Using Multiplicity Automata to Identify Transducer Relations from Membership and Equivalence Queries

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1 Using Multiplicity Automata to Identify Transducer Relations from Membership and Equivalence Queries Jose Oncina Dept. Lenguajes y Sistemas Informáticos - Universidad de Alicante oncina@dlsi.ua.es September 2008

2 Motivation Usually a transduction is viewed as a string to string function f ("My red car") = "mi coche rojo" A particular type of transductions is the Subsequential Transductions are based on a DFA We have algorithms to deal with this type of transductions The OSTIA algorithm: from input-output pairs The Vilar algorithm: from MAT Sometimes we have to cope with ambiguities f ("My red car") = "Mi coche ( rojo + colorado + encarnado)" 2/26 September 2008 Ambiguous Transducers Jose Oncina

3 Motivation Usually a transduction is viewed as a string to string function f ("My red car") = "mi coche rojo" A particular type of transductions is the Subsequential Transductions are based on a DFA We have algorithms to deal with this type of transductions The OSTIA algorithm: from input-output pairs The Vilar algorithm: from MAT Sometimes we have to cope with ambiguities f ("My red car") = "Mi coche ( rojo + colorado + encarnado)" 2/26 September 2008 Ambiguous Transducers Jose Oncina

4 Motivation Usually a transduction is viewed as a string to string function f ("My red car") = "mi coche rojo" A particular type of transductions is the Subsequential Transductions are based on a DFA We have algorithms to deal with this type of transductions The OSTIA algorithm: from input-output pairs The Vilar algorithm: from MAT Sometimes we have to cope with ambiguities f ("My red car") = "Mi coche ( rojo + colorado + encarnado)" 2/26 September 2008 Ambiguous Transducers Jose Oncina

5 Motivation Usually a transduction is viewed as a string to string function f ("My red car") = "mi coche rojo" A particular type of transductions is the Subsequential Transductions are based on a DFA We have algorithms to deal with this type of transductions The OSTIA algorithm: from input-output pairs The Vilar algorithm: from MAT Sometimes we have to cope with ambiguities f ("My red car") = "Mi coche ( rojo + colorado + encarnado)" 2/26 September 2008 Ambiguous Transducers Jose Oncina

6 Outline 1. Multiplicity Automata 2. Exact Learning 3. A bit of Algebra 4. Examples 3/26 September 2008 Ambiguous Transducers Jose Oncina

7 Outline 1. Multiplicity Automata 2. Exact Learning 3. A bit of Algebra 4. Examples 4/26 September 2008 Ambiguous Transducers Jose Oncina

8 Multiplicity Automata Multiplicity automata are essentially non deterministic stochastic automata with only one initial state and no restrictions to force the normalization Definition (Multiplicity Automata) A Multiplicity Automaton (MA) of size r, is: a set of Σ r r matrices {µ σ : σ Σ} with elements of the field K a row-vector λ = (λ 1,..., λ r ) K r a column-vector γ = (γ 1,..., γ r ) t K r The MA A defines a function f A : Σ K as: f A (x 1... x n) = λµ x1... µ xn γ 5/26 September 2008 Ambiguous Transducers Jose Oncina

9 Multiplicity Automata Example Let the MA A defined by: λ = `1 «1 1 0 µ a = µ 0 1 b = In this case: µ(x) = µ(x 1... x n) = µ x1... µ xn = «1 0 γ = 0 1 «1 α 0 1 «0 1 Where α is the number of times that a appears in x. Then f A (x) = λµ(x)γ = `1 0 1 ««α 0 = α start a, b 1 q 0 0 a 1 q 1 1 a, b 1 6/26 September 2008 Ambiguous Transducers Jose Oncina

10 The Hankel Matrix Let f : Σ K be a function. The Hankel matrix is an infinite matrix F each of its rows and columns are indexed by strings in Σ. The (x, y) entry of F (F x,y) contains the value f (xy). Example (a-count function) 0 1 ɛ a b aa ab ba bb... ɛ a b F = aa ab ba bb C A /26 September 2008 Ambiguous Transducers Jose Oncina

11 Carlyle and Paz theorem [1971] Theorem (Carlyle and Paz theorem, 1971) Let f : Σ K such that f 0 and let F be the corresponding Hankel matrix. Then, the size r of the smallest MA A such that f A f satisfies r = rank(f) (over the field) Example (a-count function) The rank is 2, F ɛ and F a are a basis. The other rows: F ɛ = (1, 0)(F ɛ, F a) t F b = (1, 0)(F ɛ, F a) t F ab = (0, 1)(F ɛ, F a) t F bb = (1, 0)(F ɛ, F a) t... F a = (0, 1)(F ɛ, F a) t F aa = ( 1, 2)(F ɛ, F a) t F ba = (0, 1)(F ɛ, F a) t 8/26 September 2008 Ambiguous Transducers Jose Oncina

