A set-theoretical investigation of Pānini's Śivasūtras

Wiebke PetersenInstitute of Language and Information

Department of Computational LinguisticsUniversity of Düsseldorf

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Pānini's Śivasūtrasanubandha

sūtras

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Pānini's Śivasūtrasanubandha

sūtras

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Phonological classes/ pratyāhāras

Phonological classes are denoted by pratyāhāras.E.g., the pratyāhāra iC denotes the set of segments in the continuous sequence starting with i and ending with au, the last element before the anubandha C.

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Phonological classes/ pratyāhāras

Phonological classes are denoted by pratyāhāras.E.g., the pratyāhāra iC denotes the set of segments in the continuous sequence starting with i and ending with au, the last element before the anubandha C.

iC

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Phonological classes/ pratyāhāras

Phonological classes are denoted by pratyāhāras.E.g., the pratyāhāra iC denotes the set of segments in the continuous sequence starting with i and ending with au, the last element before the anubandha C.

iC

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Questions around Pānini's Śivasūtras

• Are Pānini's Śivasūtras perfect?– Is the number of stop markers minimal?– Is the duplication of 'h' necessary?

• Is it possible to decide whether a set of sets has a Śivasūtras-style representa-tion?

• How to construct an optimal Śivasūtras-style representation?

.

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Basic conceptsΦ={{d,e},{b,c,d,f,g,h,i},{a,b},{f,i},{c,d,e,f,g,h,i},{g,h}}S-encodable set of sets:

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Basic conceptsΦ={{d,e},{b,c,d,f,g,h,i},{a,b},{f,i},{c,d,e,f,g,h,i},{g,h}}S-encodable set of sets:

e d M1 c i f M2 g h M3 b M4 a M5S-alphabet (A,Σ,<) of Φ:

alphabetmarker

total order on A∪Σ

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Basic conceptsΦ={{d,e},{b,c,d,f,g,h,i},{a,b},{f,i},{c,d,e,f,g,h,i},{g,h}}S-encodable set of sets:

e d M1 c i f M2 g h M3 b M4 a M5S-alphabet (A,Σ,<) of Φ:

alphabetmarker

total order on A∪Σ

{d,e}

{b,c,d,f,g,h,i}

{a,b}

{f,i}

{c,d,e,f,g,h,i}

{g,h}{d}

{c,d,f,g,h,i}

{b}

{a,b,c,d,e,f,g,h,i}

{ }

Hasse-diagram (H(Φ), )for the set of intersections of elements from Φ∪{A}:

marker

total order on A∪Σ

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Basic conceptsΦ={{d,e},{b,c,d,f,g,h,i},{a,b},{f,i},{c,d,e,f,g,h,i},{g,h}}S-encodable set of sets:

e d M1 c i f M2 g h M3 b M4 a M5S-alphabet (A,Σ,<) of Φ:

alphabetmarker

total order on A∪Σ

{d,e}

{b,c,d,f,g,h,i}

{a,b}

{f,i}

{c,d,e,f,g,h,i}

{g,h}{d}

{c,d,f,g,h,i}

{b}

{a,b,c,d,e,f,g,h,i}

{ }

Hasse-diagram (H(Φ), )for the set of intersections of elements from Φ∪{A}:

marker

total order on A∪Σ

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S-encodability and planar Hasse-diagrams

If Φ is S-encodable, then the Hasse-diagram of (H(Φ), ) is planar

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S-encodability and planar Hasse-diagrams

If Φ is S-encodable, then the Hasse-diagram of (H(Φ), ) is planar

Hasse-diagram for Pānini's pratyāhāras .

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S-encodability and planar Hasse-diagrams

K5

K3,3

Criterion of Kuratowski: A graph is planar iff it has neither K5 nor K3,3 as a minor.

part of the Hasse-diagram for Pānini's pratyāhāras.

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S-encodability and planar Hasse-diagrams

K5

K3,3

Criterion of Kuratowski: A graph is planar iff it has neither K5 nor K3,3 as a minor.

part of the Hasse-diagram for Pānini's pratyāhāras.

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X

XX

X

XX

XX

X

K5 is a minor of the Hasse-diagram for the pratyāhāras

.

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X

XX

K5 is a minor of the Hasse-diagram for the pratyāhāras

.

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X

X

K5 is a minor of the Hasse-diagram for the pratyāhāras

.

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X

K5 is a minor of the Hasse-diagram for the pratyāhāras

.

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K5 is a minor of the Hasse-diagram for the pratyāhāras

.

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We are not done yet!

plane but not S-encodable!

Φ={{d,e},{b,c,d,f,},{a,b},{b,c,d}}

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Existence of S-alphabetsAn S-alphabet of Φ exists iff 1. a plane Hasse-diagram for (H(Φ), ) exists and

2. for every b∈A the S-graph contains a node labeled with b

not S-encodableS-encodable

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Construction of S-alphabets

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Construction of S-alphabets

a

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Construction of S-alphabets

a b

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Construction of S-alphabets

a b M1

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Construction of S-alphabets

a b M1 c

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Construction of S-alphabets

a b M1 c hg

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Construction of S-alphabets

a b M1 c hg cM2

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Construction of S-alphabets

a b M1 c hg cM2 ficM2

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Construction of S-alphabets

a b M1 c hg cM2 ficM2 cM3cM2

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Construction of S-alphabets

a b M1 c hg cM2 ficM2 cM3cM2 dcM3cM2

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Construction of S-alphabets

a b M1 c hg cM2 ficM2 cM3cM2 dcM3cM2 eM4 M5

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Construction of S-alphabets

a b M1 c hg cM2 ficM2 cM3cM2 dcM3cM2 eM4 M5××

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Pānini's Śivasūtras are perfect.

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Pānini's Śivasūtras are perfect.

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Pānini's Śivasūtras are perfect.

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