Complexity Analysis of Tree Share Operations

We investigate the complexity of the tree share model of Dockins et al., which is used to reason about shared ownership of resources in e.g. concurrent programs. We obtain the precise complexity for first-order theory of the full Boolean algebra of tree shares (that is, with all tree-share constants) which is the same as first-order theory of countably atomless Boolean algebras (that is, where the only constants allowed in formulas are 0 and 1); the previous bound was nonelementary (that is, not bounded by any fixed tower of exponentials). We also prove a precise complexity bound on the first-order theory over the “relativization” multiplication operator on trees if one operand is a constant; the previous bound was non-elementary if the constant was on the right and not known to be decidable if the constant was on the left. Finally, we prove that the first-order theory over the structure that combines the Boolean algebra operators and the right-handed multiplication has a non-elementary lower bound even though the key subtheories are elementary, showing that the existing non-elementary upper bound cannot be improved. 1998 ACM Subject Classification F.1.1 Models of Computation, F.3.1 Specifying and Verifying and Reasoning about Programs, F.4.1 Mathematical Logic, F.4.3 Formal Languages