Polymorphic recursion
In computer science, polymorphic recursion (also referred to as Milner–Mycroft typability or the Milner–Mycroft calculus) refers to a recursive parametrically polymorphic function where the type parameter changes with each recursive invocation made, instead of staying constant. Type inference for polymorphic recursion is equivalent to semi-unification and therefore undecidable and requires the use of a semi-algorithm or programmer-supplied type annotations.[1]
Example
Nested datatypes
Consider the following nested datatype:
data Nested a = a :<: (Nested [a]) | Epsilon infixr 5 :<: nested = 1 :<: [2,3,4] :<: 5,6],[7],[8,9 :<: Epsilon
A length function defined over this datatype will be polymorphically recursive, as the type of the argument changes from Nested a to Nested [a] in the recursive call:
length :: Nested a -> Int length Epsilon = 0 length (_ :<: xs) = 1 + length xs
Note that Haskell normally infers the type signature for a function as simple-looking as this, but here it cannot be omitted without triggering a type error.
Higher-ranked types
Applications
Program analysis
In type-based program analysis polymorphic recursion is often essential in gaining high precision of the analysis. Notable examples of systems employing polymorphic recursion include Dussart, Henglein and Mossin's binding-time analysis[2] and the Tofte–Talpin region-based memory management system.[3] As these systems assume the expressions have already been typed in an underlying type system (not necessary employing polymorphic recursion), inference can be made decidable again.
Data structures, error detection, graph solutions
Functional programming data structures often use polymorphic recursion to simplify type error checks and solve problems with nasty "middle" temporary solutions that devour memory in more traditional data structures such as trees. In the two citations that follow, Okasaki (pp. 144–146) gives a CONS example in Haskell wherein the polymorphic type system automatically flags programmer errors.[4] The recursive aspect is that the type definition assures that the outermost constructor has a single element, the second a pair, the third a pair of pairs, etc. recursively, setting an automatic error finding pattern in the data type. Roberts (p. 171) gives a related example in Java, using a Class to represent a stack frame. The example given is a solution to the Tower of Hanoi problem wherein a stack simulates polymorphic recursion with a beginning, temporary and ending nested stack substitution structure.[5]
See also
Notes
- ↑ Henglein 1993.
- ↑ Dussart, Dirk; Henglein, Fritz; Mossin, Christian. "Polymorphic Recursion and Subtype Qualifications: Polymorphic Binding-Time Analysis in Polynomial Time". Proceedings of the 2nd International Static Analysis Symposium (SAS).
- ↑ Tofte, Mads; Talpin, Jean-Pierre (1994). "Implementation of the Typed Call-by-Value λ-calculus using a Stack of Regions". New York, NY, USA: ACM. pp. 188–201. doi:10.1145/174675.177855. ISBN 0-89791-636-0.
- ↑ Chris Okasaki (1999). Purely Functional Data Structures. New York: Cambridge. p. 144. ISBN 978-0521663502.
- ↑ Eric Roberts (2006). Thinking Recursively with Java. New York: Wiley. p. 171. ISBN 978-0471701460. https://archive.org/details/thinkingrecursiv00robe_0/page/171.
Further reading
- Meertens, Lambert (1983). "Incremental polymorphic type checking in B". ACM Symposium on Principles of Programming Languages (POPL), Austin, Texas. http://www.kestrel.edu/home/people/meertens/publications/papers/Incremental_polymorphic_type_checking_in_B.pdf.
- Mycroft, Alan (1984). "Polymorphic type schemes and recursive definitions". International Symposium on Programming. Lecture Notes in Computer Science. 167. 217–228. doi:10.1007/3-540-12925-1_41. ISBN 978-3-540-12925-7.
- Henglein, Fritz (1993). "Type inference with polymorphic recursion". ACM Transactions on Programming Languages and Systems 15 (2): 253–289. doi:10.1145/169701.169692.
- Kfoury, A. J.; Tiuryn, J.; Urzyczyn, P. (April 1993). "Type reconstruction in the presence of polymorphic recursion". ACM Transactions on Programming Languages and Systems 15 (2): 290–311. doi:10.1145/169701.169687. ISSN 0164-0925.
- Michael I. Schwartzbach (June 1995). "Polymorphic type inference". Technical Report BRICS-LS-95-3. http://www.brics.dk/LS/95/3/BRICS-LS-95-3.ps.gz.
- Emms, Martin; Leiß, Hans (1996). "Extending the type checker for SML by polymorphic recursion—A correctness proof". Technical Report 96-101. http://www.cis.uni-muenchen.de/~leiss/polyrec/polyrec.cisbericht.96-101.ps.gz.
- Richard Bird and Lambert Meertens (1998). "Nested Datatypes".
- C. Vasconcellos, L. Figueiredo, C. Camarao (2003). "Practical Type Inference for Polymorphic Recursion: an Implementation in Haskell[|permanent dead link|dead link}}]". Journal of Universal Computer Science.
- L. Figueiredo, C. Camarao. "Type Inference for Polymorphic Recursive Definitions: a Specification in Haskell".
- Hallett, J. J; Kfoury, A. J. (July 2005). "Programming Examples Needing Polymorphic Recursion". Electronic Notes in Theoretical Computer Science 136: 57–102. doi:10.1016/j.entcs.2005.06.014. ISSN 1571-0661.
External links
- Standard ML with polymorphic recursion by Hans Leiß, University of Munich
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