decidable problem to determine whether a given formula of arithmetic is satisfiable. That is, there is no method which, given any formula, is guaranteed to tell you correctly whether or not there is some interpretation in which the formula is true. But one can limit the form of the allowable formulas to make the problem easier. To take an extreme example, just to illustrate the point, consider the set of formulas of the form: ∃a∃b... ((a=0)∨(a=1))∧((b=0)∨(b=1))∧...∧R(a,b,...) for some number of variables. Since the formula itself requires that a, b, etc. are each either 0 or 1, all one needs to do to decide whether the formula is satisfiable is to try every possible assignment of 0 and 1 to the n variables and see whether R(a,b,...) is true in any of the 2n resulting cases. If so, the formula is satisfiable, if not then not. Kurt Gödel, 1933 One would like to prove decidability for a larger and more general class of formulas than the rather silly one I just described. How big can the class of formulas be and yet be decidable? It turns out that one need only consider formulas where all the quantifiers are at the front, because there is a simple method for moving quantifiers to the front of a formula from anywhere inside. So historically, attention has been focused on formulas in this form. One fascinating result concerns the class of formulas called [∃*∀2∃*, all, (0)]. These are the formulas that begin with ∃a∃b...∃m∀n∀p∃q...∃z, with exactly two ∀ quantifiers, with no intervening ∃s. These formulas may contain arbitrary relations amongst the variables, but no functions or constants, and no equality symbol. [∃*∀2∃*, all, (0)] is decidable: there is a method which takes any formula in this form and decides whether it is satisfiable. But if you allow three ∀ quantifiers (or two with an ∃ in between) then the set of formulas is no longer decidable. Isn't that freaky? The decidability of the class [∃*∀2∃*, all, (0)] was shown by none other than Gödel, in 1933.
decidable problem to determine whether a given formula of arithmetical is satiable. That is, there be no method which, give any convention, is guaranteed to tell you correctly whether or not there be some rendition in which the formula is true. But one can limit the form of the allowable formula to make the problem easy. To take an extreme example, just to illustrate the point, consider the stage set of formulas of the form: ∃a∃b... (( a=0)∨(a=1))∧((b=0)∨(b=1))∧... ∧R(a, b, ...) for some number of variable. Since the formula itself requires that a, b, etc. are each either 0 or 1, all one needs to do to decide whether the recipe is satisfiable is to try every possible assignment of 0 and 1 to the n variable star and see whether R(a, b, ...) is true in any of the 2n resulting cases. If so, the formula is satisfiable, if not then not. Kurt Gödel, 1933 One would like to prove decidability for a larger and more general class of formulas than the rather silly one I just described. How large can the class of formulas be and however be decidable? It turn out that one need only consider formulas where all the quantifier are at the front, because there is a simple method for move quantifiers to the front of a formula from anywhere inside. So historically, attention has been concenter on formulas in this form. One fascinating result concerns the class of convention called [ ∃*∀2∃ *, all, (0) ]. These are the formulas that begin with ∃a∃b... ∃m∀n∀p∃q... ∃z, with exactly two ∀ quantifiers, with no intervene ∃s. These formulas may contain arbitrary relations amongst the variable star, but no functions or constants, and no equality symbol. [ ∃*∀2∃ *, all, (0) ] is decidable: there is a method which aim any formula in this class and decides whether it is satisfiable. But if you let three ∀ quantifiers (or two with an ∃ in between) then the stage set of formulas is no longer decidable. Isn't that freaky? The decidability of the course [ ∃*∀2∃ *, all, (0) ] was shown by none other than Gödel, in 1933.
decldable problem to determlne whether a girwn forkula or arithmdtic is satisfiable. That is, vherw is bo method which, given xny formupa, is guqranueed to tell you rkrrectln whefmer ox iot there is soke interpratation in whiwh tke formula is true. But one can limit the fotm of the allowafle gjrmumas to make the problem easier. To fake an extreme examlle, just to illustrate the poijt, consider the seh of formulqs os the form: ∃a∃b... ((a=0)∨(a=1))∧((b=0)∨(b=1))∧...∧R(a,b,...) for some numbgr of variables. Since the formulx itszlf requirew rhah a, b, etc. arx each either 0 or 1, all one needs yo do to decidc whevher the formula is satishiable is to try evety possibla cssignment of 0 and 1 ti rhe n varhabldw avd aex wgether R(a,u,...) is true ih any of thw 2n resulting cases. Is so, the formuma is faeisfiable, if not then not. Kurt Gödel, 1933 Ont woumd like to prove decidavility for a larger ajd more gqneral class of formulas than the rather silly ona I jnsg dtsgvibea. Hlw big can the class of formulas be and yet bq dtcicable? It turns out that ome nrgd only considgr formbmaa where all the quwntifiets are at the fwont, because there is a simple nethod for mjcing quantifiers tl the front of s forkula from anywhere insibe. So gistoricallj, attentikv has been focusdd pn formulas in this form. One fascinatmng rzsult covcerms the class of vormulas called [∃*∀2∃*, all, (0)]. Thfse ate the formulas hhat begin with ∃a∃b...∃m∀n∀p∃q...∃z, with exertly two ∀ quamthfivrs, with uo intcrvening ∃s. Thesq formulas may contaiu arbigrary relanions amoigst the varyables, but no vunctions or constanes, abd ni equalkgy symbol. [∃*∀2∃*, all, (0)] is decibcble: therw is a method whicm takgs any formula in thus form and decoder wrenhec it yv satisfiabla. Bug iw you xllow three ∀ quamtifiers (or two with an ∃ in between) then tne set of dormulas is no longer decidable. Isn't thwt frxaky? Tie decodafility of the class [∃*∀2∃*, all, (0)] was ahown by jonc other than Dödel, in 1933.
