and formal language theory Sylvain Salvati INRIA Bordeaux - PowerPoint PPT Presentation

An example S ( x 1 y 1 x 2 y 2 ) ← P ( x 1 , x 2 ) , Q ( y 1 , y 2 ) P ( ax 1 , bx 2 ) ← P ( x 1 , x 2 ) Q ( ǫ, ǫ ) P ( ǫ, ǫ ) ← Q ( c , d ) P ( ǫ, ǫ ) Q ( cx 1 , dx 2 ) ← Q ( x 1 , x 2 ) Q ( cc , dd ) P ( a , b ) Q ( ǫ, ǫ ) ← S ( accbdd ) S ( a n c m b n d m ) ← P ( a n , b n ) , Q ( c m , d m ) The language is: { a n c m b n d m | n ∈ N ∧ m ∈ N }

The well-nestedness constraint I ( x 1 y 1 , y 2 x 2 ) ← J ( x 1 , x 2 ) , K ( y 1 , y 2 ) I ( x 1 y 1 , x 2 y 2 ) ← J ( x 1 , x 2 ) , K ( y 1 , y 2 ) A ( x 1 z 1 , z 2 x 2 y 1 , y 2 y 3 x 3 ) ← B ( x 1 , x 2 , x 3 ) C ( y 1 , y 2 , y 3 ) D ( z 1 , z 2 ) A ( z 1 x 1 , y 1 x 2 z 2 y 2 x 3 , y 3 ) ← B ( x 1 , x 2 , x 3 ) C ( y 1 , y 2 , y 3 ) D ( z 1 , z 2 )

MCFL wn and MCFL MCFL MCFL wn { a n 1 . . . a n m | n ∈ N } { w m +1 | w ∈ { a ; b } ∗ , m ∈ N }

MCFL wn and MCFL MCFL { w 1 . . . w n z n w n z n − 1 . . . z 1 w 1 z 0 w r 1 . . . w r n | n ∈ N , w i ∈ { c ; d } + , z 0 , . . . , z n ∈ D ∗ 1 } Staudacher 1993 Michaelis 2005 MCFL wn { a n 1 . . . a n m | n ∈ N } { w m +1 | w ∈ { a ; b } ∗ , m ∈ N }

MCFL wn and MCFL MCFL { w 1 . . . w n z n w n z n − 1 . . . z 1 w 1 z 0 w r 1 . . . w r n | n ∈ N , w i ∈ { c ; d } + , z 0 , . . . , z n ∈ D ∗ 1 } Staudacher 1993 Michaelis 2005 MCFL wn { w # w # w | w ∈ D ∗ 1 } { a n 1 . . . a n m | n ∈ N } Engelfriet, Skyum 1976 { w m +1 | w ∈ { a ; b } ∗ , m ∈ N }

MCFL wn and MCFL MCFL { w 1 . . . w n z n w n z n − 1 . . . z 1 w 1 z 0 w r 1 . . . w r n | n ∈ N , w i ∈ { c ; d } + , z 0 , . . . , z n ∈ D ∗ 1 } Staudacher 1993 Michaelis 2005 MCFL wn { w # w # w | w ∈ D ∗ 1 } { a n 1 . . . a n m | n ∈ N } Engelfriet, Skyum 1976 { w m +1 | w ∈ { a ; b } ∗ , m ∈ N } { w # w | w ∈ D ∗ 1 } Kanazawa, S. 2010

Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

A 2-MCFG for O 2 S ( xy ) ← Inv ( x , y ) Inv ( x 1 y 1 , y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 y 1 y 2 , x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , y 1 y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 x 2 y 1 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , x 2 y 1 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( α x 1 α, x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 α, α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( x 1 α, x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 , α x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 y 1 x 2 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( ǫ, ǫ ) ← where α ∈ { a ; b }

A 2-MCFG for O 2 S ( xy ) ← Inv ( x , y ) Inv ( x 1 y 1 , y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 y 1 y 2 , x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) well-nested binary rules Inv ( x 1 , y 1 y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 x 2 y 1 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , x 2 y 1 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( α x 1 α, x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 α, α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( x 1 α, x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 , α x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 y 1 x 2 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) non well-nested rules Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( ǫ, ǫ ) ← where α ∈ { a ; b }

