Probabilistic (k, l)-Context-Sensitive Grammar Inference with

Gibbs Sampling Applied to Chord Sequences

Henrique Barros Lopes

and Alan Freitas

Departamento de Computac¸

ao, Universidade Federal de Ouro Preto, Campus Universit

ario Morro do Cruzeiro,

Keywords:

Grammatical Inference, Computer Music, Chord Sequence Learning, Probabilistic Context-Sensitive

Grammars, Machine Learning, Algorithmic Composition, Artiﬁcial Intelligence.

Abstract:

Grammatical inference in computer music provides us with valuable models for ﬁelds such as algorithmic

composition, style modeling, and music theory analysis. Grammars with higher accuracy can lead to models

that improve the performance of various tasks in these ﬁelds. Recent studies show that Hidden Markov Models

can outperform Markov Models in terms of accuracy, but there are no signiﬁcant differences between Hidden

Markov Models and Probabilistic Context-Free Grammars (PCFGs). This paper applies a Gibbs Sampling

algorithm to infer Probabilistic (k, l)-Context-Sensitive Grammars (P(k, l)CSGs) and presents an application

of P(k, l)CSGs to model the generation of chord sequences. Our results show Gibbs Sampling and P(k, l)CSGs

can improve on PCFGs and the Metropolis-Hastings algorithm with perplexity values that are 48% lower on

average (p-value 0.0026).

1 INTRODUCTION

The inference of probabilistic grammars is a ma-

chine learning approach applied in a myriad of ﬁelds.

Works inferring grammars are used, for example, to

validate protocols (Nouri et al., 2014; Mao et al.,

2012) and to infer ﬂow diagrams (Breuker et al.,

2016). Computational linguistics is an important

ﬁeld that makes extensive use of grammatical infer-

ences (Heinz and Rogers, 2010; Jarosz, 2013; Jarosz,

2015a; Jarosz, 2015b). This ﬁeld uses inferred gram-

mars to learn patterns and relationships between let-

ters in a word or words in a phrase. These patterns

depend on the expressiveness of the inferred gram-

mar. For instance, context-sensitive grammars may

deﬁne relationships between words in English spoken

language (J

ager and Rogers, 2012).

While grammar deﬁnes rules that help people

write and speak clearly and correctly, music has hier-

archical theories, e.g., functional harmony, that guide

composers to write songs pleasant to listeners. This

similarity led researchers from linguistics and music

to use similar strategies to guide their work (Forsyth

and Bello, 2013).

https://orcid.org/0000-0002-2185-1635

https://orcid.org/0000-0002-1266-0204

In computer music, the grammatical inference is

useful to ﬁelds such as algorithmic composition, style

modeling, and musical theory analysis. In Section

2, we describe many applications of grammatical in-

ferences in computer music along with the gram-

matical classes inferred in each work. We observed

that Bayesian inference works better than other ap-

proaches to infer grammars, and the use of Probabilis-

tic (k, l)-Context-Sensitive Grammars (P(k, l)CSG)s

has the potential to improve the state of the art in

chord sequence modeling.

The Bayesian algorithms Gibbs Sampling (GS)

and Metropolis-Hastings (MH) (Johnson et al., 2007;

Tsushima et al., 2018a) have been used to infer Prob-

abilistic Context-Free Grammars (PCFG), which are

in the second level of Chomsky’s hierarchy. On the

other hand, P(k, l)CSGs have been applied to model

part-of-speech tags, as in (Shibata, 2014), who has

proposed an MH algorithm to infer both PCFGs and

P(k, l)CSGs. The experiments showed a P(k, l)CSG

with (k, l) = (1, 0) achieved smaller perplexity values

than the actual state of the art algorithms.

In Section 3, this paper proposes a blocked GS

algorithm to infer P(k, l)CSGs to model chord se-

quences. We show the details of the GS algorithm for

PCFGs and describe how it can be modiﬁed to infer a

P(k, l)CSG.

572

Lopes, H. and Freitas, A.

Probabilistic (k,l)-Context-Sensitive Grammar Inference with Gibbs Sampling Applied to Chord Sequences.

DOI: 10.5220/0010195905720579

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 2, pages 572-579

ISBN: 978-989-758-484-8

In Section 4, we describe how we conducted our

experiments by inferring PCFGs and P(k, l)CSGs to

model the Billboard data set (Burgoyne et al., 2011)

using the GS and MH algorithms. In Section 5, we

conclude our work and present suggestions for future

work.

