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Chapter 1: Problem 11

Give a criterion for a conjunctive normal form to be a tautology.

Short Answer

Expert verified
A CNF is a tautology if each clause contains both a literal and its negation.

Step by step solution

01

Understanding Conjunctive Normal Form (CNF)

A conjunctive normal form is a conjunction (AND operation) of clauses where each clause is a disjunction (OR operation) of literals. For example, \((A \lor B) \land (\lnot C \lor D)\). The entire formula is true if each clause is true.
02

Define a Tautology

A tautology is a formula that is true under any truth assignment for its variables. For a CNF to be a tautology, it must be true no matter the combination of truth values assigned to its variables.
03

Criterion for CNF Tautology

For a CNF to be a tautology, each clause must contain both a literal and its negation. For example, a clause like \((A \lor \lnot A)\) is always true regardless of the truth value of \(A\), because if \(A\) is true, \(\lnot A\) is false, and vice versa.
04

Conclusion

Every clause in the CNF must be a tautology by itself. That is, each clause must have at least one literal such that its negation is also present in the same clause. This ensures that every clause evaluates to true, making the entire CNF a tautology.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjunctive Normal Form
Conjunctive Normal Form (CNF) is a way of structuring logical expressions. It is a conjunction, or an "AND" operation, of multiple clauses. Each of these clauses is, in turn, a disjunction, meaning an "OR" operation, composed of literals. A literal is simply a variable or its negation.

Think of CNF as a combination of "ANDs" and "ORs," where you group together variables to evaluate their collective truthfulness. For instance, a CNF expression might look like: \((A \lor B) \land (\lnot C \lor D)\). Each part inside the parentheses is a clause and must be true for the whole expression to be true. This structure has practical applications in fields such as computer science and mathematics, especially in solving logical puzzles and problems.
Tautology
In logic, a tautology is a statement that remains true no matter what the truth values of its variables are. If you think of logic as a way to understand truth, then a tautology is the ultimate truth statement—always reliable and never false.

To illustrate this, consider the statement \((A \lor \lnot A)\). Here, regardless of whether \(A\) is true or false, the whole expression is true. This is because if \(A\) is true, \(\lnot A\) is false, making the OR operation true. Conversely, if \(A\) is false, \(\lnot A\) is true, again making the OR operation true.

Identifying tautologies can be crucial in simplifying complex logical expressions and verifying the reliability of logical arguments.
Logical Clauses
In the context of logic, a clause is a collection of literals connected by the "OR" operator inside a larger logical expression. Clauses serve as the building blocks in logical formulas, especially in conjunctive normal form.

Each clause must leverage its literals to ensure that it evaluates to true for the CNF formula to be valid. For example, if you have a clause like \((A \lor \lnot B)\), that clause will be true if either \(A\) is true or \(B\) is false. The goal is to construct clauses that collectively capture the desired logical conditions.

Engineering logical clauses is a crucial step in problem-solving and algorithm design, especially when automated theorem proving or circuit design is involved.
Truth Assignments
Truth assignments involve assigning truth values (true or false) to each variable within a logical expression. These assignments help determine whether the entire logic proposition evaluates to true or false under a particular configuration.

For a formula like the CNF to be a tautology, the truth assignments should result in the whole formula always being true, irrespective of the values given to variables. This means every possible combination of truth values must be considered.

Understanding truth assignments is essential for logical reasoning. It allows one to exhaustively verify the correctness of a logical formula, ensuring that no assignment of truth values leads the expression to be false.

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Most popular questions from this chapter

Consider the full language \(\mathcal{L}\) with the connectives \(\wedge, \rightarrow, \perp, \leftrightarrow \vee\) and the restricted language \(\mathcal{L}^{\prime}\) with connectives \(\wedge, \rightarrow, \perp\). Using the appropriate derivation rules we get the derivability notions \(\vdash\) and \(\vdash^{\prime}\). We define an obvious translation from \(\mathcal{L}\) into \(\mathcal{L}^{\prime}\) : \(\begin{aligned} \varphi^{+} &:=\varphi \text { for atomic } \varphi \\\\(\varphi \square \psi)^{+} &:=\varphi^{+} \square \psi^{+} \text {for } \square=\wedge, \rightarrow, \end{aligned}\) \((\varphi \vee \psi)^{+}:=\neg\left(\neg \varphi^{+} \wedge \neg \varphi^{+}\right)\), where \(\neg\) is an abbreviation, \((\varphi \leftrightarrow \psi)^{+}:=\left(\varphi^{+} \rightarrow \psi^{+}\right) \wedge\left(\psi^{+} \rightarrow \varphi^{+}\right)\), $$ (\neg \varphi)^{+}:=\varphi^{+} \rightarrow \perp . $$ Show (i) \(\vdash \varphi \leftrightarrow \varphi^{+}\), (ii) \(\quad \vdash \varphi \Leftrightarrow \vdash^{\prime} \varphi^{+}\), (iii) \(\varphi^{+}=\varphi\) for \(\varphi \in \mathcal{L}^{\prime}\). (iv) Show that the full logic, is conservative over the restricted logic, i.e. for \(\varphi \in \mathcal{L}^{\prime} \vdash \varphi \Leftrightarrow \vdash^{\prime} \varphi\).

Show that \(\mid\) and \(\downarrow\) are the only binary connectives \(\$$ such that \)\\{\$\\}$ is functionally complete.

Show that \(\\{\neg\\}\) is not a functionally complete set of connectives. Idem for \(\\{\rightarrow, \vee\\}\) (hint: show that each formula \(\varphi\) with only \(\rightarrow\) and \(\vee\) there is a valuation \(v\) such that \(\llbracket \varphi \rrbracket=1\) ).

Check which of the following sets are consistent. (a) \(\left\\{\neg p_{1} \wedge p_{2} \rightarrow p_{0}, p_{1} \rightarrow\left(\neg p_{1} \rightarrow p_{2}\right), p_{0} \leftrightarrow \neg p_{2}\right\\}\), (b) \(\left\\{p_{0} \rightarrow p_{1}, p_{1} \rightarrow p_{2}, p_{2} \rightarrow p_{3}, p_{3} \rightarrow \neg p_{0}\right\\}\), (c) \(\left\\{p_{0} \rightarrow p_{1}, p_{0} \wedge p_{2} \rightarrow p_{1} \wedge p_{3}, p_{0} \wedge p_{2} \wedge p_{4} \rightarrow p_{1} \wedge p_{3} \wedge p_{5}, \ldots\right\\}\)

Show in the system with \(\vee\) as a primitive connective $$ \begin{aligned} &\vdash(\varphi \rightarrow \psi) \leftrightarrow(\neg \varphi \vee \psi) \\ &\vdash(\varphi \rightarrow \psi) \vee(\psi \rightarrow \varphi) \end{aligned} $$
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