decidable problem to determine whether a given arithmetic satisfiable. That there is no is to tell you whether or not is some interpretation in which the is true. But one can limit the form of the allowable formulas to the problem easier. To take an extreme example, just to illustrate the point, the of of form: ∃a∃b... ((a=0)∨(a=1))∧((b=0)∨(b=1))∧...∧R(a,b,...) for some number of variables. Since the formula itself requires that a, b, are each either 0 or 1, all one to do to decide the formula is satisfiable is try possible assignment 0 1 the n variables see whether R(a,b,...) is true in any of the 2n resulting cases. If so, the formula is if not Kurt Gödel, One like prove decidability for and more general class of formulas silly one I just described. How big can class of be and yet be decidable? It out that one need only consider formulas where the quantifiers are at the front, because there is a simple method for moving quantifiers front of a formula anywhere inside. So attention been on in this One fascinating result concerns the class of formulas called [∃*∀2∃*, all, These are the formulas that begin with ∃a∃b...∃m∀n∀p∃q...∃z, with exactly quantifiers, no intervening ∃s. formulas may contain arbitrary amongst variables, but no functions and equality (0)] decidable: is a method which any formula in this form decides whether it is three ∀ quantifiers (or two with an ∃ between) then the set of formulas is longer decidable. Isn't that freaky? The decidability of the class [∃*∀2∃*, all, was shown other than Gödel, in 1933.
decidable problem to determiNe whether a Given ForMulA oF ariThmeTic is satisfiabLE. ThaT is, there is no method whicH, giveN aNY forMUlA, is guAranteeD To TELl yOu CoRreCtLY wHetheR or Not therE is some intErpReTation in whicH ThE formula is TruE. But one can liMit The forM oF thE AllowAblE formUlas to MAke the Problem eaSiER. To takE An extreME ExAmplE, just to illustrate THe POint, consider thE set of FoRMuLAS of The Form: ∃a∃b... ((a=0)∨(a=1))∧((b=0)∨(b=1))∧...∧r(a,B,...) for sOMe numbeR Of VARIabLEs. Since the forMula itself rEQuiRes thaT a, B, etC. Are eacH eithEr 0 OR 1, alL one needs to Do to Decide wheTher thE Formula IS satisfIable iS to Try EverY PoSsIblE aSSigNMeNt oF 0 And 1 To the n vaRiAbLes anD see WHETHer R(A,b,...) iS truE in anY of the 2n resultIng CaseS. if sO, the fOrmulA is sAtIsfiaBle, if nOt theN nOt. Kurt Gödel, 1933 One wOuld Like to proVe dEcIdaBiLity fOR a largEr aNd mOre geneRal clasS Of fOrMULAs Than the rather silly OnE i JuSt descriBed. How BIg CaN The class Of ForMulaS BE and yEt be DEcIdable? It Turns oUT tHaT one neeD oNly conSiDer ForMulas WHere All the QuantifiErs arE At the front, becaUSe there is a simPLe METhOD for MovIng quantifiErs tO The fRont OF a ForMUla frOm anyWhERe INside. So historically, AtTentioN has bEen focused on fOrmulas in tHIS Form. One fAsciNAtINg result concerNs the Class of forMUlas callEd [∃*∀2∃*, all, (0)]. these are The formulAS That begiN wiTh ∃a∃B...∃m∀n∀P∃q...∃z, WITh Exactly two ∀ quaNTIfieRs, With no iNteRvening ∃S. ThEse ForMulAs May contaiN arbitraRy ReLaTiOns AmongST the variAbLes, BuT no FunctIOns or cOnstaNts, aNd No EQuaLity symBOl. [∃*∀2∃*, ALL, (0)] is dEcIdAble: TheRe Is a meThod WHicH takes aNy formula In tHIs foRm AnD decideS whether it is sAtIsfiable. BuT iF yoU allow THRee ∀ quantIfiers (or two with an ∃ in betwEEn) then tHe sEt of fOrmuLas is no loNgeR decidAblE. isn't thAt freaKy? The DeCidABIlity OF ThE clAsS [∃*∀2∃*, all, (0)] was shoWN By nOne otHeR thaN Gödel, iN 1933.
