## On the Undecidability of Second-Order Unification (2000)

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Venue: | INFORMATION AND COMPUTATION |

Citations: | 33 - 16 self |

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@MISC{Levy00onthe,

author = {Jordi Levy and Margus Veanes},

title = {On the Undecidability of Second-Order Unification},

year = {2000}

}

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### Abstract

... this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is undecidable in the following three cases: (1) each second-order variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all second-order variables are ground terms of size <n, (ii) the arity of all second-order variables is <n, (iii) the number of occurrences of second-order variables is #5, (iv) there is either a single second-order variable or there are two second-order variables and no first-order variables.

### Citations

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Citation Context ..., where M is a TM and v is an input string for M is encoded effectively as a string over Σu, denoted by 〈M , v〉. The details of such an encoding are not relevant here and can be found for example in [=-=Hopcroft & Ullman 1979-=-]. The universal TM accepts 〈M , v〉 if and only if M accepts v. The following corollary is an easy consequence of Theorem 14, if the constructions there are based on Mu. Corollary 15 There is system S... |

939 |
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Citation Context ...iables. 2 Preliminary Definitions We assume that the reader is familiar with first-order unification problems, typed λ-calculus [e.g. Hindley & Seldin 1986], and basic notions in term rewriting [e.g. =-=Baader & Nipkow 1998-=-]. We briefly go through the necessary definitions. A type is an expression generated by the grammar τ ::= B | τ → τ | τ × τ, where B is a set of base types. An order of a type τ is defined as follows... |

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Citation Context ... , ln,cons(t,nil)) ? = cons(s, F (r1, . . . , rn,nil)). The precise reasons why this relationship holds are explained in detail in Section 3. Despite many similarities, word unification is decidable [=-=Makanin 1977-=-], second-order unification is undecidable [Goldfarb 1981], simultaneous rigid Eunification is undecidable [Degtyarev & Voronkov 1995], and, with the exception of some special cases [Comon 1993, Schmi... |

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Citation Context ...The precise reasons why this relationship holds are explained in detail in Section 3. Despite many similarities, word unification is decidable [Makanin 1977], second-order unification is undecidable [=-=Goldfarb 1981-=-], simultaneous rigid Eunification is undecidable [Degtyarev & Voronkov 1995], and, with the exception of some special cases [Comon 1993, Schmidt-Schauß 1995, Levy 1996, SchmidtSchauß 1998, Schmidt-Sc... |

66 | Completion of rewrite systems with membership constraints. part i: Deduction rules
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Citation Context ...der unification [Pietrzykowski 1973] and restricted forms thereof have been studied in a wide range of areas, such as natural language processing [Niehren, Pinkal & Ruhrberg 1997] and term rewriting [=-=Comon 1993-=-], among others. Although Goldfarb [1981] showed second-order unification to be undecidable in general, Makanin’s [1977] decidability result of word unification and Schulz’s [1991] extension of it hav... |

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Citation Context ...e have not discussed matching problems. It should be noted however, that the second-order matching problem was proved decidable by Huet & Lang [1978]. This result has been generalized in [Dowek 1991, =-=Dowek 1994-=-, Padovani 1996] and Comon & Jurski [1997] have shown some computational complexity results. Moreover, due to the strong connection between second-order unification and SREU, several results concernin... |

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Citation Context ...idability result of word unification and Schulz’s [1991] extension of it have been used as key tools in establishing the decidability of some important special cases of second-order unification [e.g. =-=Farmer 1988-=-]. In linear second-order unification, terms in unifiers must have exactly one occurrence of each bound variable, and in context unification, in addition to this condition, all second-order variables ... |

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Citation Context ...ultaneous rigid E-unification. We use properties underlying this reduction and additional ideas that have been used to prove the undecidability of restricted cases of simultaneous rigid Eunification [=-=Plaisted 1995-=-, Gurevich & Veanes 1999] and second-order unification [Farmer 1991], that second-order unification is undecidable in the following three cases: (Section 4) each second-order variable occurs at most t... |

