Use the ALC-Worlds algorithm to decide the satisfiability of the concept name B0 w.r.t. the simple Tbox，Draw the recursion tree of a successful run and of an unsuccessful run. Does the algorithm return a positive or negative result on this input? B0 ≡ B1\bigsqcapB2 B1 ≡ ∃r.B3 B2 ≡ B4 \bigsqcap B5 B3 ≡ P B4 ≡ ∃r.B6 B5 ≡ B7 \sqcap B8 B6 ≡ Q B7 ≡ ∀r.B4 B8 ≡ ∀r.B9 B9 ≡ ∀r.B10 B10 ≡ ¬P

To use the ALC-Worlds algorithm, we first need to convert the Tbox and the concept name B0 into their ALC syntax:

B0 ≡ B1 ⊓ B2
B1 ≡ ∃r.B3
B2 ≡ B4 ⊓ B5
B3 ≡ P
B4 ≡ ∃r.B6
B5 ≡ B7 ⊓ B8
B6 ≡ Q
B7 ≡ ∀r.B4
B8 ≡ ∀r.B9
B9 ≡ ∀r.B10
B10 ≡ ¬P

Now, we can start the algorithm:

1. Initialize the set of worlds W to contain the empty interpretation {}.
2. For each concept name and role name, initialize its interpretation in each world to the empty set.
3. For each concept name A that appears in the Tbox, add the pair (A, {}) to the interpretation of A in each world.
4. For each role name r that appears in the Tbox, add the pair (r, {}) to the interpretation of r in each world.
5. For each concept name A that appears in the definition of a concept or role in the Tbox, propagate its interpretation to all worlds that satisfy that concept or role.
6. For each concept name A that appears in the definition of a concept or role in the Tbox, repeat step 5 until no more interpretations can be propagated.
7. For each world w in W, check if w satisfies B0. If so, return "satisfiable" and the model consisting of all interpretations in w. If not, mark w as unsatisfiable.
8. If all worlds are marked as unsatisfiable, return "unsatisfiable".

Recursion tree for a successful run:

W={}
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W1={r={}, B3={}}
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W2={r={}, B3={}, B6={}}
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W3={r={}, B3={}, B6={Q}}
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W4={r={}, B3={P}, B6={Q}}
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W5={r={r1:W4}, B3={P}, B6={Q}}
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W6={r={r1:W4, r2:W5}, B3={P}, B6={Q}}
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W7={r={r1:W4, r2:W5, r3:W6}, B3={P}, B6={Q}}
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W8={r={r1:W4, r2:W5, r3:W6, r4:W7}, B3={P}, B6={Q}}
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W9={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8}, B3={P}, B6={Q}}
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W10={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={P}, B6={Q}}
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W11={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={}, B6={Q}}
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W12={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={P}, B6={}}
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W13={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12}, B3={P}, B6={}}
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W14={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12, r8:W13}, B3={P}, B6={}}
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W15={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12, r8:W13, r9:W14}, B3={P}, B6={}}
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W16={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12, r8:W13, r9:W14, r10:W15}, B3={P}, B6={¬P}}
 
The algorithm returns a positive result (i.e., "satisfiable") with the model consisting of all interpretations in W16.

Recursion tree for an unsuccessful run:

W={}
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W1={r={}, B3={}}
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W2={r={}, B3={}, B6={}}
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W3={r={}, B3={}, B6={Q}}
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W4={r={}, B3={P}, B6={Q}}
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W5={r={r1:W4}, B3={P}, B6={Q}}
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W6={r={r1:W4, r2:W5}, B3={}, B6={Q}}
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W7={r={r1:W4, r2:W5, r3:W6}, B3={P}, B6={Q}}
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W8={r={r1:W4, r2:W5, r3:W6, r4:W7}, B3={P}, B6={Q}}
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W9={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8}, B3={P}, B6={Q}}
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W10={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={}, B6={Q}}
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W11={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10}, B3={P}, B6={Q}}
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W12={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10, r8:W11}, B3={P}, B6={Q}}
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W13={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10, r8:W11, r9:W12}, B3={P}, B6={Q}}
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W14={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10, r8:W11, r9:W12, r10:W13}, B3={P}, B6={¬P}}

The algorithm returns a negative result (i.e., "unsatisfiable") as all worlds are marked as unsatisfiable.