Algebraic Specialization of Generic Functions for Recursive Types - - PowerPoint PPT Presentation

▶

Dec 27, 2023 253 likes •556 views

Algebraic Specialization of Generic Functions for Recursive Types By: Alcino Cunha, Hugo Pacheco Presented by: Joseph Tsun Kiet Man 1 Index 1. Introduction 1. Problem 2. Solutions 2. Specialization: Non-recursive Types 1. Theory 2.

SLIDE 1

Algebraic Specialization of Generic Functions for Recursive Types

By: Alcino Cunha, Hugo Pacheco

1 Presented by: Joseph Tsun Kiet Man

SLIDE 2

Index

1. Introduction
1. Problem
2. Solutions
2. Specialization: Non-recursive Types
1. Theory
2. Example
3. Specialization: Recursive Types
1. Theory
2. Example
4. Concluding remarks
5. Summary

2

SLIDE 3

Introduction | Problem

IMDB Movie Actor Year Title Director Review Box Office Name Played Year Title Role Award

Given the following data structure:

Country Value 3

SLIDE 4

Introduction | Problem

4

SLIDE 5

Introduction | Problem

𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 (𝑛𝑛𝑅𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓

Performance loss due to:

Run-time checks to determine at each node whether to

apply specific / generic behaviour.

Traversal through whole data structure.

5

SLIDE 6

Introduction | Solutions

How to solve the inefficiencies?

Restricting the power of traversal combinators.
Offsets run-time checks to compile-time.
Specialization

Generic function Transformation Specialized function 𝑑𝑑𝑑𝑑𝑑′ = 𝑡𝑑𝑛 ∘ 𝑛𝑛𝑛 𝑡𝑑𝑛 ∘ 𝑛𝑛𝑛 𝑡𝑓𝑡𝑓 ∘ 𝑓𝑓𝑓𝑓𝑓𝑆𝑡 ∘ 𝑛𝑑𝑓𝑓𝑓𝑡 wher ere 𝑛𝑑𝑓𝑓𝑓𝑡 𝐽𝑛𝐽𝐽 𝑛 𝑛 = 𝑛 𝑓𝑓𝑓𝑓𝑓𝑆𝑡 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑓 6

SLIDE 7

Specialization: Non-recursive Types

Theory

Recapitulation SYB:

7

SLIDE 8

Specialization: Non-recursive Types

Theory

Recapitulation SYB:

8

SLIDE 9

Specialization: Non-recursive Types

Theory

Point-free combinators and functions:

9

SLIDE 10

Specialization: Non-recursive Types

Theory

Algebraic Transformation Laws:

10

SLIDE 11

Specialization: Non-recursive Types

Theory

Algebraic Query Laws:

11

SLIDE 12

Specialization: Non-recursive Types

Example

𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 (𝑛𝑛𝑅𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 12

SLIDE 13

Specialization: Non-recursive Types

Example

𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 (𝑛𝑛𝑅𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 13

SLIDE 14

Specialization: Non-recursive Types

Example

𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 (𝑛𝑛𝑅𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 14

SLIDE 15

Specialization: Non-recursive Types

Example

𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 (𝑛𝑛𝑅𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 15

SLIDE 16

Specialization: Recursive Types

Theory

Apply same concept to catamorphisms and paramorphisms:

16

SLIDE 17

Specialization: Recursive Types

Theory

Catamorphisms: functions defined by iteration.

17

SLIDE 18

Specialization: Recursive Types

Theory

Paramorphisms: all functions that can be defined by primitive recursion.

18

SLIDE 19

Specialization: Recursive Types

Theory

Idea:

Rewrite the everywhere function.
Rewrite the everything function

19

SLIDE 20

Specialization: Recursive Types

Theory

Everywhere function redefined:

20

SLIDE 21

Specialization: Recursive Types

Theory

Everywhere function redefined:

21

SLIDE 22

Specialization: Recursive Types

Theory

Recap - Algebraic Transformation Laws:

22

SLIDE 23

Specialization: Recursive Types

Theory

New additional laws for catamorphisms and paramorphisms. 23

SLIDE 24

Specialization: Recursive Types

Example

Function 24

SLIDE 25

Specialization: Recursive Types

Example

Function 25

SLIDE 26

Specialization: Recursive Types

Application

26

SLIDE 27

Specialization: Performance

27

SLIDE 28

Specialization: Improvements

Improvements:

Support for non-regular data types (e.g. nested).
Added symbolic evaluation (intermediate structures are

completely removed during optimization).

28