Factor Oracle for Machine Improvisation

Factor Oracle for Machine Improvisation Jaime Arias Université de Bordeaux,
LaBRI, UMR 5800 Inria - Bordeaux Sud-Ouest Septembre 2016

Preliminaries

Preliminaries Word A word s is a finite sequence s
= s1 s2 ... sm of length |s| = m on a finite alphabet Σ. # # + # + / # + s = Factor A word x ∈ Σ∗ is a factor of s if and only if s can be written s = uxv with u, v ∈ Σ∗. Given integers i, j where 1 ≤ i ≤ j ≤ m, we denote a factor of s as s[i ... j] = si si+1 ... sj . # # + # + / # + s = s[3, 5] Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 1/10 1/10

Preliminaries Prefix A factor x of s is a prefix
of s if s = xu with u ∈ Σ∗. The ith prefix of s, denoted prefs (i), is the prefix s[1 ... i]. # # + # + / # + s = prefs (4) Suffix A factor x of s is a suffix of s if s = ux with u ∈ Σ∗. The ith suffix of s, denoted suﬀs (i), is the suffix s[i ... m]. # # + # + / # + s = suﬀs (6) Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 2/10 2/10

Preliminaries Longest Repeated Suffix (LRS) A factor x of s
is the longest repeated suffix of s if x is a suffix of s and |x| is maximal. # # + # + / # + s = lrs(s) Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 3/10 3/10

Factor Oracle

Factor Oracle Overview 0 1 a 0 2 b 0
3 b 1 4 c 0 5 a 1 6 b 2 7 c 2 8 d 0 9 a 1 10 b 2 11 c 3 b c c d d Forward Links Suffix Links Factor Oracle The factor oracle of a word s of length m is a deterministic finite automaton (Q, q0 , F, δ) where Q = {0, 1, ... , m} is the set of states, q0 = 0 is the starting state, F = Q is the set of terminal states and δ is the transition function. Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 4/10 4/10

3 b 1 4 c 0 5 a 1 6 b 2 7 c 2 8 d 0 9 a 1 10 b 2 11 c 3 b c c d d Forward Links Suffix Links Suffix Link The suffix link of a state i of the factor oracle of a word s, is equal to the state in which the longest repeated suffix (lrs) of s[1 ... i] is recognized. Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 4/10 4/10

3 b 1 4 c 0 5 a 1 6 b 2 7 c 2 8 d 0 9 a 1 10 b 2 11 c 3 b c c d d Suffix Links • s = abbcabcdabc Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 5/10 5/10

3 b 1 4 c 0 5 a 1 6 b 2 7 c 2 8 d 0 9 a 1 10 b 2 11 c 3 b c c d d Suffix Links • s = abbcabcdabc • lrs(s) = abc Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 5/10 5/10

3 b 1 4 c 0 5 a 1 6 b 2 7 c 2 8 d 0 9 a 1 10 b 2 11 c 3 b c c d d Suffix Links • s = abbcabcdabc • lrs(s) = abc • S(11) = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 5/10 5/10

Factor Oracle Algorithm - Construction Algorithm 1 Construction of a
Factor Oracle 1: function FactorOracle(p = p1p2 ... pm ) 2: Create a new oracle P with an initial state 0 3: SP (0) ← −1 4: for i ← 1, m do 5: Oracle(p = p1p2 ... pi ) ← AddLetter(Oracle(p = p1p2 ... pi−1), pi ) 6: end for 7: return Oracle(p = p1p2 ... pm) 8: end function Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 6/10 6/10

Factor Oracle 1: function FactorOracle(p = p1p2 ... pm ) 2: Create a new oracle P with an initial state 0 3: SP (0) ← −1 4: for i ← 1, m do 5: Oracle(p = p1p2 ... pi ) ← AddLetter(Oracle(p = p1p2 ... pi−1), pi ) 6: end for 7: return Oracle(p = p1p2 ... pm) 8: end function # # + # + / # + p = Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 6/10 6/10

Factor Oracle 1: function FactorOracle(p = p1p2 ... pm ) 2: Create a new oracle P with an initial state 0 3: SP (0) ← −1 4: for i ← 1, m do 5: Oracle(p = p1p2 ... pi ) ← AddLetter(Oracle(p = p1p2 ... pi−1), pi ) 6: end for 7: return Oracle(p = p1p2 ... pm) 8: end function # # + # + / # + p = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 6/10 6/10

