1 research outputs found
Decidable Horn Systems with Difference Constraints Arithmetic
This paper tackles the problem of the existence of solutions for recursive
systems of Horn clauses with second-order variables interpreted as integer
relations, and harnessed by quantifier-free difference bounds arithmetic. We
start by proving the decidability of the problem "does the system have a
solution ?" for a simple class of Horn systems with one second-order variable
and one non-linear recursive rule. The proof relies on a construction of a tree
automaton recognizing all cycles in the weighted graph corresponding to every
unfolding tree of the Horn system. We constrain the tree to recognize only
cycles of negative weight by adding a Presburger formula that harnesses the
number of times each rule is fired, and reduce our problem to the universality
of a Presburger-constrained tree automaton. We studied the complexity of this
problem and found it to be in \textsc{NEXPtime} with an \textsc{EXPtime}-hard
lower bound. Second, we drop the univariate restriction and consider
multivariate second-order Horn systems with a structural restriction, called
\emph{flatness}. This more general class of Horn systems is found to be
decidable, within the same complexity bounds. Finally, we encode the
reachability problem for Alternating Branching Vector Addition Systems (ABVASS)
using Horn systems and prove that, for flat ABVASS, this problem is in
co-\textsc{NEXPtime}.Comment: This paper has been withdrawn by the author due to a crucial error in
Lemma