On the Construction of ivas utra -Alphabets Wiebke Petersen - PowerPoint PPT Presentation

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras General problem of S-sortability Given a set of classes, order the elements of the classes (without duplications) in a linear order (in a list) such that each single class forms a continuous interval with respect to that order. The target orders are called S-orders A set of classes is S-sortable if it has an S-order On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras General problem of Śivas¯ utra -alphabets (S-alphabets) Given a set of classes, find an S-order of the elements of the classes. Interrupt this list by markers ( anubandhas ) such that each single class can be denoted by a sound-marker-pair ( praty¯ ah¯ ara ). Note that every S-order becomes a Śivas¯ utra -alphabet (S-alphabet) by adding a marker ( anubandha ) behind each element. Given the set of classes {{ a , b } , { a , b , c } , { a , b , c , d }} , the order a b c d is one of its S-orders and a M 1 b M 2 c M 3 d M 4 is one of its S-alphabets. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Some more Examples S-sortable example The set of classes: {{ d , e } , { a , b } , { b , c , d , f , g , h , i } , { f , i } , { c , d , e , f , g , h , i } , { g , h }} is S-sortable; one of its S-orders is a b c g h f i d e non-S-sortable example The set of classes: {{ a , b } , { b , c } , { a , c }} is not S-sortable. non-S-sortable example The set of classes: {{ d , e } , { a , b } , { b , c , d } , { b , c , d , f }} is not S-sortable. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Some more Examples S-sortable example The set of classes: {{ d , e } , { a , b } , { b , c , d , f , g , h , i } , { f , i } , { c , d , e , f , g , h , i } , { g , h }} is S-sortable; one of its S-orders is a b c g h f i d e non-S-sortable example The set of classes: {{ a , b } , { b , c } , { a , c }} is not S-sortable. non-S-sortable example The set of classes: {{ d , e } , { a , b } , { b , c , d } , { b , c , d , f }} is not S-sortable. a b c d e or e d c b a On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Visualize relations set of classes ( A , Φ) : A = { a , b , c , d , e , f , g , h , i } Φ = {{ d , e } , { a , b } , { b , c , d , f , g , h , i } , { f , i } , { c , d , e , f , g , h , i } , { g , h }} { } h i g d f { d } { f , i } { g , h } a b c d e f g h i e { d , e } ×× b { d , e } { b , c , d , f , g , h , i } ×××××××× c { a , b } { b } ×× { c , d , f , g , h , i } { f , i } × × a { c , d , e , f , g , h , i } ××××××× { c , d , e , f , g , h , i } { b , c , d , f , g , h , i } { a , b } { g , h } ×× { a , b , c , d , e , f , g , h , i } concept lattice of ( A , Φ) formal context of ( A , Φ) On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Visualize relations set of classes ( A , Φ) : A = { a , b , c , d , e , f , g , h , i } Φ = {{ d , e } , { a , b } , { b , c , d , f , g , h , i } , { f , i } , { c , d , e , f , g , h , i } , { g , h }} h i g d f a b c d e f g h i e { d , e } ×× b { b , c , d , f , g , h , i } ×××××××× c { a , b } ×× { f , i } × × a { c , d , e , f , g , h , i } ××××××× { g , h } ×× concept lattice of ( A , Φ) formal context of ( A , Φ) On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Visualize relations {{ d , e } , { a , b } , { b , c , d } , { b , c , d , f }} {{ a , b } , { b , c } , { a , c }} {{ d , e } , { a , b } , { b , c , d , f , g , h , i } , { f , i } , { c , d , e , f , g , h , i } , { g , h }} d b h i g d f a b c e b c c a e f a On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Main theorem of S-sortability Example: S-sortable h i A set of classes is S-sortable without g d f duplications if one of the following e equivalent statements is true: b c Its concept lattice is 1 a Hasse-planar and for any element a there is a node labeled a in the S-graph. The concept lattice of the 2 e g enlarged set of classes is c i f h d Hasse-planar. a b The Ferrers-graph of the 3 enlarged set of classes is bipartite. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Main theorem of S-sortability Example: not S-sortable A set of classes is S-sortable without duplications if one of the following d b equivalent statements is true: c Its concept lattice is 1 e f a Hasse-planar and for any element a there is a node labeled a in the S-graph. The concept lattice of the 2 e a f enlarged set of classes is d c b Hasse-planar. The Ferrers-graph of the 3 enlarged set of classes is bipartite. