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Diagram Rewriting: Examples and Theory Yves Lafont CNRS Institut de Mathmatiques de Luminy Universit de la Mditerrane (Aix-Marseille 2) Hagenberg, 14 July 2008 Introduction Planar diagrams are 2-dimensional words. Terms can

SLIDE 1

Diagram Rewriting: Examples and Theory

Yves Lafont CNRS – Institut de Mathématiques de Luminy Université de la Méditerranée (Aix-Marseille 2) Hagenberg, 14 July 2008

SLIDE 2

Introduction

◮ Planar diagrams are 2-dimensional words. ◮ Terms can be encoded as diagrams (Burroni 91). ◮ Diagrams are related to proof nets / interaction nets. ◮ A word reduction can be seen as a planar diagram. ◮ Many other examples (braids, knots, circuits, . . . ).

SLIDE 3

Planar diagrams

Inputs/outputs:

φ

Sequential and parallel composition:

φ ψ ψ φ

Laws of associativity, units, and interchange:

= φ ψ φ ψ φ ψ =

SLIDE 4

Classification of interpretations

Basic case (control flow): + (disjoint union) f : p → q (p = {1, . . . , p} = 1 + · · · + 1) Classical case (data flow): × (Cartesian product) f : Bp → Bq (B = {0, 1} = 1 + 1, Bp = B × · · · × B) Linear case: ⊕ (direct sum) f : Zp

2 → Zq 2

(Z2 = {0, 1}, Zp

2 = Z2 ⊕ · · · ⊕ Z2)

Quantum case: ⊗ (tensor product) f : B⊗p → B⊗q (B = C2 = C ⊕ C, B⊗p = B ⊗ · · · ⊗ B)

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Example 1: S (finite permutations)

Generator: Relations:

= =

Theorem:

◮ Any finite permutation is a product of transpositions:

· · · · · ·

◮ Two diagrams define the same permutation if and only if

they are equivalent modulo the above relations.

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Canonical forms

Grammar for canonical forms:

is void or · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

is

Lemma:

◮ Any permutation corresponds to a unique canonical form. ◮ Any diagram reduces to a canonical form by these rules:

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Proof of the lemma

By double induction on the width (wires) and the size (gates). There are four cases:

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

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Diagrams versus words

Theorem: The symmetric group Sn is presented by the generators σi (for 1 ≤ i < n) and the following relations: σ2

i = 1,

σiσi+1σi = σi+1σiσi+1, σiσj = σjσi (for i+1 < j). Diagrammatic interpretation of relation σ2

i = 1:

= · · · · · · · · · · · ·

Diagrammatic interpretation of relation σiσj = σjσi:

= · · · · · · · · · · · · · · · · · ·

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Rewriting

Theorem: This rewrite system is noetherian and confluent.

◮ Termination is straightforward. ◮ Confluence follows from the previous results.

What are the critical peaks?

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Confluence

Confluence of critical peaks:

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Global conflicts

Here, there is one global conflict:

φ · · · · · · · · · · · · · · · · · · · · ·

where φ is

It suffices indeed to consider the case where φ is canonical:

· · · φ · · · · · · ψ · · · · · · · · · · · · φ · · · · · · · · · φ′ · · · · · · ∗ ∗ φ′ · · · · · · φ′ φ

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Example 2: F (finite maps)

Generators: Relations:

= = = = = = =

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Rewrite rules for F

Termination is proved by using some polynomial interpretation.

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The 68 critical peaks for F

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Example 3: Fop (theory of structural gates)

Generators: Relations:

= = = = = = =

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Terms versus diagrams

Theorem (Burroni 91): Any finite equational theory yields a finite presentation. Theorem (Lafont 95): Any finite convergent left linear term rewrite system yields a finite convergent diagram rewrite system. The non linear case is more difficult because of critical peaks.

