Generalizations and variants of associativity for variadic functions: a survey

Generalizations and variants of associativity for variadic functions: a survey

53rd ISFE

Jean-Luc Marichal

in collaboration with Bruno Teheux University of Luxembourg

Associative binary operations

LetX be a nonempty set F:X² →X is associativeif

F(F(x,y),z) = F(x,F(y,z))

Example: F(x,y) =x+y on X =R

Associative binary operations

Extension to a function with an indefinite arity F: S

n>2Xⁿ→X

F(x1, . . . ,xn) = F(F(x1, . . . ,xn−1),xn) n>3

Example

F(x₁,x₂) = x₁+x₂

F(x₁,x₂,x₃) = F(F(x₁,x₂),x₃) = x₁+x₂+x₃ etc.

Notation

X =alphabet

Elements of X: letters (x,y,z, . . .∈X) The set

X^∗ = [

n>0

Xⁿ

is the set of all tuples onX, calledstrings over X (x,y,z, . . .∈X^∗)

Convention: X⁰ ={ε}, whereε= the empty string

Notation

X^∗ is endowed with concatenation (ε= neutral element) x∈Xⁿ andy ∈X ⇒ xyε=xy ∈Xⁿ⁺¹ Repeated strings

xⁿ = x· · ·x

| {z }

n

, x⁰ = ε

Length of a string

|x| = n ⇐⇒ x∈Xⁿ

|ε| = 0, |x| = 1

Notation

LetY be a nonempty set n-ary function

F:Xⁿ→Y

∗-ary functionor variadic function F:X^∗→Y n-ary part of F

F_n = F|_Xn

Default value of F

F(ε) = F0(ε)

Associativity

For any map F: S

n>2Xⁿ→X

Binary associativity : F(F(xy)z) = F(x F(yz)) Induction formula : F(xz) = F(F(x)z) |xz|>3 Proposition

Binary associativity + induction formula m

F(xyz) = F(xF(y)z) |y|>2, |xz|>1

−→ Extension to functionsF defined onX^∗ = S

n>0Xⁿ ?

Associativity

Definitions

•A variadic operationonX is a map F:X^∗ →X ∪ {ε}

•A variadic operationF:X^∗→X∪ {ε}is said to be associativeif F(xyz) = F(xF(y)z), x,y,z∈X^∗

We say thatF is ε-standardif

F(x) = ε ⇔ x = ε

Associativity

F(xyz) = F(xF(y)z), x,y,z∈X^∗

Theorem

Anε-standard operationF:X^∗ →X ∪ {ε} is associative iff (i) Binary associativity + induction formula

(ii) F1◦F1 =F1

(iii) F₁◦F₂ =F₂

(iv) F₂(xy) =F₂(F₁(x)y) =F₂(x F₁(y))

Observation: F₁ =id_X can always be considered

Associativity

Example

F:R^∗ →R∪ {ε}

Fn(x1· · ·xn) = kxk₂ = q

x₁²+· · ·+x_n² n>2

F₀(ε) = ε F₁(x) = x or F₁(x) =

√

x² or · · ·

Associativity for string functions

Definition. Astring function overX is a functionF:X^∗ →X^∗ Examples(data processing tasks)

F(x) = sorting the letters of x in alphabetic order F(x) = transforming a string x into upper case

F(x) = removing from x all occurrences of a given letter F(x) = removing from x all repeated occurrences of letters

F(associativity) = ass/ocia/ti/vi/t/y = asocitvy Each of these tasks satisfies

F(xyz) = F(xF(y)z)

Associativity for string functions

Definition. A string functionF:X^∗→X^∗ is said to be associativeif

F(xyz) = F(xF(y)z), x,y,z∈X^∗

→generalizes associativity for operationsF:X^∗ →X ∪ {ε}

Associativity for string functions

Note. Settingx=z=εin the identity F(xyz) = F(xF(y)z) we obtainF(y) = F(F(y))

