Weighted automata coalgebraically

Transcription

1 Weighted automata coalgebraically Filippo Bonchi 4 Michele Boreale 5 Marcello Bonsangue,2 Jan Rutten,3 Alexandra Silva Centrum Wiskunde en Informatica 2 LIACS - Leiden University 3 Radboud Universiteit Nijmegen 4 INRIA Saclay - LIX, École Polytechnique 5 Università di Firenze Braunschweig, August 200 Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August 200 / 24

2 Motivation Weighted automata are transition systems with many applications: speech processing, image recognition, information theory,... Interesting questions: correct notion of equivalence, algorithms for minimization,... Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

3 Motivation Weighted automata are transition systems with many applications: speech processing, image recognition, information theory,... Interesting questions: correct notion of equivalence, algorithms for minimization,... Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

4 Motivation (cont d) Modelling systems as coalgebras: The type of the system determines a canonical notion of equivalence and a universe of behaviours (final coalgebra). Goals of this talk: Show how to model weighted automata as coalgebras in two different settings; 2 Show the canonical equivalences derived in each; 3 An algorithm for minimization. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

5 Motivation (cont d) Modelling systems as coalgebras: The type of the system determines a canonical notion of equivalence and a universe of behaviours (final coalgebra). Goals of this talk: Show how to model weighted automata as coalgebras in two different settings; 2 Show the canonical equivalences derived in each; 3 An algorithm for minimization. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

6 Weighted automata A weighted automaton with input alphabet A is a pair (X, o, t ), where X is a set of states; o : X K is an output function associating to each state its output weight and t : X (K X ) A is the transition relation that associates a weight to each transition. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

7 Example K = R, A = {a, b} a,/ x a, a, x 2 x 3 Convenient matrix representation (for X finite): 0 0 x x 2 x 3 O X = ( ) T Xa = 3 0 T Xb = Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

8 Example K = R, A = {a, b} a,/ x a, a, x 2 x 3 Convenient matrix representation (for X finite): 0 0 x x 2 x 3 O X = ( ) T Xa = 3 0 T Xb = Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

9 Example K = R, A = {a, b} a,/ x a, a, x 2 x 3 Convenient matrix representation (for X finite): 0 0 x x 2 x 3 O X = ( ) T Xa = 3 0 T Xb = Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

10 Example K = R, A = {a, b} a,/ x a, a, x 2 x 3 Convenient matrix representation (for X finite): 0 0 x x 2 x 3 O X = ( ) T Xa = 3 0 T Xb = Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

11 Equivalence I : weighted bisimilarity Definition Let (X, o, t ) be a weighted automaton. An equivalence relation R X X is a weighted bisimulation if for all (x, x 2 ) R, it holds that: o(x ) = o(x 2 ), 2 a A, x X, t(x )(a)(x ) = t(x 2 )(a)(x ). x [x ] R x [x ] R [x] R : equivalence class of x with respect to R. The definition only makes sense if the automaton has finite branching. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

12 Equivalence I : weighted bisimilarity Definition Let (X, o, t ) be a weighted automaton. An equivalence relation R X X is a weighted bisimulation if for all (x, x 2 ) R, it holds that: o(x ) = o(x 2 ), 2 a A, x X, t(x )(a)(x ) = t(x 2 )(a)(x ). x [x ] R x [x ] R [x] R : equivalence class of x with respect to R. The definition only makes sense if the automaton has finite branching. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

13 Example a,/ x i, x j R x o(x i ) = o(x j ) = 2 [x i ] R = [x j ] R = X a, a, x 2 x 3 R = { x i, x j x i, x j X} We have to check for all states that the sum of the weights of all the transitions with the same label is the same. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

14 Example a,/ x i, x j R x o(x i ) = o(x j ) = 2 [x i ] R = [x j ] R = X a, a, x 2 x 3 R = { x i, x j x i, x j X} We have to check for all states that the sum of the weights of all the transitions with the same label is the same. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

15 Example a,/ x i, x j R x o(x i ) = o(x j ) = 2 [x i ] R = [x j ] R = X a, a, x 2 x 3 R = { x i, x j x i, x j X} We have to check for all states that the sum of the weights of all the transitions with the same label is the same. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

16 Example a,/ x i, x j R x o(x i ) = o(x j ) = 2 [x i ] R = [x j ] R = X a, a, x 2 x 3 R = { x i, x j x i, x j X} We have to check for all states that the sum of the weights of all the transitions with the same label is the same. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

