A coalgebraic classification of power series Marcello Bonsangue - - PowerPoint PPT Presentation

▶
a coalgebraic classification of
SMART_READER_LITE
LIVE PREVIEW

A coalgebraic classification of power series Marcello Bonsangue - - PowerPoint PPT Presentation

Sep 17, 2022 829 likes •1.32k views

CMCS - April 1, 2012, Tallin A coalgebraic classification of power series Marcello Bonsangue (with Joost Winter and Jan Rutten) Background Kleene 1956: Finite automata are regular expressions a b a,b b*a(a+b)* x y 0 1 A coalgebraic

slide-1
SLIDE 1

A coalgebraic classification of power series

Marcello Bonsangue

(with Joost Winter and Jan Rutten)

CMCS - April 1, 2012, Tallin

slide-2
SLIDE 2

4/3/2012

Background

A coalgebraic classification of power series Slide 2

Kleene 1956: Finite automata are regular expressions

y x a 1 a,b b

b*a(a+b)*

slide-3
SLIDE 3

4/3/2012

Background

A coalgebraic classification of power series Slide 3

Schützenberger 1961: Languages are non commutative series and operations on regular expressions are rationals. Addition = union Subtraction = take the coefficients from a field Product = concatenation Division = star a* = 1 + a + a2+ a3+ … = 1  1 - a L = {,ab,aab} 1 + ab + aab

slide-4
SLIDE 4

4/3/2012

Background

A coalgebraic classification of power series Slide 4

Chomsky, Schützenberger 1963: Algebraic system of equations have power series as solutions. S  SS | aSb | bSa x = xx + ay + bz y = yb z = za

slide-5
SLIDE 5

4/3/2012

Background

A coalgebraic classification of power series Slide 5

Fliess1971: Solutions of algebraic equations in one variable are algebraic streams. 1 – (X+2) + 4X2 = 0

slide-6
SLIDE 6

4/3/2012

Background

A coalgebraic classification of power series Slide 6

Rutten1999: Streams and power series are solutions of behavioral differential equations.

  • (x) = 0

x’ = x +1 x = 0 1 2 3 4 5 6 7 …

slide-7
SLIDE 7

4/3/2012

Starting point

A coalgebraic classification of power series Slide 7

Bonchi, Boreale, Milius, Rot, Rutten, Silva, Winter

slide-8
SLIDE 8

4/3/2012

Part I

Finite state power series

A coalgebraic classification of power series Slide 8

slide-9
SLIDE 9

4/3/2012

Moore automata

y x a s

X  S x XA

t b a b

x y

a

x

b s

SA*

x y

a

y

a

y

a t

A coalgebraic classification of power series Slide 9

slide-10
SLIDE 10

4/3/2012

Moore automata, coalgebraically

X

S x (SA*)A

S x [[-]]A  f [[-]]

SA*

S x XA f(x) = <s,> and (a) = y

  • (x) = s

xa = y Behavioral equation

A coalgebraic classification of power series Slide 10

slide-11
SLIDE 11

4/3/2012

(Formal) power series

(S,+,0,.,1) Semirings Power series A*  S = SA*  = s0 + s1a + s2b + s3a2 + s4ab + s5ba + … =  (,w) w

wA*

Polynomials A* fs S = SA*

A coalgebraic classification of power series Slide 11

slide-12
SLIDE 12

4/3/2012

Examples

(B,,0,,1) Boolean semiring (N,+,0,.,1) Natural numbers (R,min, ,+,0) Tropical semiring (R,+,0,.,1) Real numbers (F2,+,0,.,1) Binary field (arithmetic modulo 2)

A coalgebraic classification of power series Slide 12

slide-13
SLIDE 13

4/3/2012

Finite state power series

X S x SfsA*A  f [[-]] SfsA* S x XA Finite S x (SA*)A  SA*

Behaviour of finite Moore automata

A coalgebraic classification of power series Slide 13

slide-14
SLIDE 14

4/3/2012

Example

y x a 0.2 0.8 b a b

(R,+,0,.,1) Real numbers [[x]]() = 0.2 [[x]](wb) = 0.8 [[x]](wa) = 0.8 [[x]] = 0.2 + 0.8a + 0.2 b + 0.8a2 + 0.2ab + 0.8ba +0.2b2 + …