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 2nd condition: terminology 2nd condition A set of classes ( A , Φ) is S-sortable without duplications if and only if the concept lattice of the enlarged set of classes ( A , ˜ Φ) is Hasse-planar. Enlarging a set of classes means adding all singleton sets: � a ∈ A} ˜ Φ = Φ ∪ {{ a } � Hasse-planarity: {{ a , b } , { a , c } , { b , c }} a c a b c b planar, but not Hasse-planar On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 2nd condition: Hasse-planar ⇒ S-sortable e h g c i f d a b {{ d , e } , { a , b } , { b , c , d , f , g , h , i } , { f , i } , { c , d , e , f , g , h , i } , { g , h }} On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 2nd condition: S-sortable ⇒ Hasse-planar { a , b , c , d , e , f , g , h , i } b 4 { b , c , d , f , g , h , i } b b { c , d , e , f , g , h , i } 3 b { a , b } b { c , d , f , g , h , i } b { d , e } 2 b b b b 1 { b } { g , h } { f , i } { d } b 0 a c g e i b h f d h i g d f b e b c a b On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 2nd condition: evaluation − It is of no help in the construction of S-alphabets with minimal number of markers. − The planarity of a graph is difficult to check. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Main theorem of S-sortability Example: S-sortable h i A set of classes is S-sortable without g d f duplications if one of the following e equivalent statements is true: b c Its concept lattice is 1 a Hasse-planar and for any element a there is a node labeled a in the S-graph. The concept lattice of the Examples: not S-sortable 2 enlarged set of classes is Hasse-planar. d b The Ferrers-graph of the a b c c 3 enlarged set of classes is e f a bipartite. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 1st condition: proof 2nd condition → 1st condition Each S-order of the enlarged set of classes ( A , ˜ Φ) is trivially an S-order of the original set of classes ( A , Φ) . On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 1st condition: proof 1st condition → 2nd condition h i e g d f h g c i f d e a b c b a e a f d b d c b c e f a On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty sequence and choose a walk through the S-graph: h i g d f While moving upwards do nothing. e b While moving c downwards along an a edge add a new marker to the sequence unless its last element is already a marker. If a labeled node is reached, add the labels in arbitrary order to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i sequence and choose a walk g d f through the S-graph: e While moving upwards b do nothing. c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i sequence and choose a walk g d f through the S-graph: e e While moving upwards b do nothing. c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i sequence and choose a walk g d f through the S-graph: e e While moving upwards b do nothing. c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, e add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i sequence and choose a walk g d d f through the S-graph: e e While moving upwards b do nothing. c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, ed add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i sequence and choose a walk g d d f through the S-graph: e e While moving upwards • b do nothing. c c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 c add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i i i sequence and choose a walk g d f f d f through the S-graph: e e While moving upwards • b do nothing. c c c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfi add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h i i sequence and choose a walk g d f d f through the S-graph: e • e While moving upwards • b do nothing. c c c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfiM 2 add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h h h i i sequence and choose a walk g g g d f d f through the S-graph: e • e While moving upwards • b do nothing. c c c c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfiM 2 gh add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h h i i sequence and choose a walk g g d