## Third Order Matching is Decidable (1999)

Venue: | Annals of Pure and Applied Logic |

Citations: | 49 - 0 self |

### BibTeX

@INPROCEEDINGS{Dowek99thirdorder,

author = {Gilles Dowek},

title = {Third Order Matching is Decidable},

booktitle = {Annals of Pure and Applied Logic},

year = {1999},

pages = {2--10},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed -calculus, i.e. to solve the equation a = b where a and b are simply typed -terms and b is ground. The decidability of this problem is still open. We prove the decidability of the particular case in which the variables occurring in the problem are at most third order. Introduction The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed -calculus i.e. to solve the equation a = b where a and b are simply typed -terms and b is ground. Pattern matching algorithms are used to check if a proposition can be deduced from another by elimination of universal quantifiers or by introduction of existential quantifiers. In automated theorem proving, elimination of universal quantifiers and introduction of existential quantifiers are mixed and full unification is required, but in proof-checking and semi-aut...

### Citations

1161 | Lambda Calculus: its Syntax and Semantics - Barendregt - 1984 |

315 | Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem
- Bruijn
- 1972
(Show Context)
Citation Context ...ff-conversion (i.e. bound variable renaming) and we consider that bound variables are renamed to avoid capture during substitutions. A rigorous presentation would use, for instance, de Bruijn indices =-=[2]-=-. Obviously, if t is a term of type T , x is a variable of type U and u a term of type U then the term t[x / u] has type T . In the same way if a term t has type T and t reduces to u then u has type T... |

173 |
Introduction to Combinators and Calculus
- Hindley, Seldin
- 1986
(Show Context)
Citation Context ... reduces to u then u has type T . Proposition 1 The fij-reduction relation is strongly normalizable and confluent on typed terms, and thus each term has a unique normal form. Proof See, for instance, =-=[4]-=-. Proposition 2 Let t be a normal well-typed term of type U 1 ! ::: ! U n ! U (U atomic), the term t has the form t = y 1 : U 1 : ::: ym : Um :(x u 1 ::: u p ) where msn and x is a constant, an instan... |

172 |
A unification Algorithm for Typed !-calculus
- Huet
- 1975
(Show Context)
Citation Context ...em proving, these rules can be applied separately and thus pattern matching can be used instead of unification. Higher order matching is conjectured decidable in [6] and the problem is still open. In =-=[5]-=- [6] [7] Huet has given a semi-decision algorithm and shown that in the particular case in which the variables occurring in the term a are at most second order this algorithm terminates, and thus that... |

151 | Proving and applying program transformations expressed with second-order patterns - Huet, Lang - 1978 |

127 | Unification under a mixed prefix
- Miller
- 1992
(Show Context)
Citation Context ... for variables in a substitution. In Huet's unification algorithm [5] [6] these local variables are always kept in the head of the terms in common abstractions. In Miller's mixed prefixes terminology =-=[8]-=-, constants are universal variables declared to the left hand side of the instantiable variables and local variables are universal variables declared to the right hand side of all the instantiable var... |

54 |
Higher-order unification revisited: Complete sets of transformations
- Snyder, Gallier
- 1989
(Show Context)
Citation Context ...interpolation problem \Phi(\Psi; oe) is constructed from a pair ! \Psi; oe ? where \Psi is an arbitrary problem and oe a solution to \Psi, can be compared to the one used in the completeness proof of =-=[9]-=- in which a problem in solved form is constructed from such a pair. 4 A Decision Procedure Theorem Third Order Matching is Decidable Proof A decision procedure is obtained by considering the problem \... |

29 |
Explicit definitions and linguistic dominoes
- Gorn
- 1967
(Show Context)
Citation Context ...he problem admits no such set. At last we discuss the problems that occur when we try to generalize the proof given here to higher order matching. 1 Trees and Terms 1.1 Trees Definitions 1 (Following =-=[3]-=-) An occurrence is a list of strictly positive integers ff =! s 1 ; :::; s n ?. The number n is called the length of the occurrence ff. A tree domain D is a non empty finite set of occurrences such th... |

17 |
The clausal theory of types
- Wolfram
- 1990
(Show Context)
Citation Context ...g in the term a are at most second order this algorithm terminates, and thus that second order matching is decidable. In [10] Statman has reduced the conjecture to the -definability conjecture and in =-=[11]-=- Wolfram has given an always terminating algorithm whose completeness is conjectured. We prove in this paper that third order matching is decidable i.e. we give an algorithm that decides if a matching... |

12 |
Completeness, Invariance and *-definability
- Statman
- 1982
(Show Context)
Citation Context ...algorithm and shown that in the particular case in which the variables occurring in the term a are at most second order this algorithm terminates, and thus that second order matching is decidable. In =-=[10]-=- Statman has reduced the conjecture to the -definability conjecture and in [11] Wolfram has given an always terminating algorithm whose completeness is conjectured. We prove in this paper that third o... |

5 |
R'esolution d' ' Equations dans les Langages d'Ordre 1,2, ..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris VII
- Huet
- 1976
(Show Context)
Citation Context ...roof-checking and semi-automated theorem proving, these rules can be applied separately and thus pattern matching can be used instead of unification. Higher order matching is conjectured decidable in =-=[6]-=- and the problem is still open. In [5] [6] [7] Huet has given a semi-decision algorithm and shown that in the particular case in which the variables occurring in the term a are at most second order th... |

2 |
The set of unifiers in typed calculus as regular expression, Rewriting techniques and
- Zaionc
- 1985
(Show Context)
Citation Context ...(x a z : T:z) = a : T:a The substitutions x / o : T:s : T ! T :(s ::: (s o) ::: ) are solutions to this problem and they cannot be obtained as instances of a finite number of solutions. Remark In [6] =-=[12]-=-, the similar examples (x z : T:z) = a and (x z : T:z) = b(a) are considered. So in contrast with second order matching [6] [7] there is no (always terminating) algorithm that enumerates a complete se... |