Prenex normal formFrom Wikipedia, the free encyclopediaContents1 Conjunctive normal form 11.1 Examples and Non-Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Conversion into CNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Converting from rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Disjunctive normal form 52.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Negation normal form 73.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Prenex normal form 94.1 Conversion to prenex form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1.1 Conjunction and disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1.2 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.3 Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.5 Intuitionistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Use of prenex form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13iii CONTENTS4.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Chapter 1Conjunctive normal formIn Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction ofclauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs. As a normal form, it is usefulin automated theorem proving. It is similar to the product of sums form used in circuit theory.All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one-literal clauses and conjunctions of a single clause, respectively.As in the disjunctive normal form (DNF), the onlypropositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used aspart of a literal, which means that it can only precede a propositional variable or a predicate symbol.In automated theorem proving, the notion "clausal normal form" is often used in a narrower sense, meaning a par-ticular representation of a CNF formula as a set of sets of literals.1.1 Examples and Non-ExamplesAll of the following formulas in the variables A, B, C, D, and E are in conjunctive normal form: A (B C) (A B) (B C D) (D E) A B A BThe last formula is in conjunctive normal form because it can be seen as the conjunction of the two single-literalclauses A and B . Incidentally, the last two formulas are also in disjunctive normal form.The following formulas are not in conjunctive normal form: (B C) (A B) C A (B (D E)).Every formula can be equivalently written as a formula in conjunctive normal form. In particular this is the case forthe three non-examples just mentioned; they are respectively equivalent to the following three formulas, which are inconjunctive normal form: B C (A C) (B C) A (B D) (B E).12 CHAPTER 1. CONJUNCTIVE NORMAL FORM1.2 Conversion into CNFEvery propositional formula can be converted into an equivalent formula that is in CNF. This transformation is basedon rules about logical equivalences: the double negative law, De Morgans laws, and the distributive law.Since all logical formulae can be converted into an equivalent formula in conjunctive normal form, proofs are oftenbased on the assumption that all formulae are CNF. However, in some cases this conversion to CNF can lead to anexponential explosion of the formula. For example, translating the following non-CNF formula into CNF produces aformula with 2nclauses:(X1 Y1) (X2 Y2) (Xn Yn).In particular, the generated formula is:(X1X2 Xn)(Y1X2 Xn)(X1Y2 Xn)(Y1Y2 Xn) (Y1Y2 Yn).This formula contains 2nclauses; each clause contains either Xi or Yi for each i .There exist transformations into CNF that avoid an exponential increase in size by preserving satisability rather thanequivalence.[1][2] These transformations are guaranteed to only linearly increase the size of the formula, but introducenew variables. For example, the above formula can be transformed into CNF by adding variablesZ1, . . . , Zn asfollows:(Z1 Zn) (Z1 X1) (Z1 Y1) (Zn Xn) (Zn Yn).An interpretation satises this formula only if at least one of the new variables is true. If this variable is Zi , thenboth Xi and Yi are true as well. This means that every model that satises this formula also satises the original one.On the other hand, only some of the models of the original formula satisfy this one: since the Zi are not mentionedin the original formula, their values are irrelevant to satisfaction of it, which is not the case in the last formula. Thismeans that the original formula and the result of the translation are equisatisable but not equivalent.An alternative translation, the Tseitin transformation, includes also the