Summary Polymorphism with Type Qualifiers in System F Q arxiv.org
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One Line
System F <:Q is a language that uses higher-rank bounded polymorphism and type qualifiers to classify program values and introduces qualifier polymorphism.
Slides
Slide Presentation (10 slides)
Key Points
- The document discusses the concept of type qualifiers in programming languages and their role in enriching type systems to enforce program invariants.
- Type qualifiers provide additional information about values, allowing for more precise control over their behavior.
- Type qualifiers can be used to annotate the types of function arguments and results, indicating properties such as immutability.
- System F <:Q is a new calculus that combines higher-rank bounded polymorphism with type qualifiers.
- System F <:Q can be applied to scenarios where type qualifiers naturally arise, such as reference immutability, function coloring, and capture checking.
- The authors propose a design recipe for constructing qualifier-polymorphic enrichment systems, including System F <:Q, with desirable properties such as higher-rank qualifier and type polymorphism.
- The paper explores practical qualifier systems for reference immutability, function coloring, and capture tracking, presenting syntax, evaluation rules, and typing rules for each system.
- The paper emphasizes the usefulness of these qualifier systems in enforcing safety constraints and discusses their soundness and ability to enforce immutability, function coloring, and tracking capture.
Summaries
18 word summary
System F <:Q combines higher-rank bounded polymorphism with type qualifiers to classify program values and proposes qualifier polymorphism.
100 word summary
This document introduces System F <:Q, a new calculus that combines higher-rank bounded polymorphism with type qualifiers. It explores scenarios where type qualifiers naturally arise, such as reference immutability and function coloring. The document emphasizes the importance of type systems in classifying program values and highlights the natural subtyping relation induced by type qualifiers. It proposes System F <:Q as a qualified extension of System F <: and discusses qualifier polymorphism and its interaction with type polymorphism. The document concludes by proposing the addition of qualifier polymorphism to existing qualifier systems without losing the natural lattice structure of type qualifiers.
185 word summary
This document explores polymorphism with type qualifiers in System F <:Q, introducing the concept of type qualifiers as a mechanism for enriching type systems. The authors propose System F <:Q, a new calculus that combines higher-rank bounded polymorphism with type qualifiers. They discuss scenarios where type qualifiers naturally arise, such as reference immutability, function coloring, and capture checking. The document emphasizes the importance of type systems in classifying program values and demonstrates how type qualifiers provide additional information about values for more precise control over their behavior. It highlights the natural subtyping relation induced by type qualifiers and mentions the exploration of more exotic qualifiers in different languages. The authors also discuss qualifier polymorphism and its interaction with type polymorphism, proposing System F <:Q as a qualified extension of System F <: . They present a design recipe for constructing qualifier-polymorphic enrichment systems and discuss the desirable properties of System F <:Q. The document concludes by re-examining existing qualifier systems, proposing the addition of qualifier polymorphism without losing the natural lattice structure of type qualifiers, and mentioning mechanized soundness proofs in the Coq proof assistant.
522 word summary
This document explores the concept of polymorphism with type qualifiers in System F <:Q. It introduces the idea of type qualifiers as a mechanism for enriching type systems and enforcing program invariants. The authors propose a new calculus called System F <:Q that combines higher-rank bounded polymorphism with type qualifiers. They demonstrate how this polymorphic system can be applied to scenarios where type qualifiers naturally arise, such as reference immutability, function coloring, and capture checking.
The authors emphasize the importance of type systems in classifying program values. They discuss the issue of mutability in the context of a function like toLowerCase and introduce the concept of type qualifiers to annotate the type of the function's argument and result. Type qualifiers provide additional information about values, allowing for more precise control over their behavior.
The document highlights the natural subtyping relation induced by type qualifiers. Different examples of type qualifiers are discussed, such as "const," "mutable," "noexcept," and "throws." The authors mention that different languages are exploring more exotic qualifiers to encode properties like locality, linearity, exclusivity, and synchronicity.
The authors acknowledge that qualifier polymorphism is still understudied compared to type qualifiers themselves. They demonstrate how the type signature of a function like toLowerCase can be expressed using qualifier polymorphism. They also discuss the interaction between qualifier polymorphism and type polymorphism.
To address these issues, the authors propose System F <:Q as a qualified extension of System F <: . They present a design recipe for constructing qualifier-polymorphic enrichment systems and explain the desirable properties of System F <:Q.
The document concludes by re-examining existing qualifier systems in the literature and proposing that qualifier polymorphism can be added without losing the natural lattice structure of type qualifiers. They discuss other related work and mention that their soundness proofs are mechanized in the Coq proof assistant.
In summary, this document explores polymorphism with type qualifiers in System F <:Q and demonstrates its practical applications in various qualifier systems. It presents the necessary syntax, evaluation rules, and typing rules for each system and discusses their soundness and ability to enforce safety constraints. The document emphasizes the usefulness of these systems in enforcing immutability, function coloring, and tracking capture.
