# MPI-I-97-2-006

## Third-order matching in $\lambda\rightarrow$-Curry is undecidable

### Vorobyov, Sergei

**MPI-I-97-2-006**. May** **1997, 17 pages. | Status:** **available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:

Given closed untyped $\lambda$-terms $\lambda x1... xk.s$

and $t$, which can be assigned some types $S1->...->Sk->T$ and $T$

respectively in the Curry-style systems of type assignment

(essentially due to R.~Hindley) $\lambda->$-Curry [Barendregt 92],

$\lambda^{->}_t$ [Mitchell 96], $TA_\lambda$ [Hindley97], it is

undecidable whether there exist closed terms $s1,...,sk$ of types

$S1,...,Sk$ such that $s[s1/x1,...,sk/xk]=_{\beta\eta}t$, even if the

orders of $si$'s do not exceed 3. This undecidability result should be

contrasted to the decidability of the third-order matching in the

Church-style simply typed lambda calculus with a single constant base

type [Dowek 92]. The proof is by reduction from the recursively

inseparable sets of invalid and finitely satisfiable sentences of the

first-order theory of binary relation [Trakhtenbrot 53, Vaught 60].

Acknowledgement:** **

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**URL to this document: **http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1997-2-006

**BibTeX**
`@TECHREPORT{``Vorobyov97-2-006``,`

` AUTHOR = {Vorobyov, Sergei},`

` TITLE = {Third-order matching in $\lambda\rightarrow$-Curry is undecidable},`

` TYPE = {Research Report},`

` INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-97-2-006},`

` MONTH = {May},`

` YEAR = {1997},`

` ISSN = {0946-011X},`

`}`