Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Laguages over Large Alphabets - Traditionally automata theory is flat, based on small alphabets, e.g. { a , b } - In verification, for example, we have sequences over a huge state-space like B n for very large n - Or we want to have languages over numbers or vectors - We use symbolic automata with a modest number of states - We do not want to enumerate all transitions but represent them symbolically using predicates on the alphabet - We will use inequalities (intervals) for numbers or Boolean functions for Boolean vectors V ERIMAG O Maler - IE Mens 10 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Symbolic Automata a 11 [ 0 , 10 ) [ 0 , 50 ) , [ 70 , 100 ) a 31 , a 33 q 1 q 3 a 13 A = (Σ , Σ , ψ, Q , δ, q 0 , F ) [ 10 , 30 ) a 01 - Q finite set of states, [ 0 , 50 ) [ 30 , 100 ) [ 50 , 70 ) - q 0 initial state, a 12 a 32 q 0 - F accepting states, [ 50 , 100 ) a 02 - Σ large concrete alphabet, [ 20 , 100 ) - δ ⊆ Q × 2 Σ × Q a 21 a 22 [ 0 , 20 ) q 2 q 4 a 41 - Σ finite alphabet (symbols) Σ - ψ q : Σ → Σ q , q ∈ Q [ [ a 01 ] ] = [ 0 , 50 ) Σ = [ 0 , 100 ) ⊆ R A is complete and deterministic [ [ a ] ] = { a ∈ Σ | ψ ( a ) = a } if ∀ q ∈ Q { [ [ a ] ] | a ∈ Σ q } forms a partition of Σ w = 20 · 40 · 60 + V ERIMAG O Maler - IE Mens 11 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning using Evidences and Representatives Let Σ be a subset of R - To characterize continuations of u , ask queries about u · a for a finite sample of Σ (evidence) u - Evidence can be a fixed set, random, or a result of binary search ? ? ? ? - Form evidence compatible partitions - All evidences within a partition block a 1 a 2 ... a 3 a k behave the same | Σ - Estimate boundaries using split , binary [ [ a ] ] µ ( a ) ˆ µ ( b ) ˆ [ [ b ] ] p search ,. . . evidences - Associate a symbol to each partition block µ ( u ) · a i | a i ∈ [ µ ( u · a ) = { ˆ [ a ] ] } - Choose one evidence as the representative for representatives each new symbol µ ( u · a ) = ˆ ˆ µ ( u ) · ˆ µ ( a ) V ERIMAG O Maler - IE Mens 12 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Symbolic Observation Table - T = (Σ , Σ , S , R , ψ, E , f , µ, ˆ µ ) - Prefixes are symbolic words E - Symbols represent sets of letters (“fat” edges) ε a - Suffixes are concrete words (distinguish states) ε − + S a 1 + − - Fill in the table according to the representatives − − a 2 a 1 a 3 + − ε a 1 a 4 − + a 1 a 2 R a 1 a 5 − − a 1 a 2 a 2 a 6 + − a 3 a 5 a 6 a 4 a 1 a 3 a 1 a 4 a 1 a 5 a 2 a 6 V ERIMAG O Maler - IE Mens 13 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Symbolic Observation Table - T = (Σ , Σ , S , R , ψ, E , f , µ, ˆ µ ) E - ψ = { ψ s } s ∈ S , ψ s : Σ → Σ s semantics ε a - [ [ a ] ] = { a ∈ Σ | ψ ( a ) = a } ε − + - µ : Σ → 2 Σ evidences S a 1 + − - µ ( ε ) = { ε } , µ ( s · a ) = ˆ µ ( s ) · µ ( a ) − − a 2 - ˆ µ : Σ → Σ representative a 1 a 3 + − - ˆ µ ( ε ) = ε, ˆ µ ( s · a ) = ˆ µ ( s ) · ˆ µ ( a ) a 1 a 4 − + R - f : ( S ∪ R ) · E → {− , + } classif. function a 1 a 5 − − - f ( s · e ) = f (ˆ µ ( s ) · e ) , f s ( e ) = f ( s · e ) a 2 a 6 + − V ERIMAG O Maler - IE Mens 13 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Proposition If w is a counter-example to A T then there exists an i-factorization of w such that either f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) (1) or f (ˆ µ ( s i − 1 ) · a i · v i ) � = f (ˆ µ ( s i − 1 ) · ˆ µ ( a i ) · v i ) (2) • If (1), then v i is a new distinguishing word vertical expansion - Table not closed → new state • If (2), then a i is a new evidence for a i . horizontal expansion - Evidence incompatibility → new transition / refinement V ERIMAG O Maler - IE Mens 14 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) s i = δ ( ε , u i · a i ) ε µ ( u i ) ˆ s s ′ µ ( a i ) ˆ v i v i � = V ERIMAG O Maler - IE Mens 15 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) s i = δ ( ε , u i · a i ) s · a i is a ε new state µ ( u i ) ˆ s s ′ ε ˆ µ ( a i ) ˆ µ ( u i ) v i s s ′ v i � = µ ( a i ) ˆ � = v i new v i � = V ERIMAG O Maler - IE Mens 15 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) f (ˆ µ ( s i − 1 ) · a i · v i ) � = f (ˆ µ ( s i − 1 ) · ˆ µ ( a i ) · v i ) s i = δ ( ε , u i · a i ) s · a i is a ε ε new state µ ( u i ) ˆ ˆ µ ( u i ) s s ′ s ε µ ( a i ) ˆ a i µ ( a i ) ˆ ˆ µ ( u i ) v i � = s s ′ v i v i v i � = µ ( a i ) ˆ � = v i � = new v i � = V ERIMAG O Maler - IE Mens 15 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) f (ˆ µ ( s i − 1 ) · a i · v i ) � = f (ˆ µ ( s i − 1 ) · ˆ µ ( a i ) · v i ) s i = δ ( ε , u i · a i ) s · a i is a ε ε new state refine [ [ a i ] ] µ ( u i ) ˆ µ ( u i ) ˆ s s ′ s ε µ ( a i ) ˆ a i ε µ ( a i ) ˆ µ ( u i ) ˆ v i � = µ ( u i ) ˆ s s ′ v i v i s v i ˆ µ ( a i ) a i � = µ ( a i ) ˆ � = v i � = new v i v i v i � = � = V ERIMAG O Maler - IE Mens 15 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning with a Teacher p a i a i + 1 . . . . . . | Σ error in the partitions - Equivalence is checked by an oracle (teacher) returning a minimal counter-examples (in length and lexicographically) - Choose as evidence the min element of the interval ( Σ has min) - The counter-example indicates the minimal element of a new transition (in horizontal expansion) - The partition bounds are exact and no error is introduced p [ [ a ] ] [ [ b ] ] | Σ µ ( a ) ˆ µ ( b ) ˆ V ERIMAG O Maler - IE Mens 16 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 + a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 + a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 a 0 + a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 a 0 + a 0 [ [ a 1 ] ] = [ 1 , 100 ) 1 1 a 0 a 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) Σ 1 a 0 + a 0 [ [ a 1 ] ] = [ 1 , 100 ) 1 1 − a 0 a 1 Ask Equivalence Query: V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) Σ 1 a 0 + a 0 [ [ a 1 ] ] = [ 1 , 100 ) 1 1 − a 0 a 1 Ask Equivalence Query: counterexample − 24 24 ∈ [ [ a 0 ] ] but 1 �∼ 24 − → refine a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 a 2 Ask Equivalence Query: counterexample − 24 24 ∈ [ [ a 0 ] ] but 1 �∼ 24 − → refine a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: counterexample − 24 24 ∈ [ [ a 0 ] ] but 1 �∼ 24 − → refine a 0 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: counterexample + 1 · 66 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: counterexample + 1 · 66 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 counterexample + 1 · 66 a 0 a 3 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 counterexample + 1 · 66 + a 0 a 3 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 + a 0 a 3 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 − 24 · 1 counterexample + a 0 a 3 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 − 24 · 1 counterexample + a 0 a 3 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − + a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − − a 2 Ask Equivalence Query: 1 66 − 24 · 1 counterexample + − a 0 a 3 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − a 0 a 3 − 24 · 1 counterexample 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 − 24 · 1 counterexample 24 1 a 2 a 4 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 − 24 · 1 counterexample 24 1 − − a 2 a 4 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 − 24 · 1 counterexample 24 1 − − a 2 a 4 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 Σ [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 24 1 − − a 2 a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 Σ [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 + 24 · 51 counterexample 24 1 − − a 2 a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 Σ [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 + 24 · 51 counterexample 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 + − a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 + − a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 + − a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 True [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 24 51 + − a 2 a 5 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 True [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 24 51 + − a 2 a 5 M = { ε, 1 , 24 , 1 1 , 1 66 , 24 1 , 24 51 , 1 1 1 , 1 66 1 , 24 1 1 , 24 51 1 } | M | = 11 , | MQ | = 7 , | EQ | = 5 , | S | = 3 , | R | = 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 True [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 24 51 + − a 2 a 5 M = { ε, 1 , 24 , 1 1 , 1 66 , 24 1 , 24 51 , 1 1 1 , 1 66 1 , 24 1 1 , 24 51 1 } L ∗ over (Σ ∩ N ) → | M | = 790 , | MQ | = 789 , | EQ | = 2 , | S | = 4 , | R | = 396 | M | = 11 , | MQ | = 7 , | EQ | = 5 , | S | = 3 , | R | = 