Originator: Jean-Pierre Jouannaud
Date: April 1991
Is satisfiability of lpo or rpo ordering constraints decidable in case of
The existential fragment of the first-order theory of the “recursive path
ordering” (with multiset and lexicographic “status”) is decidable when the
precedence on function symbols is total [Com90, JO91], but is
undecidable for arbitrary formulas. Is the existential fragment decidable for
The Σ4 (∃*∀*∃*∀*) fragment is
undecidable, in general [Tre92]. The positive existential fragment
for the empty precedence (that is, for homeomorphic tree embedding) is
decidable [BC93]. One might also ask whether the first-order theory
of total recursive path orderings is decidable. Related results
include the following: The existential fragment of the subterm ordering is
decidable, but its Σ2 (∃*∀*) fragment is not
[Ven87]. The first-order theory of encompassment (the
instance-of-subterm relation) is decidable [CCD93]. The
satisfiability problem for the existential fragment in the total case is
Though the first-order theory of encompassment is decidable [CCD93],
the first-order (Σ2) theory of the recursive (lexicographic status)
path ordering, assuming certain simple conditions on the precedence, is not
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