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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].
