MPI-INF/SWS Research Reports 1991-2021

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

Vorobyov, Sergei

May 1997, 17 pages.

Status: available - back from printing

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].

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  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},