clauses ZiXiYi . With these clauses,the formula implies Zi Xi Yi ; this formula is often regarded to dene Zi to be a name for Xi Yi .1.3 First-order logicIn rst 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 rst-order resolution. In resolution-based automated theorem-proving, a CNFformulaSee below for an example.1.4 Computational complexityAn important set of problems in computational complexity involves nding assignments to the variables of a booleanformula expressed in Conjunctive Normal Form, such that the formula is true. The k-SAT problem is the problem ofnding a satisfying assignment to a boolean formula expressed in CNF in which each disjunction contains at most kvariables. 3-SAT is NP-complete (like any other k-SAT problem with k>2) while 2-SAT is known to have solutionsin polynomial time. As a consequence,[3] the task of converting a formula into a DNF, preserving satisability, isNP-hard; dually, converting into CNF, preserving validity, is also NP-hard; hence equivalence-preserving conversioninto DNF or CNF is again NP-hard.Typical problems in this case involve formulas in 3CNF": conjunctive normal formwith no more than three variablesper conjunct. Examples of such formulas encountered in practice can be very large, for example with 100,000variables and 1,000,000 conjuncts.1.5. CONVERTING FROM FIRST-ORDER LOGIC 3A formula in CNF can be converted into an equisatisable formula in "kCNF (for k>=3) by replacing each conjunctwith more than k variables X1 Xk Xn by two conjuncts X1 Xk1Z and Z Xk Xnwith Z a new variable, and repeating as often as necessary.1.5 Converting from rst-order logicTo convert rst-order logic to CNF:[4]1. Convert to negation normal form.(a) Eliminate implications and equivalences:repeatedly replace P Q with P Q ; replace P Qwith (P Q) (P Q) . Eventually, this will eliminate all occurrences of and .(b) Move NOTs inwards by repeatedly applying De Morgans Law. Specically, replace (P Q) with(P) (Q) ; replace (P Q) with (P) (Q) ; and replace P with P ; replace (xP(x))with xP(x) ; (xP(x)) with xP(x) . After that, a may occur only immediately before apredicate symbol.2. Standardize variables(a) For sentences like(xP(x)) (xQ(x)) which use the same variable name twice, change the nameof one of the variables. This avoids confusion later when dropping quantiers later. For example,x[yAnimal(y)Loves(x, y)][yLoves(y, x)] is renamed to x[yAnimal(y)Loves(x, y)][zLoves(z, x)] .3. Skolemize the statement(a) Move quantiers outwards: repeatedly replace P(xQ(x)) with x(PQ(x)) ; replace P(xQ(x))with x(P Q(x)) ; replace P (xQ(x)) with x(P Q(x)) ; replace P (xQ(x)) with x(P Q(x)) . These replacements preserve equivalence, since the previous variable standardization step en-sured that x doesn't occur in P . After these replacements, a quantier may occur only in the initial prexof the formula, but never inside a , , or .(b) Repeatedly replace x1. . . xn yP(y) with x1. . . xnP(f(x1, . . . , xn)) , where f is a new n -aryfunction symbol, a so-called "skolem function". This is the only step that preserves only satisabilityrather than equivalence. It eliminates all existential quantiers.4. Drop all universal quantiers.5. Distribute ORs inwards over ANDs: repeatedly replace P (Q R) with (P Q) (P R) .As an example, the formula saying Who loves all animals, is in turn loved by someone is converted into CNF (andsubsequently into clause form in the last line) as follows (highlighting replacement rule redices in red ):Informally, the skolem function g(x) can be thought of as yielding the person by whom x is loved, while f(x) yieldsthe animal (if any) that x doesn't love. The 3rd last line from below then reads as "x doesn't love the animal f(x) ,or else x is loved by g(x) ".The 2nd last line from above, (Animal(f(x)) Loves(g(x), x)) (Loves(x, f(x)) Loves(g(x), x)) , is theCNF.1.6 Notes[1] Tseitin (1968)[2] Jackson and Sheridan (2004)[3] since one way to check a CNF for satisability is to convert it into a DNF, the satisability of which can be checked inlinear time[4] Articial Intelligence: A modern Approach [1995...] Russell and Norvig4 CHAPTER 1. CONJUNCTIVE NORMAL FORM1.7 See alsoAlgebraic normal formDisjunctive normal formHorn clauseQuineMcCluskey algorithm1.8 ReferencesPaul Jackson, Daniel Sheridan: Clause Form Conversions for Boolean Circuits. In: Holger H. Hoos, David G.Mitchell (Eds.):Theory and