f d f through the S-graph: e • e While moving upwards • c • b do nothing. c c c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfiM 2 ghM 3 add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h h i i sequence and choose a walk g g d f d f through the S-graph: e • e While moving upwards • c • b b do nothing. c c c While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfiM 2 ghM 3 b add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h h i i sequence and choose a walk g g d f d f through the S-graph: e • e While moving upwards • c • b b do nothing. c c c • a While moving a downwards along an edge add a new marker to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfiM 2 ghM 3 bM 4 a add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras S-alphabets with a minimal number of markers procedure Start with the empty h h i i sequence and choose a walk g g d f d f through the S-graph: e • e While moving upwards • c • b b do nothing. c c c • a While moving a downwards along an edge add a new marker • to the sequence unless its last element is already a marker. If a sound is reached, edM 1 cfiM 2 ghM 3 bM 4 aM 5 add the sound to the sequence, unless it has been added before. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 1st condition: evaluation + Allows the construction of S-alphabets with minimal number of markers. − The planarity of a graph is difficult to check. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Main theorem of S-sortability A set of classes is S-sortable without duplications if one of the following equivalent statements is true: Its concept lattice is 1 Hasse-planar and for any element a there is a node labeled a in the S-graph. The concept lattice of the 2 enlarged set of classes is The Ferrers-graph can be Hasse-planar. computed directly from the The Ferrers-graph of the formal context. 3 enlarged set of classes is Its bipartity can be checked bipartite. algorithmically. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 3rd condition: terminology & proof Theorem (Zschalig 2007) The concept lattice of a formal context is Hasse-planar if and only if its Ferrers-graph is bipartite. a b c d e f 0 × × 1 × × × 2 × × 3 × × × On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 3rd condition: terminology & proof Theorem (Zschalig 2007) The concept lattice of a formal context is Hasse-planar if and only if its Ferrers-graph is bipartite. a b c d e f 0 • • • × × • 1 • × × × • • 2 × × • • • • 3 • × × • • × On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 3rd condition: example a b c d e f 0 × × 1 × × × 2 × × 3 × × × 0-b 0-a 0-c 0-f 1-f 1-a 3-a b b b b b b b b b b b b b b 3-e 2-d 2-e 3-d 1-e 2-c 2-f On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 3rd condition: example 0-a 0-b 0-c 0-f 1-a 1-f 3-a 5-a 5-f 6-a 6-b 6-f 7-a 7-b 7-c 7-f 8-a 8-b 8-c 8-d 8-f 9-a b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 4-c 4-e 5-c 5-e 7-e 1-e 2-c 2-d 2-e 2-f 3-d 3-e 4-b 4-d 4-f 5-d 6-d 6-e 9-b 9-c 9-d 9-e On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras 3rd condition: evaluation − It is of no help in the construction of S-alphabets with minimal number of markers. + It can be checked easily by an algorithm. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras What does minimal mean? a.i.un . .l .k | e.o˙ n | ai.auc | hayavarat . | r . | . | ˜ . anam | jhabha˜ n | ghad . | jabagad s | lan nama˙ nan . hadhas . ada´ khaphachat .hathacat .atav | kapay | ´ sas .asar | hal | The Śivas¯ utras are minimal if it is im possible to rearrange the Sanskrit sounds in a new list with anubandhas such that each praty¯ ah¯ ara forms an interval ending before an anubandha , 1 no sound occurs twice 2 or one sound occurs twice but less anubandhas are needed. ⇒ duplicating a sound is worse than adding anubandhas On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to duplicate a sound? On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to yes no duplicate a sound? is it the best choice to duplicate ’h’? Śivas¯ utras are not minimal On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to yes no duplicate a sound? is it the best no yes choice to duplicate ’h’? given the duplication of ’h’, is the number of anubandhas minimal? Śivas¯ utras are not minimal On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to yes no duplicate a sound? is it the best no yes choice to duplicate ’h’? given the duplication of ’h’, yes no is the number of anubandhas minimal? Śivas¯ utras are Śivas¯ utras are not minimal minimal On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Main theorem on S-sortability (part 1a) If a set of classes is S-sortable, then its concept lattice is Hasse-planar. concept lattice of P¯ an .ini’s praty¯ ah¯ aras On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Criterion of Kuratowski A graph which has the graph as a minor is not planar. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Criterion of Kuratowski A graph which has the graph as a minor is not planar. ×× × On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Criterion of Kuratowski A graph which has the graph as a minor is not planar. × × On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Criterion of Kuratowski A graph which has the graph as a minor is not planar. × On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Criterion of Kuratowski A graph which has the graph as a minor is not planar. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Is it necessary to duplicate a sound? Criterion of Kuratowski A graph which has the graph as a minor is not planar. There is no S-alphabet for the set of classes given by P¯ an .ini’s praty¯ ah¯ aras without duplicated elements! On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to yes duplicate a sound? is it the best choice to duplicate ’h’? On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras h and the independent triples h l v { h , l } × × { h , v } × × { v , l } × × Altogether there exists 249 independent triples. h is included in all of them. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to yes duplicate a sound? is it the best yes choice to duplicate ’h’? given the duplication of ’h’, is the number of anubandhas minimal? On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Concept lattice of P¯ an .ini’s praty¯ ah¯ aras with duplicated h On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Concept lattice of P¯ an .ini’s praty¯ ah¯ aras with duplicated h With the Śivas¯ utras P¯ an .ini has chosen one out of nearly 12 million minimal S-alphabets! On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras � a , i , u , M 1 , { r ., l . } 1 , M 2 , {�{ e , o } 2 , M 3 � , �{ ai, au } 3 , M 4 �} 4 , h , y , v , r , M 5 , l , M 6 , ñ, m, { ˙ n, n . , n , } 5 , M 7 , jh, bh,M 8 , { gh, d . h, dh } 6 , M 9 , j , { b, g, d . , d } 7 , M 10 , { kh, ph } 8 , { ch, t .h, th } 9 , { c, t ., t } 10 , M 11 , { k , p } 11 , M 12 , { ś, s ., s } 12 , M 13 , h , M 14 � 2 ! × 2 ! × 2 ! × 2 ! × 3 ! × 3 ! × 4 ! × 2 ! × 3 ! × 3 ! × 2 ! × 3 ! {} 1 {} 2 {} 3 {} 4 {} 5 {} 6 {} 7 {} 8 {} 9 {} 10 {} 11 {} 12 = 2 × 2 × 2 × 2 × 6 × 6 × 24 × 2 × 6 × 6 × 2 × 6 = 11 943 936 On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Are P¯ an .ini’s Śivas¯ utras minimal? is it necessary to yes duplicate a sound? is it the best yes choice to duplicate ’h’? given the duplication of ’h’, yes is the number of anubandhas minimal? Śivas¯ utras are minimal On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Introduction Śivas¯ utras and praty¯ ah¯ aras S-sortability Minimality of the Śivas¯ utras Open problems The story is much more intricate We have neither shown that P¯ an .ini’s technique for the representation of sound classes is optimal nor that he has used his technique in an optimal way. not all sound classes are denoted by praty¯ ah¯ aras rules overgeneralize s¯ utra 1.3.10: yath¯ asam . khyamanude´ sah . sam¯ an¯ am On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Transfer For physical objects ‚duplicating‘ means ‚adding copies‘ Adding copies is annoying but often not impossible Ordering objects in an S-order may improve user-friendliness save time save space simplify visual representations of classifications general hierarchy tree S-sortable On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Transfer Objects in libraries, ware-houses, and stores are only nearly linearly arranged: ⇒ Second (and third) dimension can be used in order to avoid duplications On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Possible minimality criteria total list: .K e o ˙ a i u N . r . l N ai au C h y v r T . l N . ñ m ˙ n n . n M jh bh Ñ gh d .h dh S . j b g d . d