12 Carlyle and Paz theorem [1971] Theorem (Carlyle and Paz theorem, 1971) Let f : Σ K such that f 0 and let F be the corresponding Hankel matrix. Then, the size r of the smallest MA A such that f A f satisfies r = rank(f) (over the field) Example (a-count function) The rank is 2, F ɛ and F a are a basis. The other rows: F ɛ = (1, 0)(F ɛ, F a) t F b = (1, 0)(F ɛ, F a) t F ab = (0, 1)(F ɛ, F a) t F bb = (1, 0)(F ɛ, F a) t... F a = (0, 1)(F ɛ, F a) t F aa = ( 1, 2)(F ɛ, F a) t F ba = (0, 1)(F ɛ, F a) t 8/26 September 2008 Ambiguous Transducers Jose Oncina

13 Carlyle and Paz theorem [1971] Note: Let x 1 = ɛ, x 2,..., x r a basis of the Hankel matrix. The Theorem states that we can build the MA as: λ = (1, 0,..., 0); γ = (f (x 1 ),..., f (x r )) for every σ, define the ith row of the matrix µ σ as the (unique) coefficients of the row F xi σ when expressed as a linear combination of F x1,..., F xr. That is: F xi σ = rx [µ σ] i,j F xj j=1 a 1 b 1 q 0 0 q 1 1 a 1 a 2, b 1 Example (a-count function) γ = `1 «0 Fɛ a µ a = = F a a «0 1 µ 1 2 b = Fɛ b F a b «= « /26 September 2008 Ambiguous Transducers Jose Oncina

14 Carlyle and Paz theorem [1971] Note: Let x 1 = ɛ, x 2,..., x r a basis of the Hankel matrix. The Theorem states that we can build the MA as: λ = (1, 0,..., 0); γ = (f (x 1 ),..., f (x r )) for every σ, define the ith row of the matrix µ σ as the (unique) coefficients of the row F xi σ when expressed as a linear combination of F x1,..., F xr. That is: F xi σ = rx [µ σ] i,j F xj j=1 a 1 b 1 q 0 0 q 1 1 a 1 a 2, b 1 Example (a-count function) γ = `1 «0 Fɛ a µ a = = F a a «0 1 µ 1 2 b = Fɛ b F a b «= « /26 September 2008 Ambiguous Transducers Jose Oncina

15 Carlyle and Paz theorem [1971] Note: Let x 1 = ɛ, x 2,..., x r a basis of the Hankel matrix. The Theorem states that we can build the MA as: λ = (1, 0,..., 0); γ = (f (x 1 ),..., f (x r )) for every σ, define the ith row of the matrix µ σ as the (unique) coefficients of the row F xi σ when expressed as a linear combination of F x1,..., F xr. That is: F xi σ = rx [µ σ] i,j F xj j=1 a 1 b 1 q 0 0 q 1 1 a 1 a 2, b 1 Example (a-count function) γ = `1 «0 Fɛ a µ a = = F a a «0 1 µ 1 2 b = Fɛ b F a b «= « /26 September 2008 Ambiguous Transducers Jose Oncina

16 Outline 1. Multiplicity Automata 2. Exact Learning 3. A bit of Algebra 4. Examples 10/26 September 2008 Ambiguous Transducers Jose Oncina

17 Queries Definition (Equivalence query) Let f be a target function. Given a hypothesis h, an equivalence query (EQ(h)) returns: YES if h f a counterexample otherwise Definition (Membership query) Let f be a target function. Given an assignment z a membership query (MQ(z)) returns f (z) 11/26 September 2008 Ambiguous Transducers Jose Oncina

18 Queries Definition (Equivalence query) Let f be a target function. Given a hypothesis h, an equivalence query (EQ(h)) returns: YES if h f a counterexample otherwise Definition (Membership query) Let f be a target function. Given an assignment z a membership query (MQ(z)) returns f (z) 11/26 September 2008 Ambiguous Transducers Jose Oncina

19 The exact learning model Definition (Angluin, 1988) Given a target function f, a learning algorithm should return a hypothesis function h equivalent to f. In order to do so, the learner can resort to membership and equivalence queries. We say that the learner learns a class of functions C, if, for every function f C, the learner outputs a hypothesis h that is equivalent to f and does so in time polynomial in the size of a shortest representation of f and the length of the longest counterexample. 12/26 September 2008 Ambiguous Transducers Jose Oncina

