Non-linear Associative-Commutative Many-to-One Pattern...

Non-linear Associative-CommutativeMany-to-One Pattern Matching with

Sequence Variables

Manuel Krebber

June 30, 2017

Manuel Krebber | June 30, 2017 0/31

Non-linear Associative-Commutative Many-to-One Pattern Matching with Sequence Variables

1 Introduction

2 Types of Matching

3 Algorithms

4 Experiments

5 Conclusions

Introduction

Table of Contents

Function Symbols: f , g

Constant Symbols: a, b, c

Variables: x

x ,yyy

Examples: a, f (xxx , b),f (g(a, b), c)

Introduction

Variables: x

x ,yyy

Introduction

Variables: x

x ,yyy

Introduction

Variables: x

x ,yyy

Introduction

Definition: Find substitution � : X 9 T such that�̂(pattern) = subject

Example: f (xxx ,yyy)

f (a, g(b))

x 7! a

y 7! g(b)

Subject:

Pattern:

Introduction

Pattern Matching

f (a, g(b))

x 7! a

y 7! g(b)

Subject:

Pattern:

Introduction

Pattern Matching

f (a, g(b))

x 7! a

y 7! g(b)

Subject:

Pattern:

Introduction

Pattern Matching

I Functional programming languages

I Computer algebra systems (Mathematica)I Term rewriting systemsI In this case: Linear Algebra

Introduction

Applications

I Functional programming languagesI Computer algebra systems (Mathematica)

I Term rewriting systemsI In this case: Linear Algebra

Introduction

Applications

I Functional programming languagesI Computer algebra systems (Mathematica)I Term rewriting systems

I In this case: Linear Algebra

Introduction

Applications

I Functional programming languagesI Computer algebra systems (Mathematica)I Term rewriting systemsI In this case: Linear Algebra

Introduction

Applications

1 Introduction

2 Types of Matching

3 Algorithms

4 Experiments

5 Conclusions

Types of Matching

Table of Contents

I Syntactic

I Linear (xxx + y

y) vs. non-linear (xxx + x

x)I Sequence variablesI AssociativityI CommutativityI Many-to-one vs. one-to-one

Types of Matching

I SyntacticI Linear (xxx + y

I Sequence variablesI AssociativityI CommutativityI Many-to-one vs. one-to-one

Types of Matching

x)I Sequence variablesI AssociativityI Commutativity

I Many-to-one vs. one-to-one

Types of Matching

x)I Sequence variablesI AssociativityI CommutativityI Many-to-one vs. one-to-one

Types of Matching

(A⇥ B)⇥ C

=A⇥ (B ⇥ C )

Canonical VariadicForm

Types of Matching

Associativity I

(A⇥ B)⇥ C

=A⇥ (B ⇥ C )

Types of Matching

Associativity I

(A⇥ B)⇥ C

=A⇥ (B ⇥ C )

= A⇥ B ⇥ C

Types of Matching

Associativity I

X ⇥M3

(M1 ⇥M2)⇥M3

M1 ⇥M2 ⇥M3

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II

X ⇥M3

(M1 ⇥M2)⇥M3

M1 ⇥M2 ⇥M3

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II

X ⇥M3

(M1 ⇥M2)⇥M3

M1 ⇥M2 ⇥M3

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II

X ⇥M3

(M1 ⇥M2)⇥M3

M1 ⇥M2 ⇥M3

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II

Can match a sequence of terms

Notation: x

⇤⇤⇤ star variable, xxx+++ plus variablestar variables can match empty sequence

Example: �(f (a,xxx+++, d)) = f (a, b, c , d)� = {xxx+++ 7! (b, c)}

Associativity: �(fa(a,xxx , d)) = fa(a, b, c , d)� = {xxx 7! fa(b, c)}

Types of Matching

Sequence Variables

Can match a sequence of termsNotation: x

Types of Matching

Sequence Variables

Types of Matching

Sequence Variables

Types of Matching

Sequence Variables

Types of Matching

Sequence Variables

Types of Matching

Sequence Variables

Ia+ b = b + a

I Sort the arguments: d + a+ c + b ! a+ b + c + d

I Every permutation could match:n! with n = |subject|

Types of Matching

Commutativity

Ia+ b = b + a

Types of Matching

Commutativity

Ia+ b = b + a

Types of Matching

Commutativity

Synt Assoc SeqVar Comm All

Max. # matches 1�n�1m�1

� �n+m�1m�1

�n! n

NP complete no yes yes yes yes

n = |subject|, m = |pattern|

Types of Matching

Complexity

I Many patterns, one subject

I Speedup by simultaneous matchingI Use similarity between patterns

Types of Matching

Many-to-one Matching

I Many patterns, one subjectI Speedup by simultaneous matching

I Use similarity between patterns

Types of Matching

I Many patterns, one subjectI Speedup by simultaneous matchingI Use similarity between patterns

