Conjunctive normal form

In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs.

In automated theorem proving, the notion "clausal normal form" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals.

Definition

A logical formula is considered to be in CNF if it is a conjunction of one or more disjunctions of one or more literals. As in disjunctive normal form (DNF), the only propositional operators in CNF are or (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \vee } ), and (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \land } ), and not ( $\neg$ ). The not operator can only be used as part of a literal, which means that it can only precede a propositional variable.

The following is a context-free grammar for CNF:

CNF Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \,\to \,} Disjunct

\,\mid \,

Disjunct Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \,\land \,} CNF

Disjunct Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \,\to \,} Literal

\,\mid \,

Literal Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \,\lor \,} Disjunct

Literal Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \,\to \,} Variable

\,\mid \,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \,\neg \,} Variable

Where Variable is any variable.

All of the following formulas in the variables Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A,B,C,D,E,} and $F$ are in conjunctive normal form:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (A\lor \neg B\lor \neg C)\land (\neg D\lor E\lor F)}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (A\lor B)\land (C)}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (A\lor B)}
$(A)$

The following formulas are not in conjunctive normal form:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \neg (A\land B)} , since an AND is nested within a NOT
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \neg (A\lor B)\land C} , since an OR is nested within a NOT
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A\land (B\lor (D\land E))} , since an AND is nested within an OR
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A\land (B\lor (C\lor D))} , since a nested OR should be written without parenthesis

Conversion to CNF

In classical logic each propositional formula can be converted to an equivalent formula that is in CNF.^[1] This transformation is based on rules about logical equivalences: double negation elimination, De Morgan's laws, and the distributive law.

Basic algorithm

The algorithm to compute a CNF-equivalent of a given propositional formula $\phi$ builds upon $\lnot \phi$ in disjunctive normal form (DNF): step 1.^[2]
Then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot \phi _{DNF}} is converted to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi _{CNF}} by swapping ANDs with ORs and vice versa while negating all the literals. Remove all Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot \lnot } .^[1]

Conversion by syntactic means

Convert to CNF the propositional formula $\phi$ .

Step 1: Convert its negation to disjunctive normal form.^[2]

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot \phi _{DNF}=(C_{1}\lor C_{2}\lor \ldots \lor C_{i}\lor \ldots \lor C_{m})} ,^[3]

where each Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C_{i}} is a conjunction of literals $l_{i1}\land l_{i2}\land \ldots \land l_{in_{i}}$ .^[4]

Step 2: Negate Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot \phi _{DNF}} . Then shift $\lnot$ inwards by applying the (generalized) De Morgan's equivalences until no longer possible. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\phi &\leftrightarrow \lnot \lnot \phi _{DNF}\\&=\lnot (C_{1}\lor C_{2}\lor \ldots \lor C_{i}\lor \ldots \lor C_{m})\\&\leftrightarrow \lnot C_{1}\land \lnot C_{2}\land \ldots \land \lnot C_{i}\land \ldots \land \lnot C_{m}&&{\text{// (generalized) D.M.}}\end{aligned}}} whereFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\lnot C_{i}&=\lnot (l_{i1}\land l_{i2}\land \ldots \land l_{in_{i}})\\&\leftrightarrow (\lnot l_{i1}\lor \lnot l_{i2}\lor \ldots \lor \lnot l_{in_{i}})&&{\text{// (generalized) D.M.}}\end{aligned}}}

Step 3: Remove all double negations.

Example

Convert to CNF the propositional formula Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi =((\lnot (p\land q))\leftrightarrow (\lnot r\uparrow (p\oplus q)))} .^[5]

The (full) DNF equivalent of its negation is^[2]
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot \phi _{DNF}=(p\land q\land r)\lor (p\land q\land \lnot r)\lor (p\land \lnot q\land \lnot r)\lor (\lnot p\land q\land \lnot r)}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\phi &\leftrightarrow \lnot \lnot \phi _{DNF}\\&=\lnot \{(p\land q\land r)\lor (p\land q\land \lnot r)\lor (p\land \lnot q\land \lnot r)\lor (\lnot p\land q\land \lnot r)\}\\&\leftrightarrow {\underline {\lnot (p\land q\land r)}}\land {\underline {\lnot (p\land q\land \lnot r)}}\land {\underline {\lnot (p\land \lnot q\land \lnot r)}}\land {\underline {\lnot (\lnot p\land q\land \lnot r)}}&&{\text{// generalized D.M. }}\\&\leftrightarrow (\lnot p\lor \lnot q\lor \lnot r)\land (\lnot p\lor \lnot q\lor \lnot \lnot r)\land (\lnot p\lor \lnot \lnot q\lor \lnot \lnot r)\land (\lnot \lnot p\lor \lnot q\lor \lnot \lnot r)&&{\text{// generalized D.M. }}(4\times )\\&\leftrightarrow (\lnot p\lor \lnot q\lor \lnot r)\land (\lnot p\lor \lnot q\lor r)\land (\lnot p\lor q\lor r)\land (p\lor \lnot q\lor r)&&{\text{// remove all }}\lnot \lnot \\&=\phi _{CNF}\end{aligned}}}

Conversion by semantic means

A CNF equivalent of a formula can be derived from its truth table. Again, consider the formula Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi =((\lnot (p\land q))\leftrightarrow (\lnot r\uparrow (p\oplus q)))} .^[5]

The corresponding truth table is

$p$	$q$	$r$	$\lnot$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (p\land q)}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \leftrightarrow }	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot r}	$\uparrow$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (p\oplus q)}
T	T	T	F	T	F	F	T	F
T	T	F	F	T	F	T	T	F
T	F	T	T	F	T	F	T	T
T	F	F	T	F	F	T	F	T
F	T	T	T	F	T	F	T	T
F	T	F	T	F	F	T	F	T
F	F	T	T	F	T	F	T	F
F	F	F	T	F	T	T	T	F

A CNF equivalent of $\phi$ is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\lnot p\lor \lnot q\lor \lnot r)\land (\lnot p\lor \lnot q\lor r)\land (\lnot p\lor q\lor r)\land (p\lor \lnot q\lor r)}

Each disjunction reflects an assignment of variables for which $\phi$ evaluates to F(alse).
If in such an assignment a variable $V$

is T(rue), then the literal is set to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lnot V} in the disjunction,
is F(alse), then the literal is set to $V$ in the disjunction.