  • (x) = 0.2
  • (y) = 0.8

xa = y ya = y xb = x yb = x Behavioral equations

A coalgebraic classification of power series Slide 14

slide-15
SLIDE 15

4/3/2012

Properties of SfsA*

 Decidable bisimulation

Implemented in CIRC [B.,Caltais, Goriac, Lucanu, Rutten,Silva 10]

 Several algorithms for minimization

Partition-refinement [Bonchi et al.] Brzozowski minimization: [Bonchi,B., Rutten, Silva 09] + Hansen

 Complete axiomatization

[Silva,B., Rutten 09]

A coalgebraic classification of power series Slide 15

slide-16
SLIDE 16

4/3/2012

Some closure properties of SfsA*

A coalgebraic classification of power series Slide 16

  • (x+y) = s1 + s2

x+ya = x1+y1 x+yb = … Product of x and y

  • (xy) = s1s2

xya = x1y1 xyb = …

  • (x) = s1
  • (y) = s2

… xa = x1 ya = y1 … xb = … yb = … … Sum of x and y

slide-17
SLIDE 17

4/3/2012

Part II

Rational power series

A coalgebraic classification of power series Slide 17

slide-18
SLIDE 18

4/3/2012

Weighted automata

X S x (SA*)A S x [[-]]A  f [[-]] SA* S x TXA TX t ::= x | s·t | t + t TX s1·(s2,) (s1·s2, a.s1·(a)) (s1, 1) + (s2,2) (s1+s2, a.1(a)+2(a)) Distributive law induced by F  FT

A coalgebraic classification of power series Slide 18

slide-19
SLIDE 19

4/3/2012

Weighted automata, classically

x x

1,a

y

2,a

x

4,b 1

L(x)(aab) = 8 + 24 = 32 L(x)(ab) = 8

y x 2,a 1 7 4,b 3,a 1,a

x x

2,a

y

3,a

x

4,b 1

x x

2,a

y

4,b 1

L(x) NA*

A coalgebraic classification of power series Slide 19

slide-20
SLIDE 20

4/3/2012

Two examples …

  • (s) = s
  • (a) = 0

sa = 0·s aa = 1 sb = 0·s ab = 0

s a s

[[a]] = 0 + 1a + 0b + …

1 1,a

s S, a  A [[s]] = s + 0a + 0b +...

1

A coalgebraic classification of power series Slide 20

X S x TXA

slide-21
SLIDE 21

4/3/2012

… and their Moore automata

a 1 a s s a,b a,b 1 a,b a,b b 1 a,b a,b

  • (s) = s
  • (a) = 0

sa = 0·s aa = 1 sb = 0·s ab = 0

A coalgebraic classification of power series Slide 21

S x TXA TX

slide-22
SLIDE 22

4/3/2012

A larger example …

  • (x) = 1
  • (y) = 2

xa = x ya = 2·y xb = 2·x + 3·y yb = 0·y

x 1 2,b y 3,b 1,a 2,a 2

[[x]] = 1 + 1a + 8b + 8ab + 14ba + ...

A coalgebraic classification of power series Slide 22

X N x TXA

slide-23
SLIDE 23

4/3/2012

… and its Moore automaton

  • (x) = 1
  • (y) = 2

xa = x ya = 2·y xb = 2·x + 3·y yb = 0·y

x 1 b b a a 2x+3y 8 2x+6y 8 4x+6y 16

A coalgebraic classification of power series Slide 23

N x TXA TX

slide-24
SLIDE 24

4/3/2012

Few equalities

 (TX, +, 0) is a commutative monoid  s ·

is a scalar product (s1 + s2)·t ~ s1·t + s2·t s·(t1 + t2) ~ s ·t1 + s·t2 0·t ~ 0 s ·0 ~ 0 1·t ~ t (s1·s2)·t ~ s1·(s2·t )

