♠✐♠ ❡ ❡✉ s❡r❡✐ ❡t❡r♥❛♠❡♥t❡ ❣r❛t♦✳ ❆♦s ❛♠✐❣♦s ❆❞❛♠✱ ❆✉❣✉st♦✱ ❚✐❛❣♦✱ ❈❛t❤❛r✐♥❡✱ ❉❛♥✐❡❧✱ ❑❛r♦❧✱ ❘♦❞r✐❣♦✱ ▼❛r❝❡❧❛✱ ❉❡❧❛♥♦✱ ❈❛r❧ú❝✐❛✱ ◆❛tã ❡ ❙❛✈②♦✱ ♣♦r s❡♠♣r❡ ❞❡♠♦♥str❛r❡♠ ♣r❡♦❝✉♣❛çã♦ ❡ ❢❛③❡r❡♠ ♠❡✉s ❞✐❛s ♠❛✐s ❢❡❧✐③❡s✱ ♠❡s♠♦ ❝♦♠ t♦❞❛ ❛ ❞✐stâ♥❝✐❛✳

❆❣r❛❞❡ç♦ ❛ ▼❛①tr❛❝❦✱ ♥❛ ♣❡ss♦❛ ❞❡ ❋❡❧✐♣❡ Pr♦✈❡♥③❛♥♦✱ ❡♠♣r❡s❛ q✉❡ ✜♥❛♥❝✐♦✉ ❡ q✉❡ t♦r♥♦✉ ❡ss❡ ♣r♦❥❡t♦ ❞❡ ♣❡sq✉✐s❛ ♣♦ssí✈❡❧✳ ❖❜r✐❣❛❞♦ ♣♦r t♦❞♦ ♦ s✉♣♦rt❡ ♣r❡st❛❞♦✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ à ❈❆P❊❙✱ q✉❡ ✜♥❛♥❝✐♦✉ ♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ♠❡✉ ♠❡str❛❞♦✳ P♦r ✜♠✱ ❛ t♦❞♦s q✉❡ ♥ã♦ ❢♦r❛♠ ♠❡♥❝✐♦♥❛❞♦s ❛q✉✐ ❡ q✉❡✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❞✉r❛♥t❡ ❡ss❡s ú❧t✐♠♦s ❛♥♦s✱ ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

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❘❡s✉♠♦

❚❡st❛r ❡ ❞❡♣✉r❛r s♦❢t✇❛r❡ sã♦ t❛r❡❢❛s ❞✐❢í❝❡✐s✳ ❊♠ ❣❡r❛❧✱ é ♣r❡❝✐s♦ ❡s♣❡r❛r q✉❡ ♦ ✢✉①♦ ❞❡ ❡①❡❝✉çã♦ ❝❤❡❣✉❡ ❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ❢♦r♠❛ ❛ ♣♦❞❡r t❡stá✲❧❛✳ ◆❡ss❡ s❡♥t✐❞♦✱ ♠✉✐t♦ ❡s❢♦rç♦ ❢♦✐ ❡♠♣r❡❣❛❞♦ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ té❝♥✐❝❛s✱ t❛✐s ❝♦♠♦ ❡①❡❝✉çã♦ s✐♠❜ó❧✐❝❛ ❡ ❢✉③③ t❡st✐♥❣✱ ♣❛r❛ ❣❡r❛çã♦ ❛✉t♦♠át✐❝❛ ❞❡ ❝❛s♦s ❞❡ t❡st❡ ♣❛r❛ ❛♥❛❧✐s❛r ❢✉♥çõ❡s ❞❡ ✐♥t❡r❡ss❡✳ ❊♥tr❡t❛♥t♦✱ t❛✐s té❝♥✐❝❛s ♣♦ss✉❡♠ ❧✐♠✐t❛çõ❡s✳ ❯♠❛ ❧✐♠✐t❛çã♦ q✉❡ ❝❤❛♠❛ ❛ ♥♦ss❛ ❛t❡♥çã♦ é ♦ ❢❛t♦ ❞❡ q✉❡✱ ❛té ♦♥❞❡ s❛❜❡♠♦s✱ ♥❡♥❤✉♠❛ té❝♥✐❝❛ ❛t✉❛❧ r❡❧❛❝✐♦♥❛ ❛rr❛♥❥♦s ♣❛ss❛❞♦s ❝♦♠♦ ❡♥tr❛❞❛s ♣❛r❛ ❢✉♥çõ❡s ❝♦♠ ♦✉tr❛s ❡♥tr❛❞❛s q✉❡ r❡♣r❡s❡♥t❡♠ s❡✉s t❛♠❛♥❤♦s✳ ■ss♦ ❢❛③ ❝♦♠ q✉❡ ♦s ❝❛s♦s ❞❡ t❡st❡ ❣❡r❛❞♦s✱ ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s✱ ♣♦ss❛♠ ❧❡✈❛r ❛ ❛❝❡ss♦s ✐♥✈á❧✐❞♦s à ♠❡♠ór✐❛ q✉❡ ♥ã♦ ❛❝♦♥t❡❝❡r✐❛♠ ❡♠ ❡①❡❝✉çõ❡s r❡❛✐s ❞♦ ♣r♦❣r❛♠❛ s❡♥❞♦ t❡st❛❞♦✳ ◆❡ss❛ ❞✐ss❡rt❛çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❞✉❛s ❛♥á❧✐s❡s ❡stát✐❝❛s ♣❛r❛ ✐♥❢❡rê♥❝✐❛ ❞❡ t❛♠❛♥❤♦s ❞❡ ❛rr❛♥❥♦s✱ ❜❡♠ ❝♦♠♦ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❣❡r❛çã♦ ❞❡ ❝❛s♦s ❞❡ t❡st❡ ❝❛♣❛③ ❞❡ ❣❡r❛r t❡st❡s s❡❣✉r♦s ✉s❛♥❞♦ t❛✐s ❛♥á❧✐s❡s✳ ❈♦♠♦ ❢♦r♠❛ ❞❡ ❛✈❛❧✐❛r ❛ ❡❢❡t✐✈✐❞❛❞❡ ❞❛ ♥♦ss❛ té❝♥✐❝❛✱ r❡❛❧✐③❛♠♦s ❞♦✐s ❡①♣❡r✐♠❡♥t♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✱ ❛s ❛♥á❧✐s❡s ❡stát✐❝❛s ❢♦r❛♠ ❝❛♣❛③❡s ❞❡ r❡❧❛❝✐♦♥❛r ✸✹✳✻✪ ❞♦s t❛♠❛♥❤♦s ❞❡ ❛rr❛♥❥♦s r❡❝❡❜✐❞♦s ❝♦♠♦ ♣❛râ♠❡tr♦ ♣♦r ❢✉♥çõ❡s ❞♦s ♣r♦❣r❛♠❛s ❝♦♥t✐❞♦s ♥♦ ❙P❊❈ ❈P❯✷✵✵✻ ✭✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❡♥❝❤♠❛r❦s ❝♦♥t❡♥❞♦ ♣r♦❣r❛♠❛s r❡❛✐s✮✳ ◆♦ s❡❣✉♥❞♦✱ ♥ós ✉t✐❧✐③❛♠♦s ❛ ❢❡rr❛♠❡♥t❛ ❆s②♠♣t✉s✱ q✉❡ r❡❛❧✐③❛ ❛♥á❧✐s❡ ❛✉t♦♠át✐❝❛ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s ❡ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❝♦♠♦ ✉♠ ❡st✉❞♦ ♣r❡❧✐♠✐♥❛r ❞❡st❡ ♠❡str❛❞♦✱ s♦❜r❡ ♦s ❝❛s♦s ❞❡ t❡st❡ ❣❡r❛❞♦s ♣❡❧❛ ♥♦ss❛ té❝♥✐❝❛✳ ❆s②♠♣t✉s ❢♦✐ ❝❛♣❛③ ❞❡ ✐♥❢❡r✐r ❝♦rr❡t❛♠❡♥t❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s ❞❡s❡♥✈♦❧✈✐❞❛s ♣♦r ♥ós ❡ ❡①tr❛í❞❛s ❞♦s ❜❡♥❝❤♠❛r❦s ❞♦ P♦❧②❜❡♥❝❤✳ ■ss♦ ♠♦str❛ q✉❡ ♥♦ss❛ té❝♥✐❝❛ ❞❡ ❣❡r❛çã♦ ❞❡ ❡♥tr❛❞❛s é út✐❧ ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❛✉t♦♠át✐❝❛ ❞❡ ❢✉♥çõ❡s ❞❡ ✐♥t❡r❡ss❡✳

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(13)

❆❜str❛❝t

❙♦❢t✇❛r❡ t❡st✐♥❣ ❛♥❞ ❞❡❜✉❣❣✐♥❣ ❛r❡ ❤❛r❞ t❛s❦s✳ ■♥ ❣❡♥❡r❛❧✱ ✐t ✐s ♥❡❝❡ss❛r② t❤❛t t❤❡ ❡①❡❝✉t✐♦♥ ✢♦✇ r❡❛❝❤❡s ❛ ❢✉♥❝t✐♦♥ ✐♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ t❡st ✐t✳ ■♥ t❤✐s s❡♥s❡✱ ♠✉❝❤ ❡✛♦rt ✇❛s ❡♠♣❧♦②❡❞ ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❡❝❤♥✐q✉❡s✱ s✉❝❤ ❛s s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❛♥❞ ❢✉③③ t❡st✐♥❣✱ t♦ ❛✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡r❛t❡ t❡st ❝❛s❡s ✐♥ ❛ ✇❛② t♦ ❛♥❛❧②③❡ ❢✉♥❝t✐♦♥s ♦❢ ✐♥t❡r❡st✳ ❍♦✇❡✈❡r✱ s✉❝❤ t❡❝❤♥✐q✉❡s ❤❛✈❡ ❧✐♠✐t❛t✐♦♥s✳ ❆ ❧✐♠✐t❛t✐♦♥ ✇❤✐❝❤ ❝❛t❝❤❡s ♦✉r ❛tt❡♥t✐♦♥ ✐s t❤❡ ❢❛❝t t❤❛t✱ t♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ ♥♦ ❝✉rr❡♥t t❡❝❤♥✐q✉❡ r❡❧❛t❡s ❛rr❛②s ♣❛ss❡❞ ❛s ✐♥♣✉ts ❢♦r ❢✉♥❝t✐♦♥s ✇✐t❤ ♦t❤❡r ✐♥♣✉ts t❤❛t r❡♣r❡s❡♥t t❤❡✐r s✐③❡s✳ ❚❤✉s✱ ❡①✐st✐♥❣ t❡❝❤♥✐q✉❡s✱ ✐♥ s♦♠❡ ❝❛s❡s✱ ♠❛② ❣❡♥❡r❛t❡ t❡st ❝❛s❡s ✇❤✐❝❤ r❡s✉❧t ✐♥ ✐♥✈❛❧✐❞ ♠❡♠♦r② ❛❝❝❡ss❡s t❤❛t ✇♦✉❧❞ ♥♦t ❤❛♣♣❡♥ ✐♥ ❛ r❡❛❧ ❡①❡❝✉t✐♦♥ ♦❢ t❤❡ ♣r♦❣r❛♠ ❜❡✐♥❣ t❡st❡❞✳ ■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ✇❡ ♣r❡s❡♥t t✇♦ st❛t✐❝ ❛♥❛❧②s❡s ❢♦r ✐♥❢❡r❡♥❝❡ ♦❢ ❛rr❛② s✐③❡s✱ ❛s ✇❡❧❧ ❛s ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❣❡♥❡r❛t✐♦♥ ♦❢ t❡st ❝❛s❡s ❝❛♣❛❜❧❡ ♦❢ ❣❡♥❡r❛t✐♥❣ s❛❢❡ t❡sts ❜② ✉s✐♥❣ s✉❝❤ ❛♥❛❧②s❡s✳ ■♥ ♦r❞❡r t♦ ❡✈❛❧✉❛t❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ ♦✉r t❡❝❤♥✐q✉❡✱ ✇❡ ♣❡r❢♦r♠❡❞ t✇♦ ❡①♣❡r✐♠❡♥ts✳ ■♥ t❤❡ ✜rst ♦❢ t❤❡♠✱ ✇❡ ❢♦✉♥❞ t❤❛t t❤❡ st❛t✐❝ ❛♥❛❧②s❡s ✇❡r❡ ❛❜❧❡ t♦ ❜✐♥❞ ✸✹✳✻✪ ♦❢ t❤❡ ❛rr❛② s✐③❡s r❡❝❡✐✈❡❞ ❛s ♣❛r❛♠❡t❡r ❜② ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❙P❊❈ ❈P❯✷✵✵✻ ❜❡♥❝❤♠❛r❦ s✉✐t❡ ✭✇❤✐❝❤ ❝♦♥t❛✐♥s r❡❛❧ ✇♦r❧❞ ♣r♦❣r❛♠s✮✳ ■♥ t❤❡ s❡❝♦♥❞✱ ✇❡ ✉s❡❞ ❆s②♠♣t✉s✱ ❛ t♦♦❧ ❢♦r ❛✉t♦♠❛t✐❝ ✐♥❢❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥ ❝♦♠♣❧❡①✐t② ✇❤✐❝❤ ✇❛s ❞❡✈❡❧♦♣❡❞ ❞✉r✐♥❣ t❤✐s ♠❛st❡r✬s r❡s❡❛r❝❤ ❛s ❛ ♣r❡❧✐♠✐♥❛r② ✇♦r❦✱ ♦✈❡r t❤❡ t❡st ❝❛s❡s ❣❡♥❡r❛t❡❞ ❜② ♦✉r t❡❝❤♥✐q✉❡✳ ❆s②♠♣t✉s ✇❛s ❛❜❧❡ t♦ ❝♦rr❡❝t❧② ✐♥❢❡r t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❢✉♥❝t✐♦♥s ❜♦t❤ ✇r✐tt❡♥ ❜② ✉s ❛♥❞ ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ P♦❧②❜❡♥❝❤ ❜❡♥❝❤♠❛r❦ s✉✐t❡✳ ■t s❤♦✇s t❤❛t ♦✉r t❡❝❤♥✐q✉❡ ♦❢ ✐♥♣✉t ❣❡♥❡r❛t✐♦♥ ✐s ✉s❡❢✉❧ ❢♦r t❤❡ ❛✉t♦♠❛t✐❝ ❡①❡❝✉t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ ✐♥t❡r❡st✳