Applications of Satisability Testing, 7th International Conference, SAT 2004,Vancouver, BC, Canada, May 1013, 2004, Revised Selected Papers. Lecture Notes in Computer Science3542, Springer 2005, pp. 183198G.S. Tseitin: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Structures inConstructive Mathematics and Mathematical Logic, Part II, Seminars in Mathematics (translated from Rus-sian), pp. 115125. Steklov Mathematical Institute (1968)1.9 External linksHazewinkel, Michiel, ed. (2001), Conjunctive normal form, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4Java applet for converting to CNF and DNF, showing laws usedMayuresh S. Pardeshi, Dr. Bashirahamed F. Momin Conversion of cnf to dnf using grid computing IEEE,ISBN 978-1-4673-2816-6Mayuresh S. Pardeshi, Dr. Bashirahamed F. Momin Conversion of cnf to dnf using grid computing in parallelIEEE, ISBN 978-1-4799-4041-7Chapter 2Disjunctive normal formIn boolean logic, adisjunctivenormalform (DNF) is a standardization (or normalization) of a logical formulawhich is a disjunction of conjunctive clauses; it can also be described as an OR of ANDs, a sum of products, or (inphilosophical logic) a cluster concept. As a normal form, it is useful in automated theorem proving. A logical formulais considered to be in DNF if and only if it is a disjunction of one or more conjunctions of one or more literals. ADNF formula is in full disjunctive normal form if each of its variables appears exactly once in every clause. As inconjunctive normal form (CNF), the only propositional operators in DNF are and, or, and not. The not operator canonly be used as part of a literal, which means that it can only precede a propositional variable. For example, all ofthe following formulas are in DNF:(A B C) (D E F)(A B) Cbut and alsoA BAHowever, the following formulas are NOT in DNF:(A B)A (B (C D))Converting a formula to DNF involves using logical equivalences, such as the double negative elimination, De Mor-gans laws, and the distributive law.All logical formulas can be converted into disjunctive normal form. However, in some cases conversion to DNF canlead to an exponential explosion of the formula. For example, in DNF, logical formulas of the following form have2n terms:(X1 Y1) (X2 Y2) (Xn Yn)Any particular Boolean function can be represented by one and only one full disjunctive normal form, one of the twocanonical forms.An important variation used in the study of computational complexity is k-DNF. A formula is in k-DNF if it is inDNF and each clause contains at most k literals. Unlike the corresponding subclasses of conjunctive normal form fork>=3, there is no easy algorithm to convert an arbitrary instance of a formula in DNF to k-DNF.The following is a formal grammar for DNF:56 CHAPTER 2. DISJUNCTIVE NORMAL FORM1. disjunct (conjunct disjunct)2. disjunct conjunct3. conjunct (literal conjunct)4. conjunct literal5. literal variable6. literal variableWhere variable is any variable.2.1 See alsoAlgebraic normal formBoolean functionBoolean-valued functionConjunctive normal formHorn clauseKarnaugh mapLogical graphPropositional logicQuineMcCluskey algorithmTruth table2.2 External linksHazewinkel, Michiel, ed. (2001), Disjunctive normal form, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4Java applet for converting boolean logic expressions to CNF and DNF, showing the laws usedChapter 3Negation normal formIn mathematical logic, a formula is in negation normal form if the negation operator ( , not) is only applied tovariables and the only other allowed Boolean operators are conjunction ( , and) and disjunction ( , or).Negation normal form is not a canonical form: for example, a (b c) and (a b) (a c) are equivalent, andare both in negation normal form.In classical logic and many modal logics, every formula can be brought into this form by replacing implications andequivalences by their denitions, using De Morgans laws to push negation inwards, and eliminating double negations.This process can be represented using the following rewrite rules:(x.G) x.G(x.G) x.GG G(G1 G2) (G1) (G2)(G1 G2) (G1) (G2)A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal formby applying distributivity.3.1 Examples and counterexamplesThe following formulae are all in negation normal form:(A B) C(A (B C) C) DA BA BThe rst example is also in conjunctive normal form and the last two are in both conjunctive normal