Ś kh ph ch t .h th c t . t V k p Y ś s . s R h L total list is of minimal length; 1 sound list is of minimal length; 2 anubandha list is of minimal length; 3 total list is as short as possible while the anubandha list is 4 minimal; total list is as short as possible while the sound list is minimal; 5 On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Possible minimality criteria sound list: a i u r . l e o ai au h y v r l ñ m ˙ n n . n jh bh gh d .h dh . j b g d . d kh ph ch t .h th c t . t k p ś s . s h total list is of minimal length; 1 sound list is of minimal length; 2 anubandha list is of minimal length; 3 total list is as short as possible while the anubandha list is 4 minimal; total list is as short as possible while the sound list is minimal; 5 On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Possible minimality criteria anubandha list: ˙ N K N C T N M Ñ S . . . . Ś V Y R L total list is of minimal length; 1 sound list is of minimal length; 2 anubandha list is of minimal length; 3 total list is as short as possible while the anubandha list is 4 minimal; total list is as short as possible while the sound list is minimal; 5 On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Possible minimality criteria .K e o ˙ a i u N . r . l N ai au C h y v r T . l N . ñ m ˙ n n . n M jh bh Ñ gh d .h dh S . j b g d . d Ś kh ph ch t .h th c t . t V k p Y ś s . s R h L total list is of minimal length; 1 sound list is of minimal length; 2 anubandha list is of minimal length; 3 total list is as short as possible while the anubandha list is 4 minimal; total list is as short as possible while the sound list is minimal; 5 On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Possible minimality criteria .K e o ˙ a i u N . r . l N ai au C h y v r T . l N . ñ m ˙ n n . n M jh bh Ñ gh d .h dh S . j b g d . d Ś kh ph ch t .h th c t . t V k p Y ś s . s R h L total list is of minimal length; 1 sound list is of minimal length; 2 anubandha list is of minimal length; 3 total list is as short as possible while the anubandha list is 4 minimal; total list is as short as possible while the sound list is minimal; 5 ⇒ duplicating sounds is worse than adding anubandhas On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Principle of economy Staal 1962 Another general principle is also implicitly used by P¯ an . ini. This is the famous economy criterion [. . . ] In accordance with this principle each linguistic rule should be given in the shortest possible form, whereas the number of metalinguistic symbols should be reduced as far as possible. ⇒ 5. criterion of minimality: total list is as short as possible while the sound list is minimal On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Example: semi-formal argument Kiparsky 1991 The reasoning from economy goes like this. To be grouped together in a praty¯ ah¯ ara, sounds must make up a continuous segment of the list. Economy requires making the list as short as possible, which means avoiding repetitions of sounds, and using as few markers as possible. Consequently, if class A properly includes class B, the elements shared with B should be listed last in A; the marker that follows can then be used to form praty¯ ah¯ aras for both A and B. In this way the economy principle, by selecting the shortest grammar, determines both the ordering of sounds and the placement of markers among them. On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

Example: semi-formal argument Śivas¯ utras : .K e o ˙ a i u N . r . l N ai au C h y v r T . l N . ñ m ˙ n n . n M jh bh Ñ gh d .h dh S . j b g d . d Ś kh ph ch t .h th c t . t V k p Y ś s . s R h L aK = { a, i, u, r ., l . } , iK = { i, u, r ., l . } and uK = { u, r ., l . } ⇒ a < i < u < r .,l . but: jhL = { h, s, s ., ś, p, k, t, t ., c, th, t .h, ch, ph, kh, d, d . , g, b, j, dh, d . h, gh, bh, jh } jhR = { s, s ., ś, p, k, t, t ., c, th, t .h, ch, ph, kh, d, d . , g, b, j, dh, d . h, gh, bh, jh } jhY = { p, k, t, t ., c, th, t .h, ch, ph, kh, d, d . , g, b, j, dh, d . h, gh, bh, jh } jhŚ = { d, d . , g, b, j, dh, d . h, gh, bh, jh } and jhS . = { dh, d . h, gh, bh, jh } ⇒ h < s, s ., ś < p, k, t, t ., c, th, t .h, ch, ph, kh, d < d . , g, b, j < dh, d . h, gh, bh, jh On the Construction of Śivas¯ utra -Alphabets Wiebke Petersen

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