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{{Short description|Data abstraction problem in programming languages}}
The '''expression problem''' is a challenging problem in [[programming language]]s that concerns the extensibility and modularity of statically typed data abstractions. The goal is to define a data abstraction that is extensible both in its representations and its behaviors, where one can add new representations and new behaviors to the data abstraction, without recompiling existing code, and while retaining static type safety (e.g., no casts). ItThe exposedstatement of the problem exposes deficiencies in [[programming paradigm]]s and [[programming language]]s,. andPhilip itWadler, isone stillof notthe definitivelyco-authors solvedof Haskell, although there arehas manyoriginated proposedthe solutionsterm.
 
== History ==
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| url= https://www.cs.tufts.edu/~nr/cs257/archive/john-reynolds/procedural-data-structures.pdf
}}
</ref> Reynolds discussed two forms of Data Abstraction: User-defined Types, which are now known as [[abstract data types|Abstract Data Types]] (ADTs) (not to be confused with [[algebraic data types|Algebraic Data Types]]), and Procedural Data Structures, which are now understood as a primitive form of Objects with only one method. He argued that they are complementary, in that User-defined Types could be extended with new behaviors, and Procedural Data Structures could be extended with new representations. He also discussed related work going back to 1967. However, Reynold's conclusions based on this early analysis turned out to be completely wrong: he wrote that adding a second method to an object "is more a tour de force than a specimen of clear programming," which completely missed the Object-Oriented paradigm and its great success. He also believed the two forms of data abstraction "are inherently distinct and complementary."
 
Fifteen years later in 1990, [[William Cook (computer scientist)|William Cook]]<ref name="CookOOPvsADT">
{{cite book
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| isbn= 978-3-540-46450-1}}
</ref>
applied Reynold's seminal idea in the context of Objects and Abstract Data Types, which had both grown extensively. Cook identified the matrix of representations and behaviors that are implicit in a Data Abstraction, and discussed how ADTs are based on the behavioral axis, while Objects are based on the representation axis. He provides extensive discussion of work on ADTs and Objects that are relevant to the problem. He also reviewed implementations in both styles, discussed extensibility in both directions, and also identified the importance of static typing.
Most importantly, he discussed situations in which there was more flexibility than
Reynolds considered, including internalization and optimization of methods.
 
At ECOOP '98, [[Shriram Krishnamurthi]] et al.<ref name="Synth">
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| title=Synthesizing Object-Oriented and Functional Design to Promote Re-Use
| url=https://cs.brown.edu/~sk/Publications/Papers/Published/kff-synth-fp-oo/
}}</ref> presented a design pattern solution to the problem of simultaneously extending an expression-oriented programming language and its tool-set. They dubbed it the "expressivity problem" because they thought programming language designers could use the problem to demonstrate the expressive power of their creations. For PLT, the problem had shown up in the construction of DrScheme, now [[Racket_Racket (programming_languageprogramming language)#DrRacket_IDEDrRacket IDE|DrRacket]], and they solved it<ref name="FF">
{{cite journal
| title= Modular object-oriented programming with units and mixins
| date=1999 | doi=10.1145/291251.289432 | last1=Findler | first1=Robert Bruce | last2=Flatt | first2=Matthew | journal=ACM SigplanSIGPLAN Notices | volume=34 | pages=94–104 | doi-access=free }}</ref> via a rediscovery of [[mixin]]s.<ref name="Mixins">{{cite thesis |type= PhD ▼
| date=1999 | doi=10.1145/291251.289432 | url= http://dl.acm.org/citation.cfm?id=289432
▲| last1=Findler | first1=Robert Bruce | last2=Flatt | first2=Matthew | journal=ACM Sigplan Notices | volume=34 | pages=94–104 }}</ref> via a rediscovery of [[mixin]]s.<ref name="Mixins">{{cite thesis |type= PhD
| last= Cook | first= William | date= 1989
| title= A Denotational Semantics of Inheritance | publisher= Brown University
Line 77 ⟶ 75:
 