Types of Matching

Algorithms

Table of Contents

Algorithms - One-to-One

Table of Contents

Example: Subject f (a, b) and pattern f (xxx⇤⇤⇤,yyy⇤⇤⇤)

I Try every possible distributionI Generate (weak) compositions:

0 + 2 = 2 {xxx⇤⇤⇤ 7! (),yyy⇤⇤⇤ 7! (a, b)}1 + 1 = 2 {xxx⇤⇤⇤ 7! (a),yyy⇤⇤⇤ 7! (b)}2 + 0 = 2 {xxx⇤⇤⇤ 7! (a, b),yyy⇤⇤⇤ 7! ()}

I Backtracking

Associative and Sequence Variables

I Try every possible distribution

I Generate (weak) compositions:

I Backtracking

Pattern: fc(a, g(c , d), g(xxx),xxx ,yyy⇤⇤⇤, zzz)Subject: fc(a, b, g(b), c , g(c , d), e) Permutations: 6! = 720

Match: � = {zzz 7! e}

1. Constant terms2. Matched variables3. Terms containing variables4. Repeat step 25. Regular Variables6. Sequence Variables

Steps of Commutative Matching

Match: � = {zzz 7! e}1. Constant terms

2. Matched variables3. Terms containing variables4. Repeat step 25. Regular Variables6. Sequence Variables

Match: � = {zzz 7! e}1. Constant terms2. Matched variables

3. Terms containing variables4. Repeat step 25. Regular Variables6. Sequence Variables

Match: � = {zzz 7! e,xxx 7! b}1. Constant terms2. Matched variables3. Terms containing variables

4. Repeat step 25. Regular Variables6. Sequence Variables

Match: � = {zzz 7! e,xxx 7! b}1. Constant terms2. Matched variables3. Terms containing variables4. Repeat step 2

5. Regular Variables6. Sequence Variables

Match: � = {zzz 7! e,xxx 7! b}1. Constant terms2. Matched variables3. Terms containing variables4. Repeat step 25. Regular Variables

6. Sequence Variables

Match: � = {zzz 7! e,xxx 7! b,yyy⇤⇤⇤ 7! HcI}1. Constant terms2. Matched variables3. Terms containing variables4. Repeat step 25. Regular Variables6. Sequence Variables

Match fc(xxx+++,xxx+++,yyy⇤⇤⇤) and fc(a, a, a, b, b, c)

Solve linear diophantine equations (using Extended Euclidean Alg.):

3 = 2xa + ya

2 = 2xb + yb

1 = 2xc + yc

1 xa + xb + xc

Solutions: {x 7! {a, b}, y 7! {a, c}}{x 7! {a}, y 7! {a, b, b, c}}{x 7! {b}, y 7! {a, a, a, c}}

Commutativity + Sequence Variables

3 = 2xa + ya

2 = 2xb + yb

1 = 2xc + yc

1 xa + xb + xc

3 = 2xa + ya

2 = 2xb + yb

1 = 2xc + yc

1 xa + xb + xc

Algorithms - Many-to-One

Table of Contents

Patterns: f (1,xxx⇤⇤⇤), f (1), f (yyy , 0)

f (1))

f (1,xxx⇤⇤⇤))x

⇤⇤⇤1

f (yyy , 0))0y

Subject: f (1, 0)Match: f (yyy , 0) with {yyy 7! 1}

Discrimination Net

f (1))

f (1,xxx⇤⇤⇤))x

⇤⇤⇤1

f (yyy , 0))0y

Discrimination Net

f (1))

f (1,xxx⇤⇤⇤))x

⇤⇤⇤1

f (yyy , 0))0y

Discrimination Net

f (1))

f (1,xxx⇤⇤⇤))x

⇤⇤⇤1

f (yyy , 0))0y

Subject: f (1, 0)Match: f (1,xxx⇤⇤⇤) with {xxx⇤⇤⇤ 7! (0)}

Discrimination Net

Multi Layer Discrimination Net (MLDN)

4g(a))

3g(b))

2g(xxx))

fc(a,xxx ,xxx)H1, 1, 5I

fc(a, g(xxx), g(a))

H2, 4, 5Ifc(g(b), g(xxx))