(14)
(15)

▲✐st ♦❢ ❋✐❣✉r❡s

✶✳✶ ❆ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐t❡r❛t❡s ♦✈❡r ❛♥ ❛rr❛②✳ ❚❤❡ ❢✉♥❝t✐♦♥ r❡❝❡✐✈❡s ❛♥ ❛rr❛② ♦❢ ✐♥t❡❣❡rs ❛♥❞ ✐ts s✐③❡✳ ❚❤❡ ❛rr❛② ❛❝❝❡ss❡s ❛r❡ ❧✐♠✐t❡❞ ❜② t❤❡ s✐③❡ ❛r❣✉♠❡♥t✳ ✳ ✹

✷✳✶ ❆♥ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ t♦ ❝❛❧❝✉❧❛t❡ t❤❡nth ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡r✱ ❛♥❞ ✐ts ❈❋●✳ ❇❇✯ ✐❞❡♥t✐✜❡s ❜❛s✐❝ ❜❧♦❝❦s✳ ❚❤❡ ❛rr♦✇s ❜❡t✇❡❡♥ ❜❛s✐❝ ❜❧♦❝❦s✱ s❛② ❢r♦♠ ❇❇✵ t♦ ❇❇✶✱ ♠❡❛♥ t❤❛t t❤❡ ❡①❡❝✉t✐♦♥ ♠❛② ✢♦✇ ❢r♦♠ ♦♥❡ ❜❛s✐❝ ❜❧♦❝❦ t♦ t❤❡ ♦t❤❡r✳ ✽ ✷✳✷ ❚❤❡ r❡s✉❧t ♦❢ ❛ ❧✐✈❡♥❡ss ❛♥❛❧②s✐s ♦♥ ❢✉♥❝t✐♦♥fibonacci♦❢ ❋✐❣✉r❡ ✷✳✶✳ ❚❤❡

r❡s✉❧t ✐s s❤♦✇♥ ♦♥❧② ❢♦r t❤❡ ❜❛s✐❝ ❜❧♦❝❦ ❇❇✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❋✉♥❝t✐♦♥ fibonacci♦❢ ❋✐❣✉r❡ ✷✳✶ ✐♥ ❙❙❆ ❢♦r♠✳ ❚❤❡ ❣r❛♣❤ ✇❛s ❛✉t♦♠❛t✲

✐❝❛❧❧② ❣❡♥❡r❛t❡❞ ❜② ▲▲❱▼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✸✳✶ ❚❤❡ s②♥t❛① ♦❢ ♦✉r ❝♦r❡ ❧❛♥❣✉❛❣❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✷ ❉❛t❛ ✢♦✇ ❛♥❛❧②s✐s ❡q✉❛t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✸ ❆ ❝♦❞❡ s♥✐♣♣❡t ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❛❜str❛❝t st❛t❡ ❛❢t❡r s♦❧✈✐♥❣ ❡❛❝❤ ❝♦rr❡✲

s♣♦♥❞✐♥❣ ❡q✉❛t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✹ ❆ str✉❝t ✇❤✐❝❤ ❡♥❝❛♣s✉❧❛t❡s ❛rr❛②s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✺ ❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥new❛♥❞newpos✐s ♥♦t ✐❞❡♥t✐✜❡❞ ❜② t❤❡ ❢♦r✇❛r❞ ❛♥❛❧②s✐s✳ ✷✹

✸✳✻ ❆ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ st❛t✐❝❛❧❧② ❛❧❧♦❝❛t❡s ❛♥ ❛rr❛②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✼ ❆ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐t❡r❛t❡s ♦✈❡r ❛♥ ❛rr❛②✳ ❚❤✐s ✐s t❤❡ s❛♠❡ ❡①❛♠♣❧❡ ♦❢ ❋✐❣✉r❡ ✶✳✶✳ ✷✻ ✸✳✽ P❡r❝❡♥t❛❣❡ ♦❢ ❛rr❛②s ✇✐t❤ ❦♥♦✇♥ s✐③❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✾ P❡r❝❡♥t❛❣❡ ♦❢ ❛rr❛② ❛❝❝❡ss❡s ♣❡r❢♦r♠❡❞ ♦✈❡r ❛rr❛②s ✇✐t❤ ❦♥♦✇♥ s✐③❡s✳ ✳ ✳ ✳ ✷✽ ✸✳✶✵ ❆ ❞❛t❛ str✉❝t✉r❡ ❣r❛♣❤ ❡①❛♠♣❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✶ ❆ ❣r❛♣❤✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ ❛rr❛② s✐③❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✷ ❚❤❡ ✐♥♣✉t ❣❡♥❡r❛t✐♦♥ ❛❧❣♦r✐t❤♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✸ ❆ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐t❡r❛t❡s ♦✈❡r ❛♥ ❛rr❛②✳ ❚❤✐s ✐s t❤❡ s❛♠❡ ❡①❛♠♣❧❡ ♦❢ ❋✐❣✉r❡ ✶✳✶✳ ✸✷ ✸✳✶✹ ❚❤❡ ▲▲❱▼✬s ■❘ ♦❢ t❤❡ ❝♦❞❡ s♥✐♣♣❡t ✐♥ ❋✐❣✉r❡ ✸✳✶✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✶✺ ❈♦❞❡ ♣r♦❞✉❝❡❞ ❜② t❤❡ ✐♥♣✉t ❣❡♥❡r❛t♦r t♦ t❡st ❢✉♥❝t✐♦♥ sum✳ ▲✐♥❡s ✷✱ ✸✱ ✹

❛♥❞ ✽ ❛r❡ ♣r♦❞✉❝❡❞ ❜② t❤❡ s❧✐❝❡ ❢♦r t❤❡ ❝♦❞❡ s♥✐♣♣❡t ✐♥ ❋✐❣✉r❡ ✸✳✶✸✳ ✳ ✳ ✳ ✳ ✸✸

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✹✳✶ ▼❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✕ t❤❡ r✉♥♥✐♥❣ ❡①❛♠♣❧❡ t❤❛t ✇❡ s❤❛❧❧ ✉s❡ t♦ ❡①♣❧❛✐♥ ♦✉r ❝♦♥tr✐❜✉t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷ ●♣r♦❢ ♦✉t♣✉t ❢♦r ❛ s✐♠♣❧❡ ♣r♦❣r❛♠ ❝♦♥t❛✐♥✐♥❣ ♦✉r ❡①❛♠♣❧❡ ❢✉♥❝t✐♦♥✳ ✳ ✳ ✳ ✸✽ ✹✳✸ ❚❤❡ ♦✉t♣✉t ♣r♦❞✉❝❡❞ ❜② t❤❡ ❛♣r♦❢ ✐♥♣✉t s❡♥s✐t✐✈❡ ♣r♦✜❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✹ ❆ ❢✉♥❝t✐♦♥ t♦ ♣r✐♥t ❞✉♣❧✐❝❛t❡ ❧✐♥❡s ❝♦♥t❛✐♥✐♥❣ ❛ ❣✐✈❡♥ ❦❡②✳ ❚❤❡ s❡❝♦♥❞ ❧♦♦♣

❤❛s ❛ ❝♦♥❞✐t✐♦♥❛❧ ❡①❡❝✉t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✺ ✭❛✮ Pr♦❣r❛♠ ✇✐t❤ ❛ ♠✉❧t✐✲♣❛t❤ ❧♦♦♣✳ ✭❜✮ ❚❤❡ ❝♦st✲❣r❛♣❤ ♦❢ t❤❡ ♣r♦❣r❛♠✳

◆♦❞❡s r❡♣r❡s❡♥t ♣r♦❣r❛♠ ♣♦✐♥ts ❛♥❞ t❤❡ ❡❞❣❡s✬ ✇❡✐❣❤ts r❡♣r❡s❡♥t t❤❡ ♥✉♠✲ ❜❡r ♦❢ ❡①❡❝✉t❡❞ ✐♥str✉❝t✐♦♥s ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts✳ ✭❝✮ ❚❤❡ ❝♦st ♦❢ ❡❛❝❤ ❧♦♦♣ ✐t❡r❛t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✻ ✭❛✮ P♦❧②♥♦♠✐❛❧s ❢♦✉♥❞ ❢♦r t❤❡ ❧♦♦♣ ❛t ❧✐♥❡s ✶✽✲✷✺ ♦❢ ❋✐❣✉r❡ ✹✳✶✳ ✭❜✮ P♦❧②✲

♥♦♠✐❛❧s ❢♦✉♥❞ ❢♦r t❤❡ ❧♦♦♣ ♥❡st ❛t ❧✐♥❡s ✼✲✶✺✳ ■♥ ❡❛❝❤ ✜❣✉r❡✱ t❤❡ ✜rst ❝✉r✈❡ t❤❛t ✜ts t❤❡ ♣♦✐♥ts ✐♥ t❤❡ ✈❡r✐✜❝❛t✐♦♥ s❡t ✐s ♠❛r❦❡❞ ✐♥ ❣r❛②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✼ P❡r❝❡♥t❛❣❡ ♦❢ ❧♦♦♣s ♣❡r ❜❡♥❝❤♠❛r❦ ♦❢ ❘♦❞✐♥✐❛ t❤❛t ✇❡ ❝♦✉❧❞ ❛♥❛❧②③❡✳ ❚❤❡

