Optimised ExpTime Tableaux for 𝒮ℋℐ𝒩 over Finite Residuated Lattices

Table 3

Rule description.

Rule	Description
(¬⋈)	If   ⟨¬𝐶,⋈,𝑙⟩∈Σ(𝑥)   and   ⟨𝐶,⋈^−,∼ℓ⟩∉Σ(𝑥), then   Σ(𝑥)→Σ(𝑥)∪⟨𝐶,⋈^−,∼ℓ⟩
(⊓⊳)	If   ⟨𝐶1⊓𝐶2,⊳,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked and   {⟨𝐶1,⊳,ℓ⟩,⟨𝐶2,⊳,ℓ⟩}∉Σ(𝑥), then   Σ(𝑥)→Σ(𝑥)∪{⟨𝐶1,⊳,ℓ⟩,⟨𝐶2,⊳,ℓ⟩}
(⊔⊲)	If   ⟨𝐶1⊔𝐶2,⊲,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked and   {⟨𝐶1,⊲,ℓ⟩,⟨𝐶2,⊲,ℓ⟩}∉Σ(𝑥), then   Σ(𝑥)→Σ(𝑥)∪{⟨𝐶1,⊲,ℓ⟩,⟨𝐶2,⊲,ℓ⟩}
(⊔⊳)	If   ⟨𝐶1⊔𝐶2,⊳,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked and   {⟨𝐶1,⊳,𝑙⟩,⟨𝐶2,⊳,ℓ⟩}∩Σ(𝑥)=∅, then   Σ(𝑥)→Σ(𝑥)∪{𝐶} for some 𝐶∈{⟨𝐶1,⊳,ℓ⟩,⟨𝐶2,⊳,ℓ⟩}
(⊓⊲)	If   ⟨𝐶1⊓𝐶2,⊲,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked and   {⟨𝐶1,⊲,ℓ⟩,⟨𝐶2,⊲,ℓ⟩}∩Σ(𝑥)=∅, then   Σ(𝑥)→Σ(𝑥)∪{𝐶} for some 𝐶∈{⟨𝐶1,⊲,ℓ⟩,⟨𝐶2,⊲,ℓ⟩}
(∃⊳)	If   ⟨∃𝑅⋅𝐶,⊳,ℓ⟩∈Σ(𝑥),  𝑥 is not blocked and   𝑥 has no 𝑅-neighbour 𝑦 connected with a triple ⟨𝑃^∗,⊳,ℓ⟩,𝑃⊑^∗𝑅 and ⟨𝐶,⊳,ℓ⟩∈Σ(𝑦)  , then  create a new node 𝑦 with Σ(⟨𝑥,𝑦⟩)={⟨𝑅,⊳,ℓ⟩}, Σ(𝑦)={⟨𝐶,⊳,ℓ⟩}
(∀⊲)	If   ⟨∀𝑅⋅𝐶,⊲,ℓ⟩∈Σ(𝑥),  𝑥 is not blocked and   𝑥 has no 𝑅-neighbour 𝑦 connected with a triple ⟨𝑃^∗,⊲^−,∼ℓ⟩,𝑃⊑^∗𝑅 and ⟨𝐶,⊲,ℓ⟩∈Σ(𝑦) , then  create a new node 𝑦 with Σ(⟨𝑥,𝑦⟩)={⟨𝑅,⊲^−,∼ℓ⟩}, Σ(𝑦)={⟨𝐶,⊲,ℓ⟩}
(∀⊳)	If   ⟨∀𝑅⋅𝐶,⊳,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked, and   𝑥 has an 𝑅-neighbour 𝑦 with ⟨𝐶,⊳,ℓ⟩∉Σ(𝑦)   and   ⟨∗,⊳^−,∼ℓ⟩ is conjugated with the positive triple that connects 𝑥 and 𝑦, then   Σ(𝑦)→Σ(𝑦)∪⟨𝐶,⊳,ℓ⟩
(∃⊲)	If   ⟨∃𝑅⋅𝐶,⊲,𝑙⟩∈Σ(𝑥),  𝑥 is not indirectly blocked and   𝑥 has an 𝑅-neighbour 𝑦 with ⟨𝐶,⊳,ℓ⟩∉Σ(𝑦)   and   ⟨∗,⊲,ℓ⟩ is conjugated with the positive triple that connects 𝑥 and 𝑦, then   Σ(𝑦)→Σ(𝑦)∪⟨𝐶,⊲,ℓ⟩
(∀+)	If   ⟨∀𝑅⋅𝐶,→,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked, 𝑥 has a 𝑃-neighbour 𝑦 with, ⟨𝐶,⊳,ℓ⟩∉Σ(𝑦), and  there is some 𝑃, with Trans(𝑃), and 𝑃⊑^∗𝑅, ⟨∀𝑃⋅𝐶,⊳,ℓ⟩∉Σ(𝑦)   and   ⟨∗,⊳^−,∼ℓ⟩ is conjugated with the positive triple that connects 𝑥 and 𝑦, then   Σ(𝑦)→Σ(𝑦)∪⟨∀𝑃⋅𝐶,⊳,𝑙⟩
