2 research outputs found
Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds
Given a formula in quantifier-free Presburger arithmetic, if it has a
satisfying solution, there is one whose size, measured in bits, is polynomially
bounded in the size of the formula. In this paper, we consider a special class
of quantifier-free Presburger formulas in which most linear constraints are
difference (separation) constraints, and the non-difference constraints are
sparse. This class has been observed to commonly occur in software
verification. We derive a new solution bound in terms of parameters
characterizing the sparseness of linear constraints and the number of
non-difference constraints, in addition to traditional measures of formula
size. In particular, we show that the number of bits needed per integer
variable is linear in the number of non-difference constraints and logarithmic
in the number and size of non-zero coefficients in them, but is otherwise
independent of the total number of linear constraints in the formula. The
derived bound can be used in a decision procedure based on instantiating
integer variables over a finite domain and translating the input
quantifier-free Presburger formula to an equi-satisfiable Boolean formula,
which is then checked using a Boolean satisfiability solver. In addition to our
main theoretical result, we discuss several optimizations for deriving tighter
bounds in practice. Empirical evidence indicates that our decision procedure
can greatly outperform other decision procedures.Comment: 26 page
Reachability in Two-Dimensional Vector Addition Systems with States: One Test is for Free
Vector addition system with states is an ubiquitous model of computation with
extensive applications in computer science. The reachability problem for vector
addition systems is central since many other problems reduce to that question.
The problem is decidable and it was recently proved that the dimension of the
vector addition system is an important parameter of the complexity. In fixed
dimensions larger than two, the complexity is not known (with huge complexity
gaps). In dimension two, the reachability problem was shown to be
PSPACE-complete by Blondin et al. in 2015. We consider an extension of this
model, called 2-TVASS, where the first counter can be tested for zero. This
model naturally extends the classical model of one counter automata (OCA). We
show that reachability is still solvable in polynomial space for 2-TVASS. As in
the work Blondin et al., our approach relies on the existence of small
reachability certificates obtained by concatenating polynomially many cycles.Comment: Full version of the paper with the same title and authors in the
proceedings of CONCUR 202