❝♦rr❡❝t♥❡ss ♦❢ ❛❧❧ t❤❡s❡ r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ❝❤❡❝❦❡❞ ♠❛♥✉❛❧❧②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✽ ❆♥ ❡①❛♠♣❧❡ ♦❢ ▲▲❱▼✬s ■❘ ❛♥❞ ❛ ❝♦❞❡ s♥✐♣♣❡t ♦❢ ❛♥ ▲▲❱▼ ♣❛ss✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✾ ❚❤❡ ❛r❝❤✐t❡❝t✉r❡ ♦❢ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

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▲✐st ♦❢ ❚❛❜❧❡s

✸✳✶ ❘❡❛s♦♥s ❢♦r ✐♥❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ❢♦r✇❛r❞ ❛♥❛❧②s✐s ♦♥ r❛♥❞♦♠❧② ❝❤♦s❡♥ ❢✉♥❝✲ t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

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❈♦♥t❡♥ts

❆❣r❛❞❡❝✐♠❡♥t♦s ✐①

❘❡s✉♠♦ ①✐

❆❜str❛❝t ①✐✐✐

▲✐st ♦❢ ❋✐❣✉r❡s ①✈

▲✐st ♦❢ ❚❛❜❧❡s ①✈✐✐

✶ ■♥tr♦❞✉❝t✐♦♥ ✸

✶✳✶ P✉❜❧✐❝❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✷ ▲✐t❡r❛t✉r❡ ❘❡✈✐❡✇ ✼

✷✳✶ ❉❛t❛✲✢♦✇ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❙t❛t✐❝ ❙✐♥❣❧❡ ❆ss✐❣♥♠❡♥t ❋♦r♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❆✉t♦♠❛t✐❝ ■♥❢❡r❡♥❝❡ ♦❢ Pr♦❣r❛♠ ❈♦♠♣❧❡①② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✹ Pr♦❣r❛♠ ❙❧✐❝✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✺ ❉❛t❛ ❙tr✉❝t✉r❡ ●r❛♣❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✻ ❙②♠❜♦❧✐❝ ❊①❡❝✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✼ ❋✉③③ ❚❡st✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✽ ❋✐♥❛❧ ❘❡♠❛r❦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸ ●❡♥❡r❛t✐♦♥ ♦❢ ❚❡st ❈❛s❡s ❢♦r ▲❛♥❣✉❛❣❡s ✇✐t❤ P♦✐♥t❡r ❆r✐t❤♠❡t✐❝s ✶✾ ✸✳✶ ❆rr❛② ❙✐③❡ ■♥❢❡r❡♥❝❡ ✐♥ ❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✶ ❋♦r✇❛r❞ ❙✐③❡ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✶✳✷ ❇❛❝❦✇❛r❞ ❙✐③❡ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✷ ❚❡st ❈❛s❡ ●❡♥❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷✳✶ ❉❛t❛ ❙tr✉❝t✉r❡ ●r❛♣❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷✳✷ ■♥♣✉t ●❡♥❡r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

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✸✳✷✳✸ ❙❧✐❝✐♥❣ ❚❡❝❤♥✐q✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹ ❈❛s❡ ❙t✉❞② ✸✺

✹✳✶ ❖✈❡r✈✐❡✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷ ❆✉t♦♠❛t✐❝ ■♥❢❡r❡♥❝❡ ♦❢ ▲♦♦♣ ❈♦♠♣❧❡①✐t② t❤r♦✉❣❤ P♦❧②♥♦♠✐❛❧ ■♥t❡r♣♦✲

❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷✳✶ ■♥♣✉t ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✷ ▲♦♦♣ ❉❡♣❡♥❞❡♥❝❡ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✷✳✸ ❈♦❞❡ ■♥str✉♠❡♥t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✹ P♦❧②♥♦♠✐❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✸ ❊✈❛❧✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✸✳✶ ❆♥ ▲▲❱▼ P❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✸✳✷ ❊①♣❡r✐♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✹ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✺ ❈♦♥❝❧✉s✐♦♥ ✺✶

✺✳✶ ❈♦♥tr✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✷ ❋✉t✉r❡ ❲♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

❇✐❜❧✐♦❣r❛♣❤② ✺✸

❆♣♣❡♥❞✐① ❆ ❋✉♥❝t✐♦♥s ❢♦r t❤❡ ■♥♣✉t ●❡♥❡r❛t✐♦♥ ❊①♣❡r✐♠❡♥t ✺✾

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❈❤❛♣t❡r ✶

■♥tr♦❞✉❝t✐♦♥

❙♦❢t✇❛r❡ t❡st✐♥❣ ❛♥❞ ❞❡❜✉❣❣✐♥❣ ❛r❡ ❤❛r❞ t❛s❦s✳ ■♥ ❣❡♥❡r❛❧✱ t♦ ♣r♦♣❡r❧② ♣❡r❢♦r♠ s✉❝❤ t❛s❦s✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❛♥❛❧②③❡ ❛❧❧ ♣♦ss✐❜❧❡ ❡①❡❝✉t✐♦♥ ♣❛t❤s ✐♥ t❤❡ ♣r♦❣r❛♠✳ ■♥ t❤✐s s❡♥s❡✱ ♣r❡✈✐♦✉s ✇♦r❦s ❬❈❛❞❛r ❡t ❛❧✳✱ ✷✵✵✽❀ ●♦❞❡❢r♦✐❞ ❡t ❛❧✳✱ ✷✵✵✺❛❀ ●♦❞❡❢r♦✐❞✱ ✷✵✵✼❀ ❙❛①✲ ❡♥❛ ❡t ❛❧✳✱ ✷✵✵✾❪ ❤❛✈❡ ♣r♦♣♦s❡❞ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s t♦ ❛✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡r❛t❡ ✐♥♣✉ts ✭t❡st ❝❛s❡s✮ ❢♦r ♣r♦❣r❛♠s✳ ❖♥❡ ♦❢ t❤❡ ♠♦st ❝♦♠♠♦♥ t❡❝❤♥✐q✉❡s t♦ t❤✐s ❡♥❞ ✐s s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❬❈❛❞❛r ❛♥❞ ❙❡♥✱ ✷✵✶✸❪✳ ❯s✉❛❧❧②✱ ✐t ✐s ♣❡r❢♦r♠❡❞ ❜② ♠❡❛♥s ♦❢ ❛ ✈✐rt✉❛❧ ♠❛❝❤✐♥❡ ✇❤✐❝❤ r❡♣❧❛❝❡s t❤❡ ♦♣❡r❛t✐♦♥s t❤❛t ♠❛♥✐♣✉❧❛t❡ ❝♦♥❝r❡t❡ ✈❛❧✉❡s ❜② ♦♥❡s ✇❤✐❝❤ ♦♣❡r❛t❡ ♦✈❡r s②♠❜♦❧✐❝ ✈❛❧✉❡s✳ ❚❤❡ ♣r♦❣r❛♠ st❛rts ❡①❡❝✉t✐♥❣ ✇✐t❤ s②♠❜♦❧✐❝ ✈❛❧✉❡s ❛s ✐♥♣✉ts✳ ❲❤❡♥❡✈❡r t❤❡ ❡①❡❝✉t✐♦♥ ✢♦✇ r❡❛❝❤❡s ❛ ❜r❛♥❝❤✱ ❜♦t❤ s✐❞❡s ❛r❡ s②♠❜♦❧✐❝❛❧❧② ❡①❡❝✉t❡❞ ✏s✐♠✉❧t❛♥❡♦✉s❧②✑ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥s t♦ ❡①❡❝✉t❡ s✉❝❤ ♣❛t❤s ✭❝❛❧❧❡❞ ♣❛t❤ ❝♦♥❞✐t✐♦♥s✴❝♦♥✲ str❛✐♥ts✮ ❛r❡ st♦r❡❞✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ❡①❡❝✉t✐♦♥✱ ♦r ✇❤❡♥ ✐t ✜♥❞s ❛ ❜✉❣✱ ❛ ❝♦♥str❛✐♥t s♦❧✈❡r ✜♥❞s ❝♦♥❝r❡t❡ ✐♥♣✉t ✈❛❧✉❡s ✇❤✐❝❤ ❢♦❧❧♦✇ t❤❡ s❛♠❡ ❡①❡❝✉t✐♦♥ ✢♦✇ ❛s t❤❡ s②♠❜♦❧✐❝ ✈❛❧✉❡s ✇❤✐❝❤ ♣r♦❞✉❝❡❞ ❛ ♣❛rt✐❝✉❧❛r s❡t ♦❢ ♣❛t❤ ❝♦♥❞✐t✐♦♥s✳

❆♥♦t❤❡r ✉s❡❞ ❛♣♣r♦❛❝❤ ✐s t❤❡ ❋✉③③ ❚❡st✐♥❣ ✭♦r ❢✉③③✐♥❣✮✱ ✇❤✐❝❤ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ✾✵✬s ❬▼✐❧❧❡r ❡t ❛❧✳✱ ✶✾✾✵❪ ✇✐t❤ t❤❡ ♣✉r♣♦s❡ ♦❢ t❡st✐♥❣ ♣r♦❣r❛♠s ❛s ❛ ❜❧❛❝❦ ❜♦① ♣r♦✈✐❞✐♥❣ r❛♥❞♦♠ ✐♥♣✉ts✳ ❚❤✐s t❡❝❤♥✐q✉❡ ✇❛s ❧❛t❡r ✐♠♣r♦✈❡❞ ❜② ●♦❞❡❢r♦✐❞ ❡t ❛❧✳ ❬✷✵✵✽❪ ✇✐t❤ t❤❡ ❛❜✐❧✐t② ♦❢ ❣❡tt✐♥❣ ❛ ❢❡❡❞❜❛❝❦ ♦❢ t❤❡ ❡①❡❝✉t✐♦♥ ❢♦r s♦♠❡ ✐♥✐t✐❛❧ r❛♥❞♦♠ ✐♥♣✉ts ❛♥❞ ✜♥❞✐♥❣ ♥❡✇ ✐♥♣✉t ✈❛❧✉❡s ✇❤✐❝❤ ❢♦❧❧♦✇ ❞✐✛❡r❡♥t ♣❛t❤s✳ ❚❤❡② ❝❛❧❧❡❞ t❤✐s ♥❡✇ t❡❝❤♥✐q✉❡ ❛s ❲❤✐t❡❜♦① ❋✉③③ ❚❡st✐♥❣✳ ❆❧t❤♦✉❣❤ ❞✐✛❡r❡♥t t❡❝❤♥✐q✉❡s ❛r❡ s❤♦✇♥ t♦ ❜❡ ❡✛❡❝t✐✈❡ ♦♥ ✜♥❞✐♥❣ ❜✉❣s ♦r ❛❝❤✐❡✈✐♥❣ ❤✐❣❤ ❝♦❞❡ ❝♦✈❡r❛❣❡✱ t❤❡r❡ ❡①✐st ❧✐♠✐t❛t✐♦♥s✳ ❆ ❧✐♠✐t❛t✐♦♥ ✇❤✐❝❤ ❝❛t❝❤❡s ♦✉r ❛tt❡♥t✐♦♥ ✐s r❡❣❛r❞✐♥❣ t♦ t❤❡ s✐③❡ ♦❢ ♠❡♠♦r② r❡❣✐♦♥s✳ ❯♥❧✐❦❡ s♦♠❡ str♦♥❣❧② t②♣❡❞ ❧❛♥❣✉❛❣❡s✱ ❛❧❧♦❝❛t❡❞ ♠❡♠♦r② ✐♥ ❈ ❤❛s ♥♦ ♠❡t❛ ✐♥❢♦r♠❛t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ ❡①❛♠♣❧❡ ✐♥ ❋✐❣✉r❡ ✶✳✶✳ ■❢ ✇❡ ♣❛ss t♦ ❢✉♥❝t✐♦♥ sum ❛♥ ❛rr❛② ✇✐t❤ ✶✵ ❡❧❡♠❡♥ts✱ t❤❡