17 Equivalence II : weighted language equivalence Given a word w A, we defined its weight in a state x of a weighted automaton (X, o, t ) inductively: L(x)(ɛ) = o(x) 2 L(x)(aw) = x X t(x)(a)(x ) L(w)(x ) a,2 b,2 x 2 x 4 a, x a, L(x )(aba n ) = 2 2 n L(x )(aca n ) = 3 3 n L(x )( ) = 0 x 3 c,3 x 5 Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

18 Equivalence II : weighted language equivalence Given a word w A, we defined its weight in a state x of a weighted automaton (X, o, t ) inductively: L(x)(ɛ) = o(x) 2 L(x)(aw) = x X t(x)(a)(x ) L(w)(x ) a,2 b,2 x 2 x 4 a, x a, L(x )(aba n ) = 2 2 n L(x )(aca n ) = 3 3 n L(x )( ) = 0 x 3 c,3 x 5 Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

19 Next... Model weighted automata as coalgebras; Show how to derive the equivalences above canonically. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

20 A coalgebra primer Definition (Coalgebra) Given a functor G : C C on a category C, a G-coalgebra is an object X in C together with an arrow f : X GX. For many categories and functors, such pair (X, f ) represents a transition system, the type of which is determined by the functor G. Deterministic automata (X, o, t : X 2 X A ) Labeled transition systems (X, t : X (P ω X) A ) G(X) = 2 X A G(X) = P ω (X) A. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

21 A coalgebra primer Definition (Coalgebra) Given a functor G : C C on a category C, a G-coalgebra is an object X in C together with an arrow f : X GX. For many categories and functors, such pair (X, f ) represents a transition system, the type of which is determined by the functor G. Deterministic automata (X, o, t : X 2 X A ) Labeled transition systems (X, t : X (P ω X) A ) G(X) = 2 X A G(X) = P ω (X) A. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

22 A coalgebra primer Definition (Coalgebra) Given a functor G : C C on a category C, a G-coalgebra is an object X in C together with an arrow f : X GX. For many categories and functors, such pair (X, f ) represents a transition system, the type of which is determined by the functor G. Deterministic automata (X, o, t : X 2 X A ) Labeled transition systems (X, t : X (P ω X) A ) G(X) = 2 X A G(X) = P ω (X) A. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

23 A coalgebra primer (cont d) Definition (Coalgebra homomorphism) A G-homomorphism from a G-coalgebra (X, f ) to a G-coalgebra (Y, g) is an arrow h : X Y preserving the transition structure, i.e., such that the following diagram commutes. X h Y f g GX Gh GY Definition (Final coalgebra) A G-coalgebra (Ω, ω) is said to be final if for any G-coalgebra (X, f ) there exists a unique G-homomorphism [[ ]] G X : X Ω. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August 200 / 24

24 A coalgebra primer (cont d) Definition (Coalgebra homomorphism) A G-homomorphism from a G-coalgebra (X, f ) to a G-coalgebra (Y, g) is an arrow h : X Y preserving the transition structure, i.e., such that the following diagram commutes. X h Y f g GX Gh GY Definition (Final coalgebra) A G-coalgebra (Ω, ω) is said to be final if for any G-coalgebra (X, f ) there exists a unique G-homomorphism [[ ]] G X : X Ω. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August 200 / 24

25 A coalgebra primer (cont d) Final coalgebras can be viewed as the universe of all possible G-behaviours: the unique homomorphism [[ ]] G X : X Ω maps every state of a coalgebra X to a canonical representative of its behaviour. Definition (Behavioural equivalence) Two states x, x 2 X are G-behaviourally equivalent (x G x 2 ) iff [[x ]] G X = [[x 2]] G X. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

26 A coalgebra primer (cont d) Final coalgebras can be viewed as the universe of all possible G-behaviours: the unique homomorphism [[ ]] G X : X Ω maps every state of a coalgebra X to a canonical representative of its behaviour. Definition (Behavioural equivalence) Two states x, x 2 X are G-behaviourally equivalent (x G x 2 ) iff [[x ]] G X = [[x 2]] G X. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

27 Weighted automata as Set-coalgebras Goal: Show a functor W : Set Set such that W coincides with weighted bisimilarity. Definition (Field valuation Functor) Let K be a field. The field valuation functor K ω : Set Set is defined as follows. For each set X, K X ω is the set of functions from X to K with finite support. For each function h : X Y, K h ω : K X ω K Y ω is the function mapping each ϕ K X ω into ϕ h K Y ω defined, for all y Y, by ϕ h (y) = x h (y) ϕ(x ) Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