Factor Oracle Algorithm - Construction Algorithm 2 Incremental update of
Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 0 m = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 0 m = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 0 m = 0 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 0 m = 0 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 0 m = 0 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 0 0 m = 0 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 0 0 m = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 0 0 m = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 0 0 m = 1 π1 = 1 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 0 0 m = 1 π1 = 1 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # b 0 0 m = 1 π1 = 1 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # b 0 0 m = 1 π1 = 0 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # b 0 0 m = 1 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # b 0 0 m = 1 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # b 0 0 0 m = 1 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # b 0 0 0 m = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # b 0 0 0 m = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # b 0 0 0 m = 2 π1 = 2 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # b 0 0 0 m = 2 π1 = 2 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # b 0 0 0 m = 2 π1 = 2 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # b 0 0 0 1 m = 2 π1 = 2 k = 0 lcs(2, 1) = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # b 0 0 0 1 m = 3 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b 0 0 0 1 m = 3 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b 0 0 0 1 m = 3 π1 = 3 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b 0 0 0 1 m = 3 π1 = 3 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c 0 0 0 1 m = 3 π1 = 3 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c c 0 0 0 1 m = 3 π1 = 2 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c c 0 0 0 1 m = 3 π1 = 0 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c c 0 0 0 1 m = 3 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c c 0 0 0 1 m = 3 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c c 0 0 0 1 0 m = 3 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + b c c 0 0 0 1 0 m = 4 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 b c c 0 0 0 1 0 m = 4 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 b c c 0 0 0 1 0 m = 4 π1 = 4 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 b c c 0 0 0 1 0 m = 4 π1 = 4 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 b c c 0 0 0 1 0 m = 4 π1 = 4 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 b c c 0 0 0 1 0 1 m = 4 π1 = 4 k = 0 lcs(4, 0) = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 b c c 0 0 0 1 0 1 m = 5 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # b c c 0 0 0 1 0 1 m = 5 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # b c c 0 0 0 1 0 1 m = 5 π1 = 5 k = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # b c c 0 0 0 1 0 1 m = 5 π1 = 5 k = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # b c c 0 0 0 1 0 1 m = 5 π1 = 5 k = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # b c c 0 0 0 1 0 1 2 m = 5 π1 = 5 k = 1 lcs(5, 1) = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # b c c 0 0 0 1 0 1 2 m = 6 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + b c c 0 0 0 1 0 1 2 m = 6 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + b c c 0 0 0 1 0 1 2 m = 6 π1 = 6 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + b c c 0 0 0 1 0 1 2 m = 6 π1 = 6 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + b c c 0 0 0 1 0 1 2 m = 6 π1 = 6 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + b c c 0 0 0 1 0 1 2 2 m = 6 π1 = 6 k = 2 lcs(6, 3) = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + b c c 0 0 0 1 0 1 2 2 m = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c 0 0 0 1 0 1 2 2 m = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c 0 0 0 1 0 1 2 2 m = 7 π1 = 7 k = 4 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c 0 0 0 1 0 1 2 2 m = 7 π1 = 7 k = 4 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d 0 0 0 1 0 1 2 2 m = 7 π1 = 7 k = 4 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d d 0 0 0 1 0 1 2 2 m = 7 π1 = 4 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d d 0 0 0 1 0 1 2 2 m = 7 π1 = 0 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d d 0 0 0 1 0 1 2 2 m = 7 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d d 0 0 0 1 0 1 2 2 m = 7 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: Spσ ← 0 5: lrspσ ← 0 6: else 7: ... 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d d 0 0 0 1 0 1 2 2 0 m = 7 π1 = 0 k = −1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / b c c d d 0 0 0 1 0 1 2 2 0 m = 8 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 b c c d d 0 0 0 1 0 1 2 2 0 m = 8 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 b c c d d 0 0 0 1 0 1 2 2 0 m = 8 π1 = 8 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 b c c d d 0 0 0 1 0 1 2 2 0 m = 8 π1 = 8 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 b c c d d 0 0 0 1 0 1 2 2 0 m = 8 π1 = 8 k = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 b c c d d 0 0 0 1 0 1 2 2 0 1 m = 8 π1 = 8 k = 0 lcs(8, 0) = 0 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 b c c d d 0 0 0 1 0 1 2 2 0 1 m = 9 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # b c c d d 0 0 0 1 0 1 2 2 0 1 m = 9 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # b c c d d 0 0 0 1 0 1 2 2 0 1 m = 9 π1 = 9 k = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # b c c d d 0 0 0 1 0 1 2 2 0 1 m = 9 π1 = 9 k = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # b c c d d 0 0 0 1 0 1 2 2 0 1 m = 9 π1 = 9 k = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # b c c d d 0 0 0 1 0 1 2 2 0 1 2 m = 9 π1 = 9 k = 1 lcs(9, 1) = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # b c c d d 0 0 0 1 0 1 2 2 0 1 2 m = 10 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 m = 10 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: Create state m + 1 3: Create a new transition from m to m + 1 labeled by σ ◃ δ(m, σ) = m + 1 4: π1 ← m 5: k ← Sp(m) 6: ... 7: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 m = 10 π1 = 10 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: while k > −1 and there is no transition from k by σ do 4: Create a new transition from k to m + 1 by σ ◃ δ(k, σ) = m + 1 5: π1 ← k 6: k ← Sp(k) 7: end while 8: ... 9: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 m = 10 π1 = 10 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 m = 10 π1 = 10 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 π1 = 10 k = 2 lcs(10, 3) = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: if k = −1 then 4: ... 5: else 6: Spσ ← state that leads the transition from k by σ 7: lrspσ ← LengthCommonSuffix(π1, S(m + 1) − 1) + 1 8: end if 9: ... 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 π1 = 10 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: k ← FindBetter(m + 1, p[m + 1 − lrs(m + 1)]) 4: if k ̸= 0 then 5: lrspσ ← lrs(m + 1) + 1 6: Spσ ← k 7: end if 8: T(Spσ ) ← T(S(m + 1)) ∪ {m + 1} ◃ T(i) = {j | S(j) = i ∧ i < j ≤ m} 9: return Oracle(p = p1p2 ... pmσ) 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 π1 = 10 k = 2 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: k ← FindBetter(m + 1, p[m + 1 − lrs(m + 1)]) 4: if k ̸= 0 then 5: lrspσ ← lrs(m + 1) + 1 6: Spσ ← k 7: end if 8: T(Spσ ) ← T(S(m + 1)) ∪ {m + 1} ◃ T(i) = {j | S(j) = i ∧ i < j ≤ m} 9: return Oracle(p = p1p2 ... pmσ) 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 π1 = 10 k = 7 FindBetter(11, a) = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle 1: function AddLetter(Oracle(p = p1, p2 ... pm), σ) 2: ... 3: k ← FindBetter(m + 1, p[m + 1 − lrs(m + 1)]) 4: if k ̸= 0 then 5: lrspσ ← lrs(m + 1) + 1 6: Spσ ← k 7: end if 8: T(Spσ ) ← T(S(m + 1)) ∪ {m + 1} ◃ T(i) = {j | S(j) = i ∧ i < j ≤ m} 9: return Oracle(p = p1p2 ... pmσ) 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 3 m = 10 π1 = 10 k = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 7/10 7/10