2 GRAMMATICAL INFERENCE

2.1 Sub-regular Grammars

Markov Models (MMs) are probabilistic models de-

scribing a sequence of possible events in which each

event’s probability depends only on the state attained

in the last o events (Gagniuc, 2017). When dealing

with grammar, these events are symbols of a word,

where a word is a sequence of grammar terminal sym-

bols. Equation 1 deﬁnes a MM of order o, where X

n+1

denotes the probability of an event. These models are

equivalent to n-grams.

P(X

n+1

, . . . , X

) = P(X

n+1

, . . . , X

n−o

) (1)

Although computer music researchers often use MMs

(Bhattacharjee and Srinivasan, 2011; Nakamura et al.,

2018; Takano and Osana, 2012; Trochidis et al.,

2016), it is possible to represent the test data set with

more expressive models. The perplexity measure ρ,

deﬁned in Equation 2, informs the joint probability of

a grammar yielding each of the word w in the test set

W .

ρ = e



(−

|W |

ln(

∏

w∈W

P(w|θ))



(2)

2.2 Finite Automata

A Probabilistic Finite Automaton (PFA) is a tuple

(Q,

∑

, σ, q

, θ) where (i) Q is a ﬁnite set of states; (ii)

∑

is the grammar alphabet; (iii) (σ : QX

∑

→ Q) is a

transition function; (iv) q

is its initial state; and (v) θ

is a probability vector such that |θ| is the number of

transitions in the grammar.

Extensions of Hidden Markov Models (HMM)

and PFA, such as Mondrian Hidden Markov Mod-

els (MHMMs) (Nakano et al., 2014), and variable

Markov oracles (VMO-HMM) (Wang and Dubnov,

2017), can achieve smaller perplexity values than

MMs (Tsushima et al., 2018a). These models have

been used for chord sequences (Tsushima et al.,

2018a; Wang and Dubnov, 2017), drums pattern clas-

siﬁcation (Blaß, 2013), and musical signal processing

(Nakano et al., 2014).

2.3 Context-Free Grammars

A Probabilistic Context-Free Grammar (PCFG) is a

5-tuple (N, T, R, S, θ) where (i) N is the set of non-

terminals; (ii) T is the set of terminals; (iii) R is the set

of transition rules; (iv) S ∈ N is the initial symbol; and

(v) θ is a probability vector such that |θ| is the number

of transitions. Each rule takes the form A → α, where

A ∈ N, and α = [α

, . . . ] : α

∈ N ∪ T .

PCFGs have been used for chord sequences

(Tsushima et al., 2018a), improvising follow-ups (Ki-

tani and Koike, 2010), and harmonization (Tsushima

et al., 2017). These grammars have been inferred

with Gibbs Sampling (GS), SEQUITUR (Kitani and

Koike, 2010), and the Expectation-Maximization

(EM) algorithm (Johnson et al., 2007; Tsushima et al.,

2018a; Tsushima et al., 2017; Tsushima et al., 2018b).

Bayesian approaches like GS have been shown to

achieve smaller values of perplexity than grammars

inferred with the EM algorithm (Geman and Geman,

1984).

2.4 (k, l)-Context-Sensitive Grammars

A Probabilistic Context-Sensitive Grammar (PCSG)

is a 5-tuple (N, T, R, S, θ) where (i) N is a the set of

non-terminals; (ii) T : T ∩ N = ∅ is the set of ter-

minals; (iii) R is the set of rules; (iv) S ∈ N is the

initial symbol; and (v) θ : |θ| = |R| is a probabil-

ity vector. Each rule takes the form γ → α, where

α = [α

, . . . ] : ∀iα

∈ N ∪T and γ = [γ

, . . . ] : ∃iγ

∈ N.

In other words, a PCSG considers which symbols

are surrounding a non-terminal to decide which rule

should be used.

To avoid the problem of sequences of inﬁnite

length in PCSGs, we can use Probabilistic (k, l)-

Context-Sensitive Grammars (P(k, l)CSGs) (Shibata,

2014). A P(k, l)CSG is a PCSG where the rules fol-

low the form γ

Aγ

→ γ

αγ

such that A ∈ N, γ

∈

N ∪ T, γ

∈ N ∪ T, α ∈ N ∪ T . The variables γ

and

deﬁne a rule’s right and left contexts, respectively,

and the variables k, l indicate the maximum length of

these contexts. Thus, a P(0, 0)CSG is equivalent to a

PCFG.