✈❛❧✉❡ ♦❢ size ❤❛s t♦ ❜❡ ❧❡ss t❤❛♥ ♦r ❡q✉❛❧s t♦ ✶✵✳ ■❢ ✐t ✐s ♥♦t t❤❡ ❝❛s❡✱ ✇❡ ✇✐❧❧ ❡♥❞ ✉♣

r❡❛❝❤✐♥❣ ❛ ❜✉❣ ✇❤✐❝❤ ♠❛② ♥❡✈❡r ❤❛♣♣❡♥ ✐♥ ❛ ❛❝t✉❛❧ ❡①❡❝✉t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ ✐ts

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✹ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥

❝♦♥t❡①t ✲ ✐✳❡✳ ❛ ❢❛❧s❡ ♣♦s✐t✐✈❡✳ ❋♦r t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ ♥♦ ❡①✐st✐♥❣ t❡❝❤♥✐q✉❡ ❤❛♥❞❧❡ ✐t ♣r♦♣❡r❧②✳

1: int sum(int *A, int size) { 2: int s = 0;

3: for (int i=0; i < size; i++) {

4: s += A[i];

5: }

6: return s; 7: }

❋✐❣✉r❡ ✶✳✶✿ ❆ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐t❡r❛t❡s ♦✈❡r ❛♥ ❛rr❛②✳ ❚❤❡ ❢✉♥❝t✐♦♥ r❡❝❡✐✈❡s ❛♥ ❛rr❛② ♦❢ ✐♥t❡❣❡rs ❛♥❞ ✐ts s✐③❡✳ ❚❤❡ ❛rr❛② ❛❝❝❡ss❡s ❛r❡ ❧✐♠✐t❡❞ ❜② t❤❡ s✐③❡ ❛r❣✉♠❡♥t✳

■♥ t❤✐s ✇♦r❦ ✇❡ ❛✐♠ t♦ s♦❧✈❡ t❤✐s ❧✐♠✐t❛t✐♦♥ ❜② ✉s✐♥❣ st❛t✐❝ ❛♥❛❧②s❡s t♦ ❜✐♥❞ ♠❡t❛ ✐♥❢♦r♠❛t✐♦♥ t♦ ♠❡♠♦r② r❡❣✐♦♥s✳ ❖✉r ❤②♣♦t❤❡s❡s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

■t ✐s ♣♦ss✐❜❧❡ t♦ r❡❞✉❝❡ t❤❡ ❢❛❧s❡ ♣♦s✐t✐✈❡ r❛t❡ ♦❢ ❝✉rr❡♥t ❛✉t♦♠❛t✐❝ t❡st✲ ✐♥❣ t❡❝❤♥✐q✉❡s ✐❢ ✇❡ ❝❛♥ ❞❡t❡r♠✐♥❡ s②♠❜♦❧✐❝ ♠❡t❛ ✐♥❢♦r♠❛t✐♦♥ ❢♦r ♠❡♠♦r② r❡❣✐♦♥s ✇❤✐❝❤ ❛r❡ ✐♥♣✉ts ❢♦r t❤❡ t❡st✐♥❣ ✉♥✐t✳

■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✇❡ ♣r❡s❡♥t t✇♦ st❛t✐❝ ❛♥❛❧②s❡s t♦ ❜✐♥❞ ❛❧❧♦❝❛t❡❞ ♠❡♠♦r② r❡❣✐♦♥s ✇✐t❤ t❤❡✐r s✐③❡s✳ ❚❤❡ ✜rst ♦♥❡✱ ✇❤✐❝❤ ✇❡ ❝❛❧❧ ❢♦r✇❛r❞ s✐③❡ ❛♥❛❧②s✐s✱ ❣❡ts ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠❡♠♦r② ❛❧❧♦❝❛t✐♦♥ ✐♥str✉❝t✐♦♥s ❛♥❞ ♣r♦♣❛❣❛t❡s ✐t ❢♦r✇❛r❞✳ ❚❤❡ s❡❝♦♥❞✱ ❜❛❝❦✇❛r❞ s✐③❡ ❛♥❛❧②s✐s✱ ❣❡ts ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠❡♠♦r② ❛❝❝❡ss❡s ❛♥❞ ♣r♦♣❛❣❛t❡ ✐t ❜❛❝❦✇❛r❞✳ ❲❤✐❧❡ t❤❡ ❢♦r♠❡r ✐s ♠♦r❡ ♣r❡❝✐s❡ t♦ ♦✉r ❣♦❛❧✱ t❤❡ ❧❛tt❡r ❤❛s s❤♦✇♥ t♦ ❜❡ ❛❜❧❡ t♦ ❜✐♥❞ ♠♦r❡ ♣❛✐rs ♦❢ ❛❧❧♦❝❛t❡❞ ♠❡♠♦r② ❛♥❞ s✐③❡✳

■♥ ♦r❞❡r t♦ ♠❡❛s✉r❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ s✐③❡ ❛♥❛❧②s❡s✱ ✇❡ ❤❛✈❡ ❡①❡❝✉t❡❞ ❡①✲ ♣❡r✐♠❡♥ts ♦✈❡r t❤❡ ❜❡♥❝❤♠❛r❦s ❢♦✉♥❞ ❛t ❙P❊❈ ❈P❯✷✵✵✻✳ ❲❡ ❤❛✈❡ ❜❡❡♥ ❛❜❧❡ t♦ ❜✐♥❞ s✐③❡s ❢♦r ✷✷✳✼✪✱ ✐♥ ❛✈❡r❛❣❡✱ ♦❢ ❛❧❧ ❛rr❛②s ✐♥ ❙P❊❈ ♣❛ss❡❞ ❛s ❛r❣✉♠❡♥t t♦ ❢✉♥❝t✐♦♥s✳ ❆❧s♦✱ ✇❡ ❤❛✈❡ ❝♦✉♥t❡❞ t❤❛t ❢♦r ❛❧❧ ❛rr❛② ❛❝❝❡ss❡s ✐♥ ❙P❊❈ t❤❛t ❛r❡ ♣❡r❢♦r♠❡❞ ♦✈❡r ❛r❣✉♠❡♥ts ♦❢ ❢✉♥❝t✐♦♥s✱ ✷✽✪ ♦❢ t❤❡♠ ♣❡r❢♦r♠ ♦✈❡r ❛rr❛②s ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❜❡❡♥ ❛❜❧❡ t♦ ✜♥❞ ❛ s✐③❡✳ ■t t❡❧❧s ✉s t❤❛t ❣❡♥❡r❛t✐♥❣ ✐♥♣✉ts ❢♦r ❛rr❛②s ✉s✐♥❣ t❤❡ r❡s✉❧t ♦❢ t❤✐s ❛♥❛❧②s✐s ❛✈♦✐❞s ♠❡♠♦r② ❡rr♦rs ✐♥ ✉♣ t♦ ✷✽✪ ♦❢ ❛❧❧ ❛rr❛② ❛❝❝❡ss❡s✳

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r❛♥❞♦♠ ✈❛❧✉❡s ❢♦r ❡❛❝❤ ✐♥♣✉t✳ ❲❤❡♥ ♦✉r s✐③❡ ❛♥❛❧②s❡s ❛r❡ ❛❜❧❡ t♦ ❜✐♥❞ ❛♥ ❛rr❛② ✇✐t❤ ✐ts s✐③❡✱ ♦✉r ❛❧❣♦r✐t❤♠ ❣❡♥❡r❛t❡s ✈❛❧✉❡s ✇❤✐❝❤ ❛r❡ ❝♦♥s✐st❡♥t ❢♦r ❜♦t❤ ❛r❣✉♠❡♥ts✳

❲❡ ❤❛✈❡ ✐♠♣❧❡♠❡♥t❡❞ ♦✉r t❡❝❤♥✐q✉❡s ♦♥ t♦♣ ♦❢ t❤❡ ▲▲❱▼ ❝♦♠♣✐❧❡r ✐♥❢r❛str✉❝✲ t✉r❡ ❬▲❛tt♥❡r ❛♥❞ ❆❞✈❡✱ ✷✵✵✹❜❪✳ ❖✉r ❛♣♣r♦❛❝❤ ❤❛s s❡✈❡r❛❧ ♣♦ss✐❜❧❡ ✉s❡s✳ ❋♦r ✐♥st❛♥❝❡✱ ✇❡ ❤❛✈❡ ✉s❡❞ ✐t t♦ ✐♠♣r♦✈❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ ❛ ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t② ✐♥❢❡r❡♥❝❡ t♦♦❧✱ ❝❛❧❧❡❞ ❆s②♠♣t✉s ❬❉❡♠♦♥t✐ê ❡t ❛❧✳✱ ✷✵✶✺❪✳ ❆s②♠♣t✉s ✐s ❛ t♦♦❧ ❢♦r ❛✉t♦♠❛t✐❝ ✐♥❢❡r❡♥❝❡ ♦❢ ❧♦♦♣ ❝♦♠♣❧❡①✐t②✱ ✇❤✐❝❤ ✇❛s ❞❡✈❡❧♦♣❡❞ ❞✉r✐♥❣ t❤✐s ♠❛st❡r✬s r❡s❡❛r❝❤ ❛s ❛ ♣r❡❧✐♠✐♥❛r② ✇♦r❦✳ ■t ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ ❢♦✉r ♠❛✐♥ st❡♣s✿ ✭✶✮ st❛t✐❝ ❛♥❛❧②s✐s✱ ✭✷✮ ❝♦❞❡ ✐♥str✉♠❡♥✲ t❛t✐♦♥✱ ✭✸✮ ❞②♥❛♠✐❝ ✐♥❢♦r♠❛t✐♦♥ ❡①tr❛❝t✐♦♥ ❛♥❞ ✭✹✮ ♣♦❧②♥♦♠✐❛❧ ✐♥t❡r♣♦❧❛t✐♦♥✳ ❙✐♥❝❡ t❤✐s t❡❝❤♥✐q✉❡ ✐s ♥♦✈❡❧✱ ✇❡ ❤❛✈❡ ♣✉❜❧✐s❤❡❞ ❛ ♣❛♣❡r ❛❜♦✉t ✐t✱ ✇❤✐❝❤ ✐s ♣❛rt✐❛❧❧② r❡st❛t❡❞ ✐♥ ❈❤❛♣t❡r ✹✳ ❖♥❡ ♦❢ t❤❡ ❆s②♠♣t✉s✬ ❧✐♠✐t❛t✐♦♥s ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✐t ♥❡❡❞s t♦ ❡①❡❝✉t❡ ❛ ❢✉♥❝t✐♦♥ ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ t✐♠❡s✱ ✇✐t❤ ❞✐✛❡r❡♥t ✐♥♣✉ts✱ ✐♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ❛♥❛❧②③❡ ✐t✳ ❍♦✇❡✈❡r✱ ✐t ♠❛② ♥♦t ❤❛♣♣❡♥ ❢♦r s❡✈❡r❛❧ r❡❛s♦♥s ✭❧❛❝❦ ♦❢ ❞❛t❛✲s❡ts✱ ❤❛r❞❝♦❞❡❞ ✐♥♣✉ts ❢♦r ❢✉♥❝t✐♦♥s✱ ❢✉♥❝t✐♦♥s ❝❛❧❧❡❞ ♦♥❧② ❢♦r s♣❡❝✐✜❝ ✐♥♣✉ts✱ ❛♠♦♥❣ ♦t❤❡rs✮✳ ❲❡ ❤❛✈❡ ❡①❡❝✉t❡❞ ❆s②♠♣t✉s t♦❣❡t❤❡r ✇✐t❤ ♦✉r ✐♥♣✉t ❣❡♥❡r❛t♦r ♦♥ ❢✉♥❝t✐♦♥s t❤❛t ❤❛✈❡ ❛rr❛②s✱ ♠❛tr✐❝❡s ❛♥❞ r❡❝✉rs✐✈❡ ❞❛t❛✲str✉❝t✉r❡s ❛s ✐♥♣✉ts ❛♥❞ ❢♦✉♥❞ t❤❛t t❤❡ t❡❝❤♥✐q✉❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s ❡✛❡❝t✐✈❡ ♦♥ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ✐♥t❡r❡st ❢✉♥❝t✐♦♥s✳

