Optimised ExpTime Tableaux for 𝒮ℋℐ𝒩 over Finite Residuated Lattices

Algorithm 2

A optimized decision procedure for checking satisfiability in 𝐿-𝒮ℋℐ𝒩.

 Input: A TBox 𝒯 in NNF and a finite set of concepts 𝑋 in NNF

 Output: an and-or forest 𝐺=⟨𝑉,𝐸⟩, with 𝜎∈𝑉 as the initial node such that 𝜎.𝑠𝑡𝑎𝑡𝑢𝑠={sat, unsat}

   begin

   create a new node 𝜎 with Σ(𝜎)∶=𝑋∪𝒯 and 𝜎.𝑠𝑡𝑎𝑡𝑢𝑠 = unexpanded;

   let 𝑉∶={𝜎}, 𝐸∶=∅;

(a)    while  𝑣.status∉{sat, unsat}  do

(b)     choose an node 𝑣∈𝑉, 𝑣.𝑠𝑡𝑎𝑡𝑢𝑠  := unexpanded;

     if no 𝐿-𝒮ℋℐ𝒩-tableau rule is applicable to Σ(𝑣)  then

        𝑣.𝑠𝑡𝑎𝑡𝑢𝑠={sat};

     else if  ⟂ is applicable to Σ(𝑣)  then

        𝑣.𝑠𝑡𝑎𝑡𝑢𝑠={unsat};

     else if  𝛾  (𝛾∈𝛾₁) is applicable to Σ(𝑣) giving concept 𝑌  then

        𝑣.𝑘𝑖𝑛𝑑:= and-node, Θ∶={𝑌};

     else if  𝛾  (𝛾∈𝛾₂) is applicable to Σ(𝑣) giving concepts 𝑌₁ and 𝑌₂  then

        𝑣.𝑘𝑖𝑛𝑑:= or-node, Θ∶={𝑌₁,𝑌₂};

     else

        (i) 𝑣.𝑘𝑖𝑛𝑑:= and-node;

        (ii) for every ∃𝑅⋅𝐶∈ℒ(𝑣), apply 𝛾  (𝛾∈𝛾₃) to Σ(𝑣) giving concept 𝚝𝚛𝚊𝚗𝚜_𝑅(Σ(𝑣),𝑅)∪{𝐶}∪𝒯 and add this

        concept to Θ;

(c)    for  ∀𝑌∈Θ  do

        (i) if  𝑠𝑜𝑚𝑒  𝑤∈𝑉 has Σ(𝑤)=𝑌  then

        then add edge (𝑣,𝑤) to 𝐸;

        (ii) let 𝑤 be a new node, set Σ(𝑤)∶=𝑌, 𝑤.𝑠𝑡𝑎𝑡𝑢𝑠:= unexpanded, add 𝑤 to 𝑉, and add edge (𝑣,𝑤) to 𝐸;

(d)    if (𝑣.kind = or-node and one of the successors of 𝑣 has status sat) or (𝑣.kind = and-node and all the

     successors of 𝑣 have status sat) then

        𝑣.𝑠𝑡𝑎𝑡𝑢𝑠:= sat, propagate(𝑣);

     else if (𝑣.kind = and-node and one of the successors of 𝑣 has status unsat) or (𝑣.kind = or-node and all the

     successors of 𝑣 have status unsat) then

        𝑣.𝑠𝑡𝑎𝑡𝑢𝑠≔ unsat, propagate(𝑣);

     else

        𝑣.𝑠𝑡𝑎𝑡𝑢𝑠:= expanded;

(e)    if  𝑣.𝑠𝑡𝑎𝑡𝑢𝑠 = unsat  and  𝑣 in 𝒯, {𝜅,¬𝜅}∈Σ(𝑣)  then

        dep(𝜅,𝑣)∪dep(¬𝜅,𝑣);

     else if  𝑣.𝑠𝑡𝑎𝑡𝑢𝑠:= unexpanded  then

        if  Σ(𝑣) is of the form 𝜅₁∨𝜅₂  then

        save(𝑇_𝑠);

        add 𝜅₁ to Σ(𝑣) with dep(𝜅₁,𝑣)={𝑏}∪dep(Σ(𝑣),𝑣);

(f)       if  𝑏∈𝑆  then

         add 𝜅₂ to Σ(𝑣) with dep(𝜅₂,𝑣)=𝑏∪𝑑;

     else

        break;

(g)    If  𝜎.𝑠𝑡𝑎𝑡𝑢𝑠 = unsat  then

     return false;

     else

     return true;