We address the entailment problem for separation logic with symbolic heaps admitting list pred- icates and permissions for memory cells that are essential to express ownership of a heap region. In the permission-free case, the entailment problem is known to be in P. Herein, we design new decision procedures for solving the satisfiability and entailment problems that are parameterised by the permission theories. This permits the use of solvers dealing with the permission theory at hand, independently of the shape analysis. We also show that the entailment problem without list predicates is coNP-complete for several permission models, such as counting permissions and binary tree shares but the problem is in P for fractional permissions. Furthermore, when list predicates are added, we prove that the entailment problem is coNP-complete when the entail- ment problem for permission formulae is in coNP, assuming the write permission can be split into as many read permissions as desired. Finally, we show that the entailment problem for any Boolean permission model with infinite width is coNP-complete.

We show the decidability of model checking PA-processes against several first-order logics based upon the reachability predicate. The main tool for this result is the recognizability by tree automata of the reachability relation. The tree automata approach and the transition logics we use allow a smooth and general treatment of parameterized model checking for PA. This approach is extended to handle a quite general notion of costs of PA-steps. In particular, when costs are Parikh images of traces, we show decidability of a transition logic extended by some form of first-order reasoning over costs.

We prove the decidability of the quantifier-free, static fragment of ambient logic, with composition adjunct and iteration, which corresponds to a kind of regular expression language for semistructured data. The essence of this result is a surprising connection between formulas of the ambient logic and counting constraints on (nested) vectors of integers.Our proof method is based on a new class of tree automata for unranked, unordered trees, which may result in practical algorithms for deciding the satisfiability of a formula. A benefit of our approach is to naturally lead to an extension of the logic with recursive definitions, which is also decidable. Finally, we identify a simple syntactic restriction on formulas that improves the effectiveness of our algorithms on large examples.