❚❤❡ ♠❛✐♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♦✉r ✇♦r❦ ✐s ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ ❛ t❡st✐♥❣ ✐♥❢r❛✲str✉❝t✉r❡ ❢♦r ▼❛①tr❛❝❦✱ ❛ ❇r❛③✐❧✐❛♥ ❝♦♠♣❛♥② ✇❤✐❝❤ ❜✉✐❧❞s tr❛❝❦❡rs ❢♦r tr✉❝❦s✳ ❚❤✐s ✐♥❢r❛✲str✉❝t✉r❡ ✐s ♠❡❛♥t ❜❡ ✉s❡❞ t♦ t❡st✱ ✐♥ ❛ ❯♥✐① ❡♥✈✐r♦♥♠❡♥t✱ t❤❡ s♦❢t✇❛r❡ ❞❡✈❡❧♦♣❡❞ t♦ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ tr❛❝❦❡rs✳ ❇❡❝❛✉s❡ ♦❢ t❤❛t✱ ✇❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡ ❜♦❞✐❡s ♦❢ s♦♠❡ ❧✐❜r❛r② ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡② ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡r❛t❡❞ t♦ r❡t✉r♥ ❛ r❛♥❞♦♠ ✈❛❧✉❡ ❜❛s❡❞ ♦♥ t❤❡ ❢✉♥❝t✐♦♥✬s r❡t✉r♥ t②♣❡✳ ❲❡ ❡①❡❝✉t❡ t❤❡ t❡st ❞r✐✈❡rs ❣❡♥❡r❛t❡❞ ❢♦r ❝❤♦s❡♥ ❢✉♥❝t✐♦♥s t♦❣❡t❤❡r ✇✐t❤ ❱❛❧❣r✐♥❞ ❬◆❡t❤❡r❝♦t❡ ❛♥❞ ❙❡✇❛r❞✱ ✷✵✵✼❛❪ ✐♥ ♦r❞❡r t♦ ❝❛t❝❤ ♠❡♠♦r② ❝♦rr✉♣t✐♦♥ ❡rr♦rs✳ ❲❤❡♥ ❛♥ ❡rr♦r ✐s ❢♦✉♥❞✱ ✇❡ ❧♦❣ t❤❡ s❡❡❞ ♦❢ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥s✱ ❛❧❧♦✇✐♥❣ t❤❡ ✉s❡r t♦ ❧❛t❡r r❡♣r♦❞✉❝❡ t❤❡ t❡st✳

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✻ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥

✶✳✶ P✉❜❧✐❝❛t✐♦♥s

• ❋r❛♥❝✐s❝♦ ❉❡♠♦♥t✐ê✱ ❋✐❧✐♣❡ ❞❡ ▲✐♠❛ ❆r❝❛♥❥♦✱ ❛♥❞ ▼❛r✐③❛ ❆✳ ❙✳ ❇✐❣♦♥❤❛✳ ❯♠ ❆❧❣♦r✐t♠♦ ♣❛r❛ ❊♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❈❤❛♠❛❞❛s ❞❡ ❋✉♥çã♦✳ ❇r❛③✐❧✐❛♥ ❙②♠♣♦s✐✉♠ ♦♥ Pr♦❣r❛♠♠✐♥❣ ▲❛♥❣✉❛❣❡s ✭❙❇▲P✮✳ ✷✵✶✹✳

• ❋r❛♥❝✐s❝♦ ❉❡♠♦♥t✐ê✱ ❏✉♥✐♦ ❈❡③❛r✱ ▼❛r✐③❛ ❆✳ ❙✳ ❇✐❣♦♥❤❛✱ ❋r❡❞❡r✐❝♦ ❈❛♠♣♦s✱ ❋❡r✲ ♥❛♥❞♦ ▼❛❣♥♦ ◗✉✐♥tã♦ P❡r❡✐r❛✱ ❆✉t♦♠❛t✐❝ ■♥❢❡r❡♥❝❡ ♦❢ ▲♦♦♣ ❈♦♠♣❧❡①✐t② ❚❤r♦✉❣❤ P♦❧②♥♦♠✐❛❧ ■♥t❡r♣♦❧❛t✐♦♥✳ ❇r❛③✐❧✐❛♥ ❙②♠♣♦s✐✉♠ ♦♥ Pr♦❣r❛♠♠✐♥❣ ▲❛♥❣✉❛❣❡s ✭❙❇▲P✮✱ ♣♣✳ ✶✲✶✺✳ ✷✵✶✺✳

✕ ❚❤✐s ♣❛♣❡r ✇❛s ❝❤♦s❡♥ ❛s t❤❡ 3rd ✭t❤✐r❞✮ ❜❡st ♣❛♣❡r ♦❢ t❤❡ ❇r❛③✐❧✐❛♥ ❙②♠✲ ♣♦s✐✉♠ ♦♥ Pr♦❣r❛♠♠✐♥❣ ▲❛♥❣✉❛❣❡s ✷✵✶✺✳

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❈❤❛♣t❡r ✷

▲✐t❡r❛t✉r❡ ❘❡✈✐❡✇

❚❤✐s ❝❤❛♣t❡r ♣r❡s❡♥ts ❛ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇ ❛❜♦✉t t❤❡ s✉❜❥❡❝ts ❛♥❞ ❝♦♥❝❡♣ts r❡❧❛t❡❞ t♦ t❤✐s ✇♦r❦✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ♥❡❝❡ss❛r② ❜❛❝❦❣r♦✉♥❞ ❢♦r t❤❡ r❡❛❞❡r ✐♥ ♦r❞❡r t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ t❤✐s ❞✐ss❡rt❛t✐♦♥ ❛♥❞ ❡♥❞ ♣r❡s❡♥t✐♥❣ t❤❡ ✇♦r❦s t❤❛t ❛r❡ ♠♦st r❡❧❛t❡❞ t♦ ♦✉rs✳

✷✳✶ ❉❛t❛✲✢♦✇ ❆♥❛❧②s✐s

❚♦ ✐♥❢❡r t❤❡ s✐③❡s ♦❢ ❛rr❛②s✱ ✇❡ r❡s♦rt t♦ ❞❛t❛✲✢♦✇ ❛♥❛❧②s✐s✳ ❚♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❞❛t❛✲✢♦✇ ❛♥❛❧②s✐s✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❦♥♦✇ ✇❤❛t ✐s ❛ ❈♦♥tr♦❧ ❋❧♦✇ ●r❛♣❤ ✭❈❋●✮ ❬❆❧❧❡♥✱ ✶✾✼✵❪✳ ❆ ❈❋● ✐s ❛ ♣r♦❣r❛♠ r❡♣r❡s❡♥t❛t✐♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇❤❡r❡ ♥♦❞❡s ❛r❡ ❜❛s✐❝ ❜❧♦❝❦s ❛♥❞ t❤❡r❡ ✐s ❛♥ ❡❞❣❡ ❜❡t✇❡❡♥ t✇♦ ❜❛s✐❝ ❜❧♦❝❦s ❇❇✶ ❛♥❞ ❇❇✷ ✐❢ t❤❡ ❡①❡❝✉t✐♦♥ ❝❛♥ ✢♦✇ t♦ ❇❇✷ r✐❣❤t ❛❢t❡r t❤❡ ❡♥❞ ♦❢ ❇❇✶✳ ❆ ❜❛s✐❝ ❜❧♦❝❦ ✐s t❤❡ ♠❛①✐♠✉♠ s❡t ♦❢ ❝♦♥s❡❝✉t✐✈❡ ✐♥str✉❝t✐♦♥s ✇✐t❤ ❜❛s✐❝❛❧❧② t✇♦ ♣r♦♣❡rt✐❡s✿ ✭✐✮ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ❛ ❜❛s✐❝ ❜❧♦❝❦ ♦♥❧② st❛rts ❢r♦♠ t❤❡ ✜rst ✐♥str✉❝t✐♦♥ ✭t❤❡r❡ ❛r❡ ♥♦ ❥✉♠♣s t♦ t❤❡ ♠✐❞❞❧❡ ♦❢ ❛ ❜❛s✐❝ ❜❧♦❝❦✮ ❛♥❞ ✭✐✐✮ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ❛ ❜❛s✐❝ ❜❧♦❝❦ ♦♥❧② ❡♥❞s ✐♥ t❤❡ ❧❛st ✐♥str✉❝t✐♦♥ ✭t❤❡r❡ ❛r❡ ❛❧s♦ ♥♦ ❥✉♠♣s ❢r♦♠ t❤❡ ♠✐❞❞❧❡ ♦❢ ❛ ❜❛s✐❝ ❜❧♦❝❦✮✳ ■t ♠❡❛♥s t❤❛t ❛♥② ❜r❛♥❝❤ ✐♥str✉❝t✐♦♥ ✐s t❤❡ ❡♥❞ ♦❢ ❛ ❜❛s✐❝ ❜❧♦❝❦ ❛♥❞ t❤❡ t❛r❣❡t ♦❢ ❛♥② ❜r❛♥❝❤ ✐s t❤❡ ✜rst ✐♥str✉❝t✐♦♥ ♦❢ ❛ ❜❛s✐❝ ❜❧♦❝❦✳ ❋✐❣✉r❡ ✷✳✶ s❤♦✇s ❛ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ✐♠♣❧❡♠❡♥ts t❤❡ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ nth ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡r✱ ❛♥❞ ❛ ❣r❛♣❤✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✐ts ❈❋●✳

❉❛t❛✲✢♦✇ ❛♥❛❧②s✐s ❬❑✐❧❞❛❧❧✱ ✶✾✼✸❪ ✐s ❛ t❡❝❤♥✐q✉❡ ❢♦r ❛♥❛❧②③✐♥❣ ❛♥ ✐♥t❡r❡st ♣r♦♣❡rt② ❛t s❡✈❡r❛❧ ♣♦✐♥ts ♦❢ ❛ ♣r♦❣r❛♠✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ❛♣♣r♦①✐♠❛t❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞②♥❛♠✐❝ ❜❡❤❛✈✐♦r ♦❢ ❛ ♣r♦❣r❛♠ ❜② ❛♥❛❧②③✐♥❣ ✐t st❛t✐❝❛❧❧②✳ ❚❤❡ ♣r♦♣❡rt✐❡s ❛r❡ ❛ss♦❝✐❛t❡❞ t♦ ✈❛r✐❛❜❧❡s ❛♥❞ t❤✐s ❛ss♦❝✐❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛❜str❛❝t st❛t❡✳ ❉❛t❛✲✢♦✇ ❛♥❛❧②s❡s ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ♣r♦❣r❛♠✬s ❈❋●✱ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ t❤❡ ❞❛t❛ ✢♦✇s ✐♥ t❤❡ ♣r♦❣r❛♠✱ ❛♥❞ ❛r❡ ❡①♣r❡ss❡❞

(28)

✽ ❈❤❛♣t❡r ✷✳ ▲✐t❡r❛t✉r❡ ❘❡✈✐❡✇

1: int fibonacci(int n) { 2: if (n <= 1)

3: return n;

4:

5: int n_1 = 1, n_2 = 0, fib;

6: for (int i=2; i <= n; i++) {

7: fib = n_1 + n_2;

8: n_2 = n_1;

9: n_1 = fib;

10: }

11: return fib;

12: }

BB0: n <= 1?

