## Decidable and undecidable second-order unification problems (1998)

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Venue: | In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications (RTA’98), volume 1379 of LNCS |

Citations: | 16 - 9 self |

### BibTeX

@INPROCEEDINGS{Levy98decidableand,

author = {Jordi Levy},

title = {Decidable and undecidable second-order unification problems},

booktitle = {In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications (RTA’98), volume 1379 of LNCS},

year = {1998},

pages = {47--60}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. There is a close relationship between word unification and second-order unification. This similarity has been exploited for instance for proving decidability of monadic second-order unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higher-order unification procedures) when variables are restricted to occur at most twice. Hence a well-known open question was the decidability of second-order unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid E-unification to second-order unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid E-unification is decidable. This happens, for example, for one-variable problems where the variable occurs at most twice (because rigid E-unification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable second-order unification problems with variable occurrence restrictions. 1

### Citations

172 |
A unification Algorithm for Typed !-calculus
- Huet
- 1975
(Show Context)
Citation Context ... deciding if a problem has a solution or not, and not in finding all minimum unifiers, we can simplify Pietrzykowski's procedure, since now, all flexible-flexible equations are always solvable. Huet (=-=Huet, 1975-=-) was the first to describe one of such preunification procedures for typed -calculus. The set of transformation rules is now as follows: Simplification rule. hfx : a(t 1 ; : : : ; t n ) ? = x : a(u 1... |

166 |
The problem of solvability of equations in a free semigroup
- Makanin
- 1977
(Show Context)
Citation Context ...ccurs more than once, hence significantly narrowing the gap between decidable and undecidable second-order unification problems with variable occurrence restrictions. 1 Introduction Word unification (=-=Makanin, 1977; Sc-=-hulz, 1991), linear second-order unification (Levy, 1996), context unification (Comon, 1993; Schmidt-Schau��, 1995) and second-order unification (Pietrzykowski, 1973) are closely related problems.... |

136 |
The undecidability of the second-order unification problem
- Goldfarb
- 1981
(Show Context)
Citation Context ...instance, to prove decidability of monadic second-order unification (Farmer, 1988). Despite their similarities, word unification is decidable (Makanin, 1977), second-order unification is undecidable (=-=Goldfarb, 1981-=-), and the question is open for linear second-order unification and context unification (although it is conjectured to be decidable). Decidability of word unification was an open question for a long t... |

119 |
An unification algorithm for typed λ-calculus
- Huet
- 1975
(Show Context)
Citation Context ..., we can simplify Pietrzykowski’s procedure (notice that all flexible-flexible equations are solvable). These decision procedures for unification problems are called pre-unification procedures. Huet (=-=Huet, 1975-=-) was the first to describe a pre-unification procedure for typed λcalculus. The set of transformation rules for second-order pre-unification can be easily derived from either Huet’s higher-order pre-... |

67 | Completion of rewrite systems with membership constraints. Part II: Constraint solving
- Comon
- 1998
(Show Context)
Citation Context ...ond-order unification problems with variable occurrence restrictions. 1 Introduction Word unification (Makanin, 1977; Schulz, 1991), linear second-order unification (Levy, 1996), context unification (=-=Comon, 1993; Sc-=-hmidt-Schau��, 1995) and second-order unification (Pietrzykowski, 1973) are closely related problems. The relationship between word unification and linear second-order unification becomes clear wh... |

34 | On the undecidability of second-order unification
- Levy, Veanes
(Show Context)
Citation Context ...ween decidable and undecidable second-order unification problems with variable occurrence restrictions. 1 Introduction Word unification (Makanin, 1977; Schulz, 1991), linear second-order unification (=-=Levy, 1996), c-=-ontext unification (Comon, 1993; Schmidt-Schau��, 1995) and second-order unification (Pietrzykowski, 1973) are closely related problems. The relationship between word unification and linear second... |

32 |
A unification algorithm for second-order monadic terms
- Farmer
- 1988
(Show Context)
Citation Context ...G(x))) ? = x:G(a(F (x))). The relationship between word unification and secondorder unification is not so clear, but was used, for instance, to prove decidability of monadic second-order unification (=-=Farmer, 1988-=-). Despite their similarities, word unification is decidable (Makanin, 1977), second-order unification is undecidable (Goldfarb, 1981), and the question is open for linear second-order unification and... |

