Weak Cost Monadic Logic over Infinite Trees

Cost monadic logic has been introduced recently as a quantitative extension to monadic second-order logic. A sentence in the logic defines a function from a set of structures to N∪{∞}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. The rich theory associated with these functions has already been studied over finite words and trees. We extend the theory to infinite trees for the weak form of the logic (where second-order quantification is interpreted over finite sets). In particular, we show weak cost monadic logic is equivalent to weak cost automata, and finite-memory strategies suffice in the infinite two-player games derived from such automata. We use these results to provide a decision procedure for the logic and to show there is a function definable in cost monadic logic which is not definable in weak cost monadic logic.