A 2-MCFG for O 2 S ( xy ) ← Inv ( x , y ) Inv ( x 1 y 1 , y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 y 1 y 2 , x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) well-nested binary rules Inv ( x 1 , y 1 y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 x 2 y 1 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , x 2 y 1 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( α x 1 α, x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , x 2 α ) ← Inv ( x 1 , x 2 ) rules for constants Inv ( x 1 α, α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( x 1 α, x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 , α x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 y 1 x 2 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) non well-nested rules Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( ǫ, ǫ ) ← where α ∈ { a ; b }

A 2-MCFG for O 2 terminal rule S ( xy ) ← Inv ( x , y ) Inv ( x 1 y 1 , y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 y 1 y 2 , x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) well-nested binary rules Inv ( x 1 , y 1 y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 x 2 y 1 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , x 2 y 1 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( α x 1 α, x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( α x 1 , x 2 α ) ← Inv ( x 1 , x 2 ) rules for constants Inv ( x 1 α, α x 2 ) ← Inv ( x 1 , x 2 ) Inv ( x 1 α, x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 , α x 2 α ) ← Inv ( x 1 , x 2 ) Inv ( x 1 y 1 x 2 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) non well-nested rules Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( ǫ, ǫ ) ← initial rule where α ∈ { a ; b } Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable.

A graphical interpretation of O 2 . Graphical interpretation of the word aaabaabaabbbbbaabbabbbbaaaabbbbbbbbaaa :

A graphical interpretation of O 2 . Graphical interpretation of the word aaabaabaabbbbbaabbabbbbaaaabbbbbbbbaaa : The words in O 2 are precisely the words that are represented as closed curves: babbababbabbabbababbaaabbbabbaaaabbabbbaba

Parsing with the grammar Rule Inv ( ax 1 a , x 2 ) ← Inv ( x 1 , x 2 ) Inv ( abaabaaababbbabaaabbbabbbbaaaba , babbbbaaaaaababbaab ) Inv ( baabaaababbbabaaabbbabbbbaaab , babbbbaaaaaababbaab )

Parsing with the grammar Rule: Inv ( x 1 y 1 , y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( baabaaababbbabaaabbbabbbbaaab , babbbbaaaaaababbaab ) Inv ( baabaaaba , bbaab ) Inv ( bbbabaaabbbabbbbaaab , babbbbaaaaaaba )

Parsing with the grammar Rule Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( bbbabaaabbbabbbbaaab , babbbbaaaaaaba ) Inv ( babb , ba ) Inv ( bbbabaaabbbabbbbaaab , bbaaaaaa )

Parsing with the grammar Rule: Inv ( x 1 b , bx 2 ) ← Inv ( x 1 , x 2 ) Inv ( bbbabaaabbbabbbbaaab , bbaaaaaa ) Inv ( bbbabaaabbbabbbbaaa , baaaaaa )

Parsing with the grammar Rule: Inv ( bx 1 , bx 2 ) ← Inv ( x 1 , x 2 ) Inv ( bbbabaaabbbabbbbaaa , baaaaaa ) Inv ( bbabaaabbbabbbbaaa , aaaaaa )

Parsing with the grammar Rule: Inv ( x 1 y 1 , y 2 x 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( bbabaaabbbabbbbaaa , aaaaaa ) Inv ( aaa , aaa ) Inv ( bbabaaabbbabbbb , aaa )

Parsing with the grammar Rule: Inv ( bx 1 b , x 2 ) ← Inv ( x 1 , x 2 ) Inv ( bbabaaabbbabbbb , aaa ) Inv ( babaaabbbabbb , aaa )

Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 1: w 1 or w 2 equal ǫ :

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 1: w 1 or w 2 equal ǫ : w.l.o.g., w 1 � = ǫ , then by induction hypothesis, for any v 1 and v 2 different from ǫ such that w 1 = v 1 v 2 , Inv ( v 1 , v 2 ) is derivable then: Inv ( v 1 , v 2 ) Inv ( ǫ, ǫ ) Inv ( x 1 x 2 , y 1 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( v 1 v 2 = w 1 , ǫ )

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 2: w 1 = s 1 w ′ 1 s 2 and w 2 = s 3 w ′ 2 s 4 and for i , j ∈ { 1; 2; 3; 4 } , s.t. i � = j , { s i ; s j } ∈ {{ a ; a } ; { b ; b }} :

