In the formal study of security protocols and access control systems, fragments of intuitionistic logic play a vital role. These are required to be efficient, and are typically disjunction-free. In this paper, we study the complexity of adding disjunction to these subsystems. Our lower bound results show that very little needs to be added to disjunction to get co-NP-hardness, while our upper bound results show that even a system with conjunction, disjunction, and restricted forms of negation and implication is in co-NP. Our upper bound proofs also suggest parameters which we can bound to obtain PTIME algorithms.
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