P(k, l)CSGs have been inferred with the

Metropolis-Hastings (MH) algorithm (Shibata,

2014). The GS algorithm could be applied to these

grammars despite its low performance. On the other

hand, GS has the potential to use all the data in the

training data set to estimate probabilities.

Probabilistic (k,l)-Context-Sensitive Grammar Inference with Gibbs Sampling Applied to Chord Sequences

573

3 INFERRING PCFGs AND PCSGs

WITH Gibbs SAMPLING

This paper proposes the use of a GS algorithm to infer

P(k, l)CSGs. The GS algorithm can use all the train-

ing data so that we might achieve lower perplexity

values for our grammar (see Equation 2). The source

code for this work is available in our repository

In Section 3.1, we brieﬂy discuss the GS algo-

rithm. Section 3.2 discusses the general GS algo-

rithm proposed by (Johnson et al., 2007) for inferring

PCFGs. In Section 3.3, we use this algorithm as a ref-

erence for a GS algorithm for inferring P(k, l)CSGs.

3.1 Gibbs Sampling

GS (Geman and Geman, 1984) is a Monte Carlo

Markov Chain (MCMC) method that allows us to ob-

tain samples from sets with multivariate probability

distributions. Algorithm 1 describes the method. Sup-

pose we want k samples from a multivariate random

variable X. For each of these k samples (Line 1), we

generate some initial value X

(Line 2) and iteratively

update this value with new samples (Line 3). The ini-

tial values X

can be determined randomly or by some

other algorithm.

In Line 4, we determine the current probability

distribution for the vector component x

∈ X

. Be-

cause we have not sampled values for x

such that

k > j yet, note the conditional probability p considers

for the indices k < j and X

i−1

for k > j. In Line 5,

we sample the component x

from this distribution d.

Algorithm 1: Gibbs Sampling.

Data: k

Result: {X

, . . . , X

}

1 for i=1 to k do

2 X

← initial();

3 for j=1 to n = |X| do

4 d ← p(x

, . . . , x

j−1

, x

i−1

j+1

, . . . , x

i−1

);

5 x

(i)

← sample(d);

6 end

7 end

The sample sequence X

, X

, ..., X

forms an ergodic

Markov Chain, where its stationary state represents

the target distribution p.

https://github.com/henriqueblopes/grammatical-inferences

3.2 Gibbs Sampling for PCFGs

Given the training data, the GS inference of PCFGs

(Johnson et al., 2007) deﬁnes a target distribution over

a set of syntactic trees. The GS algorithm runs until

the target distribution p is known, and θ represents the

training set appropriately.

Note that the sequence α of a PCFG can be any

combination of symbols from the set N ∪ T , making

R inﬁnite. However, we need to estimate θ

, ∀r ∈ R∗

to perform the Bayesian inference of a PCFG. That is,

we need R to be ﬁnite.

Theorem 1 (Chomsky’s Normal Form Theorem).

Any context-free language can be yielded by a gram-

mar whose rules are of the form A → BC or A → a

such that A, B,C ∈ N and a ∈ T .

With Theorem 1, we ensure that if N and T are

ﬁnite, then R is ﬁnite. Thus, θ is ﬁnite. Each word a

grammar yields has at least one syntactic tree. An

applied rule sequence from the grammar’s starting

symbol until the complete word yielding deﬁnes this

tree. We can use Bayes’ rule to calculate the syntactic

tree’s t

probability, given a word w

as in Equation 3

(Johnson et al., 2007).

P(t

|θ, w

) =

P(w

|θ,t

)P(t

|θ)

P(w

|θ)

(3)

Let W be the set of all words in the training data

set. If we calculate the product in Equation 4 for

P(t

|θ, w

) and all w

∈ W , we will have the joint prob-

ability P(T |θ,W ) = P(t

, ...t

|θ,W ), which is the

joint probability of all trees in the set T = {t

, ...t

}

that yields the set W . The GS algorithm can use the

distribution P(T|θ,W ) as a target distribution.