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Citation Context ...b.mpg.de ∗ This work was partially supported by the project MODELOGOS (TIC97-0579-C02-01) funded by the CICYT, and the ESPRIT Basic Research Action CCL. This paper is based on preliminary results in [=-=Levy 1998-=-, Veanes 1998, Levy & Veanes 1998]. 1Proposed running head: Second-Order Unification Corresponding author: Margus Veanes, Max-Planck-Institut für Informatik, Im Stadtwald, 66123 Saarbrücken, Germany.... |

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Citation Context ...d the work of Schubert [1997]. Schubert showed the undecidability of SOU for simple equations. This result was an important tool in establishing the undecidability of certain type inference problems [=-=Schubert 1997-=-, Schubert 1998]. His undecidability proof is by reduction from Minsky machines and the construction is quite technical. Although our proof was developed independently, the resulting set of equations ... |

11 | The undecidability of unification in third-order logic - Huet - 1973 |

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9 |
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Citation Context ...this paper we have not discussed matching problems. It should be noted however, that the second-order matching problem was proved decidable by Huet & Lang [1978]. This result has been generalized in [=-=Dowek 1991-=-, Dowek 1994, Padovani 1996] and Comon & Jurski [1997] have shown some computational complexity results. Moreover, due to the strong connection between second-order unification and SREU, several resul... |

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9 |
Filtrage d’ordre superieur
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Citation Context ...iscussed matching problems. It should be noted however, that the second-order matching problem was proved decidable by Huet & Lang [1978]. This result has been generalized in [Dowek 1991, Dowek 1994, =-=Padovani 1996-=-] and Comon & Jurski [1997] have shown some computational complexity results. Moreover, due to the strong connection between second-order unification and SREU, several results concerning SREU, in part... |

9 | Uniform representation of recursively enumerable sets with simultaneous rigid E-unification, UPMAIL
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Citation Context ...ined form by Veanes [1996]. The Train Lemma [Gurevich & Veanes 1999] uses tree automata theoretic techniques and connections to term rewriting [e.g. Dauchet 1993] and has its roots in [Plaisted 1995, =-=Veanes 1996-=-]. These two techniques have been the main tools in proving undecidability results of restricted forms of SREU [e.g. Gurevich & Veanes 1999]. The proof of Theorem 14 is an application of these techniq... |

8 |
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Citation Context ...iables. 3List of symbols character explanation a “blank” symbol: \texttt{\char32} All other symbols used in this document are standard symbols from L ATEX. 41 Introduction Second-order unification [=-=Pietrzykowski 1973-=-] and restricted forms thereof have been studied in a wide range of areas, such as natural language processing [Niehren, Pinkal & Ruhrberg 1997] and term rewriting [Comon 1993], among others. Although... |

8 | Monadic simultaneous rigid E-unification and related problems, in
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Citation Context ... Decidability of monadic second-order unification can be proved by reduction to word equations [Farmer 1988], whereas monadic SREU is only known to reduce to a nontrivial extension of word equations [=-=Gurevich & Voronkov 1997-=-], and its decidability is an open problem. 5 One Second-Order Variable is Enough The principal result of this section is that the number of different second-order variables in SOU plays a minor role ... |

7 | Reduction of second-order unification to simultaneous rigid E-unification
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(Show Context)
Citation Context ... detail in Section 3. Despite many similarities, word unification is decidable [Makanin 1977], second-order unification is undecidable [Goldfarb 1981], simultaneous rigid Eunification is undecidable [=-=Degtyarev & Voronkov 1995-=-], and, with the exception of some special cases [Comon 1993, Schmidt-Schauß 1995, Levy 1996, SchmidtSchauß 1998, Schmidt-Schauß & Schulz 1998], the question is open for linear second-order unificatio... |

7 |
Unification of stratified second-order terms
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Citation Context ... 1977], second-order unification is undecidable [Goldfarb 1981], simultaneous rigid Eunification is undecidable [Degtyarev & Voronkov 1995], and, with the exception of some special cases [Comon 1993, =-=Schmidt-Schauß 1995-=-, Levy 1996, SchmidtSchauß 1998, Schmidt-Schauß & Schulz 1998], the question is open for linear second-order unification and context unification (although these problems are 5conjectured to be decida... |

6 | Some undecidable problems related to the Herbrand theorem - Gurevich, Veanes - 1997 |