28 Weighted automata as Set-coalgebras Goal: Show a functor W : Set Set such that W coincides with weighted bisimilarity. Definition (Field valuation Functor) Let K be a field. The field valuation functor K ω : Set Set is defined as follows. For each set X, K X ω is the set of functions from X to K with finite support. For each function h : X Y, K h ω : K X ω K Y ω is the function mapping each ϕ K X ω into ϕ h K Y ω defined, for all y Y, by ϕ h (y) = x h (y) ϕ(x ) Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

29 Weighted automata as Set-coalgebras Weighted automata are coalgebras for the functor W = K (K ω ) A : Set Set. Every function f : X W(X) consists of a pair of functions o, t with o : X K and t : X (K X ω ) A. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

30 Canonical equivalence Proposition The functor W has a final coalgebra. Theorem Let (X, o, t ) be a weighted automaton and let x, x 2 be two states in X. Then, x and x 2 are weighted bisimilar iff x W x 2, i.e., [[x ]] W X = [[x 2]] W X. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

31 Canonical equivalence Proposition The functor W has a final coalgebra. Theorem Let (X, o, t ) be a weighted automaton and let x, x 2 be two states in X. Then, x and x 2 are weighted bisimilar iff x W x 2, i.e., [[x ]] W X = [[x 2]] W X. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

32 Weighted automata as Vect-coalgebras Goal: Coalgebraic characterization of weighted language equivalence. Introduce linear weighted automata (as coalgebras); Show the canonical equivalence is weighted language equivalence; Show a construction from weighted automata to linear weighted automata. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

33 Weighted automata as Vect-coalgebras Goal: Coalgebraic characterization of weighted language equivalence. Introduce linear weighted automata (as coalgebras); Show the canonical equivalence is weighted language equivalence; Show a construction from weighted automata to linear weighted automata. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

34 Linear weighted automata Definition (LWA) A linear weighted automaton (LWA, for short) with input alphabet A over the field K is a coalgebra for the functor L = K id A : Vect Vect. A LWA is a pair (V, o, t ), where: V is a vector space; 2 o : V K is a linear map associating to each state its output weight; 3 t : V V A is a linear map that for each input a A associates a next state (i.e., a vector) in V. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

35 Linear weighted automata Definition (LWA) A linear weighted automaton (LWA, for short) with input alphabet A over the field K is a coalgebra for the functor L = K id A : Vect Vect. A LWA is a pair (V, o, t ), where: V is a vector space; 2 o : V K is a linear map associating to each state its output weight; 3 t : V V A is a linear map that for each input a A associates a next state (i.e., a vector) in V. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

36 Equivalence The final L-coalgebra K A carries a vector space structure: the sum of two languages σ, σ 2 K A is the language σ + σ 2 defined for each word w A as σ + σ 2 (w) = σ (w) + σ 2 (w); Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

37 Equivalence The final L-coalgebra K A carries a vector space structure: the sum of two languages σ, σ 2 K A is the language σ + σ 2 defined for each word w A as σ + σ 2 (w) = σ (w) + σ 2 (w); the product of a language σ for a scalar k K is kσ defined as kσ(w) = k σ(w); Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

38 Equivalence The final L-coalgebra K A carries a vector space structure: the sum of two languages σ, σ 2 K A is the language σ + σ 2 defined for each word w A as σ + σ 2 (w) = σ (w) + σ 2 (w); the product of a language σ for a scalar k K is kσ defined as kσ(w) = k σ(w); the element 0 of K A is the language mapping each word into the 0 of K. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

39 Equivalence The final L-coalgebra K A carries a vector space structure: the sum of two languages σ, σ 2 K A is the language σ + σ 2 defined for each word w A as σ + σ 2 (w) = σ (w) + σ 2 (w); the product of a language σ for a scalar k K is kσ defined as kσ(w) = k σ(w); the element 0 of K A is the language mapping each word into the 0 of K. The empty function emp : K A K and the derivative function der : K A (K A ) A are defined for all σ K A, a A as emp(σ) = σ(ɛ) der(σ)(a) = σ a where σ a :A K denotes the a-derivative of σ that is defined for all w A as σ a (w) = σ(aw). Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

40 Equivalence The final L-coalgebra K A carries a vector space structure: the sum of two languages σ, σ 2 K A is the language σ + σ 2 defined for each word w A as σ + σ 2 (w) = σ (w) + σ 2 (w); the product of a language σ for a scalar k K is kσ defined as kσ(w) = k σ(w); the element 0 of K A is the language mapping each word into the 0 of K. The empty function emp : K A K and the derivative function der : K A (K A ) A are defined for all σ K A, a A as emp(σ) = σ(ɛ) der(σ)(a) = σ a where σ a :A K denotes the a-derivative of σ that is defined for all w A as σ a (w) = σ(aw). Proposition The maps emp : K A K and der : K A (K A ) A are linear. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