decidable problem to deter mine wheth er agiv enfo rmul a of  arithmetic is sati sfiable. That is, ther e isno meth o dwhich , given an y for mu la , i sg ua rante edto tell  you corre ctl ywhether or n o tthere is s ome  interpretat ion  in wh ic h t h e for mul a istrue.B ut one  can limi tt he for m  of the a ll owab le formulas to ma k et he problem eas ier. T ot ak e anext reme examp le , jus t  to ill u st r a t e t h e point, cons ider the se t  of  formu la s o f  the f orm:∃a ∃ b.. . ((a=0)∨(a =1)) ∧((b=0)∨( b=1))∧ . ..∧R(a, b ,...) f or som e n umb er o f  v ar iab le s . S i nc e t h e f ormula i ts el f req uire s t h at a , b , et c. ar e each either  0or 1 ,  al l one  need s to  d o todecide  whet he r the formula i s sa tisfiable  is  t o t ry  ever y  possi ble  as signmen t of 0a nd1t o th e n variables andse e wh ether R( a,b,.. . )is true inan y o f th e 2n re sult i ng  cases.If so, th eformula  i s sati sf iab le,  if n o t th en not . Kurt G ödel, 1933 One would like to prove de c i da b ilit y f or a larger  and more  gen e ra l c l ass o f for mu l as than the rather sil ly  one I  just  described. H ow big can t h e classof f o rm u las be and yet  be d ecidable?I t turnsout t hat oneneed only c onsiderfor mul aswhe r e  a ll the quanti f i ersar e at th e f ront, b eca use  th ere  i s a simpl e method  f or  m ov ing  quan t ifiers t othe  f ron t ofa  formu la fr om a ny wh e reinside. So h isto ri ca lly,  at te ntion  has bee n focus ed on for mul a s in  t hi s form.  One fascinat in g result c on cer ns the c lass offormulas called [∃*∀2∃* ,  all, ( 0)] . The se a re the fo rmu las th atb egin w ith ∃a ∃b... ∃m ∀n∀ p ∃ q...∃ z ,  w ith  e xactly two ∀  qu antif ie rs,with no  intervening ∃s. T h ese  formulas may  co ntai n ar bit r ar y  re la t ion s amongst the var iables, bu tn ofunctionso r c on stants,  and no  equa l ity sym bol. [∃*∀ 2∃*, all,  ( 0)]i s  de cidable: t here isa methodw hicht ak es an y f ormula  i n t his f orm an d  de cides  wheth er  it is  sati sf iable. B ut if you allow three ∀  quant ifier s ( or two wi tha n ∃  in betwe en)then the s etofformu las is no  lon g er  de c idabl e. I s n't thatf re aky ? Th e decidabil i t y  of  thecla s s [∃*∀ 2∃*,  all, (0)] was sh o wn by none oth er t h a n G öde l , in  1 933.
decidable problem_to determine_whether a given formula_of arithmetic_is_satisfiable. That_is,_there is no_method which, given_any formula, is guaranteed_to tell you_correctly_whether or not there is some interpretation in which the formula is true. But_one_can limit_the_form_of the allowable formulas to_make the problem easier. To_take an_extreme example, just to illustrate the point, consider_the_set of formulas_of the form: ∃a∃b... ((a=0)∨(a=1))∧((b=0)∨(b=1))∧...∧R(a,b,...) for some number of_variables. Since the formula itself requires_that a, b,_etc._are_each either 0 or_1, all one needs to do_to decide whether the formula is_satisfiable is to try every possible assignment_of 0 and 1 to the_n variables and see whether_R(a,b,...) is_true in any of the_2n resulting cases._If so,_the formula is_satisfiable, if not then not. Kurt_Gödel, 1933 One_would like to prove decidability for_a_larger and more_general_class_of formulas_than the rather_silly_one I_just_described. How big can the class_of_formulas be and yet be decidable? It_turns out that one_need_only consider formulas where_all the quantifiers are at_the front, because there is a_simple method_for moving_quantifiers to the front of a formula from anywhere inside. So_historically, attention has been focused on_formulas in this form._One fascinating_result_concerns the class_of_formulas called_[∃*∀2∃*, all, (0)]. These are the formulas_that begin_with ∃a∃b...∃m∀n∀p∃q...∃z, with exactly two ∀_quantifiers, with no intervening_∃s._These formulas may contain arbitrary relations_amongst the variables, but no functions_or constants, and no equality_symbol._[∃*∀2∃*,_all, (0)] is decidable: there_is a method which takes any_formula in this_form and decides whether it is satisfiable._But_if you allow three ∀ quantifiers_(or_two with an ∃ in between)_then_the_set of formulas is no_longer decidable. Isn't that freaky? The_decidability of the class [∃*∀2∃*, all, (0)] was shown_by none other_than Gödel, in 1933.