20 The MA Learning Algorithm [Beimel et all, 00] The idea is to work with a finite version of the Hankel matrix. Algorithm 1. initialize the matrix to null 2. build a MA using the matrix and making membership queries if necessary 3. ask an equivalence query 4. if the answer is YES then STOP 5. use the counterexample to add new rows an columns in the matrix 6. use membership queries to fill the holes in the matrix 7. Go to step 2 13/26 September 2008 Ambiguous Transducers Jose Oncina

21 Outline 1. Multiplicity Automata 2. Exact Learning 3. A bit of Algebra 4. Examples 14/26 September 2008 Ambiguous Transducers Jose Oncina

22 What is a Field? Definition (Field) (K, +, ) is a field if: Closure of K under + and a, b K, both a + b and a b belong to K Both + and are associative a, b, c K, a + (b + c) = (a + b) + c and a (b c) = (a b) c. Both + and are commutative a, b K, a + b = b + a and a b = b a. The operation is distributive over the operation + a, b, c K, a (b + c) = (a b) + (a c). Existence of an additive identity 0 K: a K, a + 0 = a. Existence of a multiplicative identity 1 K, 1 0: a K, a 1 = a. Existence of additive inverses a K, a K: a + ( a) = 0. Existence of multiplicative inverses a K, a 0, a 1 K: a a 1 = 1. 15/26 September 2008 Ambiguous Transducers Jose Oncina

23 Working with transducers Idea: Use the learning algorithm using: concatenation as the operator the inclusion in a (multi)set as the + operator We are going to extend this operations in order to have a Field and be able to identify a superclass of the ambiguous rational transducers 16/26 September 2008 Ambiguous Transducers Jose Oncina

24 Concatenation extension The concatenation is going to play the role of the multiplication. For each a Σ let we include in Σ its inverse (a 1 ). Example aabb aba 1 b aaa 1 b ( ab) a 1 b 1 Extended concatenation properties: Closure Associative Non Commutative (not good) Existence of a multiplicative identity (ɛ) Existence of multiplicative inverses 17/26 September 2008 Ambiguous Transducers Jose Oncina

25 Concatenation extension The concatenation is going to play the role of the multiplication. For each a Σ let we include in Σ its inverse (a 1 ). Example aabb aba 1 b aaa 1 b ( ab) a 1 b 1 Extended concatenation properties: Closure Associative Non Commutative (not good) Existence of a multiplicative identity (ɛ) Existence of multiplicative inverses 17/26 September 2008 Ambiguous Transducers Jose Oncina

26 Concatenation extension The concatenation is going to play the role of the multiplication. For each a Σ let we include in Σ its inverse (a 1 ). Example aabb aba 1 b aaa 1 b ( ab) a 1 b 1 Extended concatenation properties: Closure Associative Non Commutative (not good) Existence of a multiplicative identity (ɛ) Existence of multiplicative inverses 17/26 September 2008 Ambiguous Transducers Jose Oncina

27 Multiset inclusion extension the multiset inclusion is going to play the role the addition For each multiset x let we define its inverse ( x). Example aaa + bbb aaa aaa ( ) a + a a ( a) aaa Multiset inclusion properties: Closure: the inclusion of a multiset into another is a multiset. Associative: (x + y) + z = x + (y + z) Commutative: x + y = y + x Existence of an additive identity: x + = x Existence of additive inverses: x + ( x) = 18/26 September 2008 Ambiguous Transducers Jose Oncina

28 Multiset inclusion extension the multiset inclusion is going to play the role the addition For each multiset x let we define its inverse ( x). Example aaa + bbb aaa aaa ( ) a + a a ( a) aaa Multiset inclusion properties: Closure: the inclusion of a multiset into another is a multiset. Associative: (x + y) + z = x + (y + z) Commutative: x + y = y + x Existence of an additive identity: x + = x Existence of additive inverses: x + ( x) = 18/26 September 2008 Ambiguous Transducers Jose Oncina

29 Multiset inclusion extension the multiset inclusion is going to play the role the addition For each multiset x let we define its inverse ( x). Example aaa + bbb aaa aaa ( ) a + a a ( a) aaa Multiset inclusion properties: Closure: the inclusion of a multiset into another is a multiset. Associative: (x + y) + z = x + (y + z) Commutative: x + y = y + x Existence of an additive identity: x + = x Existence of additive inverses: x + ( x) = 18/26 September 2008 Ambiguous Transducers Jose Oncina

