max planck institut
informatik

# 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:
References to related material:

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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},