Learning Learning Re Regular gular Languages Languages over er - PowerPoint PPT Presentation

Irini-Eleftheria Irini-Eleftheria Mens Mens V ERIMAG , University of Grenoble-Alpes Learning Learning Re Regular gular Languages Languages over er Lar Large ge Alphabets Alphabets 10 October 2017 Jury Members Oded Maler Directeur de th` ese Laurent Fribourg Examinateur Dana Angluin Rapporteur Eric Gaussier Examinateur Peter Habermehl Rapporteur Frits Vaandrager Examinateur

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Model Black Box Learning Language Identification System Identification Inductive Inference 1 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion A Short Prehistory and History of Automaton Learning 1956 Edward F Moore. Gedanken-experiments on sequential machines. Defines the problem as a black box model inference. 1967 E. Mark Gold. Language identification in the limit. 1972 E. Mark Gold. System identification via state characterization. Learning finite automata is possible in finite time. He first uses the basic idea that underlies table-based methods. 1978 E. Mark Gold. Complexity of automaton identification from given data. Finding the minimal automaton compatible with a given sample is NP-hard. 1987 Dana Angluin. Learning regular sets from queries and counter-examples. The L ∗ active learning algorithm with membership and equiva- lence queries. Polynomial in the automaton size. 1993 Ronald L. Rivest and Robert E. Schapire. Inference of finite au- tomata using homing sequences. An improved version of the L ∗ algorithm using the breakpoint method to treat counter-examples. 2 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Machine Learning Model f : X → Y f ( x ) = y , ∀ ( x , y ) ∈ M a small sample Learn M = { ( x , y ) : x ∈ X , y ∈ Y } predict or identify f ( x ) for all x ∈ X Learning Regular Languages Model over large or infinite alphabets f is a language • Σ an alphabet L ⊆ Σ ∗ Learn • X = Σ ∗ set of words The model is an • Y = { + , −} symbolic automaton 3 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Types of Learning Off-line vs Online The sample M is known before The sample M is updated the learning procedure starts. during learning. Passive vs Active The sample M is given. The sample M is chosen by the learning algorithm. Learning using Queries The learning algorithm can access queries e.g., membership queries, equivalence queries, etc. ? L ( H ) ≡ L w ∈ L w ∈ Σ ∗ Hypothesis H Yes / No True / MQ ( · ) EQ ( · ) Counter-example (cex) 4 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Outline Preliminaries Regular Languages and Automata The L ∗ Algorithmic Scheme Large Alphabets Motivation Symbolic Representation of Transitions - Symbolic Automata Learning Symbolic Automata Why L ∗ cannot be applied? Our Solution The Algorithm Equivalence Queries and Counter-Examples Adaptation to the Boolean Alphabet Experimental Results Conclusion 5 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Outline Preliminaries Regular Languages and Automata The L ∗ Algorithmic Scheme Large Alphabets Motivation Symbolic Representation of Transitions - Symbolic Automata Learning Symbolic Automata Why L ∗ cannot be applied? Our Solution The Algorithm Equivalence Queries and Counter-Examples Adaptation to the Boolean Alphabet Experimental Results Conclusion 5 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Regular Languages and Automata suffixes a b ε . . . a b aa ab ba bb aaa a b ε − − − − + − − − . . . a − − + − − + − − . . . a b Σ = { a , b } − − − − + − − − . . . b − − − − + − − − . . . aa L ⊆ Σ ∗ is a language + + − + − − + + . . . ab prefixes ba − − + − − + − − . . . • Σ is an alphabet bb − − − − + − − − . . . • w = a 1 · · · a n is a word . . . . . . . . ... . . . . . . . . . . . . . . . . . . . • Σ ∗ is the set of all words + + − + − − + + . . . aba abb − − + − − + − − . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . 6 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Regular Languages and Automata suffixes a b ε . . . a b aa ab ba bb aaa a b ε − − − − + − − − . . . a − − + − − + − − . . . a b Σ = { a , b } − − − − + − − − . . . b − − − − + − − − . . . aa L ⊆ Σ ∗ is a language + + − + − − + + . . . ab prefixes ba − − + − − + − − . . . bb − − − − + − − − . . . Equivalence relation . . . . . . . . ... . . . . . . . . . . . . . . . . . . u ∼ L v iff u · w ∈ L ⇔ v · w ∈ L . + + − + − − + + . . . aba Nerode’s Theorem abb − − + − − + − − . . . L is a regular language iff ∼ L has . . . . . . . . ... . . . . . . . . . . . . . . . . . . . finitely many equivalence classes. ε ∼ b ∼ aa a ∼ ba ∼ abb ab ∼ aba Q = Σ ∗ / ∼ (states in the minimal representation of L . 6 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Regular Languages and Automata A sufficient sample that characterizes the language ε a b ε a b a b ε − − − a b a b − − + a aa ab ba bb + + − ab a b a a a b b b − − − b aba abb aa − − − a a b b aba + + − abb − − + 7 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Regular Languages and Automata A sufficient sample that characterizes the language ε a b b E ε a b a b ε − − − a a b − − + a S aa ab + + − ab a a b b b − − − aba abb aa − − − R + + − aba − − + abb S prefixes (states) boundary ( R = S · Σ \ S ) R E suffixes (distinguishing strings) f : S ∪ R × E → { + , −} classif. function f s : E → { + , −} residual functions 7 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion Regular Languages and Automata A sufficient sample that characterizes the language a ε b a E ε a b a b b ε − − − b − − + a S aa ab + + − ab b − − − a aba abb aa − − − R + + − aba − − + abb A L = (Σ , Q , q 0 , δ, F ) S prefixes (states) - Q = S boundary ( R = S · Σ \ S ) R - q 0 = [ ε ] E suffixes (distinguishing strings) - δ ([ u ] , a ) = [ u · a ] f : S ∪ R × E → { + , −} classif. function - F = { [ u ] : ( u · ε ) ∈ L } f s : E → { + , −} residual functions The minimal automaton for L 7 / 31