A coalgebraic classification of power series Slide 24

slide-25
SLIDE 25

4/3/2012

Rational power series

S x SratA*A  [[-]] SratA* Finite S x (SA*)A  SA* X f

S x TXA

TX TX t ::= x | s·t | t + t

slide-26
SLIDE 26

4/3/2012

Properties of SratA*

Decidable equivalence for many semirings [Ésik,Maletti 10]

 Yes: Natural numbers, any subsemiring of a field  No: tropical semiring, regular languages

Complete axiomatization for many semirings

 For regular languages [Krob 90][Kozen 94]  For proper semirings [Ésik, Kuich 12]  Adding equations to those of bisimulation [B.,Milius,Silva 12]

Several minimization algorithms

[Schützenberger 61], [Berstel,Reutenauer 88], [Sakarovitch 06], [Mohri 09],[Bonchi,B.,Boreale, Rutten,Silva 11]

A coalgebraic classification of power series Slide 26

[Sakarovitch 03]

slide-27
SLIDE 27

4/3/2012

Part III

Context free power series

A coalgebraic classification of power series Slide 27

slide-28
SLIDE 28

4/3/2012

Context free behavioral equations

X S x (SA*)A S x [[-]]A  f [[-]] SA*

S x TXA

TX t ::= x | s | t + t | t·t TX s (s, a.0,s) (s1, 1,v1)·(s2,2,v2) (s1·s2, a.1(a)·v2 + s1·2(a)) Distributive law induced by (Fxid)  FT

A coalgebraic classification of power series Slide 28

slide-29
SLIDE 29

4/3/2012

Example

  • (x) = 1 o(a) = 0
  • (b) = 0

xa = x·a·x aa = 1 ba = 0 xb = x·b·x ab = 0 bb = 1

x 1 xax a b

Dyck language

a xaxax a a xbx b xbxax b b

A coalgebraic classification of power series Slide 29

slide-30
SLIDE 30

4/3/2012

Context free power series

S x ScfA*A  [[-]] ScfA* Finite S x (SA*)A  SA* X f

S x TXA

TX TX t ::= x | s | t + t | t·t

A coalgebraic classification of power series Slide 30

slide-31
SLIDE 31

4/3/2012

(TX, +, 0, ·, 1) is a semiring

 (TX, +, 0) is a commutative monoid  (TX, ·, 1) is a monoid

Distributivity (t1 + t2)·t3 ~ t1·t3 + t2·t3 t1·(t2 + t3) ~ t1·t2 + t1·t3 Annihilation 0·t ~ 0 t ·0 ~ 0

A coalgebraic classification of power series Slide 31

slide-32
SLIDE 32

4/3/2012

A simple closure property

TX t ::= x | s | t + t | t·t

  • (x) = s
  • (y) = …

xa = t1 ya = … xb = t2 … … S a field Inverse of x (if s  0)

  • ( ) = s-1

( )a = -s-1· t1 · ( )b = -s-1· t2 · …

1  x 1  x 1  x 1  x 1  x

Lemma: x· = 1

1  x

A coalgebraic classification of power series Slide 32

slide-33
SLIDE 33

4/3/2012

Part IV

Algebraic power series

A coalgebraic classification of power series Slide 33

slide-34
SLIDE 34

4/3/2012

S-algebraic systems

x = a + x and x = 2 + xx no solution in NA* x = x·x two solutions, 0 and 1 Proper = in each pi occurs

  • 1. no constant from S
  • 2. no single variable from X

xi = pi pi S(A+X)* xiX Finite x = 2axb + by y = by + b Every proper S-algebraic system has a solution in SA*

A coalgebraic classification of power series Slide 34

Not proper

slide-35
SLIDE 35

4/3/2012

S-algebraic power series S

AlgA*

 = s0 + s1 a + s2 b + s3 a2 + s4 ab + s5 ba + … Strong solution of a proper S-algebraic system   SA*