Some follow-up work used the expression problem to showcase the power of programming language designs.<ref name="Scala0">
{{cite journalbook
| titlechapter=Extensible Algebraic Datatypes with Defaults
| first1=Matthias | last1=Zenger
| first2=Martin | last2=Odersky
| title=Proceedings of the sixth ACM SIGPLAN international conference on Functional programming | pages=241–252
| pages=241–252
| citeseerx=10.1.1.28.6778
| year=2001
| doi=10.1145/507635.507665
| isbn=1-58113-415-0 }}</ref><ref name="Scala">
{{cite webbook
| titlechapter=Independently extensible solutions to the expression problem
| chapter-url=https://homepages.inf.ed.ac.uk/wadler/fool/program/final/10/10_Paper.pdf
| first1=Matthias | last1=Zenger
| first2=Martin | last2=Odersky
| title=FOOL 2005
| series=Foundations of Object-Oriented Languages (FOOL)
| publisher=ACM SIGPLAN
| citeseerx=10.1.1.107.4449
| year=2005
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There are various solutions to the expression problem. Each solution varies in the amount of code a user must write to implement them, and the language features they require.
 
* [[Multiple dispatch]]<ref name="Chambers & Leavens, Multi-Methods">{{cite journal|last1=Chambers|first1=Craig|last2=Leavens|first2=Gary T.|title=Type Checking and Modules for Multi-Methods|journal= ACM Trans.Transactions on Programming Program.Languages Lang.and Syst.Systems|date=November 1995|volume=17 |issue=6|pages=805–843|doi=10.1145/218570.218571 |url=http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1036&context=cs_techreports|doi-access=free}}</ref>
* [[{{section link|Ruby syntax#Open classes|Open classes]]}}<ref name="Clifton et. al., MultiJava Open Classes">{{cite journalbook|last1=Clifton|first1=Curtis|last2=Leavens|first2=Gary T.|last3=Chambers|first3=Craig|last4=Millstein|first4=Todd |title=Proceedings of the 15th ACM SIGPLAN conference on Object-oriented programming, systems, languages, and applications |chapter=MultiJava: Modular Openopen Classesclasses and Symmetricsymmetric Multiplemultiple Dispatchdispatch for Java|journal=Oopsla '00|date=2000|pages=130–145 |doi=10.1145/353171.353181 |isbn=978-1-58113-200-7 |s2cid=7879645 |url=http://people.csail.mit.edu/dnj/teaching/6898/papers/multijava.pdf}}</ref>
* [[Coproduct]]s of [[functor]]s<ref>{{cite journal
|author = Wouter Swierstra
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|doi-access = free
}}</ref>
* [[Type class]]es<ref name="Wehr & Thiemann, JavaGI Type Classes">{{cite journal|last1=Wehr|first1=Stefan|last2=Thiemann|first2=Peter|title= JavaGI: The Interaction of Type Classes with Interfaces and Inheritance|journal= ACM Trans.Transactions Program.on Lang.Programming Syst.Languages and Systems|date=July 2011|issuevolume=33 |issue=4|pages= 1–83|doi=10.1145/1985342.1985343 |s2cid=13174506 |url=http://www.stefanwehr.de/publications/WehrThiemann2011TOPLAS.html|doi-access=free}}</ref>
* Tagless-final<ref name="Carette et al.,
Finally tagless, partially evaluated: Tagless staged interpreters for simpler typed languages">{{cite journal|last1=Carette|first1=Jacques|last2=Kiselyov|first2=Oleg|last3=Chung-chieh|first3=Shan|title=Finally Tagless, Partially Evaluated: Tagless Staged Interpreters for Simpler Typed Languages|journal=J. Funct. Program.|date=2009|volume=19 |issue=5 |pages=509–543 |doi=10.1017/S0956796809007205 |s2cid=6054319 |url=http://okmij.org/ftp/tagless-final/JFP.pdf}}</ref> / Object algebras<ref name="Oliveira & Cook, Object Algebras">{{cite journal|last1=Oliveira|first1=Bruno C. d. S.|last2=Cook|first2=William R.|title=Extensibility for the Masses: Practical Extensibility with Object Algebras|journal=Ecoop '12|date=2012|url=http://www.cs.utexas.edu/~wcook/Drafts/2012/ecoop2012.pdf}}</ref>
* Polymorphic Variants<ref name="Code Reuse Through Polymorphic Variants">{{cite journalbook|last1=Garrigue|first1=Jacques|titlechapter=Code Reuse Through Polymorphic Variants|chapter-url=https://www.math.nagoya-u.ac.jp/~garrigue/papers/variant-reuse.pdf|title=Workshop on Foundations of Software Engineering. Sasaguri, Japan, November 2000 |citeseerx=10.1.1.128.7169|year=2000}}</ref>
 