4 V ERIMAG O Maler - IE Mens 17 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning without a Teacher p a i a i + 1 . . . . . . | Σ error in the partitions - Equivalence is checked by testing random words selected using a probability distribution D - Counter-examples are not minimal we may have errors in the boundaries - Counter-examples may be missed terminate algorithm and return hypothesis after r ( ε, δ, i ) random words have been tested, none of which is a counter-example - The final hypothesis A is a good approximation of the target language L with high probability P ( d ( L , L A ) < ε ) ≥ 1 − δ (PAC learnability) V ERIMAG O Maler - IE Mens 18 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε ε − V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε ε − 13 42 68 78 92 ˆ µ ( a 1 ) µ ( a 2 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | ˆ µ ( a 1 ) µ ( a 2 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 µ ( a 1 ) ˆ µ ( a 2 ) ˆ + a 1 68 − a 2 V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 µ ( a 1 ) ˆ µ ( a 2 ) ˆ + a 1 68 − a 2 V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 µ ( a 3 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 + µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 µ ( a 3 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε ε ε a 1 27 ε − 13 42 68 78 92 Σ | 13 x ≥ 27 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 µ ( a 3 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε ε a 1 ε 27 ε − 13 42 68 78 92 Σ | 13 x ≥ 27 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε a 1 ε 27 ε − 13 42 68 78 92 Σ | 13 x ≥ 27 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − − a 2 2 18 26 46 54 13 18 − + a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 a 2 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ a 2 17 27 64 72 94 µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ a 2 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 a 2 a 4 a 2 68 72 a 2 a 5 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 − − a 2 a 4 a 2 68 72 + − a 2 a 5 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 − − a 2 a 4 a 2 68 72 + − a 2 a 5 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 + − x ≥ 45 µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 − − a 2 a 4 a 2 68 72 + − a 2 a 5 45 17 27 64 72 94 | ˆ µ ( a 4 ) µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ Ask Equivalence Query: 68 17 − − a 2 a 4 counterexample − 12 · 73 · 11 a 2 68 72 + − 45 a 2 a 5 17 27 64 72 94 add 73 as evidence of a 1 | ˆ µ ( a 4 ) µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 2 18 26 46 54 73 13 18 − + a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 68 17 − − a 2 a 4 counterexample − 12 · 73 · 11 a 2 68 72 + − 45 a 2 a 5 17 27 64 72 94 add 73 as evidence of a 1 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 68 17 − − a 2 a 4 counterexample − 12 · 73 · 11 a 2 68 72 + − 45 a 2 a 5 17 27 64 72 94 add 73 as evidence of a 1 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 13 73 + − a 1 a 6 counterexample − 12 · 73 · 11 a 2 68 17 − − 45 a 2 a 4 17 27 64 72 94 add 73 as evidence of a 1 | 68 72 + − a 2 a 5 µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x ≥ 63 x < 27 ε 11 ε ε a 1 27 ε − + 68 78 92 13 42 | x < 63 13 + − x ≥ 45 µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 | − + a 1 a 3 µ ( a 3 ) ˆ µ ( a 6 ) ˆ 13 73 + − a 1 a 6 a 2 68 17 − − 45 a 2 a 4 17 27 64 72 94 | 68 72 + − a 2 a 5 ˆ µ ( a 4 ) µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x ≥ 63 x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 | x < 63 13 + − x ≥ 45 ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 13 73 + − a 1 a 6 − 52 · 47 counterexample a 2 68 17 − − a 2 a 4 45 17 27 64 72 94 add 47 as evidence of a 2 | 68 72 + − a 2 a 5 ˆ µ ( a 4 ) µ ( a 5 ) ˆ − → refine existing transition V ERIMAG O Maler - IE Mens 19 / 22
Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x ≥ 63 x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 | x < 63 13 + − x ≥ 45 µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 13 73 + − a 1 a 6 − 52 · 47 counterexample a 2 68 17 − − a 2 a 4 45 17 27 47 64 72 94 add 47 as evidence of a 2 | 68 72 + − a 2 a 5 µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → refine existing transition V ERIMAG O Maler - IE Mens 19 / 22
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