Question

Give a trace of the execution of DPLL on the conjunction of these clauses.

S1: (￢A ∨ B ∨ E) ∧ (￢B ∨ A) ∧ (￢E ∨ A).

S2: (￢E ∨ D).

S3: (￢C ∨ ￢F ∨ ￢B).

S4: (￢E ∨ B).

S5: (￢B ∨ F).

S6: (￢B ∨ C).

Answer 1

mre of Initial clauses: [-Avbve],[7BVA], [TE VA], [TEVD], [1CU TRV7B], [78 vB], [ 16VF), [1Bue] Smitial symbols: A, B, C, D, E, F Pure symbols: only D occurs with one polarity: its pure. The clauses where D occurs is automatically satisfied as we assign D= true. So we ignore these clauses. Next clauses : [1AV BvE],[7BVA], [7Eva], [7€ v 7FV 78), [ 7EVB], [ 7DVF], [78 vc] 9 Symbols : A, B, C, E, F Model &D= true 1. Pwee symbols : No pure symbols, no unit clause either. io DPLL on A, cases - i> A= true, D= true YA = false, D= true - Apy A=true; skipping clause where À is present and . removing A from clauses. Next clauses : [bve], [70 V7FV7B], [7€ vB],[7BVF] 1 [mB vc]. Model=> D=true A=true No pure symbols no unit clause

DPLL on first symbol.ie B strue u B= false [Biz Since we are considering B=true case first remove all clauses with B because they will be Satisfiable. & delete zB from clauses. Next clauses: [ICVqf], [F], [C] Modelz Dafrees A=true; Baterie No pure symbols, but two whit clauses-[F] [c] Set catrrue Next deuses: Gif] [F]

Since for any value off one of the clause will be unsatisfiable & we backtrack to B=false

Tu Bi B= false, Remove all clauses having TB and delete it from clauses Next clauses a [E], [IE] . Model Datore A = true B=false of E one Again for any logic value owe expression/clause is unsalis fiable Backtrack to A=false JA iP> A=false Remove clauses howing 7A and remove A from clauses Next clauses: [18], [78], [7CU ZF 178], [1Ev B3, [76 Vf], [tbve] Pure symbol : E is pure symbol (since it occurs in 7 poterity) Set E= false Model D: true hi false E: false Next classes[TB], [7617807B], [7BVF], [nove] Pure Symbol: B, sef B= false Next clauses : 0, no clause left, all are satisfied % this is the final model: Dotoue A: false E:false B = false c=true or false Pitrue or false.

mre of Initial clauses: [-Avbve],[7BVA], [TE VA], [TEVD], [1CU TRV7B], [78 vB], [ 16VF), [1Bue] Smitial symbols: A, B, C, D, E, F Pure symbols: only D occurs with one polarity: its pure. The clauses where D occurs is automatically satisfied as we assign D= true. So we ignore these clauses. Next clauses : [1AV BvE],[7BVA], [7Eva], [7€ v 7FV 78), [ 7EVB], [ 7DVF], [78 vc] 9 Symbols : A, B, C, E, F Model &D= true 1. Pwee symbols : No pure symbols, no unit clause either. io DPLL on A, cases - i> A= true, D= true YA = false, D= true - Apy A=true; skipping clause where À is present and . removing A from clauses. Next clauses : [bve], [70 V7FV7B], [7€ vB],[7BVF] 1 [mB vc]. Model=> D=true A=true No pure symbols no unit clause

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Tu Bi B= false, Remove all clauses having TB and delete it from clauses Next clauses a [E], [IE] . Model Datore A = true B=false of E one Again for any logic value owe expression/clause is unsalis fiable Backtrack to A=false JA iP> A=false Remove clauses howing 7A and remove A from clauses Next clauses: [18], [78], [7CU ZF 178], [1Ev B3, [76 Vf], [tbve] Pure symbol : E is pure symbol (since it occurs in 7 poterity) Set E= false Model D: true hi false E: false Next classes[TB], [7617807B], [7BVF], [nove] Pure Symbol: B, sef B= false Next clauses : 0, no clause left, all are satisfied % this is the final model: Dotoue A: false E:false B = false c=true or false Pitrue or false.