Completeness of two fragments of a logic for conditional strategic reasoning

2024-05-19

arXiv.org

AI Summary

The strong completeness of the cooperative fragment (CCSR) of Conditional Strategic Reasoning logic (CSR)—specifically concerning responsibility and ability—remained unestablished; moreover, the full completeness of CSR has been an open problem for years. Method: We introduce a novel proof paradigm integrating modal semantic modeling, abstract game formalization, standard disjunctive theories, realizability characterizations, and derivation reduction techniques to achieve strict correspondence between axiomatic systems and semantics. Contribution/Results: We establish the first strong completeness result for CCSR. The resulting axiomatic system is not only strongly complete but also methodologically extensible: its framework provides a systematic toolkit and a new pathway toward resolving the long-standing completeness problem for full CSR, as well as advancing metatheoretic investigations of other strategic logics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Common-Sense Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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 two recent papers, introduced a Logic for Conditional Strategic Reasoning (CSR). However, its completeness is still an open problem. 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. 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.

Problem

• Proving completeness of two conditional strategic logic fragments
• Addressing open completeness problem for cooperating conditional reasoning
• Establishing completeness for liability and ability fragments of CCSR

Innovation

• Completeness proof for CCSR fragments
• Uses standard disjunctions and validity-reduction
• Applies abstract game forms realization