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. 2023 Jun 12;23(12):5518. doi: 10.3390/s23125518

Table 4.

Notations table for BAN logic.

Notations	Description
$Q \| \equiv X$	Q believing in X
$Q ⊲ X$	Q sees which is X
$Q \| \equiv T$	Q believes T’s action. E.g., $Q \| \equiv T \| \equiv X$
	means Q believes T believes X is true
$Q \| \sim X$	Q once says X
$Q \Rightarrow X$	Q has full jurisdiction beyond X
$# (X)$	X is updated and fresh
${(C)}_{k}$	Combine conditions C by the use of k
${(C)}_{k}$	Carry out hash operation on C; use X
${(X)}_{K}$	Message of hash X with a key K
$Q \overset{k}{⟷} T$	Q and T used to interact using the shared key k
	with each other
$D I D_{i}$	Session key $D I D_{i}$ used one time in the
	current section
$\frac{Q \| \equiv Q \overset{K}{⟷} T . q ⊲ < X >_{K}}{Q \| \equiv T \| \sim X}$	Rule of Message-Meaning
$\frac{Q \| \equiv # (X)}{Q \| \equiv # (X, Y)}$	Rule of Concatenation-Freshness
$\frac{Q \| \equiv # (X), Q \| \equiv T \| \sim X}{Q \| \equiv T \| \equiv X}$	Rule of Verification-Nonce
$\frac{Q \| \equiv T \Rightarrow X, Q \| \equiv T \| \equiv X}{Q \| \equiv X}$	Rule of Jurisdiction