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

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 classification of power series 4/3/2012 Slide 2

Background Schützenberger 1961: Languages are non commutative series and operations on regular expressions are rationals. L = {  ,ab,aab} 1 + ab + aab Addition = union Subtraction = take the coefficients from a field Product = concatenation a* = 1 + a + a 2 + a 3 + … = 1  Division = star 1 - a A coalgebraic classification of power series 4/3/2012 Slide 3

Background Chomsky, Schützenberger 1963: Algebraic system of equations have power series as solutions. x = xx + ay + bz S  SS | aSb | bSa y = yb z = za A coalgebraic classification of power series 4/3/2012 Slide 4

Background Fliess1971: Solutions of algebraic equations in one variable are algebraic streams. 1 – (X+2)  + 4X  2 = 0 A coalgebraic classification of power series 4/3/2012 Slide 5

Background Rutten1999: Streams and power series are solutions of behavioral differential equations. o (x) = 0 x = 0 1 2 3 4 5 6 7 … x’ = x +1 A coalgebraic classification of power series 4/3/2012 Slide 6

Starting point Bonchi, Boreale, Milius, Rot, Rutten, Silva, Winter A coalgebraic classification of power series 4/3/2012 Slide 7

Part I Finite state power series A coalgebraic classification of power series 4/3/2012 Slide 8

Moore automata a b X  S x X A a x y b s t a b x y x S A* s a a a x y y y t 4/3/2012 A coalgebraic classification of power series Slide 9

Moore automata, coalgebraically [[-]] X S A*  f S x (S A* ) A S x X A S x [[-]] A Behavioral equation o (x) = s f(x) = <s,  > and  (a) = y x a = y 4/3/2012 A coalgebraic classification of power series Slide 10

(Formal) power series ( S ,+,0, . ,1) Semirings A*  S = S  A*  Power series  = s 0 + s 1 a + s 2 b + s 3 a 2 + s 4 ab + s 5 ba + … =  (  ,w) w w  A* A*  fs S = S  A*  Polynomials 4/3/2012 A coalgebraic classification of power series Slide 11

Examples ( B ,  ,0,  ,1) Boolean semiring ( F 2 ,+,0, . ,1) Binary field (arithmetic modulo 2) ( N ,+,0, . ,1) Natural numbers ( R ,+,0, . ,1) Real numbers ( R  ,min,  ,+,0) Tropical semiring 4/3/2012 A coalgebraic classification of power series Slide 12

Finite state power series Finite [[-]] S fs  A*  S  A*  X   f S x X A S x S fs  A*  A S x ( S  A*  ) A Behaviour of finite Moore automata 4/3/2012 A coalgebraic classification of power series Slide 13

Example a Behavioral equations b a o (x) = 0.2 o (y) = 0.8 y x x a = y y a = y x b = x y b = x b 0.2 0.8 ( R ,+,0, . ,1) Real numbers [[x]](  ) = 0.2 [[x]](wb) = 0.8 [[x]](wa) = 0.8 [[x]] = 0.2 + 0.8a + 0.2 b + 0.8a 2 + 0.2ab + 0.8ba +0.2b 2 + … 4/3/2012 A coalgebraic classification of power series Slide 14

Properties of S fs  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 4/3/2012 Slide 15

Some closure properties of S fs  A*  o (x) = s 1 o (y) = s 2 … x a = x 1 y a = y 1 … x b = … y b = … … Product of x and y Sum of x and y o (x  y) = s 1  s 2 o (x+y) = s 1 + s 2 x  y a = x 1  y 1 x+y a = x 1 +y 1 x  y b = … x+y b = … A coalgebraic classification of power series 4/3/2012 Slide 16

Part II Rational power series A coalgebraic classification of power series 4/3/2012 Slide 17

Weighted automata TX t ::= x | s·t | t + t [[-]] S  A*  X TX  f S x ( S  A*  ) A S x TX A S x [[-]] A Distributive law induced by  F  FT s 1 ·(s 2 ,  ) (s 1 ·s 2 ,  a.s 1 ·  (a)) (s 1 ,  1 ) + (s 2 ,  2 ) (s 1 +s 2 ,  a.  1 (a)+  2 (a)) 4/3/2012 A coalgebraic classification of power series Slide 18

Weighted automata, classically 2,a 1,a 3,a L(x)  N  A*  x y 4,b 1 7 1,a 2,a 4,b x y x x 1 L(x)(aab) = 8 + 24 = 32 2,a 3,a 4,b x x x y 1 2,a 4,b x x y 1 L(x)(ab) = 8 4/3/2012 A coalgebraic classification of power series Slide 19

Two examples … X o (s) = s o (a) = 0 s  S , a  A s a = 0·s a a = 1 s b = 0·s a b = 0 S x TX A 1,a a 1 s s 1 0 [[a]] = 0 + 1 a + 0b + … [[s]] = s + 0a + 0b +... 4/3/2012 A coalgebraic classification of power series Slide 20