29 |
Theorem Proving using Rigid E-Unification: Equational Matings," LICS'87
- Gallier, Raatz, et al.
- 1987
(Show Context)
Citation Context ...analysis of how we could have proved decidability suggested us how we can, in fact, prove its undecidability by reduction of another undecidable unification problem: simultaneous rigid E-unification (=-=Gallier et al., 1987-=-; Degtyarev and Voronkov, 1996). This reduction is in some sense reversible, providing a decidability result for cases when simultaneous rigid E-unification is decidable. This happens, for example, fo... |

27 |
Rigid E-unification is NP-complete
- Gallier, Narendran, et al.
(Show Context)
Citation Context ...gid E-unification is decidable. This happens, for example, for one (second-order) variable problems where the variable occurs at most twice, since (non-simultaneous) rigid E-unification is decidable (=-=Gallier et al., 1988-=-). This paper proceeds as follows. In section 2 we introduce all the unification problems and some preliminary definitions and notation. In section 3 we prove undecidability of secondorder unification... |

24 |
Simple second-order languages for which unification is undecidable
- Farmer
- 1991
(Show Context)
Citation Context ...bclasses of second-order unification problems have been found. Farmer proved decidability for monadic terms [Far88], and undecidability for classes defined by a restriction on the number of variables =-=[Far91]-=-. Here we have characterized decidability for classes defined by the number of occurrences per variable. Moreover, we have stated a very close relationship between the simultaneous rigid E-unification... |

20 | Makanin’s algorithm - two improvements and a generalization, CIS-report 91-39, Centrum für Informations- und Sprachverarbeitung - Schulz - 1991 |

15 |
Designing unification procedures using transformations: A survey
- Gallier, Snyder
- 1990
(Show Context)
Citation Context ... term t where subterm at position p has been replaced by u. 2.1 Word unification It is easy to describe a complete 1 (non-terminating) procedure for word unification in terms of transformation rules (=-=Gallier and Snyder, 1990-=-). Any state of the process is represented by a pair hS; oei, where S is the problem and oe the substitution computed until that moment. We proceed by applying a substitution ae, that transforms the p... |

14 | The undecidability of simultaneous rigid eunification
- Degtyarev, Voronkov
- 1996
(Show Context)
Citation Context ...ld have proved decidability suggested us how we can, in fact, prove its undecidability by reduction of another undecidable unification problem: simultaneous rigid E-unification (Gallier et al., 1987; =-=Degtyarev and Voronkov, 1996-=-). This reduction is in some sense reversible, providing a decidability result for cases when simultaneous rigid E-unification is decidable. This happens, for example, for one (second-order) variable ... |

14 | Second-order unification and type inference for Church-style polymorphism
- Schubert
(Show Context)
Citation Context ...me equivalent. Based on the preliminary version of this paper, Veanes (Veanes, 1998) also reproduces some of the results we prove. Other result and ideas of this paper and (Veanes, 1998) appeared in (=-=Schubert, 1997-=-), although this paper contains a gap. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid E-unification is decidable. This happens, for example... |

9 | Unification of stratified second-order terms - Schmidt-Schau - 1994 |

8 |
A complete mechanization of second-order logic
- Pietrzykowski
- 1973
(Show Context)
Citation Context ...ns. 1 Introduction Word unification (Makanin, 1977; Schulz, 1991), linear second-order unification (Levy, 1996), context unification (Comon, 1993; Schmidt-Schau��, 1995) and second-order unificati=-=on (Pietrzykowski, 1973-=-) are closely related problems. The relationship between word unification and linear second-order unification becomes clear when we codify a word unification problem, like F \Delta a \Delta G ? = G \D... |

7 | Reduction of second-order unification to simultaneous rigid E-unification
- Degtyarev, Voronkov
- 1995
(Show Context)
Citation Context ...Ground rigid O-unification (for one equation) is decidable. Simultaneous ground rigid O-unification is undecidable, even if we restrict right-hand sides of assumption not to be a variable. Proof: In (=-=Degtyarev and Voronkov, 1995-=-) it is proved undecidability of simultaneous rigid E-unification by reducing the second-order unification problem to it. This proof may be easily adapted to simultaneous ground rigid O-unification. T... |

7 |
Unification of stratified second-order terms
- Schmidt-Schauß
- 1994
(Show Context)
Citation Context ...fication problems with variable occurrence restrictions. 1 Introduction Word unification (Makanin, 1977; Schulz, 1991), linear second-order unification (Levy, 1996), context unification (Comon, 1993; =-=Schmidt-Schauß, 1995-=-) and second-order unification (Pietrzykowski, 1973) are closely related problems. The relationship between word unification and linear second-order unification becomes clear when we codify a word uni... |