notesum.ai

Published at December 9

Reasoning about Strategic Abilities in Stochastic Multi-agent Systems

cs.LO

cs.FL

cs.GT

Released Date: December 9, 2024

Authors: Yedi Zhang, Fu Song¹, Taolue Chen², Xuzhi Wu³

Aff.: ¹Institute of Software and Chinese Academy of Sciences; ²Birkbeck, University of London; ³Shenzhen University

Arxiv: http://arxiv.org/pdf/2412.06509v1

Refer to caption

1. $\\|{\langle\langle{A}\rangle\rangle^{\bowtie k}\phi}\\|^{\xi}=\\|{\langle\langle{% A}\rangle\rangle^{\widehat{\bowtie}1-k}~{}\neg\phi}\\|^{\xi}$ .	2. $\\|{[[{A}]]^{\bowtie k}\phi}\\|^{\xi}=\\|{[[{A}]]^{\widehat{\bowtie}1-k}~{}\neg% \phi}\\|^{\xi}$ .
3. $\\|{\neg\langle\langle{A}\rangle\rangle^{\bowtie k}\phi}\\|^{\xi}=\\|{[[{A}]]^{% \overline{\bowtie}k}\phi}\\|^{\xi}$ .	4. $\\|{\neg[[{A}]]^{\bowtie k}\phi}\\|^{\xi}=\\|{\langle\langle{A}\rangle\rangle^{% \overline{\bowtie}k}~{}\phi}\\|^{\xi}$ .
5. $\\|{\langle\langle{{\sf Ag}}\rangle\rangle^{\bowtie k}\phi}\\|^{\xi}=\\|{[[{% \emptyset}]]^{{\bowtie}k}~{}\phi}\\|^{\xi}$ .	6. $\\|{[[{{\sf Ag}}]]^{\bowtie k}\phi}\\|^{\xi}=\\|{\langle\langle{\emptyset}\rangle% \rangle^{{\bowtie}k}~{}\phi}\\|^{\xi}$ .