41 Equivalence (cont d) (K A, emp, der ) is an L-coalgebra. Theorem (Finality) From every L-coalgebra (V, o, t ) there exists a unique L-homomorphism into (K A, emp, der ). o,t V L(V ) [[ ]] L V K A L([[ ]] L V ) L(K A ) emp,der Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

42 From weighted automata to linear weighted automata v o,t S R (R S ω) A o (. ) = v i o(s i ) t ( v n R S o,t v. v n )(a)(s j ) = v i t(s i )(a)(s j ) Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

43 Example a,/ x a, a, x 2 x 3 O X = ( ) T Xa = ( ) T Xb = ( Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24 )

44 Example V = {k x + k 2 x 2 + k 3 x 3 k, k 2, k 3 K} o (v) = o (k x + k 2 x 2 + k 3 x 3 ) = k + k 2 + k 3 a,/ x a, a, x 2 x 3 O X = ( ) T Xa = ( ) T Xb = ( Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24 )

45 Example V = {k x + k 2 x 2 + k 3 x 3 k, k 2, k 3 K} a,/ x a, a, x 2 x 3 O X = ( ) T Xa = o (v) = o (k x + k 2 x 2 + k 3 x 3 ) = k + k 2 + k 3 k = O X k 2 k 3 ( ) T Xb = ( Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24 )

46 Example V = {k x + k 2 x 2 + k 3 x 3 k, k 2, k 3 K} a,/ x a, a, x 2 x 3 O X = ( ) T Xa = o (v) = o (k x + k 2 x 2 + k 3 x 3 ) = k + k 2 + k 3 = O X k k 2 k 3 t (v)(a) = k t X (x )(a) + k 2 t X (x 2 )(a) + k 3 t X (x 3 )(a) = k (x + x 2 + x 3 ) + k 2 3x 2 + k 3 3x 3 ( ) T Xb = ( Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24 )

47 Example V = {k x + k 2 x 2 + k 3 x 3 k, k 2, k 3 K} a,/ x a, a, x 2 x 3 O X = ( ) T Xa = o (v) = o (k x + k 2 x 2 + k 3 x 3 ) = k + k 2 + k 3 = O X k k 2 k 3 t (v)(a) = k t X (x )(a) + k 2 t X (x 2 )(a) + k 3 t X (x 3 )(a) = k (x + x 2 + x 3 ) + k 2 3x 2 + k 3 3x 3 k = T Xa k 2 k 3 ( ) T Xb = ( Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24 )

48 Example V = {k x + k 2 x 2 + k 3 x 3 k, k 2, k 3 K} a,/ x a, a, x 2 x 3 O X = ( ) T Xa = o (v) = o (k x + k 2 x 2 + k 3 x 3 ) = k + k 2 + k 3 = O X k k 2 k 3 t (v)(a) = k t X (x )(a) + k 2 t X (x 2 )(a) + k 3 t X (x 3 )(a) = k (x + x 2 + x 3 ) + k 2 3x 2 + k 3 3x 3 k = T Xa k 2 k 3 t X (v)(b) = k 3x + k 2 3x + k 3 3x = T Xa ( ) T Xb = ( Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24 ) k k 2 k 3

49 In a nutshell... e S R S [[ ]] R A o,t = o,t R (R S ω) A R (R A ) A Lemma Let (X, o, t ) be a weighted automaton and (K X ω, o, t ) be the corresponding linear weighted automaton. Then for all x X, L(x) = [[x]] L K X ω. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

50 In a nutshell... e S R S [[ ]] R A o,t = o,t R (R S ω) A R (R A ) A Lemma Let (X, o, t ) be a weighted automaton and (K X ω, o, t ) be the corresponding linear weighted automaton. Then for all x X, L(x) = [[x]] L K X ω. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

51 And now the connection with the previous talk... Solving linear systems of behavioural differential equations; Characterizing the final homomorphism by rational streams; Minimal automaton. Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

52 Conclusions Weighted automata characterized as coalgebras; Two canonical notions of equivalence; An algorithm for minimization. In the paper: A different algorithm for minimization; Stream-like calculus for automata with input alphabet A > Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24

53 Conclusions Weighted automata characterized as coalgebras; Two canonical notions of equivalence; An algorithm for minimization. In the paper: A different algorithm for minimization; Stream-like calculus for automata with input alphabet A > Alexandra Silva (CWI) Weighted automata coalgebraically Braunschweig, August / 24