The paper presents the rules for subqualification, subtyping, and typing in System F <:Q, highlighting the changes and additions required to accommodate type qualifiers. It also introduces the concept of subqualification and presents the subtyping rules and typing rules for System F <:Q.
The paper proves soundness theorems for System F <:Q and introduces a prediction lemma relating the free variables of values to the qualifier annotated on their types. It also provides a mechanization of System F <:Q and its derived calculi.
In conclusion, the paper presents a recipe for modeling higher-rank polymorphism, subtyping, and subqualification in systems with type qualifiers using the free lattice generated from an underlying qualifier lattice. It shows how the recipe can be applied to solve problems related to reference immutability, function coloring, and capture tracking. The paper also discusses the relationship between type qualifiers and effect systems and suggests future work on modeling free complemented distributive lattice systems with subqualification.
522 word summary
This document explores the concept of polymorphism with type qualifiers in System F <:Q. It introduces the idea of type qualifiers as a mechanism for enriching type systems and enforcing program invariants. The authors propose a new calculus called System F <:Q that combines higher-rank bounded polymorphism with type qualifiers. They demonstrate how this polymorphic system can be applied to scenarios where type qualifiers naturally arise, such as reference immutability, function coloring, and capture checking.
The authors emphasize the importance of type systems in classifying program values. They discuss the issue of mutability in the context of a function like toLowerCase and introduce the concept of type qualifiers to annotate the type of the function's argument and result. Type qualifiers provide additional information about values, allowing for more precise control over their behavior.
The document highlights the natural subtyping relation induced by type qualifiers. Different examples of type qualifiers are discussed, such as “const,” “mutable,” “noexcept,” and “throws.” The authors mention that different languages are exploring more exotic qualifiers to encode properties like locality, linearity, exclusivity, and synchronicity.
The authors acknowledge that qualifier polymorphism is still understudied compared to type qualifiers themselves. They demonstrate how the type signature of a function like toLowerCase can be expressed using qualifier polymorphism. They also discuss the interaction between qualifier polymorphism and type polymorphism.
To address these issues, the authors propose System F <:Q as a qualified extension of System F <: . They present a design recipe for constructing qualifier-polymorphic enrichment systems and explain the desirable properties of System F <:Q.
The document concludes by re-examining existing qualifier systems in the literature and proposing that qualifier polymorphism can be added without losing the natural lattice structure of type qualifiers. They discuss other related work and mention that their soundness proofs are mechanized in the Coq proof assistant.
In summary, this document explores polymorphism with type qualifiers in System F <:Q and demonstrates its practical applications in various qualifier systems. It presents the necessary syntax, evaluation rules, and typing rules for each system and discusses their soundness and ability to enforce safety constraints. The document emphasizes the usefulness of these systems in enforcing immutability, function coloring, and tracking capture.
The paper presents the rules for subqualification, subtyping, and typing in System F <:Q, highlighting the changes and additions required to accommodate type qualifiers. It also introduces the concept of subqualification and presents the subtyping rules and typing rules for System F <:Q.
The paper proves soundness theorems for System F <:Q and introduces a prediction lemma relating the free variables of values to the qualifier annotated on their types. It also provides a mechanization of System F <:Q and its derived calculi.
In conclusion, the paper presents a recipe for modeling higher-rank polymorphism, subtyping, and subqualification in systems with type qualifiers using the free lattice generated from an underlying qualifier lattice. It shows how the recipe can be applied to solve problems related to reference immutability, function coloring, and capture tracking. The paper also discusses the relationship between type qualifiers and effect systems and suggests future work on modeling free complemented distributive lattice systems with subqualification.
1253 word summary
The document discusses the concept of type qualifiers in programming languages and explores the idea of polymorphism over type qualifiers. Type qualifiers provide a mechanism for enriching type systems to enforce additional program invariants. The authors propose a new calculus called System F <:Q that combines higher-rank bounded polymorphism with type qualifiers. They demonstrate how this polymorphic system can be applied to various scenarios where type qualifiers naturally arise, such as reference immutability, function coloring, and capture checking.
The authors emphasize the importance of type systems in classifying the values that a program reduces to. They use the example of a function toLowerCase, which takes in a string as an argument and returns a string as a result. They discuss two ways to address the issue of mutability in this context: viewing the modification of the argument as a property of the function or as a property of the argument itself. They adopt the latter view and introduce the concept of type qualifiers, which can be used to annotate the type of the function's argument and result. For example, the type qualifier "const" can be used to indicate that the argument is immutable. They explain that type qualifiers provide additional information about the values, allowing for more precise control over their behavior.
The authors highlight the natural subtyping relation induced by type qualifiers. For example, a mutable string can be considered a subtype of an immutable string. They mention that this subtyping relation is well-known and has been integrated into existing type systems with subtyping. They discuss various examples of type qualifiers, such as "const," "mutable," "noexcept," and "throws." They also mention that different languages are exploring more exotic qualifiers to encode properties like locality, linearity, exclusivity, and synchronicity.
