Completeness of two fragments of a logic for conditional strategic reasoning

Classical logics for strategic reasoning, such as Coalition Logic and Alternating-time Temporal Logic, formalize absolute strategic reasoning about the unconditional strategic abilities of agents to achieve their goals. Goranko and Ju in [5, 6] introduced a Logic for Conditional Strategic Reasoning (CSR). CSR is a powerful logic, and its completeness is still an open problem. In this paper, we do some useful work towards showing the completeness of CSR. CSR has three featured operators, and one of them has the following reading: for some action of A that guarantees the achievement of her goal, B has an action to guarantee the achievement of his goal. This operator makes good sense when A is cooperating with B, and the logic about this operator is called Logic for Cooperating Conditional Strategic Reasoning (CCSR). In this paper, we prove the completeness of two fragments of CCSR: the liability fragment and the ability fragment. The key ingredients of our proof approach include standard disjunctions, the validity-reduction condition of standard disjunctions, abstract game forms and their realization, and the derivability-reduction condition of standard disjunctions. The approach has good potential to be applied to the completeness of CSR and other strategic logics.