Generating Good Generators for Inductive Relations POPL 2018 - PowerPoint PPT Presentation

Generating Good Generators for Inductive Relations POPL 2018 Leonidas Lampropoulos 1 Zoe Paraskevopoulou 2 Benjamin Pierce 1 1 University of Pennsylvania 2 Princeton University

Generating Good Generators for Inductive Relations in a property-based random testing tool for Coq 1 / 21

�� Testing with QuickChick ∀ ( x : A ) . P ( x ) More confidence! ✓ P ( a )? ✗ Counterexample! 2 / 21

�� Testing with QuickChick ∀ ( x : A ) .Q ( x ) ⇒ P ( x ) More confidence! ✓ ✓ Q ( a )? P ( a )? ✗ Counterexample! ✗ 3 / 21

�� Sparse Preconditions Q 4 / 21

�� ∀ �� → �� → • Random data generator for �� • Type inference function • � �� as a function • Decidability for � �� 5 / 21

�� ∀ �� → �� → More confidence? 6 / 21

� � � � � �� λ � ( N → N ) → N .x 6 � λ � N �� λ � N �� λ � N �� λ � ( N → N ) → N �� λ � N �� λ � N �� λ � N �� λ � ( N → N ) → ( N → N ) �� λ � N �� λ � N �� λ � N �� λ � N → N � λ � N �� λ � N → N �� λ � N �� λ � N �� λ � N �� λ � N → N �� 7 / 21

�� Testing with Good Generators ∀ ( x : A ) .Q ( x ) ⇒ P ( x ) More confidence! ✓ �� such that Q ( � ) P ( a )? Counterexample! ✗ 8 / 21

�� Counterexample Q 9 / 21

�� ∀ ( x : A ) .Q ( x ) ⇒ P ( x ) �� is good if Soundness x ∈ range ( gen ) ⇒ Q ( x ) and Completeness x ∈ Q ( x ) ⇒ range ( gen ) 10 / 21

Generate only well-typed terms! �� → �� → � �� such that e ∈ range ( gen _ term Γ t ) ⇒ Γ ⊢ e : t Make sure that all of them can be generated! Γ ⊢ e : t ⇒ e ∈ range ( gen _ term Γ t ) 11 / 21

� �� := �� ← �� ← �� ← �� Testing an Optimising Compiler by Generating Random Lambda Terms. Michal H. Palka, Koen Claessen, Alejandro Russo, and John Hughes. AST ’11 � � �� ← �� ← �� ← �� ← �� ← �� 12 / 21

Generating Good Generators �� 1 → � 2 → �� 2 � � 1 → �� and ∀ a 1 , range ( gen _ A 2 a 1 ) ≡ { a 2 | Q a 1 a 2 } 13 / 21

Generating Good Generators �� → �� → �� → �� → �� → �� 14 / 21

�� Γ � �� := 15 / 21

�� Γ � �� := � �� ∀ Γ � �� Γ � �� → Γ �� 15 / 21

Generating Good Generators for Inductive Relations POPL 2018 - PowerPoint PPT Presentation

Generating Good Generators for Inductive Relations POPL 2018 Leonidas Lampropoulos 1 Zoe Paraskevopoulou 2 Benjamin Pierce 1 1 University of Pennsylvania 2 Princeton University Generating Good Generators for Inductive Relations in a

The One-Quarter Fraction Need two generating relations. E.g. a 2 6 2 design, with generating

GENERATING FUNCTIONS AND RECURRENCE RELATIONS Generating Functions Recurrence Relations Suppose

Generators and defining relations for the ring of differential operators on a smooth affine

Descriptions Sol Swords Centaur Technology, Inc. ACL2 Workshop 2020 Paper:

So what are hammers (and counterexample generators) good for? Talk outline 1. Sledgehammer

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> b x 2 w 3 x 3 Neuroscience 101 CMPSCI 689 Subhransu Maji (UMASS) 3 /19 CMPSCI 689

Inductive Definitions with Inference Rules 1 / 25 Outline Introduction Specifying inductive

Inductive types in Coq Wessel van Staal November 23, 2012 Inductive types Inductive nattree :

Synthesis, Verification, and Synthesis, Verification, and Inductive Learning Inductive Learning

Good Neighbourly Relations Committee Update on work on the Jacques Cartier Bridge (Montreal

More than 40 years generating energy Company Grupel began its activity in 1976 where its main

Inductive Types for Free Representing Nested Inductive Types using W-types Michael Abbott (U.