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Example 4: L(Z2) (linear boolean maps)

Generators:

y x+y x y y x x x x x x

Reversible gates:

x+y y x+y x x y x y y x+y x x+y x y x y

Decomposition:

= = = =

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Rewrite rules for L(Z2)

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Example 5: GL(Z2) (linear boolean permutations)

Generators: Relations:

= = = = = =

Rules:

Diagram Rewriting: Examples and Theory

Yves Lafont CNRS – Institut de Mathématiques de Luminy Université de la Méditerranée (Aix-Marseille 2) Hagenberg, 14 July 2008

Introduction

◮ Planar diagrams are 2-dimensional words. ◮ Terms can be encoded as diagrams (Burroni 91). ◮ Diagrams are related to proof nets / interaction nets. ◮ A word reduction can be seen as a planar diagram. ◮ Many other examples (braids, knots, circuits, . . . ).

Planar diagrams

Inputs/outputs:

φ

Sequential and parallel composition:

φ ψ ψ φ

Laws of associativity, units, and interchange:

= φ ψ φ ψ φ ψ =

Classification of interpretations

Basic case (control flow): + (disjoint union) f : p → q (p = {1, . . . , p} = 1 + · · · + 1) Classical case (data flow): × (Cartesian product) f : Bp → Bq (B = {0, 1} = 1 + 1, Bp = B × · · · × B) Linear case: ⊕ (direct sum) f : Zp

2 → Zq 2

(Z2 = {0, 1}, Zp

2 = Z2 ⊕ · · · ⊕ Z2)

Quantum case: ⊗ (tensor product) f : B⊗p → B⊗q (B = C2 = C ⊕ C, B⊗p = B ⊗ · · · ⊗ B)

Example 1: S (finite permutations)

Generator: Relations:

= =

Theorem:

◮ Any finite permutation is a product of transpositions:

· · · · · ·

◮ Two diagrams define the same permutation if and only if

they are equivalent modulo the above relations.

Canonical forms

Grammar for canonical forms:

is void or · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

is

Lemma:

◮ Any permutation corresponds to a unique canonical form. ◮ Any diagram reduces to a canonical form by these rules:

Proof of the lemma

By double induction on the width (wires) and the size (gates). There are four cases:

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

Diagrams versus words

Theorem: The symmetric group Sn is presented by the generators σi (for 1 ≤ i < n) and the following relations: σ2

i = 1,

σiσi+1σi = σi+1σiσi+1, σiσj = σjσi (for i+1 < j). Diagrammatic interpretation of relation σ2

i = 1:

= · · · · · · · · · · · ·

Diagrammatic interpretation of relation σiσj = σjσi:

= · · · · · · · · · · · · · · · · · ·

Rewriting

Theorem: This rewrite system is noetherian and confluent.

◮ Termination is straightforward. ◮ Confluence follows from the previous results.

What are the critical peaks?

Confluence

Confluence of critical peaks:

Global conflicts

Here, there is one global conflict:

It suffices indeed to consider the case where φ is canonical:

Example 2: F (finite maps)

Generators: Relations:

= = = = = = =

Rewrite rules for F

Termination is proved by using some polynomial interpretation.

The 68 critical peaks for F

Example 3: Fop (theory of structural gates)

Generators: Relations:

= = = = = = =

Terms versus diagrams

Theorem (Burroni 91): Any finite equational theory yields a finite presentation. Theorem (Lafont 95): Any finite convergent left linear term rewrite system yields a finite convergent diagram rewrite system. The non linear case is more difficult because of critical peaks.

Example 4: L(Z2) (linear boolean maps)

Generators:

y x+y x y y x x x x x x

Reversible gates:

x+y y x+y x x y x y y x+y x x+y x y x y

Decomposition:

= = = =

Rewrite rules for L(Z2)

Example 5: GL(Z2) (linear boolean permutations)

Generators: Relations:

Rules:

References

◮ Albert Burroni, Higher dimensional word problem (TCS

1993)

◮ Yves Lafont, Towards an algebraic theory of Boolean

circuits (JPAA 2003)

◮ Yves Guiraud, Termination Orders for 3-Dimensional

Rewriting (JPAA 2006)