2 3, 4 0 q4 1, 2, 3, 4 start q0 2, 3, 4 0 1 q6 0, 1 0, 2, 3, 4 q1 3, 4 q3 0, 1 1 q5 2 q2 0, 1, 2, 3, 4 0, 1, 2, 3, 4 Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion The L ∗ Algorithmic Scheme ∗ Active learning using queries ε Learner Teacher a b L ⊆ Σ ∗ Initialize a b ? a b ∈ L w MQ ( · ) aa ab Fill in Table a b + / − aba abb EQ ( · ) ∗ D. Angluin. Learning regular sets from queries and counter-examples , 1987. 8 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion The L ∗ Algorithmic Scheme ∗ Active learning using queries a ε Learner Teacher b a L ⊆ Σ ∗ Initialize a b b ? b ∈ L w MQ ( · ) aa ab Fill in Table + / − a aba abb Make ? L ( H ) = L Hypothesis H EQ ( · ) 2 3, 4 0 q4 1, 2, 3, 4 start q0 2, 3, 4 0 1 q6 0, 1 0, 2, 3, 4 q1 3, 4 q3 0, 1 1 q5 2 q2 0, 1, 2, 3, 4 0, 1, 2, 3, 4 ∗ D. Angluin. Learning regular sets from queries and counter-examples , 1987. 8 / 31

Introduction Preliminaries Large Alphabets Learning Symbolic Automata Counter-examples Booleans Experimental Results Conclusion The L ∗ Algorithmic Scheme ∗ Active learning using queries a ε Learner Teacher b a L ⊆ Σ ∗ Initialize a b ? b ∈ L w MQ ( · ) aa ab Fill in Table b + / − a aba abb a a Make ? L ( H ) = L Hypothesis H EQ ( · ) 2 3, 4 0 q4 1, 2, 3, 4 start q0 2, 3, 4 0 1 q6 0, 1 0, 2, 3, 4 q1 3, 4 q3 0, 1 1 q5 2 q2 0, 1, 2, 3, 4 0, 1, 2, 3, 4 counter-example Treat cex True (cex) Return H ∗ D. Angluin. Learning regular sets from queries and counter-examples , 1987. 8 / 31