P(T |θ,W ) =

∏

P(t

, θ) (4)

As each tree probability depends on the rules’ proba-

bilities, ﬁnding the distribution P(T |θ,W ) is the same

as ﬁnding the best θ that represents W . With the target

distribution P(T |θ,W ), we need to achieve the follow-

ing goals for a PCFG with GS:

• Calculate the prior probabilities θ;

• Sample from P(T |θ,W );

• Calculate the posterior probabilities of θ.

We take the prior probabilities as a Dirichlet distri-

bution (Johnson et al., 2007) because it is conjugate

to the PCFG’s tree distribution. Which means that

if a prior distribution over trees θ is a Dirichlet dis-

tribution, then the posterior distribution is a Dirichlet

distribution (Johnson et al., 2007). This property fa-

cilitates the posteriors’ calculations.

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

574

The inside algorithm calculates the inside table

(Johnson et al., 2007). This table represents the prob-

abilities of a given non-terminal yielding each sub-

word of w

. A sample of P(T |θ,W ) is then obtained

by sampling a tree t ∈ T

yielding w

for each i, which

can be done with Johnson’s Sample function (Johnson

et al., 2007). This function makes use of the inside ta-

ble to randomly draw a triplet ( j, B,C), where j is the

division point of w

, B represents a rule for sampling

t, and C represents another rule for sampling t.

With the inside table and the sample function, we

can perform the GS for our PCFG. The parameters are

the prior θ, a PCFG, the maximum number of itera-

tions, and a training data set T D. We ﬁrst initialize

a tree set. Then, for each w

∈ W , we calculate the

inside table and sample a tree. We update the proba-

bilities by calculating the posterior distribution using

the counter of each rule used to sample trees and re-

peat these steps according to the maximum number of

iterations. At this point, the algorithm only updates

the vector θ after sampling all trees. This variation is

known as blocked Gibbs Sampling.

3.3 Gibbs Sampling for P(k, l)CSGs

Let w denote a word yielded by a PCFG rule sequence

, r

, ..., r

. Then the tree probability is the product of

all rule probabilities. In a P(k, l)CSG, we deﬁne the

context-sensitive probabilities as in Equation 5 (Shi-

bata, 2014).

P(γ

Aγ

→ γ

αγ

) = P(A → α|γ

, γ

) (5)

Equation 5 allows a PCFG to be transformed into a

P(k, l)CSG by calculating the probabilities of each

rule marginalized by the contexts γ

and γ

. This

transformation is important because it allows us to

use Theorem 1 to turn the rule set ﬁnite. Therefore,

we can generate a P(k, l)CSG with all possible rules

given by T and N.

The sampling of P(T |θ,W ) in a P(k, l)CSG is sim-

ilar to the sampling in PCFGs. The main difference is

that when we use a rule of the form γ

Aγ

→ γ

BCγ

we must take into account the right context of B given

by the sequence Cγ

d−1

, where γ

d−1

is the context γ

without its last symbol. Another difference is that we

must know whether a rule with the left or right context

is valid. For instance, at the beginning of the algo-

rithm, the starting symbol S does not have a context.

So we must forbid rules of the form γ

Sγ

→ γ

BCγ

at this step.

Algorithm 2 (Shibata, 2014; Johnson et al., 2007)

shows the steps to do the sampling. The algorithm

requires an initial word w, the left-hand side γ

Aγ

the initial rule, a start position I for yielding rules, an

ﬁnal position K for yielding rules, the inside table IT,

a maximum left context length k and a maximum right

context length l.

Algorithm 2: Sample Tree-KL.

Data: w, γ

Aγ

, I, K, IT , k, l

Result: R

1 begin

2 if K-I=1 then

3 R ← R ∪ {γ

Aγ

→ γ

};

4 else

5 cd ← categorical distribution(IT );

6 (J, B,C) ← sample(cd);

7 (γ

, γ

) ← draw context(k,l);

8 AP ← update yield();

9 (γ

, γ

) ← adjust context(AP);

10 R ← R ∪ {γ

Aγ

→ γ

BCγ

};

11 R ← sample tree KL(w, γ

BCγ

r−1

, I,

I + J, IT , k, l);

12 AP ← update yield();

13 (γ

, γ

) ← adjust context(AP);

14 R ← sample tree KL(w, γ

Cγ

I + J + 1, K, IT , k, l);

15 end

16 end

In Line 2, Algorithm 2 veriﬁes if the starting and end-

ing positions consist of a terminal yield. If the po-

sitions represent a terminal yield, R includes the rule

Aγ

→ γ

, where w

is the terminal to be yielded

(Line 3). If the positions do not represent a terminal

yield, Line 5 calculates a categorical distribution from

IT . Line 6 samples two non-terminals (B,C) and a

position J that the recursive calls to the function will

use to calculate the new starting and ending positions.

