Predicate calculus

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Problem: Reduce the predicate logic formula to prenex normal form. Is the formula on the set \( M=\{1,2\} \) : 1) satisfiable, 2) refutable, 3) generally valid, 4) unsatisfiable? Calculate the truth value of the formula on the set \( M \) with the following predicates: \begin{tabular}{|c|c|c|} \hline\( x \) & 1 & 2 \\ \hline\( P(x) \) & 1 & 0 \\ \hline\( R(x) \) & 0 & 1 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline\( Q(x, y) \) & 1 & 2 \\ \hline 1 & 1 & 0 \\ \hline 2 & 0 & 0 \\ \hline \end{tabular} \[ \forall x P(x) \rightarrow(R(x) \rightarrow \exists y Q(x, y)) \text {. } \]

6.5.1 Predicate calculus

3 $

Problem: Establish the truth of the logical expression using two methods: 1) by defining quantifiers and 2) by the method of concretization. a) \( \forall x \exists y(A(y) \vee B(x))=\exists x A(x) \vee \forall x B(x) \), b) \( \forall x \exists y P(x, y) \Rightarrow \exists x \exists y P(x, y) \).

6.5.2 Predicate calculus

4.49 $

Problem: Determine the truth values of formulas in a given interpretation: a) \( \exists x \forall y(P(x) \Rightarrow Q(y)) \), b) \( \exists x(P(x) \& Q(x)) \) \[ D=\{a, b\} \] \begin{tabular}{|c|c|c|} \hline\( x \) & \( P(x) \) & \( Q(x) \) \\ \hline\( a \) & 1 & 0 \\ \hline\( b \) & 0 & 1 \\ \hline \end{tabular}

6.5.3 Predicate calculus

1.25 $

Problem: Write the following statements in the language of predicate logic: "For any three numbers, if their product is odd, then all three numbers are odd".

6.5.4 Predicate calculus

0.75 $

Problem: Indicate all subformulas, as well as the scope of quantifications, free and bound occurrences of all variables in the following formulas: \[ S(t, w) \vee \exists x \forall w[(Q(x, w) \rightarrow P(x)) \rightarrow R(w)] \text {. } \]

6.5.5 Predicate calculus

1.25 $

Problem: Perform skolem normal form of the following formulas presented in preliminary form: \[ \forall t \exists x \forall w \exists y[(P(t, w) \& Q(x, y) \rightarrow S(y)) \rightarrow R(x)] . \]

6.5.6 Predicate calculus

1.25 $

Problem: Convert the following formulas to clausal form: \[ R(t, w) \vee \neg \exists x \forall w[\neg(P(w) \vee S(x)) \rightarrow \neg Q(w)] \text {. } \]

6.5.7 Predicate calculus

3 $

Problem: Write the following sentences as predicate logic formulas and transform them into clausal form: «A student loves logic or philosophy if and only if there is a teacher who loves both logic and philosophy».

6.5.8 Predicate calculus

3 $

Problem: Determine the most general unifier and the corresponding general example for the following set of terms or show that the set is non-unifiable. \[ \left\{L_{i}\right\}=\{h(f(a), g(y, z), y), h(x, g(b, u), c)\} . \]

6.5.9 Predicate calculus

2.5 $

Problem: Perform skolemization of the following formulas presented in preliminary form: \[ \forall y \forall z \exists x \forall w[(T(x, w) \vee P(x, y) \& S(x, z)) \rightarrow R(x, w)] . \]

6.5.11 Predicate calculus

1.25 $

Problem: Convert the following formulas to clausal form: \[ Q(x, w) \vee \neg \exists x \forall w(P(x, w) \& Q(x, z)) \vee \neg S(z, w) \text {. } \]

6.5.12 Predicate calculus

2.5 $

Problem: Write the following sentences as predicate logic formulas and transform them into clausal form: "If there are no intelligent machines, and if every ideal machine is an intelligent machine, then an ideal machine does not exist".

6.5.13 Predicate calculus

3 $

Problem: Determine the most general unifier and the corresponding general example for the following set of terms or show that the set is non-unifiable. \[ \left\{L_{i}\right\}=\{g(f(b), f(x), a), g(y, v, b)\} . \]

6.5.14 Predicate calculus

2.5 $

Problem: An identically correct formula of predicate logic is the formula: 1. \( \exists x A(x) \Rightarrow \forall x B(x) \); 2. \( \forall x A(x) \Rightarrow \exists x A(x) \); 3. \( \forall x(A(x) \vee B(x)) \Rightarrow \forall x B(x) \); 4. \( \forall x A(x) \Rightarrow \exists x(A(x) \wedge B(x)) \).

6.5.15 Predicate calculus

0.75 $

Problem: Write the following argument in predicates and prove its validity using the resolution method. The premise: "no freshman likes sophomores. Everyone living in Vasyuki is a sophomore." Conclusion: "Not a single freshman likes anyone living in Vasyuki".

6.5.10 Predicate calculus

3.75 $