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How formal concept lattices solve a problem of ancient linguistics - - PowerPoint PPT Presentation

Apr 07, 2023 316 likes •822 views

How formal concept lattices solve a problem of ancient linguistics Wiebke Petersen Department of Computational Linguistics Institute of Language and Information University of Dsseldorf 2 Wiebke Petersen ICCS 2005 P nini's ivas

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2 Wiebke Petersen ICCS 2005

How formal concept lattices solve a problem of ancient linguistics

Wiebke Petersen

Department of Computational Linguistics Institute of Language and Information University of Düsseldorf

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Pānini's Śivasūtras

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Phonological rules

A is replaced by B if preceded by C and followed by D

  • in modern form:
  • as context-sensitive rule:
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Phonological rules

A is replaced by B if preceded by C and followed by D

  • in modern form:
  • as context-sensitive rule:

Example: final devoicing in German (Hunde - Hund) [d]→[t] / _#, [b]→[p] / _#, [g]→[k] / _#, ...

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Phonological rules

A is replaced by B if preceded by C and followed by D

  • in modern form:
  • as context-sensitive rule:

Example: final devoicing in German (Hunde - Hund) [d]→[t] / _#, [b]→[p] / _#, [g]→[k] / _#, ...

#

/_

consonantal consonantal nasal nasal voiced voiced + +         − → −         + −    

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Pānini's coding of rules

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Pānini's coding of rules

A + genitive, B + nominative, C + ablative, d + locative

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Pānini's coding of rules

A + genitive, B + nominative, C + ablative, d + locative

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Pānini's coding of rules

A + genitive, B + nominative, C + ablative, d + locative

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Pānini's coding of rules

A + genitive, B + nominative, C + ablative, d + locative

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Pānini's Śivasūtras

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Pānini's Śivasūtras

anubandha sūtras

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Pānini's Śivasūtras

anubandha 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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Minimality criteria

1. The length of the whole list is minimal. 2. The length of the sublist of the anubandhas is minimal and the length of the whole list is as short as possible. 3. The length of the sublist of the sounds is minimal and the length of the whole list is as short as possible.

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Minimality criteria

1. The length of the whole list is minimal. 2. The length of the sublist of the anubandhas is minimal and the length of the whole list is as short as possible. 3. The length of the sublist of the sounds is minimal and the length of the whole list is as short as possible.

– no duplication of h – less anubandhas

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Basic concepts

e d M1 c i f M2 g h M3 b M4 a M5

S-alphabet (A,Σ,<) of Φ:

alphabet marker total order on A∪Σ

Φ={{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 concepts

e d M1 c i f M2 g h M3 b M4 a M5

S-alphabet (A,Σ,<) of Φ:

alphabet marker total order on A∪Σ

Φ={{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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S-encodability and planar formal concept lattices

If Φ is S-encodable, then the formal concept lattice is planar

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S-encodability and planar formal concept lattices

If Φ is S-encodable, then the formal concept lattice is planar

concept lattice for Pānini's phonological classes

.

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S-encodability and planar formal concept lattices

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

part of the concept lattice for Pānini's phonological classes

.

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S-encodability and planar formal concept lattices

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

part of the concept lattice for Pānini's phonological classes

.

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X X X X X X X X X

K5 is a minor of the concept lattice for Pānini's phonological classes

.

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X X X

K5 is a minor of the concept lattice for Pānini's phonological classes

.

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X X

K5 is a minor of the concept lattice for Pānini's phonological classes

.

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X

K5 is a minor of the concept lattice for Pānini's phonological classes

.

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K5 is a minor of the concept lattice for Pānini's phonological classes

.

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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-alphabets

The following statements are equivalent: 1. is S-encodable 2. is planar

Φ={{d,e},{b,c,d,f,},{a,b},{b,c,d}}

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Existence of S-alphabets

The following statements are equivalent: 1. is S-encodable 2. is planar

Φ={{d,e},{b,c,d,f,},{a,b},{b,c,d}} Φ={{d e}

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Existence of S-alphabets

The following statements are equivalent:

S-encodable not S-encodable

1. is S-encodable 2. is planar

  • 3. the S-graph contains all attribute concepts
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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 h g

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Construction of S-alphabets

a b M1 c h g c M2

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Construction of S-alphabets

a b M1 c h g c M2 f i c M2

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Construction of S-alphabets

a b M1 c h g c M2 f i c M2 c M3 c M2

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Construction of S-alphabets

a b M1 c h g c M2 f i c M2 c M3 c M2 d c M3 c M2

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Construction of S-alphabets

a b M1 c h g c M2 f i c M2 c M3 c M2 d c M3 c M2 e M4 M5

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Construction of S-alphabets

a b M1 c h g c M2 f i c M2 c M3 c M2 d c M3 c M2 e M4 M5

× ×

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Pānini's Śivasūtras are optimal .

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Pānini's Śivasūtras are optimal .

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Pānini's Śivasūtras are optimal .