form anddisjunctive normal form, but the second example is in neither.The following formulae are not in negation normal form:A B(A B)78 CHAPTER 3. NEGATION NORMAL FORM(A B)(A C)They are however respectively equivalent to the following formulae in negation normal form:A BA BA BA C3.2 ReferencesAlan J.A. Robinson and Andrei Voronkov, Handbook of Automated Reasoning 1:203 (2001) ISBN0444829490.3.3 External linksJava applet for converting logical formula to Negation Normal Form, showing laws usedChapter 4Prenex normal formA formula of the predicate calculus is in prenex[1]normal form if it is written as a string of quantiers (referred toas the prex) followed by a quantier-free part (referred to as the matrix).Every formula in classical logic is equivalent to a formula in prenex normal form. For example, if (y) , (z) , and(x) are quantier-free formulas with the free variables shown thenxyz((y) ((z) (x)))is in prenex normal form with matrix (y) ((z) (x)) , whilex((y(y)) ((z(z)) (x)))is logically equivalent but not in prenex normal form.4.1 Conversion to prenex formEvery rst-order formula is logically equivalent (in classical logic) to some formula in prenex normal form. Thereare several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rulesdepend on which logical connectives appear in the formula.4.1.1 Conjunction and disjunctionThe rules for conjunction and disjunction say that(x) is equivalent to x( ) ,(x) is equivalent to x( ) ;and(x) is equivalent to x( ) ,(x) is equivalent to x( ) .The equivalences are valid when x does not appear as a free variable of ; if x does appear free in , it must bereplaced with another free variable.For example, in the language of rings,(x(x2= 1)) (0 = y) is equivalent to x(x2= 1 0 = y) ,910 CHAPTER 4. PRENEX NORMAL FORMbut(x(x2= 1)) (0 = x) is not equivalent to x(x2= 1 0 = x)because the formula on the left is true in any ring when the free variable x is equal to 0, while the formula on the righthas no free variables and is false in any nontrivial ring.4.1.2 NegationThe rules for negation say thatx is equivalent to xandx is equivalent to x .4.1.3 ImplicationThere are four rules for implication: two that remove quantiers from the antecedent and two that remove quantiersfrom the consequent.These rules can be derived by rewriting the implication as and applying therules for disjunction above.As with the rules for disjunction, these rules require that the variable quantied in onesubformula does not appear free in the other subformula.The rules for removing quantiers from the antecedent are:(x) is equivalent to x( ) ,(x) is equivalent to x( ) .The rules for removing quantiers from the consequent are: (x) is equivalent to x( ) , (x) is equivalent to x( ) .4.1.4 ExampleSuppose that , , and are quantier-free formulas and no two of these formulas share any free variable. Considerthe formula( x) zBy recursively applying the rules starting at the innermost subformulas, the following sequence of logically equivalentformulas can be obtained:( x) z(x( )) z(x( )) z(x( )) zx(( ) z)x(( ) z)4.2. USE OF PRENEX FORM 11x(z(( ) ))xz(( ) )This is not the only prenex form equivalent to the original formula. For example, by dealing with the consequentbefore the antecedent in the example above, the prenex formzx(( ) )can be obtained:z(( x) )z((x( )) )z(x(( ) ))zx(( ) )4.1.5 Intuitionistic logicThe rules for converting a formula to prenex form make heavy use of classical logic. In intuitionistic logic, it is nottrue that every formula is logically equivalent to a prenex formula. The negation connective is one obstacle, but not theonly one. The implication operator is also treated dierently in intuitionistic logic than classical logic; in intuitionisticlogic, it is not denable using disjunction and negation.The BHK interpretation illustrates why some formulas have no intuitionistically-equivalent prenex form. In thisinterpretation, a proof of(x) y (1)is a function which, given a concrete x and a proof of (x) , produces a concrete y and a proof of (y). In this caseit is allowable for the value of y to be computed from the given value of x. A proof ofy(x ), (2)on the other hand, produces a single concrete value of y and a function that converts any proof of x into a proofof (y). If each x satisfying can be used to construct a y satisfying but no such y can be constructed withoutknowledge of such an x then