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

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##### Fulltext (public)

1997-2-006

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##### Citation

Vorobyov, S.(1997). *Third-order matching in $\lambda\rightarrow$-Curry
is undecidable* (MPI-I-1997-2-006). Saarbrücken: Max-Planck-Institut für Informatik.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-9A2F-C

##### Abstract

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].