Non-linear Associative-Commutative Many-to-One Pattern ...hpac.rwth-aachen.de/~pauldj/talks/Manuel-MS-Thesis.pdfNon-linear Associative-Commutative Many-to-One Pattern Matching with

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

x ,yyy

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

f

g

a

b

c

Introduction

Term




Variables: x

x

x ,yyy


f

g

a

b

c

Introduction

Term




Variables: x

x

x ,yyy


f

g

a

b

c

Introduction

Term




Variables: x

x

x ,yyy


f

g

a

b

c

Introduction

Term


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

Example: f (xxx ,yyy)

f (a, g(b))

x

x

x 7! a

y

y

y 7! g(b)

Subject:

Pattern:

Introduction

Pattern Matching




f (a, g(b))

x

x

x 7! a

y

y

y 7! g(b)

Subject:

Pattern:

Introduction

Pattern Matching




f (a, g(b))

x

x

x 7! a

y

y

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

y) vs. non-linear (xxx + x

x

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

Types of Matching

Types of Matching


I SyntacticI Linear (xxx + y

y


x

x)

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

Types of Matching

Types of Matching



y


x

x)I Sequence variablesI AssociativityI Commutativity

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

Types of Matching

Types of Matching



y


x

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

Types of Matching

Types of Matching


(A⇥ B)⇥ C

⇥

⇥

A B

C

=A⇥ (B ⇥ C )

⇥

A

⇥

B C

Canonical VariadicForm

Types of Matching

Associativity I


(A⇥ B)⇥ C

⇥

⇥

A B

C

=A⇥ (B ⇥ C )

⇥

A

⇥

B C


Types of Matching

Associativity I


(A⇥ B)⇥ C

⇥

⇥

A B

C

=A⇥ (B ⇥ C )

⇥

A

⇥

B C

= A⇥ B ⇥ C

⇥

A B C


Types of Matching

Associativity I


X

X

X ⇥M3

⇥

X

X

X

M3

(M1 ⇥M2)⇥M3

⇥

⇥

M1 M2

M3

M1 ⇥M2 ⇥M3

⇥

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II


X

X

X ⇥M3

⇥

X

X

X

M3

(M1 ⇥M2)⇥M3

⇥

⇥

M1 M2

M3

M1 ⇥M2 ⇥M3

⇥

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II


X

X

X ⇥M3

⇥

X

X

X

M3

(M1 ⇥M2)⇥M3

⇥

⇥

M1 M2

M3

M1 ⇥M2 ⇥M3

⇥

M1 M2 M3

� = {XXX 7! (M1 ⇥M2)}

Types of Matching

Associativity II


X

X

X ⇥M3

⇥

X

X

X

M3

(M1 ⇥M2)⇥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

x

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

x

x




Types of Matching

Sequence Variables



x

x




Types of Matching

Sequence Variables



x

x




Types of Matching

Sequence Variables



x

x




Types of Matching

Sequence Variables



x

x




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

m

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







I Backtracking







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

x

x

⇤⇤⇤1

f (yyy , 0))0y

y

y

f (

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


Discrimination Net



f (1))

f (1,xxx⇤⇤⇤))x

x

x

⇤⇤⇤1

f (yyy , 0))0y

y

y

f (



Discrimination Net



f (1))

f (1,xxx⇤⇤⇤))x

x

x

⇤⇤⇤1

f (yyy , 0))0y

y

y

f (



Discrimination Net



f (1))

f (1,xxx⇤⇤⇤))x

x

x

⇤⇤⇤1

f (yyy , 0))0y

y

y

f (

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


Discrimination Net


Multi Layer Discrimination Net (MLDN)

5a

a

4g(a))

a

3g(b))

b

2g(xxx))

x

xx

g(

1x

x

x

x

xx

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)

5a

a

4g(a))

a

3g(b))

b

2g(xxx))

x

xx

g(

1x

x

x

x

xx

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

x

x 7!a,y

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

x

x 7!a,y

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

x

x 7!a,y

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

x

x 7!a,y

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


https://github.com/hpac/matchpy

I Python implementationI Open source on Github:


Experiments

MatchPy



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

A

A

T⇥B

B

B

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

A

A

T⇥B

B

B

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

0

20

40

60

Number of Patterns

SetupTim

e[m

s]

Setup Time

50 100 150

0

5

10

Number of Patterns

Match

Tim

e[m

s]

Match Time

matchMTOM

Experiments

Match Times


50 100 1500

5

10

15

20

Number of Patterns

Speedup

Speedup

50 100 1500

2

4

6

8

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!

Documents

Non-linear Associative-Commutative Many-to-One Pattern ...hpac.rwth-aachen.de/~pauldj/talks/Manuel-MS-Thesis.pdfNon-linear Associative-Commutative Many-to-One Pattern Matching with