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Abstract:
The Herbrand theorem plays a fundamental role in automated theorem
proving
methods based on \emph{global variable} or \emph{rigid variable}
approaches.
The kernel step in procedures based on such methods can be described
as the \emph{corroboration} problem (also called the \emph{Herbrand
skeleton} problem),
where, given a positive integer $m$, called \emph{multiplicity},
and a quantifier free formula, one seeks for a valid or provable (in
classical
first-order logic) disjunction of $m$ instantiations of that formula.
In logic with equality this problem was recently shown to be
undecidable.
The first main contribution of this paper is a logical theorem, that we
call
the \emph{Partisan Corroboration Theorem}, that enables us to show that,
for a certain interesting subclass of Horn formulas, corroboration with
multiplicity one can be reduced to corroboration with any given
multiplicity.
The second main contribution of this paper is a \emph{finite tree
automata}
formalization of a technique called \emph{shifted pairing} for proving
undecidability results via direct encodings of valid Turing machine
computations. We call it the \emph{Shifted Pairing Theorem}.
By using the Partisan Corroboration Theorem, the Shifted Pairing
Theorem,
and term rewriting techniques in equational reasoning, we improve upon
a number of recent undecidability results related to the
\emph{corroboration} problem,
the \emph{simultaneous rigid E-unification} problem and
the \emph{prenex fragment of intuitionistic logic with equality}.
