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

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons45677

Vorobyov,  Sergei
Programming Logics, MPI for Informatics, Max Planck Society;

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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].