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Normal forms for planar string diagrams Antonin Delpeuch, Jamie - - PowerPoint PPT Presentation
Normal forms for planar string diagrams Antonin Delpeuch, Jamie - - PowerPoint PPT Presentation
Normal forms for planar string diagrams Antonin Delpeuch, Jamie Vicary SYCO 2 Background: word problem in higher categories ? = Background: word problem in higher categories ? = Theorem (Makkai, 2005) The word problem for cells of a
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Background: word problem in higher categories
= ?
Theorem (Makkai, 2005)
The word problem for cells of a finitely generated strict n-category is decidable. Idea of the proof: the configuration space of a given diagram is finite.
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Background: word problem in higher categories
= ?
Theorem (Makkai, 2005)
The word problem for cells of a finitely generated strict n-category is decidable. Idea of the proof: the configuration space of a given diagram is finite. As an algorithm, this is vastly inefficient.
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The planar case
u v . . . . . . . . . . . . . . . u v . . . . . . . . . . . . . . . ≃
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The planar case
u v . . . . . . . . . . . . . . . u v . . . . . . . . . . . . . . . →R
L←
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The planar case
u v . . . . . . . . . . . . . . . u v . . . . . . . . . . . . . . . →R
L←
Theorem
→R is convergent (terminating and confluent) on connected diagrams.
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Confluence of right exchanges
Lemma
→R is locally confluent.
u v w u v w u v w u v w u v w u v w
R R R R R R
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Confluence of right exchanges
Lemma
→R is locally confluent.
u v w u v w u v w u v w u v w u v w
R R R R R R
=
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Termination
u v v u u v v u u v Termination fails in general, but:
Theorem
→R terminates in O(n3) for connected diagrams of size n.
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Proof of termination
First, in the case of linear diagrams:
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Proof of termination
First, in the case of linear diagrams:
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Proof of termination
First, in the case of linear diagrams:
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Proof of termination
First, in the case of linear diagrams:
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Proof of termination
First, in the case of linear diagrams: collapsible funnel
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams:
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Upper bound on derivation length
From this decomposition, we obtain inductively the following bound:
Lemma
→R terminates on linear graphs of length n in O(n3). The bound is attained for spiral-shaped diagrams: Number of steps for a spiral with n vertices: n
3
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General case
Connected graph G
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General case
Connected graph G Spanning tree G ′
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General case
Connected graph G Spanning tree G ′ Linear envelope
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General case
Connected graph G Spanning tree G ′ Linear envelope O(n3) exchanges, each of them taking O(n) time to perform: word problem solved in O(n4).
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Direct algorithm to compute normal forms
By induction on the number of edges. First case: the diagram has a leaf.
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Direct algorithm to compute normal forms
By induction on the number of edges. First case: the diagram has a leaf. ◮ remove this leaf;
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Direct algorithm to compute normal forms
By induction on the number of edges. First case: the diagram has a leaf. ◮ remove this leaf; ◮ normalize the diagram recursively;
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Direct algorithm to compute normal forms
By induction on the number of edges. First case: the diagram has a leaf. ◮ remove this leaf; ◮ normalize the diagram recursively; ◮ add the leaf back at the unique height making the diagram normalized.
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Direct algorithm to compute normal forms
Second case: the diagram has a face
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Direct algorithm to compute normal forms
Second case: the diagram has a face ◮ Find an eliminable edge in the face and remove it;
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Direct algorithm to compute normal forms
Second case: the diagram has a face ◮ Find an eliminable edge in the face and remove it;
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Direct algorithm to compute normal forms
Second case: the diagram has a face ◮ Find an eliminable edge in the face and remove it; ◮ normalize the graph recursively;
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Direct algorithm to compute normal forms
Second case: the diagram has a face ◮ Find an eliminable edge in the face and remove it; ◮ normalize the graph recursively; ◮ add the edge back.
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Direct algorithm to compute normal forms
Second case: the diagram has a face ◮ Find an eliminable edge in the face and remove it; ◮ normalize the graph recursively; ◮ add the edge back. Each step removes one edge and requires linear time in the number
- f vertices, so word problem solved in O(nm).
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Linear time solution
Theorem
Isotopy of connected planar maps can be decided in linear time (Hopcroft and Wong, 1974)
? =
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Linear time solution
Theorem
Isotopy of connected planar directed maps can be decided in linear time.
? =
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Linear time solution
Theorem
Isotopy of connected string diagrams can be decided in linear time.
? =
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Disconnected case
Theorem
Isotopy of string diagrams can be decided in quadratic time.
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References
- A. Delpeuch and J. Vicary.