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Characterization of state merging strategies which ensure identification in the limit from complete data (II) Cristina Bibire

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State of the art A new approach Motivation Results Further Research Bibliography

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State of the Art 1967 - Gold was the first one to formulate the process of learning formal languages. 1973 - Trakhtenbrot and Barzdin described a polynomial time algorithm (TB algorithm) for constructing the smallest DFA consistent with a completely labelled training set (a set that contains all the words up to a certain length). 1978 - Gold rediscovers the TB algorithm and applies it to the discipline of grammatical inference (uniformly complete samples are not required). He also specifies the way to obtain indistinguishable states using the so called state characterization matrices. 1992 - Oncina and Garcia propose the RPNI (Regular Positive and Negative Inference) algorithm. 1992 - Lang describes TB algorithm and generalize it to produce a (not necessarily minimum) DFA consistent with a sparsely labelled tree. The algorithm (Traxbar) can deal with incomplete data sets as well as complete data sets.

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State of the Art 1997 - Lang and Pearlmutter organized the Abbadingo One contest. The competition presented the challenge of predicting, with 99% accuracy, the labels which an unseen FSA would assign to test data given training data consisting of positive and negative examples. Price was able to win the Abbadingo One Learning Competition by using an evidence-driven state merging (EDSM) algorithm. Essentially, Price realized that an effective way of choosing which pair of nodes to merge next within the APTA would simply involve selecting the pair of nodes whose sub-tree share the most similar labels. As a post-competition work, Lang proposed W-EDSM. In order to improve the running time of the EDSM algorithm, we only consider merging nodes that lie within a fixed sized window from the root node of the APTA An alternative windowing method (Blue-fringe algorithm) is also described by Lang, Pearlmutter and Price. It uses a red and blue colouring scheme to provide a simple but effective way of choosing the pool of merge candidates at each merge level in the search.

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State of the Art TB algorithm Gold’s algorithm RPNI Traxbar EDSM W-EDSM Blue-fringe

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State of the Art L=Any string without an odd number of consecutive 0’s after an odd number of consecutive 1’s 0 0 42 1 3 0 0, 1 11 1 1 0 0

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TB algorithm 00000 0000 0 0 λ 000 1 0001 0010 00110100 0101 01100111 1000 1001 10101011 1100 1110 1111 001 010 011100 101 1101110001 10 1101 0 0 0 0 0 000 0 0 0 00 0 1 1 1 1 1 1 1 1 1 1 1 1 00001 1111011111 1 0 1 (0,λ) 1101

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TB algorithm λ 1 1000 1001 10101011 1100 11011110 1111 100 101 110111 10 111 0 0 0 0 0 0 01 1 1 1 1 1 0 1 0 111101111111100 11101 1100011001 1011010111101001010110010 1001110000 1000111010 11011 0 0 0 0 0 00 1 1 1 1 1 11 1 (11,λ) 1101

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TB algorithm λ 1 1 0 1 0 1000 1001 10101011 100 10110 0 0 0 1 1 0 1 1011010111101001010110010 1001110000 10001 0 0 0 1 1 1 1 (1000,10) λ 1 1 0 1 0 1001 10101011 100 10110 0 0 1 1 1 1011010111101001010110010 10011 0 0 0 1 1 1 0 (1001,1)

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TB algorithm (1001,1) λ 1 1 0 1 0 100 10 0 1 1 10110101111010010101 10101011 101 0 1 0 0 1 1 0 (1010,101) (1011,101) λ 1 1 0 1 0 100 10 0 1 1 101 0,1 0