(∃+)	If   ⟨∃𝑅⋅𝐶,⊲,ℓ⟩∈Σ(𝑥),  𝑥 is not indirectly blocked, 𝑥 has a 𝑃-neighbour 𝑦 with, ⟨𝐶,⊳,ℓ⟩∉Σ(𝑦), and  there is some 𝑃, with Trans(𝑃), and 𝑃⊑^∗𝑅, ⟨∃𝑃⋅𝐶,⊲,ℓ⟩∉Σ(𝑦)   and   ⟨∗,⊲,∼ℓ⟩ is conjugated with the positive triple that connects 𝑥 and 𝑦, then   Σ(𝑦)→Σ(𝑦)∪⟨∃𝑃⋅𝐶,⊲,ℓ⟩
(≥⊳)	If   ⟨≥𝑝⋅𝑅,⊳,𝑙⟩∈Σ(𝑥),  𝑥 is not blocked, and  there are no 𝑝  𝑅-neighbours 𝑦1,…,ℓ𝑝, connected to 𝑥 with a triple ⟨∃𝑃^∗,⊳,𝑙⟩∉Σ(𝑦), 𝑃⊑^∗𝑅, and   𝑦𝑖≠𝑦𝑗 for 1≤𝑖≤𝑗≤𝑝, then  create 𝑝 new nodes 𝑦1,…,ℓ𝑝, with Σ(⟨𝑥,𝑦𝑖⟩)={⟨𝑅,⊳,ℓ⟩} and 𝑦𝑖≠𝑦𝑗 for 1≤𝑖≤𝑗≤𝑝
(≤⊲)	If   ⟨≤𝑝⋅𝑅,⊲,ℓ⟩∈Σ(𝑥),  𝑥 is not blocked, then  apply (∃+) rule for the triple ⟨≥(𝑝+1)𝑅,⊲^−,∼ℓ⟩
(≤⊲)	If   ⟨≤𝑝⋅𝑅,⊲,ℓ⟩∈Σ(𝑥),  𝑥 is not blocked, then  apply (∃+) rule for the triple ⟨≥(𝑝+1)𝑅,⊲^−,∼ℓ⟩
(≤⊳)	If   ⟨≤𝑝⋅𝑅,⊳,𝑙⟩∈Σ(𝑥),  𝑥 is not indirectly blocked, and  there are 𝑝+1  𝑅-neighbours 𝑦1,…,𝑦𝑝+1 connected to 𝑥 with a triple ⟨𝑃^∗,⊳^′,∼ℓ𝑖⟩, 𝑃⊑^∗𝑅, and  which is conjugated with 𝑃^∗,⊳^−,∼ℓ, and there are two of them 𝑦, 𝑧, with no 𝑦≠𝑧, and   𝑦 is neither a root node nor an ancestor of 𝑧, then   Σ(𝑧)→Σ(𝑧)∪Σ(𝑦), and  if  𝑧 is an ancestor of 𝑥, then Σ(⟨𝑧,𝑥⟩)→Σ(⟨𝑧,𝑥⟩)∪𝙸𝚗𝚟(Σ(⟨𝑥,𝑦⟩)) else Σ(⟨𝑥,𝑧⟩)→Σ(⟨𝑥,𝑧⟩)∪Σ(⟨𝑥,𝑦⟩), and   Σ(⟨𝑥,𝑦⟩)→∅, and  Set 𝑢≠𝑧 for ∀𝑢 with 𝑢≠𝑦
(≥⊲)	If   ⟨≥𝑝⋅𝑅,⊲,ℓ⟩∈Σ(𝑥),𝑥 is not indirectly blocked, then  apply (≤⊳) rule for the triple ⟨≤(𝑝−1)𝑅,⊲^−,∼ℓ⟩
(≤𝑟⊳)	If   ⟨≤𝑝⋅𝑅,⊳,ℓ⟩∈Σ(𝑥), and  there are 𝑝+1  𝑅-neighbours 𝑦1,…,𝑦𝑝+1 connected to 𝑥 with a triple ⟨𝑃^∗,⊳^′,∼ℓ𝑖⟩, 𝑃⊑^∗𝑅, and  conjugated with ⟨𝑃^∗,⊳^−,∼ℓ⟩, and there are two of them 𝑦, 𝑧, both root nodes, with no 𝑦≠𝑧, then   Σ(𝑧)→Σ(𝑧)∪Σ(𝑦), and  for all edges ⟨𝑦,𝑤⟩: (1) if  the edge ⟨𝑧,𝑤⟩ does not exist, create it with Σ(⟨𝑧,𝑤⟩)→∅; (2) Σ(⟨𝑧,𝑤⟩)→Σ(⟨𝑧,𝑤⟩)∪Σ(⟨𝑦,𝑤⟩), and  for all edges ⟨𝑤,𝑦⟩: (1) if  the edge ⟨𝑤,𝑧⟩ does not exist, create it with Σ(⟨𝑤,𝑧⟩)→∅; (2) Σ(⟨𝑤,𝑧⟩)→Σ(⟨𝑤,𝑧⟩)∪Σ(⟨𝑤,𝑦⟩)   and  Set Σ(𝑦)=∅,   and remove all edges to/from 𝑦, and  set 𝑢≠𝑧 for ∀𝑢 with 𝑢≠𝑦 and set 𝑦=𝑧
(≥𝑟⊲)	⟨≥𝑝⋅𝑅,⊲,ℓ⟩∈Σ(𝑥), then  apply (≤𝑟⊳) rule for the triple ⟨≤(𝑝−1)𝑅,⊲^−,∼ℓ⟩