6 | On unification problems in restricted secondorder languages - J, Veanes - 1998 |

6 | Decidability of bounded second order unification - Schmidt-Schauß - 2004 |

5 | Rigid reachability - Ganzinger, Jacquemard, et al. - 1998 |

5 |
Logic with equality: Partisan corroboration, and shifted pairing
- Gurevich, Veanes
- 1999
(Show Context)
Citation Context ... E-unification. We use properties underlying this reduction and additional ideas that have been used to prove the undecidability of restricted cases of simultaneous rigid Eunification [Plaisted 1995, =-=Gurevich & Veanes 1999-=-] and second-order unification [Farmer 1991], that second-order unification is undecidable in the following three cases: (Section 4) each second-order variable occurs at most twice and there are only ... |

4 | The diophantiness of recursively enumerable sets - Matiyasevic - 1970 |

4 |
The relation between second-order unification and simultaneous rigid E-unification
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(Show Context)
Citation Context ...This work was partially supported by the project MODELOGOS (TIC97-0579-C02-01) funded by the CICYT, and the ESPRIT Basic Research Action CCL. This paper is based on preliminary results in [Levy 1998, =-=Veanes 1998-=-, Levy & Veanes 1998]. 1Proposed running head: Second-Order Unification Corresponding author: Margus Veanes, Max-Planck-Institut für Informatik, Im Stadtwald, 66123 Saarbrücken, Germany. email: veane... |

4 |
The undecidability of simultaneous rigid E-unification’, Theoretical Computer Science
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(Show Context)
Citation Context ...a, b, d, g}. ✷ 4 From SREU to SOU We apply the results in the previous section to show that there is a polynomial time reduction of SREU to SOU. The converse reduction, from SOU to SREU, is given in [=-=Degtyarev & Voronkov 1996-=-] and has also polynomial time complexity. We use a restricted form of SREU that is polynomial time equivalent to SREU and allows us to apply Lemma 2 in a direct manner. Let R be a system of reachabil... |

4 | Decidability of bounded second order unification - Schmidt-Schau - 1999 |

4 |
Introduction to Combinatorics and λ-Calculus
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(Show Context)
Citation Context ...le, or there are two second-order variables and no first-order variables. 2 Preliminary Definitions We assume that the reader is familiar with first-order unification problems, typed λ-calculus [e.g. =-=Hindley & Seldin 1986-=-], and basic notions in term rewriting [e.g. Baader & Nipkow 1998]. We briefly go through the necessary definitions. A type is an expression generated by the grammar τ ::= B | τ → τ | τ × τ, where B i... |

3 | Simultaneous rigid E-unification and other decision problems related to the Herbrand theorem
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(Show Context)
Citation Context ...istic logic and to several fundamental classical decision problems related to Herbrand’s theorem, carry over to second-order unification. The most recent survey discussing such relations is given by [=-=Voronkov 1998-=-]. The decidability of monadic SREU is currently an open problem, only some special cases are known to be decidable [Degtyarev, Matiyasevich & Voronkov 1996, Gurevich & Voronkov 1997, Cortier, Ganzing... |

3 | Decidable fragments of simultaneous rigid reachability - Cortier, Ganzinger, et al. - 1999 |

3 | Herbrand’s theorem and equational reasoning: Problems and solutions - Degtyarev, Gurevich, et al. - 1996 |

2 | The undecidability of the unification problem for thirdorder languages - Lucchesi - 1972 |

2 | Decidability of the unification problem for second-order languages with unary function symbols - Zhezherun - 1979 |

2 | Introduction to Combinatorics and *-Calculus - Hindley, Seldin - 1986 |

2 |
Rewriting and tree automata, in H. Comon
- Dauchet
- 1993
(Show Context)
Citation Context ...ed [1995] in the context of SREU, and then in a refined form by Veanes [1996]. The Train Lemma [Gurevich & Veanes 1999] uses tree automata theoretic techniques and connections to term rewriting [e.g. =-=Dauchet 1993-=-] and has its roots in [Plaisted 1995, Veanes 1996]. These two techniques have been the main tools in proving undecidability results of restricted forms of SREU [e.g. Gurevich & Veanes 1999]. The proo... |