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Polymorphism Dynamic Typing Favonia | Nick Benton | Bob Harper 1 - - PowerPoint PPT Presentation
Polymorphism Dynamic Typing Favonia | Nick Benton | Bob Harper 1 - - PowerPoint PPT Presentation
Polymorphism Dynamic Typing Favonia | Nick Benton | Bob Harper 1 Polymorphism T C E R R O C Dynamic Typing Favonia | Nick Benton | Bob Harper 1 Is this the shortest PLunch? Bob and I wanted to make homework 2 Is this the shortest
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Is this the shortest PLunch?
Bob and I wanted to make homework
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Is this the shortest PLunch?
Bob and I wanted to make homework and so we had a JFP Theoretical Pearl.
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source TARGET type var. “top” type
System F to PCF
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source TARGET type var. “top” type
System F to PCF
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source TARGET type var. “top” type But keep other types!
System F to PCF
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System F source
System F to PCF
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System F PCF with dyn
source TARGET
System F to PCF
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System F PCF with dyn
source TARGET
System F to PCF
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System F to PCF
Keep everything except variables
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System F to PCF
Keep everything except variables
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Type Application
type app
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Type Application
type app
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Type Application
type app ?
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Type Application
type app ?
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Type Application
type app ?
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Type Application
type app ?
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Embed and Project
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Type Application
type app
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Type Application
type app Idea: lift the projection to handle arbitrary type operators.
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Type Application
type app Idea: lift the projection to handle arbitrary type operators.
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Type Application
type app Idea: lift the projection to handle arbitrary type operators. *Mayer-Wand style
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Embed and Project 2.0
Find all the t!
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Embed and Project 2.0
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Problem Statement
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Problem Statement
(Intended) homework assignment: no run-time errors
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Problem Statement
(Intended) homework assignment: no run-time errors Igarashi, Pierce and Wadler [2001] showed correctness in a minimal core calculus for Java with generics, relying on the class table.
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Problem Statement
(Intended) homework assignment: no run-time errors Igarashi, Pierce and Wadler [2001] showed correctness in a minimal core calculus for Java with generics, relying on the class table.
Our difficulty: composition of types
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System F types System F terms PCF terms relations for type variables that respect obs. equiv.
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System F types System F terms PCF terms relations for type variables that respect obs. equiv. no J here!
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Logical Relation Recipe
weakening and exchange STEP 1 the relation itself respects obs. equiv. (and the properties you care) STEP 2 composition of types/relations STEP 3 fundamental lemma! STEP 4
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Logical Relation Recipe
weakening and exchange STEP 1 the relation itself respects obs. equiv. (and the properties you care) STEP 2 composition of types/relations STEP 3 fundamental lemma! STEP 4
?
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Compositionality
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Compositionality
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Compositionality
not so clear if it's “iff.”
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Compositionality
FAILED ATTEMPT
implies(?)
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Compositionality
Assume We want
FAILED ATTEMPT
implies(?)
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Compositionality
Assume We have We want
FAILED ATTEMPT
implies(?)
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Compositionality
Assume We have We want
FAILED ATTEMPT
implies(?)
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Compositionality
Assume We have We want
FAILED ATTEMPT
implies(?)
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Logical Relation Recipe
weakening and exchange STEP 1 the relation itself respects obs. equiv. (and the properties you care) STEP 2 quasi-composition of types/relations STEP 3 fundamental lemma! STEP 4
Thus, compiled programs give the same numbers!
(no run-time errors)
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Concluding Notes
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Concluding Notes
A nice trick to deal with embedding/projection!
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Concluding Notes
We used refinements to make but rejected by POPL. :-( A nice trick to deal with embedding/projection!
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Concluding Notes
We used refinements to make but rejected by POPL. :-( A nice trick to deal with embedding/projection! We have 3 different proofs with different setups.
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Concluding Notes
We used refinements to make but rejected by POPL. :-( Thunks to preserve values: A nice trick to deal with embedding/projection! We have 3 different proofs with different setups.
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Concluding Notes
We used refinements to make but rejected by POPL. :-( Thunks to preserve values: You can find open problems by TAing! A nice trick to deal with embedding/projection! We have 3 different proofs with different setups.