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Gold’s algorithm 100010011010 100 101 110 1011 1 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ}{λ}{λ}{λ}{0,1} (0,λ) dist? No λ 0 1 0 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ}{λ}{λ}{λ}{0,1} {(0,λ)} (0,λ) dist? No λ 0 1 0 (1,λ) dist? Yes add 1 to S 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {0,1}{0,10,11} {(0,λ)} (0,λ) dist? No (1,λ) dist? Yes (10,λ) dist? (10,1) dist? Yes add 10 to S add 1 to S λ 0 1 0 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {0,1}{0,10,11}{0,11,100,101} {(0,λ)} (0,λ) dist? No (1,λ) dist? Yes (10,λ) dist? (10,1) dist? Yes add 10 to S add 1 to S (11,λ) dist?No λ 0 1 0 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {0,1}{0,10,11}{0,11,100,101} {(0,λ)} {(0,λ),(11,λ)} (0,λ) dist? No (1,λ) dist? Yes (10,λ) dist? (10,1) dist? Yes add 10 to S add 1 to S (11,λ) dist?No (100,λ) dist? (100,1) dist? (100,10) dist? Yes add 100 to S λ 0 1 0 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 λ 0 1 0 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001} {(0,λ)} {(0,λ),(11,λ)} (101,λ) dist? Yes add 101 to S (101,1) dist? (101,10) dist? (101,100) dist? 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 λ 0 1 0 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {λ,1,10,100,101} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001} {0,11,1000,1001,1010, 1011} {(0,λ)} {(0,λ),(11,λ)} (1000,10) dist? No 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 λ 0 1 0 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {λ,1,10,100,101} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001} {0,11,1000,1001,1010, 1011} {(0,λ)} {(0,λ),(11,λ)} {(0,λ),(11,λ),(10 00,10)} (1000,10) dist? (1001,1) dist? No 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 λ 0 1 0 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {λ,1,10,100,101} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001} {0,11,1000,1001,1010, 1011} {(0,λ)} {(0,λ),(11,λ)} {(0,λ),(11,λ),(10 00,10),(1001,1)} (1000,10) dist? (1001,1) dist? No (1010,101) dist? No 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 λ 0 1 0 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {λ,1,10,100,101} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001} {0,11,1000,1001,1010, 1011} {(0,λ)} {(0,λ),(11,λ)} {(0,λ),(11,λ),(10 00,10),(1001,1),( 1010,101)} (1000,10) dist? (1001,1) dist? No (1010,101) dist? No (1011,101) dist?No 1

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Gold’s algorithm 100010011010 100 101 110 1011 1 λ 0 1 0 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {λ,1,10,100,101} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001} {0,11,1000,1001,1010, 1011} {(0,λ)} {(0,λ),(11,λ)} {(0,λ),(11,λ),(10 00,10),(1001,1),( 1010,101),(1011, 101)} (1000,10) dist? (1001,1) dist? No (1010,101) dist? No (1011,101) dist?No 1

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Gold’s algorithm S SΣ-S list {λ} {λ,1} {λ,1,10} {λ,1,10,100} {λ,1,10,100,101} {0,1}{0,10,11}{0,11,100,101}{0,11,101,1000,1001}{0,11,1000,1001,1010,1011} {(0,λ)} {(0,λ),(11,λ)} {(0,λ),(11,λ),(1000,10),(1001,1), (1010,101),(1011,101)} λ 1 1 0 1 0 10010 0 1 1 101 0,1 0 10101

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RPNI 10001001 100110 1011 1 0 0 0 0 0 1 10000 λ 0 1 0 1 λ 1 1 0 1 0 100 10 0 1 0 K Fr

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Traxbar A variation of the Trakhtenbrot and Barzdin algorithm was implemented by Lang. The modifications made to the algorithm were needed to maintain consistency with incomplete training sets. For instance, unlabeled nodes and missing transitions in the APTA needed to be considered. The simple extensions added to the Trakhtenbrot and Barzdin algorithm are as follows. If node q 2 is to be merged with node q 1 then: labels of labelled nodes in the sub-tree rooted at q 2 must be copied over their respective unlabeled nodes in the sub-tree rooted at q 1 ; transitions in any of the nodes in the sub-tree rooted at q 2 that do not exist in their respective node in the sub-tree rooted at q 1 must be copied in. As a result of these changes, the Traxbar algorithm will produce a (not necessarily minimum size) DFA that is consistent with the training set.

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Traxbar 10011010 100 101 110 1011 1 00 0 0 0 0 1 1 1 1001010100 1011 10110 0 0 λ 0 1 0 1 1000 (0,λ) (11,λ) 0 1001 1010 100101 110 10 1 0 0 0 0 0 1 1 1 10010 10100 1011 10110 0 0 1 1 1000 11 λ,0 0 1000 1001 1010 100101 10 00 0 0 1 1 10010 10100 1011 10110 0 0 1 λ,0,11, 110 0 1 0 11

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Traxbar 1000 1001 1010 100101 10 00 0 0 1 1 10010 10100 1011 10110 0 0 1 λ,0,11, 110 0 1 0 11 (1000,10)(1001,1) 1010 101 10,10 00 0 1 1 10100 1011 10110 0 0 1001 100 0 10010 1 λ,0,11, 110 0 1 0 11 0 0 1010 101 10,10 00,10 010 0 1 1 10100 1011 10110 0 0 100 λ,0,11, 110 0 1, 1001 0 11 0 0 1 (1010,101) (1011,101)

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Traxbar 101,1010, 10100,10 11,10110 10,100 0,1001 0 1 100 λ,0,11, 110 0 1, 1001 0 11 0 0 1 0,1