… and their Moore automata TX o (s) = s o (a) = 0 s a = 0·s a a = 1 S x TX A s b = 0·s a b = 0 a,b a,b a,b 1 a a,b s a 0 1 a,b s 0 b 0 a,b 0 0 1 4/3/2012 A coalgebraic classification of power series Slide 21

A larger example … X o (x) = 1 o (y) = 2 x a = x y a = 2·y x b = 2·x + 3·y y b = 0·y N x TX A 2,b 2,a 1,a 3,b y x 1 2 [[x]] = 1 + 1a + 8b + 8ab + 14ba + ... 4/3/2012 A coalgebraic classification of power series Slide 22

… and its Moore automaton TX o (x) = 1 o (y) = 2 x a = x y a = 2·y N x TX A x b = 2·x + 3·y y b = 0·y b a a 2x+6y x 2x+3y b 1 8 8 0 4x+6y 16 4/3/2012 A coalgebraic classification of power series Slide 23

Few equalities  (TX, +, 0) is a commutative monoid  s · is a scalar product (s 1 + s 2 )·t ~ s 1 ·t + s 2 ·t s·(t 1 + t 2 ) ~ s ·t 1 + s·t 2 0·t ~ 0 s ·0 ~ 0 1·t ~ t (s 1 ·s 2 )·t ~ s 1 ·(s 2 ·t ) A coalgebraic classification of power series 4/3/2012 Slide 24

Rational power series Finite [[-]] S rat  A*  S  A*  X TX f   S x TX A S x S rat  A*  A S x ( S  A*  ) A TX t ::= x | s·t | t + t 4/3/2012

Properties of S rat  A*  [Sakarovitch 03] 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 4/3/2012 Slide 26

Part III Context free power series A coalgebraic classification of power series 4/3/2012 Slide 27

Context free behavioral equations TX t ::= x | s | t + t | t·t [[-]] S  A*  X TX  f S x TX A S x ( S  A*  ) A S x [[-]] A Distributive law induced by  (Fxid)  FT s (s,  a.0,s) (s 1 ,  1 , v 1 )·(s 2 ,  2 , v 2 ) (s 1 ·s 2 ,  a.  1 (a)· v 2 + s 1 ·  2 (a)) 4/3/2012 A coalgebraic classification of power series Slide 28

Example Dyck language o (x) = 1 o (a) = 0 o (b) = 0 x a = x·a·x a a = 1 b a = 0 x b = x·b·x a b = 0 b b = 1 xaxax a a a xax 0 a b x b 0 xbxax b 1 b xbx 0 0 4/3/2012 A coalgebraic classification of power series Slide 29

Context free power series Finite [[-]] S cf  A*  S  A*  X TX f   S x TX A S x S cf  A*  A S x ( S  A*  ) A TX t ::= x | s | t + t | t·t 4/3/2012 A coalgebraic classification of power series Slide 30

(TX, +, 0, ·, 1) is a semiring  (TX, +, 0) is a commutative monoid  (TX, ·, 1) is a monoid Distributivity (t 1 + t 2 )·t 3 ~ t 1 ·t 3 + t 2 ·t 3 t 1 ·(t 2 + t 3 ) ~ t 1 ·t 2 + t 1 ·t 3 Annihilation 0·t ~ 0 t ·0 ~ 0 A coalgebraic classification of power series 4/3/2012 Slide 31

A simple closure property TX t ::= x | s | t + t | t·t S a field Inverse of x (if s  0) o (x) = s o (y) = … 1 o ( ) = s -1  x x a = t 1 y a = … 1 1 ( ) a = -s -1 · t 1 ·   x b = t 2 … x x 1 … 1  ( ) b = -s -1 · t 2 ·  x x 1 …  Lemma: x· = 1 x 4/3/2012 A coalgebraic classification of power series Slide 32

Part IV Algebraic power series A coalgebraic classification of power series 4/3/2012 Slide 33

S -algebraic systems Finite p i  S  (A+X)*  x i  X x i = p i Proper = in each p i occurs x = 2axb + by 1. no constant from S y = by + b 2. no single variable from X Every proper S -algebraic system has a solution in S  A*  Not proper no solution in N  A*  x = a + x and x = 2 + xx x = x·x two solutions, 0 and 1 A coalgebraic classification of power series 4/3/2012 Slide 34

Alg  A*  S- algebraic power series S   S  A*   = s 0 + s 1 a + s 2 b + s 3 a 2 + s 4 ab + s 5 ba + … Strong solution of a proper S -algebraic system A coalgebraic classification of power series 4/3/2012 Slide 35

Alg  A*  = S cf  A*  S x = 2axb + by o (x) = s 0 o (y) = 0 o (b) = 0 y = by + b x a = 2·x·b y a = 0 b a = 0 x b = y y b = y+1 b b = 1 Using Greibach normal form x = a(y+2) + 3b o (x) = 1 o (y) = 2 y = 4a(x+1) x a = y y a = 4x x b = 3 y b = 0 A coalgebraic classification of power series 4/3/2012 Slide 36