H2, 3Ifc

fc(a, g(a), g(a)) ) H1, 1, 1, 2, 2, 4, 4, 5I

Many-to-One for Commutative

Multi Layer Discrimination Net (MLDN)

4g(a))

3g(b))

2g(xxx))

fc(a,xxx ,xxx)H1, 1, 5I

fc(a, g(xxx), g(a))

H2, 4, 5Ifc(g(b), g(xxx))

H2, 3Ifc

fc(a, g(a), g(a)) ) H1, 1, 1, 2, 2, 4, 4, 5I

1. Match subject arguments with nested DN

2. Build bipartite graph from matches3. Enumerate all maximum matchings (Hopcroft-Karp, Uno)4. Combine substitutions from matching5. Match sequence variables

1. Match subject arguments with nested DN2. Build bipartite graph from matches

3. Enumerate all maximum matchings (Hopcroft-Karp, Uno)4. Combine substitutions from matching5. Match sequence variables

1. Match subject arguments with nested DN2. Build bipartite graph from matches3. Enumerate all maximum matchings (Hopcroft-Karp, Uno)

4. Combine substitutions from matching5. Match sequence variables

1. Match subject arguments with nested DN2. Build bipartite graph from matches3. Enumerate all maximum matchings (Hopcroft-Karp, Uno)4. Combine substitutions from matching

5. Match sequence variables

1. Match subject arguments with nested DN2. Build bipartite graph from matches3. Enumerate all maximum matchings (Hopcroft-Karp, Uno)4. Combine substitutions from matching5. Match sequence variables

f (a, b)

f (a, a)

f (yyy , b)

f (xxx ,yyy)

Subjectfc(f (a, b), f (a, a))

Patternfc(f (yyy , b), f (xxx ,yyy))

{yyy 7! a}

x 7!a,y

y 7!b}

{xxx 7! a,yyy 7! a}

Bipartite Graph

f (a, b)

f (a, a)

f (yyy , b)

f (xxx ,yyy)

Patternfc(f (yyy , b), f (xxx ,yyy))

{yyy 7! a}

x 7!a,y

y 7!b}

{xxx 7! a,yyy 7! a}

Bipartite Graph

f (a, b)

f (a, a)

f (a,xxx)

f (xxx ,yyy)

Patternfc(f (a,xxx), f (xxx ,yyy))

{xxx 7! b}

x 7!a,y

y 7!b}

{xxx 7! a,yyy 7! a}

Bipartite Graph

f (a, b)

f (a, a)

f (a,xxx)

f (xxx ,yyy)

Patternfc(f (a,xxx), f (xxx ,yyy))

{xxx 7! b}

x 7!a,y

y 7!b}

{xxx 7! a,yyy 7! a}

Bipartite Graph

Experiments

Table of Contents

I Python implementation

I Open source on Github:https://github.com/hpac/matchpy

Experiments

MatchPy

I Python implementationI Open source on Github:

https://github.com/hpac/matchpy

Experiments

MatchPy

I ⇠ 200 patterns for BLAS/LAPACK kernels, e.g. ↵↵↵⇥A

I Supported operations:

Operation Symbol Arity Properties

Multiplication ⇥ variadic associativeAddition + variadic associative, commutativeTransposition T unaryInversion �1 unaryInverse Transposition �T unary

Experiments

Linear Algebra Example

I ⇠ 200 patterns for BLAS/LAPACK kernels, e.g. ↵↵↵⇥A

I Supported operations:

Operation Symbol Arity Properties

Multiplication ⇥ variadic associativeAddition + variadic associative, commutativeTransposition T unaryInversion �1 unaryInverse Transposition �T unary

Experiments

Linear Algebra Example

50 100 150

Number of Patterns

SetupTim

Setup Time

50 100 150

Number of Patterns

Match Time

matchMTOM

Experiments

Match Times

50 100 1500

Number of Patterns

Speedup

50 100 1500

Number of PatternsNumber

ofSubjects

Break Even

Experiments

Speedup

Conclusions

Table of Contents

I Generalized discrimination nets

I Many-to-one matching

Isequence variables

Iseparate associativity/commutativity

I Open source implementation

Conclusions

Contributions

I Generalized discrimination netsI Many-to-one matching

Isequence variables

Conclusions

Contributions

Isequence variables

Conclusions

Contributions

Isequence variables

Conclusions

Contributions

Isequence variables

Conclusions

Contributions

Project is on GitHub:https://github.com/hpac/matchpy

Conclusions

Repository

Thank you for your attention!

Non-linear Associative-Commutative Many-to-One Pattern...

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