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EDSM The general idea of the EDSM approach is to avoid bad merges by selecting the pair of nodes within the APTA which has the highest score. It is expected that the scoring will indicate the correctness of each merge, since on average, a merge that survives more label comparisons is more likely to be correct. A post-competition version of the EDSM algorithm as described by Lang, Pearlmutter and Price is included below. Evaluate all possible pairings of nodes within the APTA. Merge the pair of nodes which has the highest calculated score. Repeat the steps above until no other nodes within the APTA can be merged. The score is calculated by assigning one point for each overlapping label node within the sub-tree rooted at the nodes considered for merging

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Windowed-EDSM To improve the running time of the EDSM algorithm, it is suggested that we only consider merging nodes that lie within a fixed sized window from the root node of the APTA. In breadth-first order, create a window of nodes starting from the root of the APTA. Evaluate all possible merge pairs within the window. Merge the pair of nodes which has the highest number of matching labels within its sub-trees. If the merge reduces the size of the window, in breadth-first order, include the number of nodes needed to regain the fixed size of the window. If no merge is possible within the given window, increase the size of the window by a factor of 2. Terminate when no merges are possible. The recommended size of the window is twice the number of states in the target DFA.

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Blue-Fringe An alternative windowing method to that used by the W-EDSM algorithm is also described by Lang, Pearlmutter and Price. It uses a red and blue colouring scheme to provide a simple but effective way of choosing the pool of merge candidates at each merge level in the search. The Blue-fringe windowing method aids in the implementation of the algorithm and improves on its running time. Colour the root of the APTA red. Colour the non-red children of each red node blue. Evaluate all possible pairings of red and blue nodes. Promote the first blue node which is distinguishable from each red node. Otherwise, merge the pair of nodes which have the highest number of matching labels within their sub-trees. Terminate when there are no blue nodes to promote and no possible merges to perform.

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A new approach Goal: to design an algorithm capable of incrementally learn new information without forgetting previously acquired knowledge and without requiring access to the original set of samples. We denote by: = the set of all automata having the alphabet Let be any of the algorithms defined so far (TB, Gold, RPNI, Traxbar, EDSM, W-EDSM, Blue-fringe)

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A new approach 0 1 1 0 1 0 3 2 0 0 0 1 0 1

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0 1 1 0 1 0 3 2 0 0 0 1 0 1

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0 1 1 0 1 0 3 2 0 0 0 0 1

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0 1 1 0 1 0 3 2 0 0 0 1

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0 1 1 0 1 0 3 2 0 0 0 1 1 0 1 0 3 2 0 0 1 1 (q,1) q

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A new approach 0 1 1 0 1 0 3 2 0 0 1 0 1 1 0 1 0 3 2 0 0 1

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Motivation Q: Why do we need this algorithm? 1.In many practical applications, acquisition of a representative training data is expensive and time consuming. 2.It might be the case that a new sample is introduced after several days, months or even years 3.We might have lost the initial database

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Results Lemma 1 It is not always true that: Lemma 2 Let such that. It is not always true that:Lemma 3 Let such that. It is not always true that:

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Results Sketch of the Proof (for Lemma1 and Lemma2): Sketch of the Proof (for Lemma3):

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Further Research o To determine the complexity of the algorithm and to test it on large/sparse data o To determine how much time and resources we save using this algorithm instead of the classical ones o To design an algorithm to deal with new introduced negative samples o To find the answer to the question: when is the automaton created with this method weakly equivalent with the one obtained with the entire sample? o To improve the software in order to be able to deal with new samples

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Bibliography Colin de la Higuera, José Oncina, Enrique Vidal. “Identification of DFA: Data-Dependent versus Data-Independent Algorithms” Rajesh Parekh, Vasant Honavar. “Learning DFA from Simple Examples” Michael J. Kearns, Umesh V. Vazirani “An Introduction to Computational Theory” J. Oncina, P. Garcia. “A polynomial algorithm to infer regular languages” Dana Angluin. “Inference of Reversible Languages” P. Garcia, A. Cano, J. Ruiz. “A comparative study of two algorithms for automata identification” P. Garcia, A. Cano, J. Ruiz. “Inferring subclasses of regular languages faster using RPNI and forbidden configurations” K.J. Lang, B.A. Pearlmutter, R.A. Price. “Results of the Abbadingo One DFA Learning Competition and a New Evidence-Driven State Merging Algorithm”

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Bibliography Takashi Yokomori. “Grammatical Inference and Learning”. M. Sebban, J.C. Janodet, E. Tantini. ”BLUE : a Blue-Fringe Procedure for Learning DFA with Noisy Data” Kevin J. Lang. “Random DFA’s can be Approximately Learned from Sparse Uniform Examples” P. Dupont, L. Miclet, E. Vidal. “What is the search space of the regular inference?” Sara Porat, Jerome A. Feldman. “Learning Automata from Ordered Examples” Orlando Cicchello, Stefan C. Kremer. “Inducing Grammars from Sparse Data Sets: A Survey of Algorithms and results”

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