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Thierry Coquand and Arnaud Spiwack. Constructively finite? In Contribuciones cient´ ıficas en honor de Mirian Andr´ es G´
- mez,
Some Varieties of Constructive Finiteness Erik Parmann University - - PowerPoint PPT Presentation
Some Varieties of Constructive Finiteness Erik Parmann University - - PowerPoint PPT Presentation
Some Varieties of Constructive Finiteness Erik Parmann University of Bergen Presented at: TYPES 2014 May 12, 2014 Thierry Coquand and Arnaud Spiwack. Constructively finite? In Contribuciones cient ficas en honor de Mirian Andr es G
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Enumerated (Kuratowski finite) Bounded Size Noetherian Streamless
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Enumerated (Kuratowski finite) Bounded Size Noetherian Streamless Sum and Product
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Enumerated (Kuratowski finite) Bounded Size Noetherian Streamless Sum and Product Sum and Product
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Enumerated (Kuratowski finite) Bounded Size Noetherian Streamless Sum and Product Sum and Product Sum and Product (decidable eq)
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Enumerated (Kuratowski finite) Bounded Size Noetherian Streamless Sum and Product Sum and Product Sum and Product (decidable eq) Sum and Product (decidable eq)
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Streamless is closed under product (given decidable equality): . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . .
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Producing n equal elements: . . .
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Producing n equal elements: . . .
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Producing n equal elements: . . .
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Producing n equal elements: . . .
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Producing n equal elements: . . . . . .
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Producing n equal elements: . . . . . .
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Producing n equal elements: . . . . . .
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Producing n equal elements: . . . . . .
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Producing n equal elements: . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . .
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=
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Streamless is closed under product (given decidable equality): . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . . Duplicates?
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Streamless is closed under product (given decidable equality): . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . . Duplicates?
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Streamless is closed under product (given decidable equality): . . . . . . . . . =
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Streamless is closed under product (given decidable equality): . . . . . . . . . . . .
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Streamless is closed under product (given decidable equality): . . . . . . . . . . . .
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Streamless and function extensionality implies decidable equality: 1
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= 2
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Streamless and function extensionality implies decidable equality: 1
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= 2 1 1 1 1 1 1 1 1 1 . . .
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Streamless and function extensionality implies decidable equality: 1
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= 2 1 1 1 1 1 1 1 1 1 . . .
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Streamless and function extensionality implies decidable equality: 1
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= 2 1 1 2 1 1 1 1 1 1 . . .
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Streamless and function extensionality implies decidable equality: 1
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= 2 1 1 2 1 1 1 1 1 1 . . .
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Streamless and function extensionality implies decidable equality: 1
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= 2 1 1 1 1 1 1 1 . . . 2 1 1 1
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TODO: Can it work withouth decidable equality?
With Markov’s Principle? If one of the sets are Noetherian?
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TODO: Can it work withouth decidable equality?
With Markov’s Principle? If one of the sets are Noetherian?
Look into natural definitions of finiteness which does not give decidable equality with function extensionality.
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Thanks, http://folk.uib.no/epa095/
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Definition (Markov’s Principle, MP) For any decidable predicate: ¬¬∃n : N, P(n) → ∃n : N, P(n).
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Definition (Limited Principle of Omniscience (LPO)) For any decidable predicate P, we have (∀n : N, P(n)) ∨ (∃n : N, ¬P(n)). Definition (Weak Limited Principle of Omniscience (WLPO)) For any decidable predicate P, we have (∀n : N, P(n)) ∨ (¬∀n : N, P(n)).
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Fact (MP ∧ WLPO) ⇐ ⇒ LPO
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Definition (Eventually always false (Eaf)) ∃n : N, ∀m : N, m ≥ n → f (m) = 0. 1 1 1 . . .
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Definition (Bounded(f )) ∃n : N, ∀k : N, NrOf1f k ≤ n. E.g with n = 5 : 1 1 1 . . .
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Definition (Sb) ∃n : N, (∀k : N, NrOf1f k ≤ n ∧ ¬∀k : N, NrOf1f k ≤ n − 1) E.g with n = 5: 1 1 1 . . .
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Eaf Bounded LPO ∃n : N, ∀m : N, m ≥ n → f (m) = 0. ∃n : N, ∀k : N, NrOf1f k ≤ n. Definition (Limited Principle of Omniscience, LPO) (∀n : N, P(n)) ∨ (∃n : N, ¬P(n)).
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Eaf Bounded LPO Sb MP WLPO ∃n : N, ∀m : N, m ≥ n → f (m) = 0. ∃n : N, (∀k : N, NrOf1f k ≤ n ∧ ¬∀k : N, NrOf1f k ≤ n − 1). ∃n : N, ∀k : N, NrOf1f k ≤ n.
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Eaf Bounded LPO ∃n : N, ∀m : N, m ≥ n → f (m) = 0. ∃n : N, ∀k : N, NrOf1f k ≤ n. Definition (Markov’s Principle, MP) ¬¬∃n : N, P(n) → ∃n : N, P(n).
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