The Mathematics behind the Property of Associativity An invitation - PowerPoint PPT Presentation

The Mathematics behind the Property of Associativity An invitation to study the many variants of associativity Bruno Teheux and Jean-Luc Marichal University of Luxembourg

Associativity for binary functions X , Y ≡ non-empty sets F : X × X → X is associative if F ( x , F ( y , z )) = F ( F ( x , y ) , z ) Associativity enables us to define expressions like F ( x , y , z , t ) = F ( F ( F ( x , y ) , z ) , t ) = F ( x , F ( F ( y , z ) , t )) = · · · n ≥ 2 X n → X : x ∈ X n �→ F ( x 1 , . . . , x n ) Define F : �

Notation We regard n -tuples x in X n as n-strings over X 0-string: ε 1-strings: x , y , z , . . . n -strings: x , y , z , . . . | x | = length of x X ∗ := � X n n ≥ 0 We endow X ∗ with concatenation ( X ∗ is a free monoid) Any F : X ∗ → Y is called a variadic function , and we set F n := F | X n . We assume F ( x ) = ε ⇐ ⇒ x = ε

Associativity for variadic operations F : X ∗ → X ∪ { ε } is called a variadic operation . Definition. F : X ∗ → X ∪ { ε } is associative if ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z ) Examples. · the sum x 1 + · · · + x n , · the minimum x 1 ∧ . . . ∧ x n , · variadic extensions of binary associative functions. F 1 may differ from the identity map!

Associativity for string functions Definition. F : X ∗ → X ∗ is associative if ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z )

Associativity for string functions ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z ) Examples. · sorting in alphabetical order · letter removing, duplicate removing

Associativity for string functions ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z ) Examples. [. . . ] duplicate removing Input : xzu · · · in blocks of unknown length given at unknown time intervals. Output : F ( xzu · · · ) F ( x ) F ( z ) F ( u ) F F F x z u

Associativity for string functions ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z ) Examples. [. . . ] duplicate removing Input : xzu · · · in blocks of unknown length given at unknown time intervals. Output : F ( xzu · · · ) F ( x ) F ( z ) F ( u ) F F F x z u

Associativity for string functions ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z ) Examples. [. . . ] duplicate removing Input : xzu · · · in blocks of unknown length given at unknown time intervals. Output : F ( xzu · · · ) F ( x ) F ( z ) F ( u ) F F F x z u “Highly” distributed algorithms

Associativity for string functions ∀ xyz ∈ X ∗ F ( xyz ) = F ( x F ( y ) z ) Proposition. (1) If F , G : X ∗ → X ∗ are associative, then F = G ⇐ ⇒ ( F 1 = G 1 and F 2 = G 2 ) (2) G : X 2 → X is associative if and only if it admits a variadic associative extension F : X ∗ → X ∪ { ε } ( i.e. , F 2 = G ).

Preassociative variadic functions Definition. We say that F : X ∗ → Y is preassociative if F ( y ) = F ( y ′ ) F ( xyz ) = F ( xy ′ z ) ⇒ Examples. F n ( x ) = x 2 1 + · · · + x 2 ( X = Y = R ) n F n ( x ) = | x | ( X arbitrary , Y = N ) Slogan. Preassociativity is a composition-free version of associativity. Fact. For F : X ∗ → Y ker( F ) is a congruence on X ∗ F is preassociative ⇐ ⇒

Associative and preassociative functions Proposition. Let F : X ∗ → X ∗ . F is associative ⇐ ⇒ F is preassociative and F ◦ F = F . Proposition Let F : X ∗ → ran ( F ) be preassociative and g : ran ( F ) → Z If g is one-to-one or constant, then g ◦ F is preassociative. Problem. Let F : X ∗ → Y be preassociative. For which g is g ◦ F preassociative? Hard! Characterize [ker( F )) in the congruence lattice of X ∗ .

Associative and preassociative functions Theorem. (AC) Let F : X ∗ → Y . The following conditions are equivalent. (i) F is preassociative. (ii) F = f ◦ H where H : X ∗ → X ∗ is associative and f : ran ( H ) → Y is one-to-one.

Associative and preassociative functions Theorem. (AC) Let F : X ∗ → Y . The following conditions are equivalent. (i) F is preassociative. (ii) F = f ◦ H where H : X ∗ → X ∗ is associative and f : ran ( H ) → Y is one-to-one. Proof. Define g ( F ( x )) ∈ x / ker( F ) , F X ∗ ran ( F ) H := g ◦ F , g then F = F ◦ H .

Factorizations lead to axiomatizations of function classes A three step technique: (Binary) Start with a class associative functions F : X 2 → X , (Source) Axiomatize all their associative extensions F : X ∗ → X ∪ { ε } , (Target) Use factorization theorem to weaken this axiomatization to capture preassociativity. The methodology will be used for other factorization results.

