The Decision Problem for Linear Tree Constraints

We present new results on a constraint satisfaction problem arising from the inference of resource types. Linear constraints were introduced by Hofmann and Jost in the context of type-based amortized resource analysis by the potential method [5] where it was applied to functional programs. The constraint systems appearing in this system and subsequent ones has finitely many variables and can be reduced to linear programming. Later, Hofmann and Rodriguez extended type-based amortized resource analysis to object-oriented programs [8],[6] which led to constraints involving infinite lists or trees whose entries are numerical variables. Therefore, a straightforward reduction to linear programming is no longer an option. However, the constraint systems exhibit enough regularity that a heuristic procedure developed by Hofmann and Rodriguez allowed to find solutions in many cases. In the case of infinite lists the constraint systems can be simply described as follows. One has finitely many unknowns ranging over infinite lists (sequences) of nonnegative rational numbers including ∞. Terms and constraints are built from those by addition (+) and comparison (≤) understood pointwise, and the tail function (tl) that removes the first element of a sequence. Furthermore, constraint systems may contain ordinary linear arithmetic constraints over the nonnegative rationals including infinity where the head function (hd) maps sequences to numbers. For example, the following constraint system is solvable and has the (not unique) solution y a constant list and x an exponentially decreasing list and z a Fibonacci list with an additional linear summand. We have the (arithmetic and list) constraints