Completeness of two fragments of a logic for conditional strategic reasoning

Abstract 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 ($\textsf{CSR}$). $\textsf{CSR}$ 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 $\textsf{CSR}$. $\textsf{CSR}$ 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 ($\textsf{CCSR}$). In this paper, we prove the completeness of two fragments of $\textsf{CCSR}$: 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 $\textsf{CSR}$ and other strategic logics.