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

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. Example 3. Specialization: Recursive Types 1. Theory 2. Example 4. Concluding remarks 2 5. Summary

Introduction | Problem Given the following data structure: IMDB Movie Actor Name Played Year Title Box Office Director Review Country Value Year Title Role Award 3

Introduction | Problem 4

Introduction | Problem 𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 ( 𝑛𝑛𝑅 𝑆𝑆𝑆𝑆𝑆𝑆 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

Introduction | Solutions How to solve the inefficiencies? Restricting the power of traversal combinators. • Offsets run-time checks to compile-time. • Specialization • Transformation Generic function Specialized function 𝑑𝑑𝑑𝑑𝑑 ′ = 𝑡𝑑𝑛 ∘ 𝑛𝑛𝑛 𝑡𝑑𝑛 ∘ 𝑛𝑛𝑛 𝑡𝑓𝑡𝑓 ∘ 𝑓𝑓𝑓𝑓𝑓𝑆𝑡 ∘ 𝑛𝑑𝑓𝑓𝑓𝑡 wher ere 𝑛𝑑𝑓𝑓𝑓𝑡 𝐽𝑛𝐽𝐽 𝑛 𝑛 = 𝑛 6 𝑓𝑓𝑓𝑓𝑓𝑆𝑡 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑓

Specialization: Non-recursive Types Theory Recapitulation SYB: 7

Specialization: Non-recursive Types Theory Recapitulation SYB: 8

Specialization: Non-recursive Types Theory Point-free combinators and functions: 9

Specialization: Non-recursive Types Theory Algebraic Transformation Laws: 10

Specialization: Non-recursive Types Theory Algebraic Query Laws: 11

Specialization: Non-recursive Types Example 𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 ( 𝑛𝑛𝑅 𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓 ) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 12

Specialization: Non-recursive Types Example 𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 ( 𝑛𝑛𝑅 𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓 ) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 13

Specialization: Non-recursive Types Example 𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 ( 𝑛𝑛𝑅 𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓 ) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 14

Specialization: Non-recursive Types Example 𝑑𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑑𝑓𝑓𝑑𝑓 ( 𝑛𝑛𝑅 𝑆𝑆𝑆𝑆𝑆𝑆 𝑡𝑓𝑡𝑓 ) wher ere 𝑡𝑓𝑡𝑓 𝑆𝑓𝑓𝑓𝑓𝑆 𝑓 = 𝑚𝑓𝑑𝑓𝑑𝑓 𝑓 Rewrite: 15

Specialization: Recursive Types Theory Apply same concept to catamorphisms and paramorphisms: 16

Specialization: Recursive Types Theory Catamorphisms: functions defined by iteration. 17

Specialization: Recursive Types Theory Paramorphisms: all functions that can be defined by primitive recursion. 18

Specialization: Recursive Types Theory Idea: Rewrite the everywhere function. • Rewrite the everything function • 19

Specialization: Recursive Types Theory Everywhere function redefined: 20

Specialization: Recursive Types Theory Everywhere function redefined: 21

Specialization: Recursive Types Theory Recap - Algebraic Transformation Laws: 22

Specialization: Recursive Types Theory New additional laws for catamorphisms and paramorphisms. 23

Specialization: Recursive Types Example Function 24

Specialization: Recursive Types Example Function 25

Specialization: Recursive Types Application 26

Specialization: Performance 27

Specialization: Improvements Improvements: Support for non-regular data types (e.g. nested). • Added symbolic evaluation (intermediate structures are • completely removed during optimization). 28

Specialization: Conclusion Specialization comes near handwritten code. • Specialized code can be optimized by compiler • optimization methods. Specialization by rewriting generic function to • specialized functions. 29