Optimised ExpTime Tableaux for 𝒮ℋℐ𝒩 over Finite Residuated Lattices

Table 1

Semantics of 𝐿-𝒮ℋℐ𝒩 concepts and 𝐿-𝒮ℋℐ𝒩 roles.

Constructor	Syntax	Semantics
Top	⊤	⊤^ℐ(𝑥)=1
Bottom	⟂	⟂^ℐ(𝑥)=0
Conjunction	𝐶⊓𝐷	(𝐶⊓𝐷)^ℐ(𝑥)=𝐶^ℐ(𝑥)⊗𝐷^ℐ(𝑥)
Disjunction	𝐶⊔𝐷	(𝐶⊔𝐷)^ℐ(𝑥)=𝐶^ℐ(𝑥)⊕𝐷^ℐ(𝑥)
General negation	¬𝐶	(¬𝐶)^ℐ(𝑥)=∼𝐶^ℐ(𝑥)
Implication	𝐶→𝐷	(𝐶→𝐷)^ℐ(𝑥)=𝐶^ℐ(𝑥)⇒𝐷^ℐ(𝑥)
Exists restriction	∃𝑠⋅𝐶	∃(𝑠⋅𝐶)^ℐ(𝑥)=∨𝑦∈△^ℐ{𝑠^ℐ(𝑥,𝑦)⇒𝐶^ℐ(𝑦)}
Value restriction	∀𝑠⋅𝐶	∀(𝑠⋅𝐶)^ℐ(𝑥)=∧𝑦∈△^ℐ{𝑠^ℐ(𝑥,𝑦)⊗𝐶^ℐ(𝑦)}
At-most restriction	(≤𝑛𝑅)^ℐ(𝑥)	(≤𝑛𝑅)^ℐ(𝑥)=∧𝑦1,…,𝑦𝑛+1∨^𝑛+1𝑖=1{∼𝑅^ℐ(𝑥,𝑦𝑖)}
At-least restriction	(≥𝑛𝑅)^ℐ(𝑥)	(≥𝑛𝑅)^ℐ(𝑥)=∨𝑦1,…,𝑦𝑛∧^𝑛𝑖=1{𝑅^ℐ(𝑥,𝑦𝑖)}