On the soundness, completeness and applicability of the logic of knowledge and communicative commitments in multi-agent systems

Abstract

Benthem’s correspondence theory is one of the most important tools of the theory of modal logics developed in the last three decades. Correspondence theory, a subfield of the model theory, reflects a systematic study of relations between classes of frames and modal language. In this paper, we use correspondence theory for modal logics to solve a problem not addressed yet in the literature, namely the soundness and completeness of a logic combining two different, yet related modalities: agents’ knowledge and commitments. The paper proves the soundness and completeness of this logic called CTLKC+. This work is highly significant as it proves that combining the two agents’ modalities resulted in a consistent logic that can be used to design reliable systems. The methodology is as follows: we develop a set of reasoning postulates (axioms) that reflect the interaction between agents’ knowledge and social commitments in multi-agent system (MAS) using the CTLKC+ logic and correspond them to certain classes of frames. In particular, we first give a name, formalization and meaning for each postulate. Then, we correspond the postulates to certain classes of frames and provide the required proofs. Thereafter, we present a discussion that illustrates the importance of the proposed postulates in MASs using a concrete application example called the NetBill protocol taken from the business domain. Finally, we show how the postulates were addressed in the literature. The existence of such a correspondence allows us to prove that the logic generated by any subset of these postulates is sound and complete with respect to models that are based on the corresponding frames. The ultimate goal of this paper is to further assess the logic of knowledge and commitments (CTLKC+) from a new perspective (i.e., the soundness and completeness). Consequently, this work advances the literature of logics in MASs and closes a gap that has not been explored before.