Next, we need to decide which right and left con-

texts the yield will use. Line 7 randomly draws one

left and one right context from all possible contexts

according to its maximum lengths, k, and l. Line 8

stores the current state of the tree sampling on the

variable AP. Then, if necessary, Line 9 removes sym-

bols from γ

and γ

to guarantee their validity.

Using the non-terminals A, B,C, and the context

, γ

, Line 10 includes the rule γ

Aγ

→ γ

BCγ

in the

rule list R. Line 11 calls the sampling algorithm re-

cursively, passing as left-hand side the non-terminal

B surrounded by the contexts γ

and Cγ

r−1

, I as yield

starting position, and I + J as the yield ending posi-

tion. Lines 12 and 13 adjust the contexts again. This

adjustment is necessary because Line 11 has changed

the current context.

Finally, Line 14 calls the sampling algorithm re-

cursively, passing as left-hand side the non-terminal

Probabilistic (k,l)-Context-Sensitive Grammar Inference with Gibbs Sampling Applied to Chord Sequences

575

C surrounded by the contexts γ

and Cγ

r−1

, I + J + 1

as yield starting position, and K as the yield ending

position. At the end, the rules R will contain all sam-

pled rules to yield w, representing a syntactic tree.

As the tree sampling in a P(k, l)CSG only allows

left yields, there will never be a non-terminal to the

left of w

. Consequently, there will never be a termi-

nal to the right of w

while w

is yielding. So, the con-

texts γ

and γ

, where w = (w

, w

i+1

, ..., w

j−1

, w

), i <

j, are sequences containing only terminals and non-

terminals respectively, that is, γ ∈ γ

⇒ γ ∈ T and

γ ∈ γ

⇒ γ ∈ N. Therefore, the probability of A ∈ N

yielding w

, given γ

and γ

, can by calculated by

Equations 6 and 7.

Aγ

,i,i

= θ

Aγ

→γ

(6)

Aγ

,i,k

∑

Aγ

→γ

BCγ

∈R

∑

j=i+1

Aγ

→γ

BCγ

Bγ

d−1

,i, j

Cγ

, j+1,k

(7)

To construct the inside table IT , we use the inside al-

gorithm modiﬁed according to Equations 6 and 7. To

infer a P(k, l)CSG with the GS, we need to run the

GS algorithm with the sampling in Algorithm 2 and

calculate IT for a P(k, l)CSG.

4 EXPERIMENTAL RESULTS

Given the algorithms described in Sections 3.2 and

3.3, we perform experiments to compare the GS and

MH algorithms to understand the inﬂuence of the val-

ues (k, l) in a P(k,l)CSG.

4.1 Data Set

The Billboard data set (Burgoyne et al., 2011) con-

tains 890 songs. Each ﬁle contains chord sequences

split into labeled musical phrases. We extracted chord

sequences from the data set Billboard to build our

data set for inferences using the same procedure as

in (Tsushima et al., 2018a): each word represents a

chord sequence, and each chord represents a gram-

mar symbol. After the extraction, we obtained a data

set of 7217 words to infer our grammar. This data set

allows us to extract the musical phrases to get words

with a length smaller than the entire song. The ad-

vantage of this approach is to obtain a data set with

a higher number of words. The source code for this

work is available in our repository

https://github.com/henriqueblopes/grammatical-inferences

4.2 Inference Details

The total number of unique chords was 738. We used

the same procedure as in (Tsushima et al., 2018a) to

simplify the number |N| of non-terminals N in our

grammar. We transposed all phrases to C major and

identiﬁed the 20 most frequent chords in the data set.

We then switched the other chords to a terminal node

referred to as “Other”. We split the data set into a

training data set T D with 80% of the phrases, and a

validation data set V D with the other 20%.

Table 1 summarizes the combinations of factors in

our experiment. Due to the high computational cost of

GS, we limited the factors |T D| ≤ 3000, and |N| < 6

in our experiments.

Table 1: Factors in our Experiment.