F = F ◦F

Preassociativity

LetY be a nonempty set

Definition. We say that F:X^∗ →Y ispreassociativeif F(y) = F(y⁰) ⇒ F(xyz) = F(xy⁰z)

Examples. F:R^∗→R∪ {ε}

F0 = ε, Fn(x) = x1+· · ·+xn

F₀ = ε, F_n(x) = x₁²+· · ·+x_n² = kxk²₂

F₀ = ε, F_n(x) = g(x₁+· · ·+x_n), g one-to-one

Preassociativity

F(y) = F(y⁰) ⇒ F(xyz) = F(xy⁰z)

Proposition

LetF:X^∗ →X^∗ be a string function F associative ⇔

F ◦F =F F preassociative

Preassociativity

F(y) = F(y⁰) ⇒ F(xyz) = F(xy⁰z)

Various codomains can be considered ExamplesF:X^∗ →Z

F(x) =|x|(number of letters in x)

F(x) = number of occurrences in xof a given letter, say ‘z’ F(x) = number of letters distinct fromz minus the number of occurrences of z

Note. The function that outputs the number of distinct letters in xis not preassociative:

Ifa,b∈X are distinct, thenF(a) =F(b) = 1 but 1 =F(aa)6=F(ab) = 2

Preassociativity

Theorem

LetF:X^∗ →Y. The following assertions are equivalent:

(i) F is preassociative (ii) F can be factorized into

F = f ◦H whereH:X^∗ →X^∗ is associative

f :ran(H)→Y is one-to-one

Preassociativity

Preassociative functions

Associative string functions

Barycentric associativity

Definition. A variadic operationF:X^∗→X ∪ {ε} is said to be barycentrically associative(orB-associative) if

F(xyz) = F(xF(y)^|y|z)

F(abcd) = F(F(ab)²cd) = F(F(ab)F(ab)cd) Notes.

...first considered for symmetric functions on S

n>1Rⁿ (Schimmack 1909, Kolmogoroff 1930, Nagumo 1930) ...can be considered also for string functions F:X^∗ →X^∗ F(x) = removing from x all repeated occurrences of letters

Barycentric associativity

F(xyz) = F(xF(y)^|y|z)

SupposeF:X^∗ →X∪ {ε}is B-associative and ε-standard ThenF remains B-associative if we modifyF(ε)

⇒ The value F(ε)is unimportant and we can assumeran(F)⊆X

Barycentric associativity

F(xyz) = F(xF(y)^|y|z) Example. Arithmetic mean F:R^∗→R

F(x₁· · ·x_n) = 1 n

n

X

i=1

x_i

F(x₁F(x₂x₃)²) = F x₁^x2+x₂ ^{3 x2+x}₂ ³

= ₃¹ x₁+^x2+x₂ ³ +^x2+x₂ ³

= ¹₃(x₁+x₂+x₃)

= F(x1x2x3)

Barycentric associativity

Definition. Quasi-arithmetic means

I= non-trivial real interval, possibly unbounded f:I→Rcontinuous and strictly monotonic

F:I^∗→I F(x₁· · ·x_n) = f⁻¹

1 n

n

X

i=1

f(x_i)

Note. F is B-associative

Barycentric associativity

Theorem(Kolmogoroff-Nagumo, 1930)

I= non-trivial real interval, possibly unbounded LetF:I^∗→I

The following assertions are equivalent:

(i) F is B-associative Fn symmetric Fn continuous

Fn strictly increasing in each argument Fn reflexive, i.e.,Fn(x· · ·x) =x (ii) F is a quasi-arithmetic mean

Note. One can show that reflexivity is redundant

Barycentric associativity

Further examples of real B-associative functions F_n(x) = min(x₁, . . . ,x_n)

Fn(x) = max(x1, . . . ,xn) Fn(x) = x1

F_n(x) = x_n F_n(x) = Pn

i=1 2ⁱ⁻¹ 2ⁿ−1x_i

F_n^α(x) = Pn

i=1αⁿ⁻ⁱ(1−α)ⁱ⁻¹ x_i Pn

i=1αⁿ⁻ⁱ(1−α)ⁱ⁻¹ , α∈R Takeα= 1, α= 0,α= 1/3, etc.