Factor Oracle Algorithm - Construction Algorithm 3 Length Common Suffix
Algorithm 1: function LengthCommonSuffix(π1, π2 ) 2: if S(π1) = π2 then 3: return lrs(π1) 4: else 5: while S(π1) ̸= S(π2) do 6: π2 ← S(π2) 7: end while 8: end if 9: return min(lrs(π1), lrs(π2)) 10: end function Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 8/10 8/10

Algorithm 1: function LengthCommonSuffix(π1, π2 ) 2: if S(π1) = π2 then 3: return lrs(π1) 4: else 5: while S(π1) ̸= S(π2) do 6: π2 ← S(π2) 7: end while 8: end if 9: return min(lrs(π1), lrs(π2)) 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 π1 = 10 π2 = 3 lcs(10, 3) = Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 8/10 8/10

Algorithm 1: function LengthCommonSuffix(π1, π2 ) 2: if S(π1) = π2 then 3: return lrs(π1) 4: else 5: while S(π1) ̸= S(π2) do 6: π2 ← S(π2) 7: end while 8: end if 9: return min(lrs(π1), lrs(π2)) 10: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 π1 = 10 π2 = 3 lcs(10, 3) = 1 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 8/10 8/10

Factor Oracle Algorithm - Construction Algorithm 4 Find Better Algorithm
1: function FindBetter(i, σ) 2: for all the elements j of T(i) in increasing order do 3: if lrs(j) = lrs(i) and p[j − lrs(i)] = σ then 4: return j 5: end if 6: end for 7: return 0 8: end function Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 9/10 9/10

