Variations on Noetherianness Denis Firsov, Tarmo Uustalu, Niccol` o Veltri Institute of Cybernetics at TUT January 31, 2016 1 / 22
Finiteness constructively Straightforward way of saying that the set X is finite is to ask for enumeration of its elements: Listable X = ∃ [ xs ∈ List X ] ((x : X) → x ∈ xs) Previously we showed that listability of a set implies decidable equality on its elements: Listable → DecEq : { X : Set } → Listable X → DecEq X Can we define “morally” finite sets without decidable equality? What are the alternative notions of finiteness to listability? 2 / 22
Emmy Noether In 2010 Coquand and Spiwack introduced the alternative notion of finiteness – Noetherian sets. The definition is inspired by Noetherian rings from abstract algebra. Noetherian ring is a ring that satisfies the ascending chain condition on ideals; that is, given any chain of ideals: I 1 ⊆ · · · ⊆ I k − 1 ⊆ I k ⊆ I k +1 ⊆ · · · there exists an n such that: Emmy Noether (1882-1935) I n = I n +1 = · · · . 3 / 22
Noetherian sets A set is Noetherian if whenever we are shown elements from it one after another, sooner or later we will have seen some element twice. Formally it can be expressed as two inductive rules: X) → NoethAcc’ X (x :: acc) (x : ask NoethAcc’ X acc Dup acc NoethAcc’ X acc stop The set X is Noetherian if we can prove NoethAcc’ X [] . Therefore, the synonym: NoethAcc X = NoethAcc’ X [] 4 / 22
Example: Booleans The proof that the set of Booleans is Noetherian is a tree: [] [T] [F] [T,F] [T,T] [F,T] [F,F] [T,F,T] [T,F,F] [F,T,T] [F,T,F] The proof of NoethAcc X does not reveal the elements of X or even the size of X . Any branch of a proof is not required to stop immediately after there was a duplicate. The particular branch of a proof may stop before seeing all the elements of a set, e.g. [T, T] . 5 / 22
Listable and Noetherian sets I What is the relation between Listable and NoethAcc ? Clearly, if set is listable then it is Noetherian: Lstbl → NoethAcc : { X : Set } → Listable X → NoethAcc X We mentioned that all listable sets have decidable equality. Same holds for Noetherian sets: NoethAcc → Eq : { X : Set } → NoethAcc X → DecEq X Can it be that all Noetherian sets are also listable? 6 / 22
Separating Listability from Noetherianness To prove that listability is stronger than Noetherianness we exploit that from listability you can conclude the cardinality of a set. First, we define the propositional sets: isProp X = (x 1 x 2 : X) → x 1 ≡ x 2 Then we prove that any proposition X is Noetherian. NoethProp : (X : Set) → isProp X → NoethAcc X Next, assuming that every proposition X is listable allows us to derive law of excluded middle for propositions (LEM prop ): ListableProp → LEM : ((X : Set) → isProp X → Listable X) → (X : Set) → isProp X → X + ¬ X 7 / 22
Strict Noetherianness Some questions remain: Can we define finite sets without decidable equality? What is the notion of finiteness which does not imply decidable equality? Next variation of Noetherianness we consider are “strict” Noetherian sets: X) → (x / ∈ acc) → NoethAccS’ X (x :: acc) (x : ask NoethAccS’ X acc 8 / 22
Decidable equality for strict Noetherian sets? Consider a set NotNotIn : { X : Set } → List X → Set NotNotIn { X } xs = ∃ [ x ∈ X ] ¬ ¬ x ∈ xs The set NotNotIn is strictly Noetherian. NoethAccSNotNotIn : { X : Set } → (xs : List X) → NoethAccS (NotNotIn xs) Moreover, decidable equality on the elements of NotNotIn implies general decidable equality on all sets: DEQ : ( { X : Set } (xs : List X) → DecEq (NotNotIn xs)) → { X : Set } → DecEq X 9 / 22
Current situation Listable LEM prop NoethAcc DEQ NoethAccS 10 / 22
Another variation of Noetherianness NoethExpose combines ideas from Listable and NoethAcc : X) → NoethExpose’ X (x :: acc) (x : ask NoethExpose’ X acc NoethExpose’ X (x :: acc) x : X give NoethExpose’ X acc (x : X) → x ∈ acc NoethExpose’ X acc all The distinctive property of NoethExpose : exppr : { X : Set } → X → NoethExpose X → Listable X 11 / 22
Relation of NoethExpose to others It is immediate that Listable X implies NoethExpose X (just use the constructors give and all ). We prove that NoethExpose X implies NoethAcc X . (As a corollary NoethExpose X implies also decidable equality on X .) We prove that the implication from NoethExpose X to Listable X would also imply the LEM prop . Finally, we show that given the proof that NoethAcc X implies NoethExpose X we can derive the limited principle of omniscience (LPO). 12 / 22
Current situation Listable LEM prop NoethExpose LPO NoethAcc DEQ NoethAccS 13 / 22
Game of Noetherianness Another variation on the theme is: X) → (x / ∈ acc) → NoethGame’ X (x :: acc) (x : ask NoethGame’ X acc NoethGame’ X (x :: acc) x : X give NoethGame’ X acc We know that NoethGame implies NoethAccS (therefore no decidable equality). It is still open if the opposite direction holds. 14 / 22
Current situation Listable LEM prop NoethExpose LPO NoethAcc DEQ NoethAccS ? NoethGame 15 / 22
Other notions of finiteness: Bounded sets Bounded sets: List X) → n ≤ length xs → Dup xs n : Nat (xs : Bounded X 16 / 22
Current situation Listable LEM prop LEM prop LPO NoethExpose Bounded LEM prop LPO LPO NoethAcc DEQ NoethAccS ? NoethGame 17 / 22
Other notions of finiteness: Streamless sets Streamless sets: Stream X) → Dup xs (xs : Streamless X 18 / 22
Final diagram Listable LEM prop LEM prop LPO NoethExpose Bounded LEM prop LPO LPO NoethAcc Streamless DEQ NoethAccS ? NoethGame 19 / 22
Almost-full relations: X) → AF X ( λ y z → R y z + R x y)) ((x : R is total AF X R AF X R AFEq X = AF X _ ≡ _ 20 / 22
Final diagram Listable LEM prop LEM prop LPO NoethExpose Bounded LEM prop LPO LPO AFEq NoethAcc Streamless DEQ NoethAccS ? NoethGame 21 / 22
Thank you for your attention! Questions? 22 / 22
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