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  1. P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❉▲❚ ✷✵✶✷ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  2. P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✶✴✷ Σ ✿ ❋✐♥✐t❡ ❛❧♣❤❛❜❡t ♦❢ t❡r♠✐♥❛❧ s②♠❜♦❧s ✭❡✳ ❣✳ Σ = { ❛ , ❜ , ❝ , ❞ } ✮✳ ❳ = { ① ✶ , ① ✷ , ① ✸ , . . . } ✿ ■♥✜♥✐t❡ ❛❧♣❤❛❜❡t ♦❢ ✈❛r✐❛❜❧❡s✳ ❆ ✇♦r❞ α ∈ (Σ ∪ ❳ ) + ✐s ❝❛❧❧❡❞ ❛ ♣❛tt❡r♥ ✳ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  3. P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ ❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ▼♦r♣❤✐s♠ ✿ ▼❛♣♣✐♥❣ ❤ : Γ ∗ ✶ → Γ ∗ ✷ ✇✐t❤ ❤ ( ① · ② ) = ❤ ( ① ) · ❤ ( ② ) ❀ ❤ ✐s ♥♦♥❡r❛s✐♥❣ ✐✛✱ ❢♦r ❡✈❡r② ❛ ∈ Γ ✶ ✱ ❤ ( ❛ ) � = ε ✳ ❙✉❜st✐t✉t✐♦♥ ✿ ▼♦r♣❤✐s♠ ❤ : (Σ ∪ ❳ ) ∗ → Σ ∗ ✇✐t❤ ❤ ( ❛ ) = ❛ ❢♦r ❛❧❧ ❛ ∈ Σ ✳ ▲ ❊ , Σ ( α ) := { ❤ ( α ) | ❤ ✐s ❛ s✉❜st✐t✉t✐♦♥ } ✳ ▲ ◆❊ , Σ ( α ) := { ❤ ( α ) | ❤ ✐s ♥♦♥❡r❛s✐♥❣ s✉❜st✐t✉t✐♦♥ } ✳ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  4. ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ▲ ◆❊ ✇ ✇ ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ ✉ ✈ ❛ ❜ ❝ ❛ ❜ ❝ ✳ ▲ ◆❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  5. ▲ ◆❊ ✇ ✇ ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ ✉ ✈ ❛ ❜ ❝ ❛ ❜ ❝ ✳ ▲ ◆❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  6. ▲ ◆❊ ✇ ✇ ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ ✉ ✈ ❛ ❜ ❝ ❛ ❜ ❝ ✳ ▲ ◆❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  7. ▲ ◆❊ ✇ ✇ ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ ✉ ✈ ❛ ❜ ❝ ❛ ❜ ❝ ✳ ▲ ◆❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  8. ▲ ◆❊ ✇ ✇ ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ ✉ ✈ ❛ ❜ ❝ ❛ ❜ ❝ ✳ ▲ ◆❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  9. ▲ ◆❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ▲ ◆❊ , { ❛ , ❜ , ❝ } ( α ) = { ✇ | ✇ = ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ , ✉ , ✈ ∈ { ❛ , ❜ , ❝ } + } ✳ ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

  10. ▲ ❊ ❝❝❜❛❛❝❝❜❝❜❝❝❜ ❛ ❜ ❝ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❘❡❣✉❧❛r✐t② ❛♥❞ ❈♦♥t❡①t✲❋r❡❡♥❡ss ❆ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥ ❢♦r ❘❡❣✉❧❛r✐t② ❈♦♥❝❧✉s✐♦♥ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s ✷✴✷ α = ① ✶ ❛❛ ① ✷ ① ✶ ① ✷ ❝❜ ① ✶ ❛❝❛❛❛❜❝❜❛❛❝❛❜❝❜❛❝❜❛❝ ▲ ◆❊ , { ❛ , ❜ , ❝ } ( α ) = { ✇ | ✇ = ✉ ❛❛ ✈ ✉ ✈ ❝❜ ✉ , ✉ , ✈ ∈ { ❛ , ❜ , ❝ } + } ✳ ∈ ▲ ◆❊ , { ❛ , ❜ , ❝ } ( α ) ❝❝❜❛❛❝❝❜❝❜❝❝❜ / ❉❛♥✐❡❧ ❘❡✐❞❡♥❜❛❝❤✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ ✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❘❡❣✉❧❛r ❛♥❞ ❈♦♥t❡①t✲❋r❡❡ P❛tt❡r♥ ▲❛♥❣✉❛❣❡s ❖✈❡r ❙♠❛❧❧ ❆❧♣❤❛❜❡ts

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