A coalgebraic classification of power series Slide 35

slide-36
SLIDE 36

4/3/2012

S

AlgA* = S cfA*

x = 2axb + by y = by + b

  • (x) = s0
  • (y) = 0
  • (b) = 0

xa = 2·x·b ya = 0 ba = 0 xb = y yb = y+1 bb = 1

  • (x) = 1 o(y) = 2

xa = y ya = 4x xb = 3 yb = 0 x = a(y+2) + 3b y = 4a(x+1) Using Greibach normal form

A coalgebraic classification of power series Slide 36

slide-37
SLIDE 37

4/3/2012

S

cfA* is closed under reverse

x = a(x+1)b b = b

  • (x) = 1
  • (b) = 0

xa = x·b ba = 0 xb = 0 bb = 1

  • (xR) = 1
  • (a) = 0

xR

a = 0

aa = 1 xR

b = xR·a + a

ab = 0 xR = bR(xR+1)a bR = b xR = bxRa + ba a = a reverse GNF

A coalgebraic classification of power series Slide 37

slide-38
SLIDE 38

4/3/2012

Part V

Algebraic streams

A coalgebraic classification of power series Slide 38

slide-39
SLIDE 39

4/3/2012

Algebraic streams over F

A = { X } F is a field FA*  N  F i.e. streams A stream  FA* is algebraic if there are non-null polynomials pi FA* such that p0 + p1 + p2 2 + … + pnn = 0

A coalgebraic classification of power series Slide 39

slide-40
SLIDE 40

4/3/2012

Algebraic streams are context free

1 -  + X2 = 0

  • ’ + 2 = 0

’ = 2

Taking derivative

  • () = 1

’ = · Paper folding stream  = 1 1 0 1 1 0 0 1 1 1 0 0 …

F2

A coalgebraic classification of power series Slide 40

Behavioral equation

slide-41
SLIDE 41

4/3/2012

  • (x) = 0
  • (y) = 1
  • (z) = 1
  • (w) = 1

x’ = y y’ = z z’ = z+y+x·x·w w’ = y·w The Thue-Morse stream  = 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 …

Algebraic streams are context free

X +(1+X2) + (1+X+X2+X3)2= 0

F2

A coalgebraic classification of power series Slide 41

Behavioral equations

slide-42
SLIDE 42

4/3/2012

F-algebraic = F

cf{X}*

For a perfect field F and a singleton alphabet A, a stream is algebraic if and only if is F-algebraic

A coalgebraic classification of power series Slide 42

In any field F and a singleton alphabet A, if a well behaved stream is algebraic then is context-free

[Fliess 71] [BRW 12]

slide-43
SLIDE 43

4/3/2012

Part VI

Automatic streams

A coalgebraic classification of power series Slide 43

slide-44
SLIDE 44

4/3/2012

p-Automatic streams

A coalgebraic classification of power series Slide 44

[Allouche, Sallit 03]

The n-th term of the stream is the output

  • f the state when the automata input is

the digits of n in some fixed base p

1 1 1

F2

slide-45
SLIDE 45

4/3/2012

Example: Thue-Morse again …

A coalgebraic classification of power series Slide 45

1 1 1

F2

Input binary

  • utput

1 1 1 2 10 1 3 11 4 100 1 5 101 Input binary

  • utput

6 101 7 111 1 8 1000 1 9 1001 10 1010 11 1011 1

slide-46
SLIDE 46

4/3/2012

p-automatic are context free

A coalgebraic classification of power series Slide 46

For any prime number p, a stream is p-automatic if and only if is algebraic over Fp

[Berstel,Retenauer 11]

slide-47
SLIDE 47

4/3/2012 A coalgebraic classification of power series Slide 47

The big picture

p-automatic streams Algebraic over a field S-algebraic CF behavioral equations Weighted CFG in GNF Rat behavioral equations FS behavioral equations streams power series

Perfect field p prime

fields semirings

slide-48
SLIDE 48

4/3/2012

Why?

 Clarifying the definitions  Importance of weighted systems  Towards the understanding of treatable subsets of

the final coalgebra

 Algorithmically interesting

Three coalgebraic characterizations of context free languages Slide 48