T F

BB1: return n

BB2: n_1 = 1 n_2 = 0 i = 2

BB3: i <= n

T F

BB4:

fib = n_1 + n_2 n_2 = n_1 n_1 = fib

BB6: return fib

BB5: i++

❋✐❣✉r❡ ✷✳✶✿ ❆♥ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ t♦ ❝❛❧❝✉❧❛t❡ t❤❡nth ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡r✱ ❛♥❞ ✐ts ❈❋●✳ ❇❇✯ ✐❞❡♥t✐✜❡s ❜❛s✐❝ ❜❧♦❝❦s✳ ❚❤❡ ❛rr♦✇s ❜❡t✇❡❡♥ ❜❛s✐❝ ❜❧♦❝❦s✱ s❛② ❢r♦♠ ❇❇✵ t♦ ❇❇✶✱ ♠❡❛♥ t❤❛t t❤❡ ❡①❡❝✉t✐♦♥ ♠❛② ✢♦✇ ❢r♦♠ ♦♥❡ ❜❛s✐❝ ❜❧♦❝❦ t♦ t❤❡ ♦t❤❡r✳

❛s tr❛♥s❢❡r ❢✉♥❝t✐♦♥s✳ ❆ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❝♦♠♣✉t❡s t❤❡ ❛❜str❛❝t st❛t❡ ♦❢ ❛ ✈❛r✐❛❜❧❡ ❛t ❛ ♣r♦❣r❛♠ ♣♦✐♥t ✉s✐♥❣ ❛s ✐♥♣✉t t❤❡ ✈❛r✐❛❜❧❡s ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❛♥❛❧②③❡❞ ✐♥str✉❝t✐♦♥ ❛♥❞ t❤❡ ❛❜str❛❝t st❛t❡ ❝❛❧❝✉❧❛t❡❞ ❛t t❤❡ ♥❡✐❣❤❜♦r ✐♥str✉❝t✐♦♥s✳

❆ ❝❧❛ss✐❝ ❡①❛♠♣❧❡ ♦❢ ❞❛t❛✲✢♦✇ ❛♥❛❧②s✐s ✐s t❤❡ ❧✐✈❡♥❡ss ❛♥❛❧②s✐s ✇❤✐❝❤ ❝❛❧❝✉❧❛t❡s ✇❤❛t ✈❛r✐❛❜❧❡s ❛r❡ ❛❧✐✈❡ ❛t ❛♥② ♣r♦❣r❛♠ ♣♦✐♥t✳ ❆ ✈❛r✐❛❜❧❡ ✐s ❛❧✐✈❡ ❛t ❛ ♣r♦❣r❛♠ ♣♦✐♥t ✐❢ ✐t ✇❛s ❛❧r❡❛❞② ❞❡✜♥❡❞✱ ✐s ♥♦t ♦✉t ♦❢ s❝♦♣❡ ❛♥❞ ✇✐❧❧ st✐❧❧ ❜❡ ✉s❡❞ ✐♥ ❛ ❢✉rt❤❡r ♣r♦❣r❛♠ ♣♦✐♥t✳ ❚❤✉s✱ t❤❡ ❧✐✈❡♥❡ss ❛♥❛❧②s✐s st❛rts ✇✐t❤ t❤❡ ❧❛st ✐♥str✉❝t✐♦♥✱ ✐♥ ❛ t♦♣♦❧♦❣✐❝❛❧ ♦r❞❡r✱ ❛♥❞ ❣♦ ❜❛❝❦✇❛r❞✳ ❆t t❤❡ ✐♥✐t✐❛❧ ♣r♦❣r❛♠ ♣♦✐♥t✱ ❛❢t❡r t❤❡ ❧❛st ✐♥str✉❝t✐♦♥✱ ❛❧❧ ✈❛r✐❛❜❧❡s ❛r❡ ❞❡❛❞✳ ❉✉r✐♥❣ t❤❡ ❛♥❛❧②s✐s✱ t❤❡ ✈❛r✐❛❜❧❡s ✉s❡❞ ❛s ♦♣❡r❛♥❞s ✐♥ ❛ ✐♥str✉❝t✐♦♥ ❛r❡ ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❛❧✐✈❡ ✐♥ t❤❡ ♣r♦❣r❛♠ ♣♦✐♥t r✐❣❤t ❛❜♦✈❡ ✐t✱ ✇❤✐❧❡ t❤❡ ❛ss✐❣♥❡❞ ✈❛r✐❛❜❧❡✱ ✐❢ t❤❡r❡ ✐s ♦♥❡✱ ✐s ❝♦♥s✐❞❡r❡❞ ❞❡❛❞✳

■t ✐s ♣♦ss✐❜❧❡ t♦ ❞❡s❝r✐❜❡ t❤❡ ❧✐✈❡♥❡ss ❛♥❛❧②s✐s ✉s✐♥❣ t✇♦ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s✿

JpKin = (JpKout− {v})∪vars(E) ✭✷✳✶✮

JpKout =

[

ps∈succ(p)

JpsKin ✭✷✳✷✮

(29)

✷✳✷✳ ❙t❛t✐❝ ❙✐♥❣❧❡ ❆ss✐❣♥♠❡♥t ❋♦r♠ ✾

❛♥ ❡①♣r❡ss✐♦♥✮✱ JpKin ❛♥❞ JpKout t❤❡ ❛❜str❛❝t st❛t❡ ♦❢ p ✭✐✳❡✳✱ t❤❡ s❡t ♦❢ ❛❧✐✈❡ ✈❛r✐❛❜❧❡s r✐❣❤t ❜❡❢♦r❡ ❛♥❞ r✐❣❤t ❛❢t❡r p✱ r❡s♣❡❝t✐✈❡❧②✮✱ vars(E) t❤❡ ✈❛r✐❛❜❧❡s t❤❛t ❛♣♣❡❛r ✐♥ E✱ succ(p) t❤❡ s❡t ♦❢ ♣r♦❣r❛♠ ♣♦✐♥ts ✇❤✐❝❤ ❛r❡ ✐♠♠❡❞✐❛t❡ s✉❝❝❡ss♦rs ♦❢ p✳ ◆♦t✐❝❡ t❤❛t✱

❢r♦♠ t❤❡ ❊q✉❛t✐♦♥ ✷✳✷✱ ✇❤❡♥ ❛ ♣♦✐♥t ✇✐t❤ ♠✉❧t✐♣❧❡ s✉❝❝❡ss♦rs ✐s ❢♦✉♥❞✱ ❡✳❣✳ ❛ ❜r❛♥❝❤ ✐♥str✉❝t✐♦♥ ♦❢ ❛♥ ✐❢✲t❤❡♥✲❡❧s❡✱ ✇❡ ❣❡t ❛s r❡s✉❧t t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ ❛❜str❛❝t st❛t❡s ♦❢ t❤❡ s✉❝❝❡ss♦rs✳ ❲❡ ❞♦ ✐t ❜❡❝❛✉s❡ ✐❢ ❛ ✈❛r✐❛❜❧❡ ♠❛② ❜❡ ❛❧✐✈❡ ✐♥ ❛t ❧❡❛st ♦♥❡ ♣♦ss✐❜❧❡ ♣❛t❤ ❢r♦♠ t❤❡ ♣r♦❣r❛♠ ♣♦✐♥t p t♦ t❤❡ ❡♥❞ ♦❢ t❤❡ ♣r♦❣r❛♠✱ ✐t ❤❛s t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛❧✐✈❡

r✐❣❤t ❛❢t❡r p✳ ❖t❤❡r✇✐s❡✱ s♦♠❡ ❝♦♠♣✐❧❡r ♦♣t✐♠✐③❛t✐♦♥s ❝♦✉❧❞ ♠❛❦❡ ❛ ✇r♦♥❣ ❞❡❝✐s✐♦♥✳

❋✐❣✉r❡ ✷✳✷ s❤♦✇s t❤❡ r❡s✉❧t ♦❢ ❛ ❧✐✈❡♥❡ss ❛♥❛❧②s✐s ❢♦r t❤❡ ✐♥str✉❝t✐♦♥s ♦❢ t❤❡ ❜❛s✐❝ ❜❧♦❝❦ ❇❇✹ ♦❢ ❋✐❣✉r❡ ✷✳✶✳ ❚❤❡ ✈❛❧✉❡s ❜❡t✇❡❡♥ ❝✉r❧② ❜r❛❝❦❡ts ❛r❡ t❤❡ ❛❧✐✈❡ ✈❛r✐❛❜❧❡s ✐♥ ❡❛❝❤ ♣r♦❣r❛♠ ♣♦✐♥t✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ✈❛r✐❛❜❧❡ n❴2✐s ♥♦t ❛❧✐✈❡ r✐❣❤t ❜❡❢♦r❡ t❤❡ ✐♥str✉❝t✐♦♥

n❴2 = n❴1✱ s✐♥❝❡ t❤✐s ✐♥str✉❝t✐♦♥ r❡❞❡✜♥❡s t❤❡ ✈❛r✐❛❜❧❡ ❛♥❞ ❛♥② ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t

t❤❡ ✈❛r✐❛❜❧❡ ✐s ❧♦st✳ ❖❜✈✐♦✉s❧②✱ ✐t ✐s ❛❧✐✈❡ ❛❢t❡r t❤❡ ❞❡✜♥✐t✐♦♥✳

❋✐❣✉r❡ ✷✳✷✿ ❚❤❡ r❡s✉❧t ♦❢ ❛ ❧✐✈❡♥❡ss ❛♥❛❧②s✐s ♦♥ ❢✉♥❝t✐♦♥ fibonacci ♦❢ ❋✐❣✉r❡ ✷✳✶✳

❚❤❡ r❡s✉❧t ✐s s❤♦✇♥ ♦♥❧② ❢♦r t❤❡ ❜❛s✐❝ ❜❧♦❝❦ ❇❇✹✳

■❢ ✐♥ ❛ ❞❛t❛✲✢♦✇ ❛♥❛❧②s✐s t❤❡ ❛❜str❛❝t st❛t❡ ❤❛s t♦ ❜❡ ❛ss♦❝✐❛t❡❞ t♦ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❛t ❡❛❝❤ ♣r♦❣r❛♠ ♣♦✐♥t✱ ✇❡ s❛② t❤❛t t❤❡ ❛♥❛❧②s✐s ✐s ❞❡♥s❡✳ ❖t❤❡r✇✐s❡✱ ✐❢ t❤❡ ❛❜str❛❝t st❛t❡ ♦❢ ❛ ✈❛r✐❛❜❧❡ ✐s t❤❡ s❛♠❡ ❢♦r t❤❡ ✇❤♦❧❡ ♣r♦❣r❛♠✱ t❤❡ ❛♥❛❧②s✐s ✐s s♣❛rs❡✳ ❙♣❛rs❡ ❛♥❛❧②s❡s ❛r❡ ❢❛st❡r t❤❛♥ ❞❡♥s❡ ♦♥❡s ❛♥❞ r❡q✉✐r❡ ❧❡ss s♣❛❝❡ ✐♥ ♠❡♠♦r②✳ ❖❢t❡♥✱ t❤❡r❡ ❛r❡ ✇❛②s t♦ ♠❛❦❡ ❛ ❞❡♥s❡ ❛♥❛❧②s✐s t♦ ❜❡❝♦♠❡ s♣❛rs❡ ❜② ❝❤❛♥❣✐♥❣ t❤❡ ♣r♦❣r❛♠✳

