MPI-I-98-2-005

Simultaneous rigid E-unification, or SREU for short, is a fundamental problem that arises in global methods of automated theorem proving in classical logic with equality. In order to do proof search in intuitionistic logic with equality one has to handle SREU. Furthermore, restricted forms of SREU are strongly related to word equations and finite tree automata. Higher-order unification has applications in proof theory, computational linguistics, program transformation, and also in theorem proving. It was recently shown that second-order unification has a very natural reduction to simultaneous rigid $E$-unification, which constituted probably the most transparent undecidability proof of SREU. Here we show that there is also a natural encoding of SREU in second-order unification. It follows that the problems are logspace equivalent. We exploit this connection and use finite tree automata techniques to prove that second-order unification is undecidable in more restricted cases than known before. We present a more elementary undecidability proof of second-order unification than the previously known proofs exposing that already a very small fragment of second-order unification has the universal computational power.

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Extended version of a paper accepted for LICS'98