Abstract:
We study a model of graph related formulae that we call the \emph{Constraint-Graph model}. A constraint-graph is a labeled multi-graph (a graph where loops and parallel edges are allowed), where each edge $e$ is labeled by a distinct Boolean variable and every vertex is associate with a Boolean function over the variables that label its adjacent edges. A Boolean assignment to the variables satisfies the constraint graph if it satisfies every vertex function. We associate with a constraint-graph $G$ the property of all assignments satisfying $G$, denoted $SAT(G)$. We show that the above model is quite general. That is, for every property of strings ${\cal P}$ there exists a property of constraint-graphs ${\cal P}_G$ such that ${\cal P}$ is testable using $q$ queries iff ${\cal P}_G$ is thus testable. In addition, we present a large family of constraint-graphs for which $SAT(G)$ is testable with constant number of queries. As an implication of this, we infer the testability of some edge coloring problems (e.g. the property of two coloring of the edges in which every node is adjacent to at least one vertex of each color). Another implication is that every property of Boolean strings that can be represented by a Read-twice CNF formula is testable. We note that this is the best possible in terms of the number of occurrences of every variable in a formula.