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 2: w 1 = s 1 w ′ 1 s 2 and w 2 = s 3 w ′ 2 s 4 and for i , j ∈ { 1; 2; 3; 4 } , s.t. i � = j , { s i ; s j } ∈ {{ a ; a } ; { b ; b }} : e.g. , if i = 1, j = 2, s 1 = a and s 2 = a then by induction hypothesis Inv ( w ′ 1 , w 2 ) is derivable and: Inv ( w ′ 1 , w 2 ) Inv ( ax 1 a , x 2 ) ← Inv ( x 1 , x 2 ) Inv ( aw ′ 1 a , w 2 )

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 3: the curves representing w 1 and w 2 have a non-trivial intersection point:

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 3: the curves representing w 1 and w 2 have a non-trivial intersection point: v 1 v 3 w 1 w 2 Inv ( v 1 , v 4 ) Inv ( v 2 , v 3 ) A B Inv ( v 1 v 2 = w 1 , v 3 v 4 = w 2 ) v 4 v 2

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 4: the curve representing w 1 or w 2 starts or ends with a loop:

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 4: the curve representing w 1 or w 2 starts or ends with a loop: v 1 v 2 Inv ( v 1 , ǫ ) Inv ( v 2 , w 2 ) Inv ( v 1 v 2 = w 1 , w 2 )

The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 ∈ O 2 , Inv ( w 1 , w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( | w 1 w 2 | , max( | w 1 | , | w 2 | )) . There are five cases: Case 5: w 1 and w 2 do not start or end with compatible letters, the curve representing then do not intersect and do not start or end with a loop.

Case 5 No rule other than Inv ( x 1 y 1 x 2 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) can be used.

Case 5 No rule other than Inv ( x 1 y 1 x 2 , y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) Inv ( x 1 , y 1 x 2 y 2 ) ← Inv ( x 1 , x 2 ) , Inv ( y 1 , y 2 ) can be used.

The relevance of case 5 The word abbaabaaabbbbaaaba is not in the language of the grammar only containing the well-nested rules.

The relevance of case 5: a proof is now in hand ◮ Joshi (1985) [ MIX ] represents the extreme case of the degree of free word order permitted in a language. This extreme case is linguistically not relevant. [. . . ] TAGs also cannot generate this language although for TAGs the proof is not in hand yet. Theorem (Kanazawa, S. 12) There is no 2-MCFL wn (or TAG) generating MIX or O 2 .

Solving case 5: towards geometry

Solving case 5: towards geometry

Solving case 5: towards geometry

Solving case 5: towards geometry

Solving case 5: a geometrical invariant

Solving case 5: a geometrical invariant

Solving case 5: a geometrical invariant An invariant on the Jordan curve representing w ′ 1 w ′ 2 : w ′ 1 = aw ′′ 1 a and w ′ 2 = aw ′′ w ′ 1 = aw ′′ 1 a and w ′ 2 = aw ′′ 2 a 2 b w ′ 1 = aw ′′ 1 a and w ′ 2 = bw ′′ w ′ 1 = aw ′′ 1 a and w ′ 2 = bw ′′ 2 a 2 b

Solving case 5: a geometrical invariant An invariant on the Jordan curve representing w ′ 1 w ′ 2 : w ′ 1 = aw ′′ 1 b and w ′ 2 = aw ′′ w ′ 1 = aw ′′ 1 b and w ′ 2 = bw ′′ 2 b 2 a w ′ 1 = aw ′′ w ′ w ′ 1 = aw ′′ w ′ 1 a and 2 = a 1 a and 2 = b

Solving case 5: a geometrical invariant An invariant on the Jordan curve representing w ′ 1 w ′ 2 : w ′ 1 = aw ′′ 1 b and w ′ w ′ 1 = aw ′′ 1 b and w ′ 2 = a 2 = b

Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

Jordan curves illustration from: A combinatorial introduction to topology by Michael Henle (Dover Publications).