Other approaches

Since all propositional formulas can be converted into an equivalent formula in conjunctive normal form, proofs are often based on the assumption that all formulae are CNF. However, in some cases this conversion to CNF can lead to an exponential explosion of the formula. For example, translating the non-CNF formula

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (X_{1}\wedge Y_{1})\vee (X_{2}\wedge Y_{2})\vee \ldots \vee (X_{n}\wedge Y_{n})}

into CNF produces a formula with $2^{n}$ clauses:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_1 \vee X_2 \vee \ldots \vee X_n) \wedge (Y_1 \vee X_2 \vee \ldots \vee X_n) \wedge (X_1 \vee Y_2 \vee \ldots \vee X_n) \wedge (Y_1 \vee Y_2 \vee \ldots \vee X_n) \wedge \ldots \wedge (Y_1 \vee Y_2 \vee \ldots \vee Y_n).}

Each clause contains either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_i} for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} .

There exist transformations into CNF that avoid an exponential increase in size by preserving satisfiability rather than equivalence.^[6]^[7] These transformations are guaranteed to only linearly increase the size of the formula, but introduce new variables. For example, the above formula can be transformed into CNF by adding variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_1,\ldots,Z_n} as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Z_1 \vee \ldots \vee Z_n) \wedge (\neg Z_1 \vee X_1) \wedge (\neg Z_1 \vee Y_1) \wedge \ldots \wedge (\neg Z_n \vee X_n) \wedge (\neg Z_n \vee Y_n). }

An interpretation satisfies this formula only if at least one of the new variables is true. If this variable is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_i} , then both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_i} are true as well. This means that every model that satisfies this formula also satisfies the original one. On the other hand, only some of the models of the original formula satisfy this one: since the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_i} are not mentioned in the original formula, their values are irrelevant to satisfaction of it, which is not the case in the last formula. This means that the original formula and the result of the translation are equisatisfiable but not equivalent.

An alternative translation, the Tseitin transformation, also includes the clauses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_i \vee \neg X_i \vee \neg Y_i} . With these clauses, the formula implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_i \equiv X_i \wedge Y_i} ; this formula is often regarded to "define" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_i} to be a name for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i \wedge Y_i} .

Maximum number of disjunctions

Consider a propositional formula with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} variables, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ge 1} .

There are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} possible literals: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \{ p_1, \lnot p_1, p_2, \lnot p_2, \ldots, p_n, \lnot p_n\}} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2^{2n} -1)} non-empty subsets.^[8]

This is the maximum number of disjunctions a CNF can have.^[9]

All truth-functional combinations can be expressed with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{n}} disjunctions, one for each row of the truth table.
In the example below they are underlined.

Example

Consider a formula with two variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} .

The longest possible CNF has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{(2 \times 2)} -1 = 15} disjunctions:^[9] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcl} (\lnot p) \land (p) \land (\lnot q) \land (q) \land \\ (\lnot p \or p) \land \underline{(\lnot p \or \lnot q)} \land \underline{(\lnot p \or q)} \land \underline{( p \or \lnot q)} \land \underline{( p \or q)} \land (\lnot q \or q) \land \\ (\lnot p \or p \or \lnot q) \land (\lnot p \or p \or q) \land (\lnot p \or \lnot q \or q) \land ( p \or \lnot q \or q) \land \\ (\lnot p \or p \or \lnot q \or q) \end{array}}

This formula is a contradiction. It can be simplified to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\neg p \land p)} or to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\neg q \land q)} , which are also contradictions, as well as valid CNFs.

Computational complexity

An important set of problems in computational complexity involves finding assignments to the variables of a Boolean formula expressed in conjunctive normal form, such that the formula is true. The k-SAT problem is the problem of finding a satisfying assignment to a Boolean formula expressed in CNF in which each disjunction contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem with k>2) while 2-SAT is known to have solutions in polynomial time. As a consequence,^[10] the task of converting a formula into a DNF, preserving satisfiability, is NP-hard; dually, converting into CNF, preserving validity, is also NP-hard; hence equivalence-preserving conversion into DNF or CNF is again NP-hard.

Typical problems in this case involve formulas in "3CNF": conjunctive normal form with no more than three variables per conjunct. Examples of such formulas encountered in practice can be very large, for example with 100,000 variables and 1,000,000 conjuncts.

A formula in CNF can be converted into an equisatisfiable formula in "kCNF" (for k≥3) by replacing each conjunct with more than k variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 \vee \ldots \vee X_k \vee \ldots \vee X_n} by two conjuncts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 \vee \ldots \vee X_{k-1} \vee Z} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg Z \vee X_k \lor \ldots \vee X_n} with $Z$ a new variable, and repeating as often as necessary.

First-order logic

In first order logic, conjunctive normal form can be taken further to yield the clausal normal form of a logical formula, which can be then used to perform first-order resolution. In resolution-based automated theorem-proving, a CNF formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{11}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lor}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lor}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{1n_1}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle )}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \land}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \land}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{m1}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lor}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lor}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{mn_m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle )}

,^[11] is commonly represented as a set of sets

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{11}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{1n_1}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \}}