Algorithm |N| |T D| k l

MH 2 300 0 0

MH 2 300 1 0

MH 2 3000 0 0

MH 2 3000 1 0

MH 6 300 0 0

MH 6 300 1 0

GS 2 300 0 0

GS 2 300 1 0

GS 2 3000 0 0

GS 2 3000 1 0

GS 6 300 0 0

GS 6 300 1 0

We used the inference algorithms P(k, l)CSGs with

GS described in Section 3.3 and MH (Shibata, 2014)

with k = {0, 1} and l = 0. The GS with (k, l) = (0, 0)

is one of the algorithms implemented in (Tsushima

et al., 2018a). Therefore, we will compare them to

the state of the art for the Billboards data set when

performing our experiments. We run all algorithms

in a Google Cloud virtual machine N1 series, High-

CPU type, eight cores, 7.2GB of RAM, OS Debian

GNU/Linux 10. The number of iterations was 200

and we limited the number of replicates to ﬁve for

each combination of factors due to the high compu-

tational cost of GS inferences, as discussed in more

detail in Section 4.4.

4.3 Inference Validation

We used the perplexity measure to validate the in-

ferred grammar and used the cross-validation method

dividing the data set into one different data sets per

replicate. For each replicate, we calculated the aver-

age perplexity value (Equation 2) in the test set for

each run.

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

576

4.4 Results

Figure 1 presents the box-plots for all values of per-

plexity in the y-axis for each replicate of the MH al-

gorithm. Each pair of boxplots represents executions

with (k, l) = (0, 0) and (k,l) = (0, 1).

Figure 1: Perplexity for the MH algorithm. Experiments

with a higher value of |T D| and |N| tend to achieve lower

perplexity values. Also, grammars with k = 1 led to lower

averages.

We can see that higher values of |TD| and |N| tend to

achieve lower perplexity values. Also, when N = 6,

the presence of a left context of length k = 1 led to a

smaller perplexity value.

The GS samples a large number of trees at each

iteration. So GS has a much higher run time than the

MH algorithm, which is why we have a limited num-

ber of replicates. Figure 2 compares the run time of

the GS and MH.

Figure 3 shows the perplexity values obtained

with GS. In all cases, the P(k, l)CSGs have smaller

average perplexity values than PCFGs.

As in the results for the MH algorithm,

P(k, l)CSGs have smaller perplexity values than

PCFGs using GS. The perplexity becomes smaller

when we use a larger training set. This result also

indicates the presence of the left context allowed the

grammar to better represent the test sets.

In Figure 4, we compare the perplexity values

for GS and MH. The results indicate GS can achieve

lower perplexity values than MH.

We aggregated our results and used a Wilcoxon

signed paired rank-test for a paired test for the null hy-

pothesis that the average perplexity values for GS and

MH are samples from distributions with equal medi-

ans. We obtained a p-value = 0.0026 indicating that

Figure 2: GS and MH run time.

Figure 3: Perplexity Values for the GS algorithm. Gram-

mars where (k, l) = (1, 0) tend to achieve lower perplexity

values.

GS, despite its computational cost, can achieve lower

median perplexity values than MH.

5 CONCLUSIONS

We have investigated the capability of P(k, l)CSG to

represent features related to musical chord sequences

present in the Billboards data set. We used Bayesian

approaches to infer these grammars and implemented

the ﬁrst blocked GS algorithm to infer P(k, l)CSGs

using the theory in (Shibata, 2014). Despite the high

computational cost GS, the presence of a left context

of size 1 provides smaller perplexity values than the

absence of contexts to represent the test sets.

These ﬁndings mean we could use the inferred

Probabilistic (k,l)-Context-Sensitive Grammar Inference with Gibbs Sampling Applied to Chord Sequences

577

Figure 4: Perplexity Values for the GS and MH algorithms

inferring P(1, 0)CSGs.

models to perform machine learning tasks related to

chord sequences like style classiﬁcation or the gener-

ation of musical phrases. The data suggest we could

improve our results by either investing in enough

computational power or developing cheaper algo-

rithms to infer grammars with k > 1 and l > 0.

ACKNOWLEDGMENTS

This work has been supported by the Brazilian Agen-

cies State of Minas Gerais Research Foundation –

FAPEMIG (APQ-00040-14); Coordination for the

Improvement of Higher Level Personnel – CAPES;

and National Council of Scientiﬁc and Technological

Development – CNPq (402956/2016-8).

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