An example based on Acz´ elian semigroups el 1949). H : R 2 → R is Theorem (Acz´ · continuous · one-to-one in each argument · associative if and only if H ( xy ) = ϕ − 1 ( ϕ ( x ) + ϕ ( y )) where ϕ : R → R is continuous and strictly monotone. Source class of associative variadic operations H n ( x ) = ϕ − 1 ( ϕ ( x 1 ) + · · · + ϕ ( x n ))

An example based on Acz´ elian semigroups Target axiomatization theorem Let F : R ∗ → R ∪ { ε } . The following assertions are equivalent: (i) F is preassociative and · ran ( F 1 ) = ran ( F ), · F 1 and F 2 are continuous, · F 1 and F 2 one-to-one in each argument, (ii) we have � � F n ( x ) = ψ ϕ ( x 1 ) + · · · + ϕ ( x n ) where ϕ, ψ : R → R are continuous and strictly monotone.

Transition systems A transition system over X : a b A = ( Q , q 0 , δ ) q 0 q 1 where q 0 ∈ Q is the initial state and a b δ : Q × X → Q q 2 is the transition function. b The map δ is extended to Q × X ∗ by For instance, δ ( q , ε ) := q , δ ( q 0 , ababb ) = q 2 δ ( q , x y ) := δ ( δ ( q , x ) , y )

Transition systems A transition system over X : a b A = ( Q , q 0 , δ ) q 0 q 1 where q 0 ∈ Q is the initial state and a b δ : Q × X → Q q 2 is the transition function. b The map δ is extended to Q × X ∗ by For instance, δ ( q , ε ) := q , δ ( q 0 , ababb ) = q 2 δ ( q , x y ) := δ ( δ ( q , x ) , y ) Definition. F A ( x ) := δ ( q 0 , x )

Preassociativity and transition systems F A ( x ) := δ ( q 0 , x ) Fact. If A is transition system, · F A is “half”-preassociative: F A ( x ) = F A ( y ) = ⇒ F A ( xz ) = F A ( yz ) · F A may not be preassociative: a b q 0 q 1 F A ( b ) = q 1 = F A ( ba ) a b F A ( bb ) = q 2 � = q 0 = F A ( bba ) q 2 b

Preassociativity and transition systems F A ( x ) := δ ( q 0 , x ) Definition. A transition system is preassociative if it satisfies δ ( q 0 , x ) = δ ( q 0 , y ) = ⇒ δ ( q 0 , z x ) = δ ( q 0 , z y ) Lemma. A preassociative ⇐ ⇒ F A preassociative

Preassociativity and transition systems F A ( x ) := δ ( q 0 , x ) Definition. A transition system is preassociative if it satisfies δ ( q 0 , x ) = δ ( q 0 , y ) = ⇒ δ ( q 0 , z x ) = δ ( q 0 , z y ) Lemma. A preassociative ⇐ ⇒ F A preassociative Example. X = { 0 , 1 } 1 0 0 F A ( x ) = e ⇐ ⇒ # { i | x i = 1 } is even, e o F A ( x ) = o ⇐ ⇒ # { i | x i = 1 } is odd. 1

Preassociativity and transition systems X , Q finite. Definition. For an onto F : X ∗ → Q , set q 0 := F ( ε ) , δ ( q , z ) := { F ( x z ) | q = F ( x ) } , A F := ( Q , q 0 , δ ) Generally, A F is a non-deterministic transition system. Lemma. ⇒ A F is deterministic and preassociative F is preassociative ⇐

A criterion for preassociativity ⇒ A F is deterministic and preassociative F is preassociative ⇐ For any state q of A = ( Q , q 0 , δ ), any L ⊆ 2 X ∗ and z ∈ X , set L A ( q ) := { x ∈ X ∗ | δ ( q 0 , x ) = q } z . L := { z x | x ∈ L } Proposition. Let A = ( Q , q 0 , δ ) be a transition system. The following conditions are equivalent. (i) A is preassociative, (ii) for all z ∈ X and q ∈ Q , for some q ′ ∈ Q . z . L A ( q ) ⊆ L A ( q ′ ) ,

for some q ′ ∈ Q . z . L A ( q ) ⊆ L A ( q ′ ) , Example. X = { 0 , 1 } 0 1 0 e o 1 L A ( e ) = { x | x contains an even number of 1 } L A ( o ) = { x | x contains an odd number of 1 } 0 . L A ( o ) ⊆ L A ( o ) 0 . L A ( e ) ⊆ L A ( e ) 1 . L A ( o ) ⊆ L A ( e ) 1 . L A ( e ) ⊆ L A ( o )

Associative length-based functions Definition. F : X ∗ → X ∗ is length-based if for some φ : N → X ∗ . F = φ ◦ | · | Proposition. Let F : X ∗ → X ∗ be a length-based function. The following conditions are equivalent. (i) F is associative (ii) | F ( x ) | = α ( | x | ) where α : N → N satisfies α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N

α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N 10 9 8 7 6 α ( n ) 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 n

α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N 10 9 8 7 6 α ( n ) 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 n

α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N 10 9 8 7 6 α ( n ) 5 ℓ 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 = n n 1

α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N 10 9 8 7 6 α ( n ) 5 ℓ 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 = n n 1

α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N 10 9 8 7 6 α ( n ) 5 ℓ 4 3 2 1 ℓ 0 0 1 2 3 4 5 6 7 8 9 10 = n n 1

α ( n + k ) = α ( α ( n ) + k ) , ∀ n , k ∈ N 10 9 8 7 6 α ( n ) 5 ℓ 4 3 2 1 ℓ 0 0 1 2 3 4 5 6 7 8 9 10 = n n 1

B-associativity and its variants