Inductive Programming A Unifying Framework for Analysis and Evaluation of Inductive Programming

Recursive Definitions Generating Functions Lecture 18 Generating Functions A generating

Interpreting inductive-inductive definitions as indexed inductive definitions Fredrik Nordvall

Micro Power Generators Sung Park Kelvin Yuk ECS 203 Overview Why Micro Power Generators are

A categorical semantics for inductive-inductive definitions Thorsten Altenkirch 2 Peter Morris 2

Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace William

Documentation Generators Steven J Zeil March 3, 2013 Documentation Generators Outline

National Generators Forum NGF Sponsored Rule changes Information provisions Alex Cruickshank

Generating Good Generators for Inductive Relations POPL 2018 - PowerPoint PPT Presentation

Generating Good Generators for Inductive Relations POPL 2018 Leonidas Lampropoulos 1 Zoe Paraskevopoulou 2 Benjamin Pierce 1 1 University of Pennsylvania 2 Princeton University Generating Good Generators for Inductive Relations in a

The One-Quarter Fraction Need two generating relations. E.g. a 2 6 2 design, with generating

GENERATING FUNCTIONS AND RECURRENCE RELATIONS Generating Functions Recurrence Relations Suppose

Generators and defining relations for the ring of differential operators on a smooth affine

Descriptions Sol Swords Centaur Technology, Inc. ACL2 Workshop 2020 Paper:

So what are hammers (and counterexample generators) good for? Talk outline 1. Sledgehammer

Good Relations with R Kurt Hornik David Meyer Kurt Hornik and David Meyer useR! 2008 Good

Atmospheric Water Generators Our Aim: Translating Advanced Technologies for Social Good Team

Good partners and good relations are the base of our culture Elettronica Rossoni, established in

Good industrial relations produce good socio-economic and health protection measures The example

&gt; b x 2 w 3 x 3 Neuroscience 101 CMPSCI 689 Subhransu Maji (UMASS) 3 /19 CMPSCI 689

Inductive Definitions with Inference Rules 1 / 25 Outline Introduction Specifying inductive

Inductive types in Coq Wessel van Staal November 23, 2012 Inductive types Inductive nattree :

Synthesis, Verification, and Synthesis, Verification, and Inductive Learning Inductive Learning

Good Neighbourly Relations Committee Update on work on the Jacques Cartier Bridge (Montreal

More than 40 years generating energy Company Grupel began its activity in 1976 where its main

Inductive Types for Free Representing Nested Inductive Types using W-types Michael Abbott (U.

Inductive Programming A Unifying Framework for Analysis and Evaluation of Inductive Programming

Recursive Definitions Generating Functions Lecture 18 Generating Functions A generating

Interpreting inductive-inductive definitions as indexed inductive definitions Fredrik Nordvall

Micro Power Generators Sung Park Kelvin Yuk ECS 203 Overview Why Micro Power Generators are

A categorical semantics for inductive-inductive definitions Thorsten Altenkirch 2 Peter Morris 2

Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace William

Documentation Generators Steven J Zeil March 3, 2013 Documentation Generators Outline

National Generators Forum NGF Sponsored Rule changes Information provisions Alex Cruickshank

> b x 2 w 3 x 3 Neuroscience 101 CMPSCI 689 Subhransu Maji (UMASS) 3 /19 CMPSCI 689