30 Combining concatenation and inclusion Properties: The concatenation is distributive over the inclusion: x (y + z) = x y + x z We have a Field with a non commutative multiplication. This is known as a Divisive Ring But the Carlyle an Paz theorem does not use the commutativity in the multiplication! Their theorem is also true for Divisive Rings! Then the inference algorithm can be used exactly as it is just substituting: addition by the (extended) inclusion multiplication by the (extended) concatenation 19/26 September 2008 Ambiguous Transducers Jose Oncina

31 Combining concatenation and inclusion Properties: The concatenation is distributive over the inclusion: x (y + z) = x y + x z We have a Field with a non commutative multiplication. This is known as a Divisive Ring But the Carlyle an Paz theorem does not use the commutativity in the multiplication! Their theorem is also true for Divisive Rings! Then the inference algorithm can be used exactly as it is just substituting: addition by the (extended) inclusion multiplication by the (extended) concatenation 19/26 September 2008 Ambiguous Transducers Jose Oncina

32 Outline 1. Multiplicity Automata 2. Exact Learning 3. A bit of Algebra 4. Examples 20/26 September 2008 Ambiguous Transducers Jose Oncina

33 Non ambiguous, non subsequential example f (x n ) = ( a n b n if n is odd if n is even Text books proposal: x a x a q 1 ɛ x a q 3 start q 0 ɛ x b q 2 x b x b q 4 ɛ 21/26 September 2008 Ambiguous Transducers Jose Oncina

34 The Result Applying the algorithm we obtain: x b 4 (a 4 b 4 )(a 2 b 2 ) 1 b 2 start q 1 ɛ x ɛ q 1 x ɛ q 2 x ɛ a bb q 3 aaa It can be shown that for any input we obtain a plain string Open questions: x (a 4 b 4 )(a 2 b 2 ) 1 Does there exist a general method to simplify and compare string expressions? Does there exist a method to know if a multiplicity automaton produces only plain strings? Does there exist a method to remove complex expressions in arcs and states, possibly adding more states? 22/26 September 2008 Ambiguous Transducers Jose Oncina

35 The Result Applying the algorithm we obtain: x b 4 (a 4 b 4 )(a 2 b 2 ) 1 b 2 start q 1 ɛ x ɛ q 1 x ɛ q 2 x ɛ a bb q 3 aaa It can be shown that for any input we obtain a plain string Open questions: x (a 4 b 4 )(a 2 b 2 ) 1 Does there exist a general method to simplify and compare string expressions? Does there exist a method to know if a multiplicity automaton produces only plain strings? Does there exist a method to remove complex expressions in arcs and states, possibly adding more states? 22/26 September 2008 Ambiguous Transducers Jose Oncina

36 An ambiguous example f (x n ) = nx a i The good one x ɛ + a i=0 A non equivalent one x ɛ x ɛ + a start q 0 ɛ x a q 1 ɛ + a start The second transducer does not preserve the multiplicity of the strings Note that in the ambiguous case, the membership query should return all the possible transductions. Open questions: Can we still be able to learn if only information about just one transduction is provided in each query? Does any learnable function remain learnable if the multiplicity is not taken into account? 23/26 September 2008 Ambiguous Transducers Jose Oncina q 0 ɛ

37 An ambiguous example f (x n ) = nx a i The good one x ɛ + a i=0 A non equivalent one x ɛ x ɛ + a start q 0 ɛ x a q 1 ɛ + a start The second transducer does not preserve the multiplicity of the strings Note that in the ambiguous case, the membership query should return all the possible transductions. Open questions: Can we still be able to learn if only information about just one transduction is provided in each query? Does any learnable function remain learnable if the multiplicity is not taken into account? 23/26 September 2008 Ambiguous Transducers Jose Oncina q 0 ɛ

38 Another ambiguous example Applying the algorithm: f (x n ) = 2nX a i i=0 x ɛ + a 2 start q 0 ɛ x ɛ q 1 x ɛ q 2 P 2 i=0 ai P 4 i=0 ai x a 2 24/26 September 2008 Ambiguous Transducers Jose Oncina

39 Conclusions We have proposed a learning algorithm that: Can identify any rational fuction with output built up with no empty-transitions extended concatenations extended multiset inclusions It uses membership and equivalence queries As a special case, it identifies any ambiguous rational transducer (with finite output) It works in polynomial time (perhaps there is a problem in the parsing) 25/26 September 2008 Ambiguous Transducers Jose Oncina

40 Any Questions? 26/26 September 2008 Ambiguous Transducers Jose Oncina

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