1: function FindBetter(i, σ) 2: for all the elements j of T(S(i)) in increasing order do 3: if lrs(j) = lrs(i) and p[j − lrs(i)] = σ then 4: return j 5: end if 6: end for 7: return 0 8: end function Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 9/10 9/10

1: function FindBetter(i, σ) 2: for all the elements j of T(S(i)) in increasing order do 3: if lrs(j) = lrs(i) and p[j − lrs(i)] = σ then 4: return j 5: end if 6: end for 7: return 0 8: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 i = 11 σ = a FindBetter(11, a) = Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 9/10 9/10

1: function FindBetter(i, σ) 2: for all the elements j of T(S(i)) in increasing order do 3: if lrs(j) = lrs(i) and p[j − lrs(i)] = σ then 4: return j 5: end if 6: end for 7: return 0 8: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 i = 11 σ = a FindBetter(11, a) = T(S(11)) = {7} Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 9/10 9/10

1: function FindBetter(i, σ) 2: for all the elements j of T(S(i)) in increasing order do 3: if lrs(j) = lrs(i) and p[j − lrs(i)] = σ then 4: return j 5: end if 6: end for 7: return 0 8: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 i = 11 σ = a FindBetter(11, a) = T(S(11)) = {7} j = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 9/10 9/10

1: function FindBetter(i, σ) 2: for all the elements j of T(S(i)) in increasing order do 3: if lrs(j) = lrs(i) and p[j − lrs(i)] = σ then 4: return j 5: end if 6: end for 7: return 0 8: end function # # + # + / # + p = 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 m = 10 i = 11 σ = a FindBetter(11, a) = 7 T(S(11)) = {7} j = 7 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 9/10 9/10

Factor Oracle Algorithm - Improvisation Algorithm 5 FO-Generate function Require:
Oracle P = p1, p2 ... pm in active state i, a generated sequence v, and a continuation parameter 0 ≤ q ≤ 1. 1: Generate uniformly distribute random number u 2: if u < q then 3: i ← i + 1 v ← vpi 4: else 5: Choose at random a symbol σ ∈ {σj | δ(S(i), σj ) ̸= ⊥} 6: i ← δ(S(i), σ) v ← vσ 7: end if 8: return Sequence v Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 10/10 10/10

Oracle P = p1, p2 ... pm in active state i, a generated sequence v, and a continuation parameter 0 ≤ q ≤ 1. 1: Generate uniformly distribute random number u 2: if u < q then 3: i ← i + 1 v ← vpi 4: else 5: Choose at random a symbol σ ∈ {σj | δ(S(i), σj ) ̸= ⊥} 6: i ← δ(S(i), σ) v ← vσ 7: end if 8: return Sequence v 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 v = cabc q = 0.5 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 10/10 10/10

Oracle P = p1, p2 ... pm in active state i, a generated sequence v, and a continuation parameter 0 ≤ q ≤ 1. 1: Generate uniformly distribute random number u 2: if u < q then 3: i ← i + 1 v ← vpi 4: else 5: Choose at random a symbol σ ∈ {σj | δ(S(i), σj ) ̸= ⊥} 6: i ← δ(S(i), σ) v ← vσ 7: end if 8: return Sequence v 0 1 2 # 3 # 4 + 5 6 # 7 + 8 / 9 10 # 11 + b c c d d 0 0 0 1 0 1 2 2 0 1 2 2 v = cabca q = 0.5 Jaime Arias - Equipe PoSET, Inria Bordeaux Sud-Ouest, LaBRI Factor Oracle for Machine Improvisation 10/10 10/10

Thank you for your attention!

Factor Oracle for Machine Improvisation Jaime Arias Université de Bordeaux,
LaBRI, UMR 5800 Inria - Bordeaux Sud-Ouest Septembre 2016

References Gerard Assayag and Shlomo Dubnov. Using Factor Oracles for
Machine Improvisation. Soft Computing, 8(9), sep 2004. Shlomo Dubnov, Gérard Assayag, and Arshia Cont. Audio Oracle: a New Algorithm for Fast Learning of audio Structures. In International Computer Music Conference, 2007. Arnaud Lefebvre, Thierry Lecroq, and Joël Alexandre. An Improved Algorithm for Finding Longest Repeats with a Modified Factor Oracle. J. Autom. Lang. Comb., 8(4):647–657, 2003. Greg Surges and Shlomo Dubnov. Feature Selection and Composition Using PyOracle. In AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, 2013.

Factor Oracle for Machine Improvisation

Factor Oracle for Machine Improvisation

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