A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A ′ and D ′ inside J such that − AD = − → − → A ′ D ′ , then there are two points B and C pairwise distinct from A and D such that A , B , C , and D appear in that order on one of the arcs going from A to D and − AD = − → → BC . A ′ D ′ A D

A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A ′ and D ′ inside J such that − AD = − → − → A ′ D ′ , then there are two points B and C pairwise distinct from A and D such that A , B , C , and D appear in that order on one of the arcs going from A to D and − AD = − → → BC . A ′ D ′ A D

A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A ′ and D ′ inside J such that − AD = − → − → A ′ D ′ , then there are two points B and C pairwise distinct from A and D such that A , B , C , and D appear in that order on one of the arcs going from A to D and − AD = − → → BC . B C A ′ D ′ A D F E G H

A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A ′ and D ′ inside J such that − AD = − → − → A ′ D ′ , then there are two points B and C pairwise distinct from A and D such that A , B , C , and D appear in that order on one of the arcs going from A to D and − AD = − → → BC . Applying this Theorem solves case 5.

A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A ′ and D ′ inside J such that − AD = − → − → A ′ D ′ , then there are two points B and C pairwise distinct from A and D such that A , B , C , and D appear in that order on one of the arcs going from A to D and − AD = − → → BC . Applying this Theorem solves case 5.

A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A ′ and D ′ inside J such that − AD = − → − → A ′ D ′ , then there are two points B and C pairwise distinct from A and D such that A , B , C , and D appear in that order on one of the arcs going from A to D and − AD = − → → BC . Applying this Theorem solves case 5.

Winding number Let wn ( J , z ) be the winding number of a closed curve around z . illustration from: A combinatorial introduction to topology by Michael Henle (Dover Publications).

An interesting Lemma � C → C − { 0 } Let exp : . e 2 i π z z → Lemma AB such that − � → Given an simple arc AB = k ∈ N , we have: � wn ( exp ( AB ) , 0) = k

Translation becomes rotation � C → C − { 0 } exp : . e 2 i π z z → B, C B C J I A, D O G, H I, J E, F A D F E G H

Translation becomes rotation � C → C − { 0 } exp : . e 2 i π z z → B, C B C J I A, D O I, J E, F A D F E

An interesting characterization Lemma AD such that − � → Given an simple arc AD = 1 , we have: BC such that − AD = − → → � � AD contains a proper subarc BC iff ◮ � exp ( AD ) is not a Jordan curve.

Jordan curves and winding numbers illustration from: A combinatorial introduction to topology by Michael Henle (Dover Publications). Theorem: There is k ∈ {− 1; 1 } such that the winding number of Jordan curve around a point in its interior is k , its winding number around a point in its exterior is 0.

Proving the characterization Lemma AD such that − � → Given an simple arc AD = 1 , we have: BC such that − AD = − → → � � � ◮ AD contains a proper subarc BC iff exp ( AD ) is not a Jordan curve. Proof

Proving the characterization Lemma AD such that − � → Given an simple arc AD = 1 , we have: BC such that − AD = − → → � � � ◮ AD contains a proper subarc BC iff exp ( AD ) is not a Jordan curve. Proof � BC such that − � AD = − → → ◮ by 1-periodicity of exp , if AD contains a proper subarc BC , � then exp ( AD ) is not a Jordan curve,

Proving the characterization Lemma AD such that − � → Given an simple arc AD = 1 , we have: BC such that − AD = − → → � � � ◮ AD contains a proper subarc BC iff exp ( AD ) is not a Jordan curve. Proof � BC such that − � AD = − → → ◮ by 1-periodicity of exp , if AD contains a proper subarc BC , � then exp ( AD ) is not a Jordan curve, � ◮ if exp ( AD ) is not a Jordan curve: ◮ take the closed curve C obtained by removing the closed subcurves of � exp ( AD ) that have a negative winding number,

Proving the characterization Lemma AD such that − � → Given an simple arc AD = 1 , we have: BC such that − AD = − → → � � � ◮ AD contains a proper subarc BC iff exp ( AD ) is not a Jordan curve. Proof � BC such that − � AD = − → → ◮ by 1-periodicity of exp , if AD contains a proper subarc BC , � then exp ( AD ) is not a Jordan curve, � ◮ if exp ( AD ) is not a Jordan curve: ◮ take the closed curve C obtained by removing the closed subcurves of � exp ( AD ) that have a negative winding number, ◮ take a proper closed subcurve D of C that is minimal for inclusion,