Barycentric preassociativity

Definition. We say that F:X^∗ →Y isbarycentrically preassociative(orB-preassociative) if

F(y) = F(y⁰)

|y| = |y⁰|

⇒ F(xyz) = F(xy⁰z)

Notes

...inspired from the following property by de Finetti (1931) F(y) = F(u^|y|) ⇒ F(xyz) = F(xu^|y|z) (|y|, |xz| > 1) Preassociativity ⇒ B-preassociativity

The value F(ε) is unimportant

Barycentric preassociativity

B-preassociative functions Preassociative functions

Associative string functions

Barycentric preassociativity

F(y) = F(y⁰)

|y| = |y⁰|

⇒ F(xyz) = F(xy⁰z)

Interpretations

Decision making : if we express an indifference when comparing two profiles, then this indifference is preserved when adding identical pieces of information to these profiles Aggregation function theory: the aggregated value of a series of numerical values remains unchanged when modifying a bundle of these values without changing their partial aggregation

Barycentric preassociativity

F(y) = F(y⁰)

|y| = |y⁰|

⇒ F(xyz) = F(xy⁰z)

LetF:X^∗ →X^∗

F associative ⇔

F(x) =F(F(x)) F preassociative

Proposition

LetF:X^∗ →X ∪ {ε}

F B-associative ⇒

F(x) =F(F(x)^|x|) F B-preassociative The converse holds wheneverF(x)∈X for all x6=ε

Barycentric preassociativity

B-preassociative functions

B-associative variadic operations

Barycentric preassociativity

Definition. Quasi-arithmetic pre-means

f:I→Randf_n:R→Rcontinuous and strictly increasing (n>1) F:I^∗→R

F(x₁· · ·x_n) = f_n 1

n

n

X

i=1

f(x_i)

Note. F is B-preassociative

Barycentric preassociativity

Quasi-arithmetic pre-means F(x₁· · ·x_n) = f_n

1 n

n

X

i=1

f(x_i)

F quasi-arithmetic pre-mean F_n reflexive ∀n

⇔ F quasi-arithmetic mean

Non-reflexive examples

fn(x) =nx and f(x) =x ⇒ F(x) =Pn i=1x_i fn(x) =e^nx and f(x) = lnx ⇒ F(x) =Qn

i=1xi

Barycentric preassociativity

Theorem

I= non-trivial real interval, possibly unbounded LetF:I^∗→R

The following assertions are equivalent:

(i) F is B-preassociative Fn symmetric Fn continuous

Fn strictly increasing in each argument (ii) F is a quasi-arithmetic pre-mean function

Selected references

B. de Finetti. Sul concetto di media.Giornale dell’ Instituto Italiano degli Attari 2(3):369–396, 1931.

M. Grabisch, J.-L. Marichal, R. Mesiar, and E. Pap.Aggregation functions.

Cambridge University Press, Cambridge, 2009.

A. N. Kolmogoroff. Sur la notion de la moyenne.Atti Accad. Naz. Lincei, 12(6):388–391, 1930.

E. Lehtonen, J.-L. Marichal, B. Teheux. Associative string functions.

Asian-European Journal of Mathematics, 7(4):1450059 (18 pages), 2014.

J.-L. Marichal, P. Mathonet, and J. Tomaschek. A classification of

barycentrically associative polynomial functions.Aequat. Math., in press, 2015.

J.-L. Marichal and B. Teheux. Associative and preassociative functions.

Semigroup Forum, 89(2):431–442, 2014.

J.-L. Marichal and B. Teheux. Barycentrically associative and preassociative functions.Acta Mathematica Hungarica, 145(2):468–488, 2015.

M. Nagumo. ¨Uber eine klasse der mittelwerte. (German).Japanese Journ. of Math., 7:71–79, 1930.

R. Schimmack. Der Satz vom arithmetischen Mittel in axiomatischer Begr¨undung.Math. Ann.68:125–132, 1909.