== Example ==
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<syntaxhighlight lang="c#" line="1">
public interface IEvalExp
{
int Eval();
}
 
public class Lit : IEvalExp
{
publicinternal Lit(int n)
{
N = n;
}
 
publicinternal int N { get; }
 
public int Eval()
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}
 
public class Add : IEvalExp
{
publicinternal Add(IEvalExp left, IEvalExp right)
{
Left = left;
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}
 
publicinternal IEvalExp Left { get; }
 
publicinternal IEvalExp Right { get; }
 
public int Eval()
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}
 
public static class ExampleOne
{
public static IEvalExp AddOneAndTwo() => new Add(new Lit(1), new Lit(2));
public static int EvaluateTheSumOfOneAndTwo() => AddOneAndTwo().Eval();
}
</syntaxhighlight>
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<syntaxhighlight lang="c#" line="1">
public class Mult : IEvalExp
{
publicinternal Mult(IEvalExp left, IEvalExp right)
{
Left = left;
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}
 
publicinternal IEvalExp Left { get; }
 
publicinternal IEvalExp Right { get; }
 
public int Eval()
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</syntaxhighlight>
 
However, if we wish to add a new function over the type (a new method in C# terminology), for example to pretty print an expression, we have to change the {{code|IEvalExp}} interface and then modify all the classes that implement the interface. Another possibility is to create a new interface that extends the {{code|IEvalExp}} interface and then create sub-types for {{code|Lit}}, {{code|Add}} and {{code|Mult}} classes, but the expression returned in {{code|ExampleOne.AddOneAndTwo()}} has already been compiled so we will not be able to use the new function over the old type. The problem is reversed in functional programming languages like [[F Sharp (programming language)|F#]] where it is easy to add a function over a given type, but extending or adding types is difficult.
 
=== Problem Solutionsolution using Objectobject Algebraalgebra ===
Let us redesign the original library with extensibility in mind using the ideas from the paper ''Extensibility for the Masses.''<ref name="Oliveira & Cook, Object Algebras" />
 
<syntaxhighlight lang="c#" line="1">
public interface ExpAlgebra<T>
{
T Lit(int n);
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}
 
public class ExpFactory : ExpAlgebra<IEvalExp>
{
public IEvalExp Lit(int n)
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}
 
public static class ExampleTwo<T>
{
public static T AddOneToTwo(ExpAlgebra<T> ae) => ae.Add(ae.Lit(1), ae.Lit(2));
Line 228:
 
<syntaxhighlight lang="c#" line="1">
public interface IPrintExp : IEvalExp
{
string Print();
}
 
public class PrintableLit : Lit, IPrintExp
{
publicinternal PrintableLit(int n) : base(n)
{
N = n;
}
 
publicinternal int N { get; }
 
public string Print()
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}
 
public class PrintableAdd : Add, IPrintExp
{
publicinternal PrintableAdd(IPrintExp left, IPrintExp right) : base(left, right)
{
Left = left;
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}
 
publicinternal new IPrintExp Left { get; }
 
publicinternal new IPrintExp Right { get; }
 
public string Print()
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}
 
public class PrintFactory : ExpFactory, ExpAlgebra<IPrintExp>
{
public IPrintExp Add(IPrintExp left, IPrintExp right)
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}
 
public static class ExampleThree
{
publicinternal static int Evaluate() => ExampleTwo<IPrintExp>.AddOneToTwo(new PrintFactory()).Eval();
publicinternal static string Print() => ExampleTwo<IPrintExp>.AddOneToTwo(new PrintFactory()).Print();
}
</syntaxhighlight>
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