The authors acknowledge that while type qualifiers themselves are well-explored, qualifier polymorphism is still understudied. They explain that sometimes parametric polymorphism is not necessary when subtyping is present. They demonstrate how the type signature of a function like toLowerCase can be expressed using qualier polymorphism. They also discuss the interaction between qualier polymorphism and type polymorphism, raising questions about the range of type variables and the merging of qualiers.
To address these issues, the authors propose System F <:Q as a qualied extension of System F <: . They present a design recipe for constructing qualier-polymorphic enrichment systems and explain the desirable properties of System F <:Q , such as higher-rank qualifier and type polymorphism, natural subtyping with qualifier variables, and easy meets and joins over qualiers. They also discuss how System F <:Q can be used to model reference immutability, function coloring, and capture tracking.
The authors conclude by re-examining existing qualifier systems in the literature in light of their observations developed while developing System F <:Q . They highlight the limitations of current qualifier systems and propose that it is possible to add qualifier polymorphism without losing the natural lattice structure of type qualifiers. They argue that a design recipe based on free lattices can be used to construct qualifier systems with subqualification and polymorphism. They discuss other related work and mention that their soundness proofs are mechanized in the Coq proof assistant.
The document discusses the concept of polymorphism with type qualifiers in System F Q. It presents the typing rules for System F <:Q, focusing on preservation and progress theorems. The document highlights the usefulness of such a system, particularly in enforcing safety constraints.
The authors then generalize qualifiers to general lattices and explain how to tweak the recipe used to construct System F <:Q to support countable bounded qualifier lattices. They provide the syntax and subqualification rules for this extended system, emphasizing the need for evaluation rules to simplify qualifier expressions.
Next, the document explores the application of the design recipe in three practical qualifier systems. It starts by examining the qualier system for reference immutability, where references can be either mutable or immutable. The authors explain how to assign qualifiers to references and present the syntax, evaluation rules, and typing rules for System F <:QM. They also discuss the soundness of the system and its ability to enforce immutability safety.
The document then moves on to function colouring, another qualifier system that assigns colors to functions based on their restrictions and capabilities. The authors describe how to assign qualifiers to synchronous and asynchronous functions and present the modified syntax, evaluation rules, and typing rules for System F <:QA. They highlight the ability of the system to reason about different function colors and discuss its connection to effect systems.
Lastly, the document introduces the concept of tracking capture, where values are qualified based on what they capture. It discusses the motivation behind capture tracking, such as reasoning about side effects, and presents the assignment of qualifiers to different variables. The authors propose an extension of System F <:Q called System F <:QC to model capture tracking, including the assignment of qualifiers to types, syntax for tracking variables, and subtyping rules.
In summary, the document explores polymorphism with type qualifiers in System F Q and demonstrates its practical applications in various qualifier systems. It presents the necessary syntax, evaluation rules, and typing rules for each system and discusses their soundness and ability to enforce safety constraints. The document emphasizes the usefulness of these systems in enforcing immutability, function coloring, and tracking capture.
The paper discusses the concept of polymorphism with type qualifiers in System F <:Q. It introduces the notion of a qualier abstraction, which enables polymorphism over type qualiers in addition to simple type polymorphism. The paper presents the rules for subqualication, subtyping, and typing in System F <:Q, highlighting the changes and additions required to accommodate type qualifiers.
In System F <:Q, term application now has three arguments: a function, a qualifier, and an argument. Term abstractions can be viewed as a combination of a qualifier abstraction followed by a term abstraction. The paper also introduces the concept of subqualification, which adjusts subqualification to account for qualifier variables bound by term binders as well as qualifier variables bound by qualifier binders. The paper presents the subtyping rules and typing rules for System F <:Q.
The paper discusses how values in System F <:Q are qualified with the free variables that they close over. It checks that the tag on the value super-qualifies the free variables that the value captures. The paper also highlights the changes in the rules for term application typing and variable typing, explaining how term abstractions are a combination of qualifier and term abstractions.
The paper proves soundness theorems for System F <:Q and introduces a prediction lemma relating the free variables of values to the qualifier annotated on their types. The paper also provides a mechanization of System F <:Q and its derived calculi using existing mechanizations of System F <: and inspiration from other mechanizations.
The paper compares its approach to other existing qualifier systems and discusses how they differ in their treatment of type qualifiers and their interaction with qualified type variables. The paper also discusses related work on languages with type qualifier systems and implementing type qualifiers.
In conclusion, the paper presents a recipe for modeling higher-rank polymorphism, subtyping, and subqualification in systems with type qualifiers using the free lattice generated from an underlying qualifier lattice. It shows how the recipe can be applied to solve problems related to reference immutability, function coloring, and capture tracking. The paper also discusses the relationship between type qualifiers and effect systems and suggests future work on modeling free complemented distributive lattice systems with subqualification.