Bounded quantification

<h2 id="overview">Overview</h2>
<p>The purpose of bounded quantification is to allow for <a href="/facts/Polymorphic_function/S3cCKsVT">polymorphic functions</a> to depend on some specific behaviour of objects instead of <a href="/facts/Type_inheritance/a7R1xv8n">type inheritance</a>. It assumes a record-based model for object classes, where every class member is a record element and all class members are named functions. Object attributes are represented as functions that take no argument and return an object. The specific behaviour is then some function name along with the types of the arguments and the return type. Bounded quantification considers all objects with such a function. An example would be a polymorphic min function that considers all objects that are comparable to each other.
</p>
<h3>F-bounded quantification</h3>
<p><i>F</i>-bounded quantification or recursively bounded quantification, introduced in 1989, allows for more precise typing of functions that are applied on recursive types. A recursive type is one that includes a function that uses it as a type for some argument or its return value.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="example">Example</h2>
<p>This kind of type constraint can be expressed in <a href="/facts/Java_(programming_language)/9ScgFyAL">Java</a> with a generic interface. The following example demonstrates how to describe types that can be compared to each other and use this as typing information in <a href="/facts/Polymorphic_function/S3cCKsVT">polymorphic functions</a>. The Test.min function uses simple bounded quantification and does not ensure the objects are mutually comparable, in contrast with the Test.fMin function which uses F-bounded quantification.
</p><p>In mathematical notation, the types of the two functions are
</p>
min: ∀ T, ∀ S ⊆ {compareTo: T → int}. S → S → S
fMin: ∀ T ⊆ Comparable[T]. T → T → T
<p>where
</p>
Comparable[T] = {compareTo: T → int}
interface Comparable<T> {

int compareTo(T other);

}

public class Integer implements Comparable<Integer> {

@Override
    public int compareTo(Integer other) {
        // ...
    }
}

public class String implements Comparable<String> {

@Override
    public int compareTo(String other) {
        // ...
    }

}

public class Test {
    
    public static void main(String[] args) {
        final String a = min("cat", "dog");
        final Integer b = min(10, 3);
        final Comparable c = min("cat", 3); // Throws ClassCastException at runtime
        final String str = fMin("cat", "dog");
        final Integer i = fMin(10, 3);
        // final Object o = fMin("cat", 3); // Does not compile
    }
    
    public static <S extends Comparable> S min(S a, S b) {
        if (a.compareTo(b) <= 0) {
            return a;
        } else {
            return b;
        }
    }
    
    public static <T extends Comparable<T>> T fMin(T a, T b) {
        if (a.compareTo(b) <= 0) {
            return a;
        } else {
            return b;
        }
    }

}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Covariance_and_contravariance_(computer_science)/NzBlXxF3">Covariance and contravariance (computer science)</a></li>
<li><a href="/facts/Curiously_recurring_template_pattern/ASIdfGNr">Curiously recurring template pattern</a></li>
<li><a href="/facts/Wildcard_(Java)/NAIe5pnk">Wildcard (Java)</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Luca_Cardelli/bQjxEQhA">Cardelli, Luca</a>; <a href="/facts/Peter_Wegner_(computer_scientist)/XYjwa29v">Wegner, Peter</a> (December 1985). <a href="http://lucacardelli.name/Papers/OnUnderstanding.A4.pdf">"On Understanding Types, Data Abstraction, and Polymorphism"</a> (PDF). <i><a href="/facts/ACM_Computing_Surveys/18Itftvp">ACM Computing Surveys</a></i>. 17 (4): 471–523. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.695">10.1.1.117.695</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F6041.6042">10.1145/6041.6042</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0360-0300">0360-0300</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:2921816">2921816</a>.</li>
<li>Peter S. Canning, <a href="/facts/William_Cook_(computer_scientist)/PWPluAgM">William R. Cook</a>, Walter L. Hill, <a href="/facts/John_C._Mitchell/8tzw9FNg">John C. Mitchell</a>, and William Olthoff. <a href="http://dl.acm.org/citation.cfm?id=99392">"F-bounded polymorphism for object-oriented programming"</a>. In <i>Conference on Functional Programming Languages and Computer Architecture</i>, 1989.</li>
<li><a href="/facts/Benjamin_C._Pierce/m9DKUgML">Benjamin C. Pierce</a> "Intersection types and bounded polymorphism". <i>Lecture Notes in Computer Science</i> 664, 1993.</li>
<li><a href="/facts/Gilad_Bracha/eJEXUGIW">Gilad Bracha</a>, <a href="/facts/Martin_Odersky/fWmg60WX">Martin Odersky</a>, David Stoutamire, and <a href="/facts/Philip_Wadler/ShX91nId">Philip Wadler</a>. "Making the future safe for the past: Adding genericity to the Java programming language". In <i>Object-Oriented Programming: Systems, Languages, Applications</i> (OOPSLA). ACM, October 1998.</li>
<li>Andrew Kennedy and <a href="/facts/Don_Syme/2xyvKfh1">Don Syme</a>. "Design and Implementation of Generics for the .NET Common Language Runtime". In <i>Programming Language Design and Implementation</i>, 2001.</li>
<li><a href="/facts/Benjamin_C._Pierce/m9DKUgML">Pierce, Benjamin C.</a> (2002). <i>Types and Programming Languages</i>. MIT Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-262-16209-8., Chapter 26: Bounded quantification</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.c2.com/cgi/wiki?BoundedPolymorphism">Bounded Polymorphism</a> at the <a href="/facts/Portland_Pattern_Repository/bL2cwnTn">Portland Pattern Repository</a></li>
<li><a href="http://www.cs.washington.edu/research/projects/cecil/www/Vortex-Three-Zero/doc-cecil-lang/cecil-spec-86.html">"F-bounded Polymorphism"</a> in <i>The Cecil Language: Specification and Rationale</i></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>F-bounded polymorphism for object-oriented programming. Canning, Cook, Hill, Olthof and Mitchell. http://dl.acm.org/citation.cfm?id=99392 <a href="/wiki/William_Cook_(computer_scientist)" target="_blank">/wiki/William_Cook_(computer_scientist)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Bounded quantification open-in-new

Bounded quantification