✷✳✷ ❙t❛t✐❝ ❙✐♥❣❧❡ ❆ss✐❣♥♠❡♥t ❋♦r♠

(30)

✶✵ ❈❤❛♣t❡r ✷✳ ▲✐t❡r❛t✉r❡ ❘❡✈✐❡✇

♣♦ss✐❜❧❡ t❤❛t ✇❡ ❤❛✈❡ ❞✐✛❡r❡♥t ✈❡rs✐♦♥s ♦❢ t❤❡ s❛♠❡ ♦r✐❣✐♥❛❧ ✈❛r✐❛❜❧❡ ✐♥ ❞✐✛❡r❡♥t s✐❞❡s ♦❢ ❛ ❜r❛♥❝❤✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ ♥❡❡❞ t♦ ❤❛✈❡ ❛ ✇❛② t♦ ✐❞❡♥t✐❢② t❤❡ ✈❛❧✉❡ r❡❛❝❤✐♥❣ ❛ ✉s❡✳ ❚❤✐s ✐s ❞♦♥❡ ❜② P❍■ ✭φ✮ ❢✉♥❝t✐♦♥s✳ ❆φ ❢✉♥❝t✐♦♥ ✐s ♦♥❧② ❛ ❝♦♥❝❡♣t ✭✐t ❞♦❡s ♥♦t ❡①✐st ✐♥ t❤❡

❝♦♥❝r❡t❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥✮ ✇❤✐❝❤ r❡❝❡✐✈❡s ❛ ❧✐st ♦❢ ✈❡rs✐♦♥s ♦❢ ❛ ✈❛r✐❛❜❧❡ ✲ ♦♥❡ ❢♦r ❡❛❝❤ ❞✐✛❡r❡♥t ♣❛t❤ r❡❛❝❤✐♥❣ t❤❡ ♣♦✐♥t ✲ ❛♥❞ r❡t✉r♥s t❤❡ ♣r♦♣❡r ✈❛❧✉❡ ❢♦r ❛ ❣✐✈❡♥ ❡①❡❝✉t✐♦♥✳ ❋✐❣✉r❡ ✷✳✸ s❤♦✇s t❤❡ ❡①❛♠♣❧❡ ♦❢ ❋✐❣✉r❡ ✷✳✶ ✐♥ ❙❙❆ ❢♦r♠✳ ■t ✐s ✇♦rt❤ t♦ ♠❡♥t✐♦♥ t❤❛t ❛❧❧ t❤❡ ❛♥❛❧②s❡s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ❛r❡ ♣❡r❢♦r♠❡❞ ♦✈❡r ▲▲❱▼✬s ✐♥t❡r♠❡❞✐❛t❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ ❙❙❆ ❢♦r♠✳

entry:

%cmp = icmp sle i32 %n, 1

br i1 %cmp, label %if.then, label %if.end

T F

if.then: br label %return if.end:

br label %for.cond

return:

%retval.0 = phi i32 [ %n, %if.then ], [ %fib.0, %for.end ] ret i32 %retval.0

for.cond:

%fib.0 = phi i32 [ undef, %if.end ], [ %add, %for.inc ] %n_2.0 = phi i32 [ 1, %if.end ], [ %n_1.0, %for.inc ] %n_1.0 = phi i32 [ 0, %if.end ], [ %add, %for.inc ] %i.0 = phi i32 [ 2, %if.end ], [ %inc, %for.inc ] %cmp1 = icmp slt i32 %i.0, %n

br i1 %cmp1, label %for.body, label %for.end

T F

for.body:

%add = add nsw i32 %n_1.0, %n_2.0 br label %for.inc

for.end: br label %return

for.inc:

%inc = add nsw i32 %i.0, 1 br label %for.cond

❋✐❣✉r❡ ✷✳✸✿ ❋✉♥❝t✐♦♥fibonacci♦❢ ❋✐❣✉r❡ ✷✳✶ ✐♥ ❙❙❆ ❢♦r♠✳ ❚❤❡ ❣r❛♣❤ ✇❛s ❛✉t♦♠❛t✲

✐❝❛❧❧② ❣❡♥❡r❛t❡❞ ❜② ▲▲❱▼✳

✷✳✸ ❆✉t♦♠❛t✐❝ ■♥❢❡r❡♥❝❡ ♦❢ Pr♦❣r❛♠ ❈♦♠♣❧❡①②

(31)

✷✳✸✳ ❆✉t♦♠❛t✐❝ ■♥❢❡r❡♥❝❡ ♦❢ Pr♦❣r❛♠ ❈♦♠♣❧❡①② ✶✶

♣r♦♣♦s❡❞ ❛ t❡❝❤♥✐q✉❡ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ ❡①❡❝✉t✐♥❣ t❤❡ t❛r❣❡t ♣r♦❣r❛♠ ♦✈❡r ✇♦r❦❧♦❛❞s ✇✐t❤ ❞✐✛❡r❡♥t ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ ❛♥❞ tr❛❝❦✐♥❣ ❤♦✇ ♠❛♥② t✐♠❡s ❡❛❝❤ ♣r♦❣r❛♠ ❧♦❝❛t✐♦♥ ✇❛s ❡①❡❝✉t❡❞✳ ❚❤❡② ✉s❡ ♣♦❧②♥♦♠✐❛❧ r❡❣r❡ss✐♦♥ t♦ ✜t t❤❡ ❞❛t❛ ✐♥t♦ ❛ ❧✐♥❡❛r ♦r ♣♦✇❡r✲❧❛✇ ♠♦❞❡❧✳ ❍♦✇❡✈❡r✱ t❤❡ ✉s❡r ❤❛s t♦ s♣❡❝✐❢②✱ ❢♦r ❡❛❝❤ ✇♦r❦❧♦❛❞✱ t❤❡ ✈❛❧✉❡ ♦❢ ❢❡❛t✉r❡s ✲ ❛ ❢❡❛t✉r❡ ✐s ❛♥ ✐♥♣✉t ♣r♦♣❡rt② ✇❤✐❝❤ ❛✛❡❝ts t❤❡ ❛❧❣♦r✐t❤♠ ❡①❡❝✉t✐♦♥✱ ❡✳❣✳ t❤❡ s✐③❡ ♦❢ ❛♥ ❛rr❛② ♦r t❤❡ ❤❡✐❣❤t ♦❢ ❛ tr❡❡✳ ❖✉r t❡❝❤♥✐q✉❡ ✐s ❛❜❧❡ t♦ ❛✉t♦♠❛t✐❝❛❧❧② ✐♥❢❡r ❧♦♦♣s✬ ✐♥♣✉ts❀ ❤❡♥❝❡✱ ✐t ❞♦❡s ♥♦t r❡q✉✐r❡ t❤✐s t②♣❡ ♦❢ ✉s❡r ✐♥t❡r✈❡♥t✐♦♥✳

❩❛♣❛r❛♥✉❦s ❛♥❞ ❍❛✉s✇✐rt❤ ❬✷✵✶✷❪ ♣r♦♣♦s❡❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❛❧❣♦r✐t❤♠✐❝ ♣r♦✜❧❡r✳ ❚❤❡✐r ❛♣♣r♦❛❝❤ ❝♦♥s✐sts ✐♥ ❣r♦✉♣✐♥❣ t❤❡ ❜❛s✐❝ ❜❧♦❝❦s ♦❢ ❛ ❧♦♦♣ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ♠❛❦❡ ❛ ❝②❝❧❡ ✐♥ t❤❡ ❝❛❧❧✲❣r❛♣❤ ✐♥t♦ t❤❡ s♦ ❝❛❧❧❡❞ r❡♣❡t✐t✐♦♥ ♥♦❞❡s✳ ❚❤♦s❡ ♥♦❞❡s ❛r❡ t❤❡♥ ❝♦♠❜✐♥❡❞ ✐♥ ✉♥✐ts t❤❛t t❤❡② ❤❛✈❡ ♥❛♠❡❞ ❛❧❣♦r✐t❤♠s✳ ❚❤❡ t❡❝❤♥✐q✉❡ ✐s ❛❜❧❡✱ ❢♦r ❡①❛♠♣❧❡✱ t♦ ✐❞❡♥t✐❢② ✐❢ ❛♥ ❛❧❣♦r✐t❤♠ ✐s ♠♦❞✐❢②✐♥❣ ♦r tr❛✈❡rs✐♥❣ ❛ ❧✐st ♦r ❛♥ ❛rr❛②✳ ■♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠✱ t❤❡② r❡tr✐❡✈❡ t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉ts ❛♥❞ s♦♠❡ ♣❡r❢♦r♠❛♥❝❡ ♠❡tr✐❝s ❢♦r ❡❛❝❤ ❡①❡❝✉t✐♦♥ ♦❢ t❤❡ r❡♣❡t✐t✐♦♥ ♥♦❞❡s✳ ❚❤✐s ♠♦❞✉s ♦♣❡r❛♥❞✐ ❧❡❛❞s t♦ ❛ s✐❣♥✐✜❝❛♥t ♦✈❡r❤❡❛❞✱ s✐♥❝❡ t❤❡ ❛♥❛❧②③❡r ✐t❡r❛t❡s ♦✈❡r t❤❡ ❡♥t✐r❡ ❞❛t❛ str✉❝t✉r❡ t♦ ❝❛❧❝✉❧❛t❡ ✐ts s✐③❡✳ ❚❤❡ ❛✉t♦♠❛t✐❝ r❡❝♦♥str✉❝t✐♦♥ ♦❢ ❞❛t❛✲str✉❝t✉r❡s ✐s st✐❧❧ ❛♥ ✐♥❝✐♣✐❡♥t ❛r❡❛ ♦❢ r❡s❡❛r❝❤✳ ❚❤❡r❡❢♦r❡✱ ❩❛♣❛r❛♥✉❦s ❡t ❛❧✳ ❤❛✈❡ ✐♠♣❧❡♠❡♥t❡❞ ❛ ♣r♦t♦t②♣❡ ✇❤✐❝❤✱ ✉♣ t♦ t❤✐s ♣♦✐♥t✱ ❝❛♥ ❛♥❛❧②③❡ ♦♥❧② t♦② ❡①❛♠♣❧❡s✳ ❲❡ ❝❛♥♥♦t r❡❝♦♥str✉❝t r❡❝✉rs✐✈❡ ❞❛t❛✲ str✉❝t✉r❡s ❛s ❩❛♣❛r❛♥✉❦s ❞♦❡s❀ ❤♦✇❡✈❡r✱ ♦✉r ❛♣♣r♦❛❝❤ ✐s ❛❜❧❡ t♦ ✐♥❢❡r t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ♠♦st ♦❢ t❤❡ ❧♦♦♣s ✐♥ ❛ r❡❛❧✲✇♦r❧❞ ❜❡♥❝❤♠❛r❦ s✉✐t❡✳

❚❤❡ ✇♦r❦ t❤❛t ✐s ♠♦r❡ r❡❧❛t❡❞ t♦ t❤❡ t❡❝❤♥✐q✉❡ ❞❡s❝r✐❜❡❞ ✐♥ ❈❤❛♣t❡r ✹ ✐s ❈♦♣♣❛ ❡t ❛❧✳ ❬✷✵✶✷❪✬s ✐♥♣✉t s❡♥s✐t✐✈❡ ♣r♦✜❧❡r✳ ❚❤✐s ✇♦r❦ ❤❛s ♠❛t❡r✐❛❧✐③❡❞ ✐ts❡❧❢ ✐♥t♦ ❛ t♦♦❧ ❝❛❧❧❡❞ aprof✳ ❈♦r❡ t♦aprof✬s ✇♦r❦ ✐s t❤❡ ♥♦t✐♦♥ ♦❢ ❘❡❛❞ ▼❡♠♦r② ❙✐③❡ ✭❘▼❙✮✳ ❚❤✐s

