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#### The Relation Between Second-Order Unification and Simultaneous Rigid E-Unification

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##### Citation

Veanes, M. (1998). The Relation Between Second-Order Unification and Simultaneous Rigid
E-Unification. In V. Pratt (*Proceedings of the 13th Annual
IEEE Symposium on Logic in Computer Science (LICS-98)* (pp. 264-275). Los Alamitos, USA: IEEE.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-38A1-E

##### Abstract

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 as well. Furthermore,
restricted forms of SREU are strongly related to word equations
and finite tree automata.
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.
So second-order unification plays the same fundamental role as SREU in
automated reasoning in logic with equality.
We exploit this connection and use
finite tree automata techniques to
present a very elementary undecidability proof of second-order unification,
by reduction from the halting problem for Turing machines.
It follows from that proof that second-order unification is undecidable
for all nonmonadic second-order term languages having
at least two second-order variables with sufficiently high arities.