formula (1) will not be equivalent to formula (2).The rules for converting a formula to prenex form that do fail in intuitionistic logic are:(1) x( ) implies (x) ,(2) x( ) implies (x) ,(3) (x) implies x( ) ,(4) (x) implies x( ) ,(5) x implies x ,(x does not appear as a free variable of in (1) and (3); x does not appear as a free variable of in (2) and (4)).4.2 Use of prenex formSome proof calculi will only deal with a theory whose formulae are written in prenex normal form. The concept isessential for developing the arithmetical hierarchy and the analytical hierarchy.Gdel's proof of his completeness theorem for rst-order logic presupposes that all formulae have been recast inprenex normal form.12 CHAPTER 4. PRENEX NORMAL FORM4.3 See alsoHerbrandizationSkolemizationArithmetical hierarchy4.4 Notes[1] The term 'prenex' comes from the Latin praenexus tied or bound up in front, past participle of praenectere .4.5 ReferencesHinman, P. (2005), Fundamentals of Mathematical Logic, A K Peters, ISBN 978-1-56881-262-54.6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 134.6 Text and image sources, contributors, and licenses4.6.1 Text Conjunctive normal formSource: https://en.wikipedia.org/wiki/Conjunctive_normal_form?oldid=672056976 Contributors: Bryan Derk-sen, Robert Merkel, BenBaker, Arvindn, Toby Bartels, B4hand, Michael Hardy, Aquatopia~enwiki, Ldo, Thesilverbail, Jleedev, Giftlite,CyborgTosser, Niteowlneils, Macrakis, Gubbubu, Wzwz, Ary29, Erik Garrison, MementoVivere, ESkog, Cherlin, Obradovic Goran,Poromenos, Oleg Alexandrov, Linas, Jacobolus, Eclecticos, Graham87, BD2412, Tizio, Salix alba, Mike Segal, Mathbot, Jrtayloriv,Fresheneesz, Masnevets, Chobot, Dresdnhope, YurikBot, Hairy Dude, GrinBot~enwiki, SmackBot, Jpvinall, Rotemliss, Mhss, Bluebot,PrimeHunter, Jon Awbrey, Danlev, Ylloh, CBM, Andkore, Myasuda, Simeon, Gregbard, Blaisorblade, Thijs!bot, Hermel, A3nm, Ay-dos~enwiki, Mikhail Dvorkin, AntiSpamBot, Policron, Ross Fraser, TXiKiBoT, Mx2323, Jamelan, AlleborgoBot, IsleLaMotte, Tesi1700,Dardasavta, Alejandrocaro35, DumZiBoT, Addbot, Yobot, IM Serious, LilHelpa, Snikeris, Hobsonlane, D'ohBot, Mikolasj, EmausBot,Frostyandy2k, Bobogoobo, Tijfo098, Cognominally, Hofmic, OSDevLabs, Jamesx12345, Jochen Burghardt, Tleo and Anonymous: 73 Disjunctive normal formSource: https://en.wikipedia.org/wiki/Disjunctive_normal_form?oldid=669934430 Contributors: Bryan Derk-sen, BenBaker, Toby Bartels, B4hand, Sarrazip, Altenmann, Tobias Bergemann, DavidCary, Bnn, Brona, CyborgTosser, Macrakis,Wzwz, MementoVivere, ZeroOne, EmilJ, Haham hanuka, Linas, A3r0, Graham87, BD2412, Tizio, Fresheneesz, Roboto de Ajvol, Grin-Bot~enwiki, Ajm81, Mhss, Bluebot, Jon Awbrey, Ben Spinozoan, CBM, Simeon, Gregbard, Dougher, Smerdis, Batenka~enwiki, Kundu,Policron, Jamelan, Tvdm, Alejandrocaro35, Hans Adler, Addbot, Linket, Groovenstein, Doulos Christos, Gryllida, Igor Yalovecky, DiegoGrez Bot, Jiri 1984, Intervallic, Ref1fois and Anonymous: 28 Negationnormalform Source: https://en.wikipedia.org/wiki/Negation_normal_form?oldid=665947556 Contributors: Vkuncak, Sil-versh, Olathe, Macrakis, Starblue, Obradovic Goran, Pearle, Oleg Alexandrov, Linas, Kbdank71, Mets501, CRGreathouse, CBM,Gregbard, Cydebot, Christian75, Amikake3, Brian Geppert, Addbot, Luckas-bot, Tomdwrightenator, Kejia, Mogism, Ayush3292 andAnonymous: 6 Prenex normal form Source: https://en.wikipedia.org/wiki/Prenex_normal_form?oldid=622113385 Contributors: The Anome, MichaelHardy, AugPi, Charles Matthews, Dysprosia, Greenrd, Pfortuny, Gandalf61, Lockeownzj00, 4pq1injbok, Oleg Alexandrov, Joriki, Linas,MattGiuca, BD2412, Reetep, Jayme, SmackBot, Mhss, Esoth~enwiki, CRGreathouse, CBM, Thijs!bot, Jakob.scholbach, Toobaz, Alle-borgoBot, SieBot, IsleLaMotte, PixelBot, Addbot, SamatBot, Coreyoconnor, AnomieBOT, Omnipaedista, Epiglottisz, WikitanvirBot andAnonymous: 214.6.2 Images File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007 File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License:Public domain Contributors:Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham) File:Wiktionary-logo-en.svg Source:https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License:Publicdomain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk contribs),based on original logo tossed together by Brion Vibber4.6.3 Content license Creative Commons Attribution-Share Alike 3.0