♠❡tr✐❝ r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ ♠❡♠♦r② ❧♦❝❛t✐♦♥s ✇❤✐❝❤ ❛r❡ r❡❛❞ ❜❡❢♦r❡ t❤❡② ❤❛✈❡ ❜❡❡♥ ✇r✐tt❡♥ ✐♥s✐❞❡ ❛ ❢✉♥❝t✐♦♥✳ ❆♣r♦❢ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ❛s ❛ ❱❛❧❣r✐♥❞ ❬◆❡t❤❡r❝♦t❡ ❛♥❞ ❙❡✇❛r❞✱ ✷✵✵✼❜❪ ❡①t❡♥s✐♦♥✳ ❲❡ ❜❡❧✐❡✈❡ t❤❛t aprof ✐s t❤❡ ♠♦st ♣r❛❝t✐❝❛❧ t♦♦❧ ❛✈❛✐❧❛❜❧❡

♥♦✇❛❞❛②s t♦ ✐♥❢❡r t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❣❡♥❡r❛❧ ♣✉r♣♦s❡ ♣r♦❣r❛♠s✳ ◆❡✈❡rt❤❡❧❡ss✱ ✐t ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt❝♦♠✐♥❣s✿ ✭✐✮ t❤❡ ❣r❛♥✉❧❛r✐t② ♦❢ r❡s✉❧ts ✐s ❛t t❤❡ ❢✉♥❝t✐♦♥✱ ♥♦t ❛t t❤❡ ❧♦♦♣✱ ❧❡✈❡❧❀ ✭✐✐✮ ✉s❡rs ❤❛✈❡ t♦ ✜t ❡q✉❛t✐♦♥ ❜② ❤❛♥❞ ✐♥ aprof✬s r❡s✉❧ts t♦ ✜♥❞ t❤❡ ❝♦♠♣❧❡①✐t②

♦❢ ❛ ❢✉♥❝t✐♦♥❀ ❛♥❞ ✭✐✐✐✮ r❡s✉❧ts ❛r❡ ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ ❘▼❙✱ ✇❤✐❝❤ ♠❛② ♥♦t ❜❡ s✐❣♥✐✜❝❛♥t t♦ t❤❡ ❞❡✈❡❧♦♣❡r✳ ❖✉r t❡❝❤♥✐q✉❡ ✐s ❝❛♣❛❜❧❡ ♦❢ ❛❞❞r❡ss✐♥❣ t❤❡s❡ ❞r❛✇❜❛❝❦s✳

(32)

✶✷ ❈❤❛♣t❡r ✷✳ ▲✐t❡r❛t✉r❡ ❘❡✈✐❡✇

t♦ ♣r♦✈❡ ♣r♦♣❡rt✐❡s ❛❜♦✉t t❤❡ ♣r♦❣r❛♠✬s ❝♦♠♣❧❡①✐t②✿ t❤❡r❡ ❛r❡ ♥♦ ❣✉❛r❛♥t❡❡s t❤❛t ✇❡ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ♦❜s❡r✈❡ ❡✈❡r② ♣♦ss✐❜❧❡ ❡①❡❝✉t✐♦♥ ♣❛t❤ ✇✐t❤✐♥ t❤❡ ♣r♦❣r❛♠ ❝♦❞❡✳ ❚❤❡ ✉♣s✐❞❡ ✐s ♣r❡❝✐s✐♦♥✿ ♦✉r ❛♣♣r♦❛❝❤ ✐s ❛❜❧❡ t♦ r❡❛s♦♥ ❛❜♦✉t t②♣✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ ❢❡❛t✉r❡s s✉❝❤ ❛s ❞②♥❛♠✐❝❛❧❧② ❛❧❧♦❝❛t❡❞ ♠❡♠♦r②✱ ♠✉❧t✐♣❧❡ ♣❛t❤s ✐♥ ❧♦♦♣s✱ ♥♦♥✲str✉❝t✉r❡❞ ❝♦♥tr♦❧ ✢♦✇ ❣r❛♣❤s ❛♥❞ ♣♦✐♥t❡r ❛r✐t❤♠❡t✐❝s✳ ❙♦ ❢❛r✱ t❤❡s❡ r❡❛❧✲✇♦r❧❞ ❝♦♥str✉❝ts ❤❛✈❡ ❜❡❡♥ ❝❤❛❧❧❡♥❣✐♥❣ ❛❞✈❡rs❛r✐❡s t♦ t❤❡ ♣✉r❡❧② st❛t✐❝ ❛♥❛❧②s❡s✳

✷✳✹ Pr♦❣r❛♠ ❙❧✐❝✐♥❣

Pr♦❣r❛♠ s❧✐❝✐♥❣✱ ❛s ❞❡✜♥❡❞ ❜② ❲❡✐s❡r ❬✶✾✽✶❪✱ ❝♦♥s✐sts ♦❢ s❡❣♠❡♥t✐♥❣ ❛ ♣r♦❣r❛♠ ❡①tr❛❝t✲ ✐♥❣ ♦♥❧② t❤❡ ✐♥str✉❝t✐♦♥s ✇❤✐❝❤ ❛✛❡❝t t❤❡ ❝♦♠♣✉t❡❞ ✈❛❧✉❡ ❛t ❛♥ ✐♥t❡r❡st ♣♦✐♥t✱ ❝❛❧❧❡❞ s❧✐❝✐♥❣ ❝r✐t❡r✐♦♥✳ ❚❤❡r❡ ❛r❡ ❜❛s✐❝❛❧❧② t✇♦ ❦✐♥❞s ♦❢ s❧✐❝✐♥❣ t❡❝❤♥✐q✉❡s✿ st❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ s❧✐❝✐♥❣✳ ❚❤❡ st❛t✐❝ t❡❝❤♥✐q✉❡s ❬❍♦r✇✐t③ ❡t ❛❧✳✱ ✶✾✽✽❀ ❉❛♥✐❝✐❝ ❡t ❛❧✳✱ ✶✾✾✺❀ ❘❡♣s✱ ✶✾✾✽❪✱ ✐♥ ❣❡♥❡r❛❧✱ st❛t✐❝❛❧❧② ❣❡♥❡r❛t❡ ❛ ❞❡♣❡♥❞❡♥❝❡ ❣r❛♣❤ ❬❖tt❡♥st❡✐♥ ❛♥❞ ❖tt❡♥st❡✐♥✱ ✶✾✽✹❪ ❢♦r ❛ ♣r♦❣r❛♠✬s ✐♥str✉❝t✐♦♥s ❛♥❞✱ st❛rt✐♥❣ ❢r♦♠ t❤❡ s❧✐❝✐♥❣ ❝r✐t❡r✐♦♥✱ tr❛✈❡rs❡ t❤❡ ❣r❛♣❤ ❣❡tt✐♥❣ t❤❡ ✐♥str✉❝t✐♦♥s ✇❤✐❝❤ ❛✛❡❝t ✐t✳ ❚❤❡ ❞②♥❛♠✐❝ ❛♣♣r♦❛❝❤❡s ❬❑♦r❡❧ ❛♥❞ ▲❛s❦✐✱ ✶✾✽✽✱ ✶✾✾✵❀ ❆❣r❛✇❛❧ ❛♥❞ ❍♦r❣❛♥✱ ✶✾✾✵❀ ❑❛♠❦❛r ❡t ❛❧✳✱ ✶✾✾✸❪ ✉s❡ r✉♥t✐♠❡ ✐♥❢♦r♠❛t✐♦♥ t♦ ❝♦♠♣✉t❡ t❤❡ ✐♥str✉❝t✐♦♥s ✇❤✐❝❤ ❛✛❡❝t t❤❡ s❧✐❝✐♥❣ ❝r✐t❡r✐♦♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ❡①❡❝✉t✐♦♥✳ ❉②♥❛♠✐❝ s❧✐❝❡s t❡♥❞ t♦ ❜❡ s♠❛❧❧❡r ❛♥❞ ♠♦r❡ s✐❣♥✐✜❝❛♥t t❤❛♥ t❤❡ st❛t✐❝ ♦♥❡s ❢♦r s♦♠❡ ♣r♦❣r❛♠s✴✐♥♣✉ts ❛♥❞ ❛r❡ ✉s❡❢✉❧ t♦ r❡❞✉❝❡ t❤❡ s❡❛r❝❤ s♣❛❝❡ ♦❢ ❛ ❜✉❣ ❦♥♦✇♥ t♦ ❤❛♣♣❡♥ ❢♦r ❛ ❣✐✈❡♥ ✐♥♣✉t✳ ❆♥♦t❤❡r t❡❝❤♥✐q✉❡ ✇❤✐❝❤ ❛✐♠s t♦ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ ❛ s❧✐❝❡ ✐s ❝❛❧❧❡❞ ❝♦♥❞✐t✐♦♥❡❞ s❧✐❝✐♥❣ ❬❉❡ ▲✉❝✐❛ ❡t ❛❧✳✱ ✶✾✾✻❪✳ ❋✐rst❧②✱ ✐t ❝♦♠♣✉t❡s ❛ r❡❞✉❝❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣r♦❣r❛♠ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② t❤❡ r❡❛❝❤❛❜❧❡ ✐♥str✉❝t✐♦♥s ❣✐✈❡♥ ❛♥ ✐♥✐t✐❛❧ ✐♥♣✉t s❡t ✲ ✉s✐♥❣✱ ❢♦r ✐♥st❛♥❝❡✱ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❬❏❛✛❛r ❡t ❛❧✳✱ ✷✵✶✷❪✳ ❚❤❡♥✱ t❤❡ s❧✐❝✐♥❣ ✐s ♣❡r❢♦r♠❡❞ ✐♥ t❤❡ s♦ ❝❛❧❧❡❞ ❝♦♥❞✐t✐♦♥❡❞ ♣r♦❣r❛♠✳ ❚❤❡ ❞✐✛❡r❡♥t ♣r♦❣r❛♠ s❧✐❝✐♥❣ t❡❝❤♥✐q✉❡s ❛r❡ ✉s❡❞ ❢♦r ❞❡❝❛❞❡s ✐♥ ❞✐✛❡r❡♥t ❛r❡❛s ♦❢ ❝♦♠♣✉t❡r s❝✐❡♥❝❡ ❢♦r s❡✈❡r❛❧ ❡♥❞s✱ ❛s ❞❡❜✉❣❣✐♥❣✱ r❡❢❛❝✲ t♦r✐♥❣✱ ♣❛r❛❧❧❡❧✐s♠ ❛♥❞ ❝♦♠♣✐❧❡r ♦♣t✐♠✐③❛t✐♦♥s✳ ❲❡ ❤❛✈❡ ✉s❡❞ st❛t✐❝ ♣r♦❣r❛♠ s❧✐❝✐♥❣ ✇❤❡♥ ❣❡♥❡r❛t✐♥❣ t❡st ❝❛s❡s ✐♥ ♦r❞❡r t♦ r❡✉s❡ ❛ ♣r♦❣r❛♠✬s ✐♥str✉❝t✐♦♥s ✇❤✐❝❤ ❝❛❧❝✉❧❛t❡ t❤❡ s✐③❡ ♦❢ ❛♥ ❛rr❛②✳

✷✳✺ ❉❛t❛ ❙tr✉❝t✉r❡ ●r❛♣❤

Referências

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